\(\int \frac {1}{(d+e x)^3 (a+c x^4)^3} \, dx\) [414]

Optimal result

Integrand size = 17, antiderivative size = 2204 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^3} \, dx=-\frac {e^{11}}{2 \left (c d^4+a e^4\right )^3 (d+e x)^2}-\frac {12 c d^3 e^{11}}{\left (c d^4+a e^4\right )^4 (d+e x)}+\frac {c x \left (7 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-6 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+10 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )^2}+\frac {c e^4 \left (12 a d^2 e^3 \left (3 c d^4-a e^4\right )+x \left (3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^4 \left (a+c x^4\right )}-\frac {\sqrt {c} e^9 \left (55 c^2 d^8-40 a c d^4 e^4+a^2 e^8\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^5}-\frac {\sqrt {c} e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )^4}-\frac {3 \sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \left (c d^4+a e^4\right )^3}-\frac {3 c^{3/4} d e^8 \left (15 c^2 d^8-16 a c d^4 e^4+a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (11 c d^4-5 a e^4\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^5}-\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )+9 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (10 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+21 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )^3}+\frac {3 c^{3/4} d e^8 \left (15 c^2 d^8-16 a c d^4 e^4+a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (11 c d^4-5 a e^4\right )\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^5}+\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )+9 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (10 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+21 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )^3}+\frac {6 c d^2 e^{11} \left (13 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^5}-\frac {3 c^{3/4} d e^8 \left (15 c^2 d^8-16 a c d^4 e^4+a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (11 c d^4-5 a e^4\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^5}+\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-9 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (10 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )-21 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )^3}+\frac {3 c^{3/4} d e^8 \left (15 c^2 d^8-16 a c d^4 e^4+a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (11 c d^4-5 a e^4\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^5}-\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-9 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (10 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )-21 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )^3}-\frac {3 c d^2 e^{11} \left (13 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (c d^4+a e^4\right )^5} \]

[Out]

-1/2*e^11/(a*e^4+c*d^4)^3/(e*x+d)^2-12*c*d^3*e^11/(a*e^4+c*d^4)^4/(e*x+d)+1/32*c*x*(7*d*(3*a^2*e^8-12*a*c*d^4*
e^4+c^2*d^8)-6*e*(a^2*e^8-12*a*c*d^4*e^4+3*c^2*d^8)*x+10*c*d^3*e^2*(-5*a*e^4+3*c*d^4)*x^2)/a^2/(a*e^4+c*d^4)^3
/(c*x^4+a)+1/8*c*(2*a*d^2*e^3*(-3*a*e^4+5*c*d^4)+x*(d*(3*a^2*e^8-12*a*c*d^4*e^4+c^2*d^8)-e*(a^2*e^8-12*a*c*d^4
*e^4+3*c^2*d^8)*x+2*c*d^3*e^2*(-5*a*e^4+3*c*d^4)*x^2))/a/(a*e^4+c*d^4)^3/(c*x^4+a)^2+1/4*c*e^4*(12*a*d^2*e^3*(
-a*e^4+3*c*d^4)+x*(3*d*(a^2*e^8-10*a*c*d^4*e^4+5*c^2*d^8)-e*(a^2*e^8-26*a*c*d^4*e^4+21*c^2*d^8)*x+4*c*d^3*e^2*
(-5*a*e^4+7*c*d^4)*x^2))/a/(a*e^4+c*d^4)^4/(c*x^4+a)+6*c*d^2*e^11*(-3*a*e^4+13*c*d^4)*ln(e*x+d)/(a*e^4+c*d^4)^
5-3/2*c*d^2*e^11*(-3*a*e^4+13*c*d^4)*ln(c*x^4+a)/(a*e^4+c*d^4)^5-1/4*e^5*(a^2*e^8-26*a*c*d^4*e^4+21*c^2*d^8)*a
rctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/a^(3/2)/(a*e^4+c*d^4)^4-3/16*e*(a^2*e^8-12*a*c*d^4*e^4+3*c^2*d^8)*arctan(x^
2*c^(1/2)/a^(1/2))*c^(1/2)/a^(5/2)/(a*e^4+c*d^4)^3-1/2*e^9*(a^2*e^8-40*a*c*d^4*e^4+55*c^2*d^8)*arctan(x^2*c^(1
/2)/a^(1/2))*c^(1/2)/(a*e^4+c*d^4)^5/a^(1/2)+1/256*c^(3/4)*d*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2)
)*(-63*a^2*e^8+252*a*c*d^4*e^4-21*c^2*d^8+10*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(11/4)/(a*e^4+c*d^4
)^3*2^(1/2)-1/256*c^(3/4)*d*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-63*a^2*e^8+252*a*c*d^4*e^4-21*
c^2*d^8+10*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(11/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/128*c^(3/4)*d*arctan
(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(63*a^2*e^8-252*a*c*d^4*e^4+21*c^2*d^8+10*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^
(1/2))/a^(11/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/128*c^(3/4)*d*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(63*a^2*e^8-252*a*
c*d^4*e^4+21*c^2*d^8+10*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(11/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/32*c^(3
/4)*d*e^4*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-9*a^2*e^8+90*a*c*d^4*e^4-45*c^2*d^8+4*d^2*e^2*(
-5*a*e^4+7*c*d^4)*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^4*2^(1/2)-1/32*c^(3/4)*d*e^4*ln(a^(1/4)*c^(1/4)*x*2^(
1/2)+a^(1/2)+x^2*c^(1/2))*(-9*a^2*e^8+90*a*c*d^4*e^4-45*c^2*d^8+4*d^2*e^2*(-5*a*e^4+7*c*d^4)*a^(1/2)*c^(1/2))/
a^(7/4)/(a*e^4+c*d^4)^4*2^(1/2)+1/16*c^(3/4)*d*e^4*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(9*a^2*e^8-90*a*c*d^4*
e^4+45*c^2*d^8+4*d^2*e^2*(-5*a*e^4+7*c*d^4)*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^4*2^(1/2)+1/16*c^(3/4)*d*e^
4*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(9*a^2*e^8-90*a*c*d^4*e^4+45*c^2*d^8+4*d^2*e^2*(-5*a*e^4+7*c*d^4)*a^(1/2
)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^4*2^(1/2)-3/8*c^(3/4)*d*e^8*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2)
)*(15*c^2*d^8-16*a*c*d^4*e^4+a^2*e^8-2*d^2*e^2*(-5*a*e^4+11*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^5*2^
(1/2)+3/8*c^(3/4)*d*e^8*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(15*c^2*d^8-16*a*c*d^4*e^4+a^2*e^8-2
*d^2*e^2*(-5*a*e^4+11*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^5*2^(1/2)+3/4*c^(3/4)*d*e^8*arctan(-1+c^(1
/4)*x*2^(1/2)/a^(1/4))*(15*c^2*d^8-16*a*c*d^4*e^4+a^2*e^8+2*d^2*e^2*(-5*a*e^4+11*c*d^4)*a^(1/2)*c^(1/2))/a^(3/
4)/(a*e^4+c*d^4)^5*2^(1/2)+3/4*c^(3/4)*d*e^8*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(15*c^2*d^8-16*a*c*d^4*e^4+a^
2*e^8+2*d^2*e^2*(-5*a*e^4+11*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^5*2^(1/2)

Rubi [A] (verified)

Time = 2.13 (sec) , antiderivative size = 2204, normalized size of antiderivative = 1.00, number of steps used = 46, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.882, Rules used = {6874, 1868, 1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 266} \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^3} \, dx=\frac {6 c d^2 \left (13 c d^4-3 a e^4\right ) \log (d+e x) e^{11}}{\left (c d^4+a e^4\right )^5}-\frac {3 c d^2 \left (13 c d^4-3 a e^4\right ) \log \left (c x^4+a\right ) e^{11}}{2 \left (c d^4+a e^4\right )^5}-\frac {12 c d^3 e^{11}}{\left (c d^4+a e^4\right )^4 (d+e x)}-\frac {e^{11}}{2 \left (c d^4+a e^4\right )^3 (d+e x)^2}-\frac {\sqrt {c} \left (55 c^2 d^8-40 a c e^4 d^4+a^2 e^8\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^9}{2 \sqrt {a} \left (c d^4+a e^4\right )^5}-\frac {3 c^{3/4} d \left (15 c^2 d^8-16 a c e^4 d^4+2 \sqrt {a} \sqrt {c} e^2 \left (11 c d^4-5 a e^4\right ) d^2+a^2 e^8\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^8}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^5}+\frac {3 c^{3/4} d \left (15 c^2 d^8-16 a c e^4 d^4+2 \sqrt {a} \sqrt {c} e^2 \left (11 c d^4-5 a e^4\right ) d^2+a^2 e^8\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^8}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^5}-\frac {3 c^{3/4} d \left (15 c^2 d^8-16 a c e^4 d^4-2 \sqrt {a} \sqrt {c} e^2 \left (11 c d^4-5 a e^4\right ) d^2+a^2 e^8\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^8}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^5}+\frac {3 c^{3/4} d \left (15 c^2 d^8-16 a c e^4 d^4-2 \sqrt {a} \sqrt {c} e^2 \left (11 c d^4-5 a e^4\right ) d^2+a^2 e^8\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^8}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^5}-\frac {\sqrt {c} \left (21 c^2 d^8-26 a c e^4 d^4+a^2 e^8\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{4 a^{3/2} \left (c d^4+a e^4\right )^4}+\frac {c \left (12 a d^2 \left (3 c d^4-a e^4\right ) e^3+x \left (4 c e^2 \left (7 c d^4-5 a e^4\right ) x^2 d^3+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right ) d-e \left (21 c^2 d^8-26 a c e^4 d^4+a^2 e^8\right ) x\right )\right ) e^4}{4 a \left (c d^4+a e^4\right )^4 \left (c x^4+a\right )}-\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+9 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+9 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-9 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-9 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^4}-\frac {3 \sqrt {c} \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{16 a^{5/2} \left (c d^4+a e^4\right )^3}+\frac {c \left (2 a d^2 \left (5 c d^4-3 a e^4\right ) e^3+x \left (2 c e^2 \left (3 c d^4-5 a e^4\right ) x^2 d^3+\left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right ) d-e \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) x\right )\right )}{8 a \left (c d^4+a e^4\right )^3 \left (c x^4+a\right )^2}-\frac {c^{3/4} d \left (10 \sqrt {a} \sqrt {c} d^2 \left (3 c d^4-5 a e^4\right ) e^2+21 \left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (10 \sqrt {a} \sqrt {c} d^2 \left (3 c d^4-5 a e^4\right ) e^2+21 \left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (10 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )-21 \left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right )\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{128 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (10 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )-21 \left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right )\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{128 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )^3}+\frac {c x \left (10 c e^2 \left (3 c d^4-5 a e^4\right ) x^2 d^3+7 \left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right ) d-6 e \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) x\right )}{32 a^2 \left (c d^4+a e^4\right )^3 \left (c x^4+a\right )} \]

[In]

Int[1/((d + e*x)^3*(a + c*x^4)^3),x]

[Out]

-1/2*e^11/((c*d^4 + a*e^4)^3*(d + e*x)^2) - (12*c*d^3*e^11)/((c*d^4 + a*e^4)^4*(d + e*x)) + (c*x*(7*d*(c^2*d^8
- 12*a*c*d^4*e^4 + 3*a^2*e^8) - 6*e*(3*c^2*d^8 - 12*a*c*d^4*e^4 + a^2*e^8)*x + 10*c*d^3*e^2*(3*c*d^4 - 5*a*e^
4)*x^2))/(32*a^2*(c*d^4 + a*e^4)^3*(a + c*x^4)) + (c*(2*a*d^2*e^3*(5*c*d^4 - 3*a*e^4) + x*(d*(c^2*d^8 - 12*a*c
*d^4*e^4 + 3*a^2*e^8) - e*(3*c^2*d^8 - 12*a*c*d^4*e^4 + a^2*e^8)*x + 2*c*d^3*e^2*(3*c*d^4 - 5*a*e^4)*x^2)))/(8
*a*(c*d^4 + a*e^4)^3*(a + c*x^4)^2) + (c*e^4*(12*a*d^2*e^3*(3*c*d^4 - a*e^4) + x*(3*d*(5*c^2*d^8 - 10*a*c*d^4*
e^4 + a^2*e^8) - e*(21*c^2*d^8 - 26*a*c*d^4*e^4 + a^2*e^8)*x + 4*c*d^3*e^2*(7*c*d^4 - 5*a*e^4)*x^2)))/(4*a*(c*
d^4 + a*e^4)^4*(a + c*x^4)) - (Sqrt[c]*e^9*(55*c^2*d^8 - 40*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a
]])/(2*Sqrt[a]*(c*d^4 + a*e^4)^5) - (Sqrt[c]*e^5*(21*c^2*d^8 - 26*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/
Sqrt[a]])/(4*a^(3/2)*(c*d^4 + a*e^4)^4) - (3*Sqrt[c]*e*(3*c^2*d^8 - 12*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*
x^2)/Sqrt[a]])/(16*a^(5/2)*(c*d^4 + a*e^4)^3) - (3*c^(3/4)*d*e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4 + a^2*e^8 + 2*Sq
rt[a]*Sqrt[c]*d^2*e^2*(11*c*d^4 - 5*a*e^4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4
+ a*e^4)^5) - (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) + 9*(5*c^2*d^8 - 10*a*c*d^4*e^4 +
a^2*e^8))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) - (c^(3/4)*d*(10*Sqr
t[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4) + 21*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8))*ArcTan[1 - (Sqrt[2]*c^(1
/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^3) + (3*c^(3/4)*d*e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4 + a^2
*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(11*c*d^4 - 5*a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(
3/4)*(c*d^4 + a*e^4)^5) + (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) + 9*(5*c^2*d^8 - 10*a*
c*d^4*e^4 + a^2*e^8))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) + (c^(3/4
)*d*(10*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4) + 21*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8))*ArcTan[1 + (S
qrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^3) + (6*c*d^2*e^11*(13*c*d^4 - 3*a*e^4)*Log[d
+ e*x])/(c*d^4 + a*e^4)^5 - (3*c^(3/4)*d*e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4 + a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e
^2*(11*c*d^4 - 5*a*e^4))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a
*e^4)^5) + (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) - 9*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2
*e^8))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) + (c^(3/
4)*d*(10*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4) - 21*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8))*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^3) + (3*c^(3/4)*d*e^8*(15*c^
2*d^8 - 16*a*c*d^4*e^4 + a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(11*c*d^4 - 5*a*e^4))*Log[Sqrt[a] + Sqrt[2]*a^(1/
4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^5) - (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2
*(7*c*d^4 - 5*a*e^4) - 9*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqr
t[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) - (c^(3/4)*d*(10*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4)
- 21*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqr
t[2]*a^(11/4)*(c*d^4 + a*e^4)^3) - (3*c*d^2*e^11*(13*c*d^4 - 3*a*e^4)*Log[a + c*x^4])/(2*(c*d^4 + a*e^4)^5)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1868

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] -
b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Dist[1/(a*n*(p + 1))
, Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /
; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1869

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] +
Dist[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b
}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]

Rule 1890

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^{12}}{\left (c d^4+a e^4\right )^3 (d+e x)^3}+\frac {12 c d^3 e^{12}}{\left (c d^4+a e^4\right )^4 (d+e x)^2}+\frac {6 c d^2 e^{12} \left (13 c d^4-3 a e^4\right )}{\left (c d^4+a e^4\right )^5 (d+e x)}+\frac {c \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^3 \left (a+c x^4\right )^3}+\frac {c e^4 \left (3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^4 \left (a+c x^4\right )^2}+\frac {c e^8 \left (3 d \left (15 c^2 d^8-16 a c d^4 e^4+a^2 e^8\right )-e \left (55 c^2 d^8-40 a c d^4 e^4+a^2 e^8\right ) x+6 c d^3 e^2 \left (11 c d^4-5 a e^4\right ) x^2-6 c d^2 e^3 \left (13 c d^4-3 a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^5 \left (a+c x^4\right )}\right ) \, dx \\ & = -\frac {e^{11}}{2 \left (c d^4+a e^4\right )^3 (d+e x)^2}-\frac {12 c d^3 e^{11}}{\left (c d^4+a e^4\right )^4 (d+e x)}+\frac {6 c d^2 e^{11} \left (13 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^5}+\frac {\left (c e^8\right ) \int \frac {3 d \left (15 c^2 d^8-16 a c d^4 e^4+a^2 e^8\right )-e \left (55 c^2 d^8-40 a c d^4 e^4+a^2 e^8\right ) x+6 c d^3 e^2 \left (11 c d^4-5 a e^4\right ) x^2-6 c d^2 e^3 \left (13 c d^4-3 a e^4\right ) x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^5}+\frac {\left (c e^4\right ) \int \frac {3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^3}{\left (a+c x^4\right )^2} \, dx}{\left (c d^4+a e^4\right )^4}+\frac {c \int \frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3}{\left (a+c x^4\right )^3} \, dx}{\left (c d^4+a e^4\right )^3} \\ & = -\frac {e^{11}}{2 \left (c d^4+a e^4\right )^3 (d+e x)^2}-\frac {12 c d^3 e^{11}}{\left (c d^4+a e^4\right )^4 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )^2}+\frac {c e^4 \left (12 a d^2 e^3 \left (3 c d^4-a e^4\right )+x \left (3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^4 \left (a+c x^4\right )}+\frac {6 c d^2 e^{11} \left (13 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^5}+\frac {\left (c e^8\right ) \int \left (\frac {3 d \left (15 c^2 d^8-16 a c d^4 e^4+a^2 e^8\right )+6 c d^3 e^2 \left (11 c d^4-5 a e^4\right ) x^2}{a+c x^4}+\frac {x \left (-e \left (55 c^2 d^8-40 a c d^4 e^4+a^2 e^8\right )-6 c d^2 e^3 \left (13 c d^4-3 a e^4\right ) x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^5}-\frac {\left (c e^4\right ) \int \frac {-9 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )+2 e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x-4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^4}-\frac {c \int \frac {-7 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+6 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x-10 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{\left (a+c x^4\right )^2} \, dx}{8 a \left (c d^4+a e^4\right )^3} \\ & = -\frac {e^{11}}{2 \left (c d^4+a e^4\right )^3 (d+e x)^2}-\frac {12 c d^3 e^{11}}{\left (c d^4+a e^4\right )^4 (d+e x)}+\frac {c x \left (7 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-6 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+10 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )^2}+\frac {c e^4 \left (12 a d^2 e^3 \left (3 c d^4-a e^4\right )+x \left (3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^4 \left (a+c x^4\right )}+\frac {6 c d^2 e^{11} \left (13 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^5}+\frac {\left (c e^8\right ) \int \frac {3 d \left (15 c^2 d^8-16 a c d^4 e^4+a^2 e^8\right )+6 c d^3 e^2 \left (11 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^5}+\frac {\left (c e^8\right ) \int \frac {x \left (-e \left (55 c^2 d^8-40 a c d^4 e^4+a^2 e^8\right )-6 c d^2 e^3 \left (13 c d^4-3 a e^4\right ) x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^5}-\frac {\left (c e^4\right ) \int \left (\frac {2 e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x}{a+c x^4}+\frac {-9 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{4 a \left (c d^4+a e^4\right )^4}+\frac {c \int \frac {21 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-12 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+10 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{32 a^2 \left (c d^4+a e^4\right )^3} \\ & = -\frac {e^{11}}{2 \left (c d^4+a e^4\right )^3 (d+e x)^2}-\frac {12 c d^3 e^{11}}{\left (c d^4+a e^4\right )^4 (d+e x)}+\frac {c x \left (7 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-6 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+10 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )^2}+\frac {c e^4 \left (12 a d^2 e^3 \left (3 c d^4-a e^4\right )+x \left (3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^4 \left (a+c x^4\right )}+\frac {6 c d^2 e^{11} \left (13 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^5}+\frac {\left (c e^8\right ) \text {Subst}\left (\int \frac {-e \left (55 c^2 d^8-40 a c d^4 e^4+a^2 e^8\right )-6 c d^2 e^3 \left (13 c d^4-3 a e^4\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^5}-\frac {\left (c e^4\right ) \int \frac {-9 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^4}+\frac {c \int \left (-\frac {12 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x}{a+c x^4}+\frac {21 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+10 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{32 a^2 \left (c d^4+a e^4\right )^3}-\frac {\left (c e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )\right ) \int \frac {x}{a+c x^4} \, dx}{2 a \left (c d^4+a e^4\right )^4}-\frac {\left (3 c d e^8 \left (22 c d^6 e^2-10 a d^2 e^6-\frac {15 c^2 d^8-16 a c d^4 e^4+a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^5}+\frac {\left (3 c d e^8 \left (22 c d^6 e^2-10 a d^2 e^6+\frac {15 c^2 d^8-16 a c d^4 e^4+a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^5} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 1338, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^3} \, dx=\frac {-\frac {128 e^{11} \left (c d^4+a e^4\right )^2}{(d+e x)^2}-\frac {3072 c d^3 e^{11} \left (c d^4+a e^4\right )}{d+e x}+\frac {8 c \left (c d^4+a e^4\right ) \left (a^3 e^{11} \left (-96 d^2+45 d e x-14 e^2 x^2\right )+c^3 d^{11} x \left (7 d^2-18 d e x+30 e^2 x^2\right )+a c^2 d^7 e^4 x \left (43 d^2-114 d e x+204 e^2 x^2\right )+a^2 c d^3 e^7 \left (288 d^3-303 d^2 e x+274 d e^2 x^2-210 e^3 x^3\right )\right )}{a^2 \left (a+c x^4\right )}+\frac {32 c \left (c d^4+a e^4\right )^2 \left (-a^2 e^7 \left (6 d^2-3 d e x+e^2 x^2\right )+c^2 d^7 x \left (d^2-3 d e x+6 e^2 x^2\right )+2 a c d^3 e^3 \left (5 d^3-6 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )}{a \left (a+c x^4\right )^2}-\frac {6 \sqrt {c} \left (7 \sqrt {2} c^{17/4} d^{17}-24 \sqrt [4]{a} c^4 d^{16} e+10 \sqrt {2} \sqrt {a} c^{15/4} d^{15} e^2+50 \sqrt {2} a c^{13/4} d^{13} e^4-176 a^{5/4} c^3 d^{12} e^5+78 \sqrt {2} a^{3/2} c^{11/4} d^{11} e^6+220 \sqrt {2} a^2 c^{9/4} d^9 e^8-960 a^{9/4} c^2 d^8 e^9+702 \sqrt {2} a^{5/2} c^{7/4} d^7 e^{10}-770 \sqrt {2} a^3 c^{5/4} d^5 e^{12}+1200 a^{13/4} c d^4 e^{13}-390 \sqrt {2} a^{7/2} c^{3/4} d^3 e^{14}+77 \sqrt {2} a^4 \sqrt [4]{c} d e^{16}-40 a^{17/4} e^{17}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac {6 \sqrt {c} \left (7 \sqrt {2} c^{17/4} d^{17}+24 \sqrt [4]{a} c^4 d^{16} e+10 \sqrt {2} \sqrt {a} c^{15/4} d^{15} e^2+50 \sqrt {2} a c^{13/4} d^{13} e^4+176 a^{5/4} c^3 d^{12} e^5+78 \sqrt {2} a^{3/2} c^{11/4} d^{11} e^6+220 \sqrt {2} a^2 c^{9/4} d^9 e^8+960 a^{9/4} c^2 d^8 e^9+702 \sqrt {2} a^{5/2} c^{7/4} d^7 e^{10}-770 \sqrt {2} a^3 c^{5/4} d^5 e^{12}-1200 a^{13/4} c d^4 e^{13}-390 \sqrt {2} a^{7/2} c^{3/4} d^3 e^{14}+77 \sqrt {2} a^4 \sqrt [4]{c} d e^{16}+40 a^{17/4} e^{17}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+1536 c d^2 e^{11} \left (13 c d^4-3 a e^4\right ) \log (d+e x)-\frac {3 \sqrt {2} c^{3/4} \left (7 c^4 d^{17}-10 \sqrt {a} c^{7/2} d^{15} e^2+50 a c^3 d^{13} e^4-78 a^{3/2} c^{5/2} d^{11} e^6+220 a^2 c^2 d^9 e^8-702 a^{5/2} c^{3/2} d^7 e^{10}-770 a^3 c d^5 e^{12}+390 a^{7/2} \sqrt {c} d^3 e^{14}+77 a^4 d e^{16}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{11/4}}+\frac {3 \sqrt {2} c^{3/4} \left (7 c^4 d^{17}-10 \sqrt {a} c^{7/2} d^{15} e^2+50 a c^3 d^{13} e^4-78 a^{3/2} c^{5/2} d^{11} e^6+220 a^2 c^2 d^9 e^8-702 a^{5/2} c^{3/2} d^7 e^{10}-770 a^3 c d^5 e^{12}+390 a^{7/2} \sqrt {c} d^3 e^{14}+77 a^4 d e^{16}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{11/4}}-384 c d^2 e^{11} \left (13 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{256 \left (c d^4+a e^4\right )^5} \]

[In]

Integrate[1/((d + e*x)^3*(a + c*x^4)^3),x]

[Out]

((-128*e^11*(c*d^4 + a*e^4)^2)/(d + e*x)^2 - (3072*c*d^3*e^11*(c*d^4 + a*e^4))/(d + e*x) + (8*c*(c*d^4 + a*e^4
)*(a^3*e^11*(-96*d^2 + 45*d*e*x - 14*e^2*x^2) + c^3*d^11*x*(7*d^2 - 18*d*e*x + 30*e^2*x^2) + a*c^2*d^7*e^4*x*(
43*d^2 - 114*d*e*x + 204*e^2*x^2) + a^2*c*d^3*e^7*(288*d^3 - 303*d^2*e*x + 274*d*e^2*x^2 - 210*e^3*x^3)))/(a^2
*(a + c*x^4)) + (32*c*(c*d^4 + a*e^4)^2*(-(a^2*e^7*(6*d^2 - 3*d*e*x + e^2*x^2)) + c^2*d^7*x*(d^2 - 3*d*e*x + 6
*e^2*x^2) + 2*a*c*d^3*e^3*(5*d^3 - 6*d^2*e*x + 6*d*e^2*x^2 - 5*e^3*x^3)))/(a*(a + c*x^4)^2) - (6*Sqrt[c]*(7*Sq
rt[2]*c^(17/4)*d^17 - 24*a^(1/4)*c^4*d^16*e + 10*Sqrt[2]*Sqrt[a]*c^(15/4)*d^15*e^2 + 50*Sqrt[2]*a*c^(13/4)*d^1
3*e^4 - 176*a^(5/4)*c^3*d^12*e^5 + 78*Sqrt[2]*a^(3/2)*c^(11/4)*d^11*e^6 + 220*Sqrt[2]*a^2*c^(9/4)*d^9*e^8 - 96
0*a^(9/4)*c^2*d^8*e^9 + 702*Sqrt[2]*a^(5/2)*c^(7/4)*d^7*e^10 - 770*Sqrt[2]*a^3*c^(5/4)*d^5*e^12 + 1200*a^(13/4
)*c*d^4*e^13 - 390*Sqrt[2]*a^(7/2)*c^(3/4)*d^3*e^14 + 77*Sqrt[2]*a^4*c^(1/4)*d*e^16 - 40*a^(17/4)*e^17)*ArcTan
[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(11/4) + (6*Sqrt[c]*(7*Sqrt[2]*c^(17/4)*d^17 + 24*a^(1/4)*c^4*d^16*e + 10
*Sqrt[2]*Sqrt[a]*c^(15/4)*d^15*e^2 + 50*Sqrt[2]*a*c^(13/4)*d^13*e^4 + 176*a^(5/4)*c^3*d^12*e^5 + 78*Sqrt[2]*a^
(3/2)*c^(11/4)*d^11*e^6 + 220*Sqrt[2]*a^2*c^(9/4)*d^9*e^8 + 960*a^(9/4)*c^2*d^8*e^9 + 702*Sqrt[2]*a^(5/2)*c^(7
/4)*d^7*e^10 - 770*Sqrt[2]*a^3*c^(5/4)*d^5*e^12 - 1200*a^(13/4)*c*d^4*e^13 - 390*Sqrt[2]*a^(7/2)*c^(3/4)*d^3*e
^14 + 77*Sqrt[2]*a^4*c^(1/4)*d*e^16 + 40*a^(17/4)*e^17)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(11/4) + 15
36*c*d^2*e^11*(13*c*d^4 - 3*a*e^4)*Log[d + e*x] - (3*Sqrt[2]*c^(3/4)*(7*c^4*d^17 - 10*Sqrt[a]*c^(7/2)*d^15*e^2
+ 50*a*c^3*d^13*e^4 - 78*a^(3/2)*c^(5/2)*d^11*e^6 + 220*a^2*c^2*d^9*e^8 - 702*a^(5/2)*c^(3/2)*d^7*e^10 - 770*
a^3*c*d^5*e^12 + 390*a^(7/2)*Sqrt[c]*d^3*e^14 + 77*a^4*d*e^16)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[
c]*x^2])/a^(11/4) + (3*Sqrt[2]*c^(3/4)*(7*c^4*d^17 - 10*Sqrt[a]*c^(7/2)*d^15*e^2 + 50*a*c^3*d^13*e^4 - 78*a^(3
/2)*c^(5/2)*d^11*e^6 + 220*a^2*c^2*d^9*e^8 - 702*a^(5/2)*c^(3/2)*d^7*e^10 - 770*a^3*c*d^5*e^12 + 390*a^(7/2)*S
qrt[c]*d^3*e^14 + 77*a^4*d*e^16)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(11/4) - 384*c*d^2*
e^11*(13*c*d^4 - 3*a*e^4)*Log[a + c*x^4])/(256*(c*d^4 + a*e^4)^5)

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 1008, normalized size of antiderivative = 0.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1008\)
risch	\(\text {Expression too large to display}\)	\(2238\)

[In]

int(1/(e*x+d)^3/(c*x^4+a)^3,x,method=_RETURNVERBOSE)

[Out]

c/(a*e^4+c*d^4)^5*((-3/16*c^2*d^3*e^2*(35*a^3*e^12+a^2*c*d^4*e^8-39*a*c^2*d^8*e^4-5*c^3*d^12)/a^2*x^7-1/16*e*c
*(7*a^4*e^16-130*a^3*c*d^4*e^12-80*a^2*c^2*d^8*e^8+66*a*c^3*d^12*e^4+9*c^4*d^16)/a^2*x^6+1/32*c*d*(45*a^4*e^16
-258*a^3*c*d^4*e^12-260*a^2*c^2*d^8*e^8+50*a*c^3*d^12*e^4+7*c^4*d^16)/a^2*x^5+(-3*a^2*c*d^2*e^15+6*a*c^2*d^6*e
^11+9*c^3*d^10*e^7)*x^4-1/16*c*d^3*e^2*(125*a^3*e^12+31*a^2*c*d^4*e^8-121*a*c^2*d^8*e^4-27*c^3*d^12)/a*x^3-3/1
6*e*(3*a^4*e^16-50*a^3*c*d^4*e^12-40*a^2*c^2*d^8*e^8+18*a*c^3*d^12*e^4+5*c^4*d^16)/a*x^2+1/32*d*(57*a^4*e^16-2
82*a^3*c*d^4*e^12-340*a^2*c^2*d^8*e^8+10*a*c^3*d^12*e^4+11*c^4*d^16)/a*x-15/4*a^3*d^2*e^15+23/4*a^2*d^6*e^11*c
+43/4*a*d^10*e^7*c^2+5/4*d^14*e^3*c^3)/(c*x^4+a)^2+3/32/a^2*(1/8*(77*a^4*d*e^16-770*a^3*c*d^5*e^12+220*a^2*c^2
*d^9*e^8+50*a*c^3*d^13*e^4+7*c^4*d^17)*(a/c)^(1/4)/a*2^(1/2)*(ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-
(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/2*(
-20*a^4*e^17+600*a^3*c*d^4*e^13-480*a^2*c^2*d^8*e^9-88*a*c^3*d^12*e^5-12*c^4*d^16*e)/(a*c)^(1/2)*arctan(x^2*(c
/a)^(1/2))+1/8*(-390*a^3*c*d^3*e^14+702*a^2*c^2*d^7*e^10+78*a*c^3*d^11*e^6+10*c^4*d^15*e^2)/c/(a/c)^(1/4)*2^(1
/2)*(ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c
)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/4*(192*a^3*c*d^2*e^15-832*a^2*c^2*d^6*e^11)/c*ln(c*x^4+a)))-
1/2*e^11/(a*e^4+c*d^4)^3/(e*x+d)^2-12*c*d^3*e^11/(a*e^4+c*d^4)^4/(e*x+d)-6*e^11*c*d^2*(3*a*e^4-13*c*d^4)/(a*e^
4+c*d^4)^5*ln(e*x+d)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)^3/(c*x^4+a)^3,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)**3/(c*x**4+a)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 2198, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^3/(c*x^4+a)^3,x, algorithm="maxima")

[Out]

-3/256*c*(sqrt(2)*(832*sqrt(2)*a^(11/4)*c^(9/4)*d^6*e^11 - 192*sqrt(2)*a^(15/4)*c^(5/4)*d^2*e^15 - 7*c^5*d^17
+ 10*sqrt(a)*c^(9/2)*d^15*e^2 - 50*a*c^4*d^13*e^4 + 78*a^(3/2)*c^(7/2)*d^11*e^6 - 220*a^2*c^3*d^9*e^8 + 702*a^
(5/2)*c^(5/2)*d^7*e^10 + 770*a^3*c^2*d^5*e^12 - 390*a^(7/2)*c^(3/2)*d^3*e^14 - 77*a^4*c*d*e^16)*log(sqrt(c)*x^
2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + sqrt(2)*(832*sqrt(2)*a^(11/4)*c^(9/4)*d^6*e^11 -
192*sqrt(2)*a^(15/4)*c^(5/4)*d^2*e^15 + 7*c^5*d^17 - 10*sqrt(a)*c^(9/2)*d^15*e^2 + 50*a*c^4*d^13*e^4 - 78*a^(3
/2)*c^(7/2)*d^11*e^6 + 220*a^2*c^3*d^9*e^8 - 702*a^(5/2)*c^(5/2)*d^7*e^10 - 770*a^3*c^2*d^5*e^12 + 390*a^(7/2)
*c^(3/2)*d^3*e^14 + 77*a^4*c*d*e^16)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4))
- 2*(7*sqrt(2)*a^(1/4)*c^(21/4)*d^17 + 10*sqrt(2)*a^(3/4)*c^(19/4)*d^15*e^2 + 50*sqrt(2)*a^(5/4)*c^(17/4)*d^13
*e^4 + 78*sqrt(2)*a^(7/4)*c^(15/4)*d^11*e^6 + 220*sqrt(2)*a^(9/4)*c^(13/4)*d^9*e^8 + 702*sqrt(2)*a^(11/4)*c^(1
1/4)*d^7*e^10 - 770*sqrt(2)*a^(13/4)*c^(9/4)*d^5*e^12 - 390*sqrt(2)*a^(15/4)*c^(7/4)*d^3*e^14 + 77*sqrt(2)*a^(
17/4)*c^(5/4)*d*e^16 + 24*sqrt(a)*c^5*d^16*e + 176*a^(3/2)*c^4*d^12*e^5 + 960*a^(5/2)*c^3*d^8*e^9 - 1200*a^(7/
2)*c^2*d^4*e^13 + 40*a^(9/2)*c*e^17)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*s
qrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(5/4)) - 2*(7*sqrt(2)*a^(1/4)*c^(21/4)*d^17 + 10*sqrt(2)*a^(3/4)*c^(
19/4)*d^15*e^2 + 50*sqrt(2)*a^(5/4)*c^(17/4)*d^13*e^4 + 78*sqrt(2)*a^(7/4)*c^(15/4)*d^11*e^6 + 220*sqrt(2)*a^(
9/4)*c^(13/4)*d^9*e^8 + 702*sqrt(2)*a^(11/4)*c^(11/4)*d^7*e^10 - 770*sqrt(2)*a^(13/4)*c^(9/4)*d^5*e^12 - 390*s
qrt(2)*a^(15/4)*c^(7/4)*d^3*e^14 + 77*sqrt(2)*a^(17/4)*c^(5/4)*d*e^16 - 24*sqrt(a)*c^5*d^16*e - 176*a^(3/2)*c^
4*d^12*e^5 - 960*a^(5/2)*c^3*d^8*e^9 + 1200*a^(7/2)*c^2*d^4*e^13 - 40*a^(9/2)*c*e^17)*arctan(1/2*sqrt(2)*(2*sq
rt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(5/4)))/(a^2*c^5*d^
20 + 5*a^3*c^4*d^16*e^4 + 10*a^4*c^3*d^12*e^8 + 10*a^5*c^2*d^8*e^12 + 5*a^6*c*d^4*e^16 + a^7*e^20) + 6*(13*c^2
*d^6*e^11 - 3*a*c*d^2*e^15)*log(e*x + d)/(c^5*d^20 + 5*a*c^4*d^16*e^4 + 10*a^2*c^3*d^12*e^8 + 10*a^3*c^2*d^8*e
^12 + 5*a^4*c*d^4*e^16 + a^5*e^20) + 1/32*(40*a^2*c^3*d^12*e^3 + 304*a^3*c^2*d^8*e^7 - 520*a^4*c*d^4*e^11 - 16
*a^5*e^15 + 6*(5*c^5*d^11*e^4 + 34*a*c^4*d^7*e^8 - 99*a^2*c^3*d^3*e^12)*x^9 + 6*(7*c^5*d^12*e^3 + 49*a*c^4*d^8
*e^7 - 91*a^2*c^3*d^4*e^11 - 5*a^3*c^2*e^15)*x^8 + (c^5*d^13*e^2 + 19*a*c^4*d^9*e^6 + 35*a^2*c^3*d^5*e^10 + 17
*a^3*c^2*d*e^14)*x^7 - 4*(c^5*d^14*e + 7*a*c^4*d^10*e^5 + 11*a^2*c^3*d^6*e^9 + 5*a^3*c^2*d^2*e^13)*x^6 + (7*c^
5*d^15 + 97*a*c^4*d^11*e^4 + 461*a^2*c^3*d^7*e^8 - 1165*a^3*c^2*d^3*e^12)*x^5 + 2*(39*a*c^4*d^12*e^3 + 293*a^2
*c^3*d^8*e^7 - 539*a^3*c^2*d^4*e^11 - 25*a^4*c*e^15)*x^4 + (5*a*c^4*d^13*e^2 + 31*a^2*c^3*d^9*e^6 + 47*a^3*c^2
*d^5*e^10 + 21*a^4*c*d*e^14)*x^3 - 8*(a*c^4*d^14*e + 5*a^2*c^3*d^10*e^5 + 7*a^3*c^2*d^6*e^9 + 3*a^4*c*d^2*e^13
)*x^2 + (11*a*c^4*d^15 + 79*a^2*c^3*d^11*e^4 + 269*a^3*c^2*d^7*e^8 - 567*a^4*c*d^3*e^12)*x)/(a^4*c^4*d^18 + 4*
a^5*c^3*d^14*e^4 + 6*a^6*c^2*d^10*e^8 + 4*a^7*c*d^6*e^12 + a^8*d^2*e^16 + (a^2*c^6*d^16*e^2 + 4*a^3*c^5*d^12*e
^6 + 6*a^4*c^4*d^8*e^10 + 4*a^5*c^3*d^4*e^14 + a^6*c^2*e^18)*x^10 + 2*(a^2*c^6*d^17*e + 4*a^3*c^5*d^13*e^5 + 6
*a^4*c^4*d^9*e^9 + 4*a^5*c^3*d^5*e^13 + a^6*c^2*d*e^17)*x^9 + (a^2*c^6*d^18 + 4*a^3*c^5*d^14*e^4 + 6*a^4*c^4*d
^10*e^8 + 4*a^5*c^3*d^6*e^12 + a^6*c^2*d^2*e^16)*x^8 + 2*(a^3*c^5*d^16*e^2 + 4*a^4*c^4*d^12*e^6 + 6*a^5*c^3*d^
8*e^10 + 4*a^6*c^2*d^4*e^14 + a^7*c*e^18)*x^6 + 4*(a^3*c^5*d^17*e + 4*a^4*c^4*d^13*e^5 + 6*a^5*c^3*d^9*e^9 + 4
*a^6*c^2*d^5*e^13 + a^7*c*d*e^17)*x^5 + 2*(a^3*c^5*d^18 + 4*a^4*c^4*d^14*e^4 + 6*a^5*c^3*d^10*e^8 + 4*a^6*c^2*
d^6*e^12 + a^7*c*d^2*e^16)*x^4 + (a^4*c^4*d^16*e^2 + 4*a^5*c^3*d^12*e^6 + 6*a^6*c^2*d^8*e^10 + 4*a^7*c*d^4*e^1
4 + a^8*e^18)*x^2 + 2*(a^4*c^4*d^17*e + 4*a^5*c^3*d^13*e^5 + 6*a^6*c^2*d^9*e^9 + 4*a^7*c*d^5*e^13 + a^8*d*e^17
)*x)

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 2200, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^3/(c*x^4+a)^3,x, algorithm="giac")

[Out]

-3/64*(23*sqrt(2)*a*c^4*d^6*e - 115*sqrt(2)*a^2*c^3*d^2*e^5 + 30*sqrt(2)*sqrt(a*c)*a*c^3*d^4*e^3 + 20*sqrt(2)*
sqrt(a*c)*a^2*c^2*e^7 - 65*(a*c^3)^(1/4)*a*c^3*d^5*e^2 + 123*(a*c^3)^(1/4)*a^2*c^2*d*e^6 - 7*(a*c^3)^(3/4)*c^2
*d^7 + 65*(a*c^3)^(3/4)*a*c*d^3*e^4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(25*sqrt(2)*a
^4*c^4*d^8*e^2 + 90*sqrt(2)*a^5*c^3*d^4*e^6 + sqrt(2)*a^6*c^2*e^10 - sqrt(2)*sqrt(a*c)*a^3*c^4*d^10 - 90*sqrt(
2)*sqrt(a*c)*a^4*c^3*d^6*e^4 - 25*sqrt(2)*sqrt(a*c)*a^5*c^2*d^2*e^8 - 80*(a*c^3)^(1/4)*a^4*c^3*d^7*e^3 - 80*(a
*c^3)^(1/4)*a^5*c^2*d^3*e^7 - 10*(a*c^3)^(3/4)*a^3*c^2*d^9*e - 148*(a*c^3)^(3/4)*a^4*c*d^5*e^5 - 10*(a*c^3)^(3
/4)*a^5*d*e^9) + 3/64*(23*sqrt(2)*a*c^4*d^6*e - 115*sqrt(2)*a^2*c^3*d^2*e^5 - 30*sqrt(2)*sqrt(a*c)*a*c^3*d^4*e
^3 - 20*sqrt(2)*sqrt(a*c)*a^2*c^2*e^7 + 65*(a*c^3)^(1/4)*a*c^3*d^5*e^2 - 123*(a*c^3)^(1/4)*a^2*c^2*d*e^6 + 7*(
a*c^3)^(3/4)*c^2*d^7 - 65*(a*c^3)^(3/4)*a*c*d^3*e^4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4
))/(25*sqrt(2)*a^4*c^4*d^8*e^2 + 90*sqrt(2)*a^5*c^3*d^4*e^6 + sqrt(2)*a^6*c^2*e^10 - sqrt(2)*sqrt(a*c)*a^3*c^4
*d^10 - 90*sqrt(2)*sqrt(a*c)*a^4*c^3*d^6*e^4 - 25*sqrt(2)*sqrt(a*c)*a^5*c^2*d^2*e^8 + 80*(a*c^3)^(1/4)*a^4*c^3
*d^7*e^3 + 80*(a*c^3)^(1/4)*a^5*c^2*d^3*e^7 + 10*(a*c^3)^(3/4)*a^3*c^2*d^9*e + 148*(a*c^3)^(3/4)*a^4*c*d^5*e^5
+ 10*(a*c^3)^(3/4)*a^5*d*e^9) + 3/128*(7*(a*c^3)^(1/4)*c^5*d^17 + 50*(a*c^3)^(1/4)*a*c^4*d^13*e^4 + 220*(a*c^
3)^(1/4)*a^2*c^3*d^9*e^8 - 770*(a*c^3)^(1/4)*a^3*c^2*d^5*e^12 + 77*(a*c^3)^(1/4)*a^4*c*d*e^16 - 10*(a*c^3)^(3/
4)*c^3*d^15*e^2 - 78*(a*c^3)^(3/4)*a*c^2*d^11*e^6 - 702*(a*c^3)^(3/4)*a^2*c*d^7*e^10 + 390*(a*c^3)^(3/4)*a^3*d
^3*e^14)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a^3*c^6*d^20 + 5*sqrt(2)*a^4*c^5*d^16*e^4 + 10*
sqrt(2)*a^5*c^4*d^12*e^8 + 10*sqrt(2)*a^6*c^3*d^8*e^12 + 5*sqrt(2)*a^7*c^2*d^4*e^16 + sqrt(2)*a^8*c*e^20) - 3/
128*(7*(a*c^3)^(1/4)*c^5*d^17 + 50*(a*c^3)^(1/4)*a*c^4*d^13*e^4 + 220*(a*c^3)^(1/4)*a^2*c^3*d^9*e^8 - 770*(a*c
^3)^(1/4)*a^3*c^2*d^5*e^12 + 77*(a*c^3)^(1/4)*a^4*c*d*e^16 - 10*(a*c^3)^(3/4)*c^3*d^15*e^2 - 78*(a*c^3)^(3/4)*
a*c^2*d^11*e^6 - 702*(a*c^3)^(3/4)*a^2*c*d^7*e^10 + 390*(a*c^3)^(3/4)*a^3*d^3*e^14)*log(x^2 - sqrt(2)*x*(a/c)^
(1/4) + sqrt(a/c))/(sqrt(2)*a^3*c^6*d^20 + 5*sqrt(2)*a^4*c^5*d^16*e^4 + 10*sqrt(2)*a^5*c^4*d^12*e^8 + 10*sqrt(
2)*a^6*c^3*d^8*e^12 + 5*sqrt(2)*a^7*c^2*d^4*e^16 + sqrt(2)*a^8*c*e^20) - 3/2*(13*c^2*d^6*e^11 - 3*a*c*d^2*e^15
)*log(abs(c*x^4 + a))/(c^5*d^20 + 5*a*c^4*d^16*e^4 + 10*a^2*c^3*d^12*e^8 + 10*a^3*c^2*d^8*e^12 + 5*a^4*c*d^4*e
^16 + a^5*e^20) + 6*(13*c^2*d^6*e^12 - 3*a*c*d^2*e^16)*log(abs(e*x + d))/(c^5*d^20*e + 5*a*c^4*d^16*e^5 + 10*a
^2*c^3*d^12*e^9 + 10*a^3*c^2*d^8*e^13 + 5*a^4*c*d^4*e^17 + a^5*e^21) + 1/32*(30*c^5*d^11*e^4*x^9 + 204*a*c^4*d
^7*e^8*x^9 - 594*a^2*c^3*d^3*e^12*x^9 + 42*c^5*d^12*e^3*x^8 + 294*a*c^4*d^8*e^7*x^8 - 546*a^2*c^3*d^4*e^11*x^8
- 30*a^3*c^2*e^15*x^8 + c^5*d^13*e^2*x^7 + 19*a*c^4*d^9*e^6*x^7 + 35*a^2*c^3*d^5*e^10*x^7 + 17*a^3*c^2*d*e^14
*x^7 - 4*c^5*d^14*e*x^6 - 28*a*c^4*d^10*e^5*x^6 - 44*a^2*c^3*d^6*e^9*x^6 - 20*a^3*c^2*d^2*e^13*x^6 + 7*c^5*d^1
5*x^5 + 97*a*c^4*d^11*e^4*x^5 + 461*a^2*c^3*d^7*e^8*x^5 - 1165*a^3*c^2*d^3*e^12*x^5 + 78*a*c^4*d^12*e^3*x^4 +
586*a^2*c^3*d^8*e^7*x^4 - 1078*a^3*c^2*d^4*e^11*x^4 - 50*a^4*c*e^15*x^4 + 5*a*c^4*d^13*e^2*x^3 + 31*a^2*c^3*d^
9*e^6*x^3 + 47*a^3*c^2*d^5*e^10*x^3 + 21*a^4*c*d*e^14*x^3 - 8*a*c^4*d^14*e*x^2 - 40*a^2*c^3*d^10*e^5*x^2 - 56*
a^3*c^2*d^6*e^9*x^2 - 24*a^4*c*d^2*e^13*x^2 + 11*a*c^4*d^15*x + 79*a^2*c^3*d^11*e^4*x + 269*a^3*c^2*d^7*e^8*x
- 567*a^4*c*d^3*e^12*x + 40*a^2*c^3*d^12*e^3 + 304*a^3*c^2*d^8*e^7 - 520*a^4*c*d^4*e^11 - 16*a^5*e^15)/((a^2*c
^4*d^16 + 4*a^3*c^3*d^12*e^4 + 6*a^4*c^2*d^8*e^8 + 4*a^5*c*d^4*e^12 + a^6*e^16)*(c*e*x^5 + c*d*x^4 + a*e*x + a
*d)^2)

Mupad [B] (verification not implemented)

Time = 12.67 (sec) , antiderivative size = 6280, normalized size of antiderivative = 2.85 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \]

[In]

int(1/((a + c*x^4)^3*(d + e*x)^3),x)

[Out]

symsum(log(root(2684354560*a^12*c^9*d^36*e^4*z^5 + 32212254720*a^18*c^3*d^12*e^28*z^5 + 32212254720*a^14*c^7*d
^28*e^12*z^5 + 2684354560*a^20*c*d^4*e^36*z^5 + 56371445760*a^17*c^4*d^16*e^24*z^5 + 56371445760*a^15*c^6*d^24
*e^16*z^5 + 12079595520*a^19*c^2*d^8*e^32*z^5 + 12079595520*a^13*c^8*d^32*e^8*z^5 + 67645734912*a^16*c^5*d^20*
e^20*z^5 + 268435456*a^11*c^10*d^40*z^5 + 268435456*a^21*e^40*z^5 + 45339770880*a^9*c^6*d^20*e^14*z^3 - 791484
82560*a^13*c^2*d^4*e^30*z^3 + 791941349376*a^12*c^3*d^8*e^26*z^3 + 1239810048*a^7*c^8*d^28*e^6*z^3 - 155544492
4416*a^11*c^4*d^12*e^22*z^3 + 83755008*a^6*c^9*d^32*e^2*z^3 + 81566760960*a^10*c^5*d^16*e^18*z^3 + 12177506304
*a^8*c^7*d^24*e^10*z^3 + 117964800*a^14*c*e^34*z^3 - 2785204224*a^6*c^6*d^18*e^13*z^2 + 8128512*a^3*c^9*d^30*e
*z^2 + 2700933120*a^10*c^2*d^2*e^29*z^2 - 543361222656*a^8*c^4*d^10*e^21*z^2 + 1048135680*a^5*c^7*d^22*e^9*z^2
+ 118499328*a^4*c^8*d^26*e^5*z^2 - 55938263040*a^7*c^5*d^14*e^17*z^2 + 123990497280*a^9*c^3*d^6*e^25*z^2 + 24
139215*a^2*c^7*d^20*e^8*z + 2819286*a*c^8*d^24*e^4*z + 10462847841*a^6*c^3*d^4*e^24*z - 5777473473*a^4*c^5*d^1
2*e^16*z - 43509753450*a^5*c^4*d^8*e^20*z - 548810316*a^3*c^6*d^16*e^12*z + 12960000*a^7*c^2*e^28*z + 194481*c
^9*d^28*z - 977636142*a^2*c^4*d^6*e^19 + 233280000*a^3*c^3*d^2*e^23 - 140556060*a*c^5*d^10*e^15 - 15169518*c^6
*d^14*e^11, z, k)*(root(2684354560*a^12*c^9*d^36*e^4*z^5 + 32212254720*a^18*c^3*d^12*e^28*z^5 + 32212254720*a^
14*c^7*d^28*e^12*z^5 + 2684354560*a^20*c*d^4*e^36*z^5 + 56371445760*a^17*c^4*d^16*e^24*z^5 + 56371445760*a^15*
c^6*d^24*e^16*z^5 + 12079595520*a^19*c^2*d^8*e^32*z^5 + 12079595520*a^13*c^8*d^32*e^8*z^5 + 67645734912*a^16*c
^5*d^20*e^20*z^5 + 268435456*a^11*c^10*d^40*z^5 + 268435456*a^21*e^40*z^5 + 45339770880*a^9*c^6*d^20*e^14*z^3
- 79148482560*a^13*c^2*d^4*e^30*z^3 + 791941349376*a^12*c^3*d^8*e^26*z^3 + 1239810048*a^7*c^8*d^28*e^6*z^3 - 1
555444924416*a^11*c^4*d^12*e^22*z^3 + 83755008*a^6*c^9*d^32*e^2*z^3 + 81566760960*a^10*c^5*d^16*e^18*z^3 + 121
77506304*a^8*c^7*d^24*e^10*z^3 + 117964800*a^14*c*e^34*z^3 - 2785204224*a^6*c^6*d^18*e^13*z^2 + 8128512*a^3*c^
9*d^30*e*z^2 + 2700933120*a^10*c^2*d^2*e^29*z^2 - 543361222656*a^8*c^4*d^10*e^21*z^2 + 1048135680*a^5*c^7*d^22
*e^9*z^2 + 118499328*a^4*c^8*d^26*e^5*z^2 - 55938263040*a^7*c^5*d^14*e^17*z^2 + 123990497280*a^9*c^3*d^6*e^25*
z^2 + 24139215*a^2*c^7*d^20*e^8*z + 2819286*a*c^8*d^24*e^4*z + 10462847841*a^6*c^3*d^4*e^24*z - 5777473473*a^4
*c^5*d^12*e^16*z - 43509753450*a^5*c^4*d^8*e^20*z - 548810316*a^3*c^6*d^16*e^12*z + 12960000*a^7*c^2*e^28*z +
194481*c^9*d^28*z - 977636142*a^2*c^4*d^6*e^19 + 233280000*a^3*c^3*d^2*e^23 - 140556060*a*c^5*d^10*e^15 - 1516
9518*c^6*d^14*e^11, z, k)*(root(2684354560*a^12*c^9*d^36*e^4*z^5 + 32212254720*a^18*c^3*d^12*e^28*z^5 + 322122
54720*a^14*c^7*d^28*e^12*z^5 + 2684354560*a^20*c*d^4*e^36*z^5 + 56371445760*a^17*c^4*d^16*e^24*z^5 + 563714457
60*a^15*c^6*d^24*e^16*z^5 + 12079595520*a^19*c^2*d^8*e^32*z^5 + 12079595520*a^13*c^8*d^32*e^8*z^5 + 6764573491
2*a^16*c^5*d^20*e^20*z^5 + 268435456*a^11*c^10*d^40*z^5 + 268435456*a^21*e^40*z^5 + 45339770880*a^9*c^6*d^20*e
^14*z^3 - 79148482560*a^13*c^2*d^4*e^30*z^3 + 791941349376*a^12*c^3*d^8*e^26*z^3 + 1239810048*a^7*c^8*d^28*e^6
*z^3 - 1555444924416*a^11*c^4*d^12*e^22*z^3 + 83755008*a^6*c^9*d^32*e^2*z^3 + 81566760960*a^10*c^5*d^16*e^18*z
^3 + 12177506304*a^8*c^7*d^24*e^10*z^3 + 117964800*a^14*c*e^34*z^3 - 2785204224*a^6*c^6*d^18*e^13*z^2 + 812851
2*a^3*c^9*d^30*e*z^2 + 2700933120*a^10*c^2*d^2*e^29*z^2 - 543361222656*a^8*c^4*d^10*e^21*z^2 + 1048135680*a^5*
c^7*d^22*e^9*z^2 + 118499328*a^4*c^8*d^26*e^5*z^2 - 55938263040*a^7*c^5*d^14*e^17*z^2 + 123990497280*a^9*c^3*d
^6*e^25*z^2 + 24139215*a^2*c^7*d^20*e^8*z + 2819286*a*c^8*d^24*e^4*z + 10462847841*a^6*c^3*d^4*e^24*z - 577747
3473*a^4*c^5*d^12*e^16*z - 43509753450*a^5*c^4*d^8*e^20*z - 548810316*a^3*c^6*d^16*e^12*z + 12960000*a^7*c^2*e
^28*z + 194481*c^9*d^28*z - 977636142*a^2*c^4*d^6*e^19 + 233280000*a^3*c^3*d^2*e^23 - 140556060*a*c^5*d^10*e^1
5 - 15169518*c^6*d^14*e^11, z, k)*((44040192*a^8*c^12*d^31*e^5 - 11010048*a^7*c^13*d^35*e + 994050048*a^9*c^11
*d^27*e^9 + 13683916800*a^10*c^10*d^23*e^13 + 42936041472*a^11*c^9*d^19*e^17 + 52628029440*a^12*c^8*d^15*e^21
+ 23429382144*a^13*c^7*d^11*e^25 - 2132803584*a^14*c^6*d^7*e^29 - 3125280768*a^15*c^5*d^3*e^33)/(1048576*(a^16
*e^32 + a^8*c^8*d^32 + 8*a^15*c*d^4*e^28 + 8*a^9*c^7*d^28*e^4 + 28*a^10*c^6*d^24*e^8 + 56*a^11*c^5*d^20*e^12 +
70*a^12*c^4*d^16*e^16 + 56*a^13*c^3*d^12*e^20 + 28*a^14*c^2*d^8*e^24)) + root(2684354560*a^12*c^9*d^36*e^4*z^
5 + 32212254720*a^18*c^3*d^12*e^28*z^5 + 32212254720*a^14*c^7*d^28*e^12*z^5 + 2684354560*a^20*c*d^4*e^36*z^5 +
56371445760*a^17*c^4*d^16*e^24*z^5 + 56371445760*a^15*c^6*d^24*e^16*z^5 + 12079595520*a^19*c^2*d^8*e^32*z^5 +
12079595520*a^13*c^8*d^32*e^8*z^5 + 67645734912*a^16*c^5*d^20*e^20*z^5 + 268435456*a^11*c^10*d^40*z^5 + 26843
5456*a^21*e^40*z^5 + 45339770880*a^9*c^6*d^20*e^14*z^3 - 79148482560*a^13*c^2*d^4*e^30*z^3 + 791941349376*a^12
*c^3*d^8*e^26*z^3 + 1239810048*a^7*c^8*d^28*e^6*z^3 - 1555444924416*a^11*c^4*d^12*e^22*z^3 + 83755008*a^6*c^9*
d^32*e^2*z^3 + 81566760960*a^10*c^5*d^16*e^18*z^3 + 12177506304*a^8*c^7*d^24*e^10*z^3 + 117964800*a^14*c*e^34*
z^3 - 2785204224*a^6*c^6*d^18*e^13*z^2 + 8128512*a^3*c^9*d^30*e*z^2 + 2700933120*a^10*c^2*d^2*e^29*z^2 - 54336
1222656*a^8*c^4*d^10*e^21*z^2 + 1048135680*a^5*c^7*d^22*e^9*z^2 + 118499328*a^4*c^8*d^26*e^5*z^2 - 55938263040
*a^7*c^5*d^14*e^17*z^2 + 123990497280*a^9*c^3*d^6*e^25*z^2 + 24139215*a^2*c^7*d^20*e^8*z + 2819286*a*c^8*d^24*
e^4*z + 10462847841*a^6*c^3*d^4*e^24*z - 5777473473*a^4*c^5*d^12*e^16*z - 43509753450*a^5*c^4*d^8*e^20*z - 548
810316*a^3*c^6*d^16*e^12*z + 12960000*a^7*c^2*e^28*z + 194481*c^9*d^28*z - 977636142*a^2*c^4*d^6*e^19 + 233280
000*a^3*c^3*d^2*e^23 - 140556060*a*c^5*d^10*e^15 - 15169518*c^6*d^14*e^11, z, k)*((402653184*a^19*c^4*d*e^38 -
134217728*a^10*c^13*d^37*e^2 - 671088640*a^11*c^12*d^33*e^6 - 536870912*a^12*c^11*d^29*e^10 + 3758096384*a^13
*c^10*d^25*e^14 + 13153337344*a^14*c^9*d^21*e^18 + 20669530112*a^15*c^8*d^17*e^22 + 18790481920*a^16*c^7*d^13*
e^26 + 10200547328*a^17*c^6*d^9*e^30 + 3087007744*a^18*c^5*d^5*e^34)/(1048576*(a^16*e^32 + a^8*c^8*d^32 + 8*a^
15*c*d^4*e^28 + 8*a^9*c^7*d^28*e^4 + 28*a^10*c^6*d^24*e^8 + 56*a^11*c^5*d^20*e^12 + 70*a^12*c^4*d^16*e^16 + 56
*a^13*c^3*d^12*e^20 + 28*a^14*c^2*d^8*e^24)) + (x*(335544320*a^19*c^4*e^39 - 201326592*a^10*c^13*d^36*e^3 - 12
75068416*a^11*c^12*d^32*e^7 - 2952790016*a^12*c^11*d^28*e^11 - 1879048192*a^13*c^10*d^24*e^15 + 4697620480*a^1
4*c^9*d^20*e^19 + 12213813248*a^15*c^8*d^16*e^23 + 13153337344*a^16*c^7*d^12*e^27 + 7784628224*a^17*c^6*d^8*e^
31 + 2483027968*a^18*c^5*d^4*e^35))/(1048576*(a^16*e^32 + a^8*c^8*d^32 + 8*a^15*c*d^4*e^28 + 8*a^9*c^7*d^28*e^
4 + 28*a^10*c^6*d^24*e^8 + 56*a^11*c^5*d^20*e^12 + 70*a^12*c^4*d^16*e^16 + 56*a^13*c^3*d^12*e^20 + 28*a^14*c^2
*d^8*e^24))) - (x*(40894464*a^7*c^13*d^34*e^2 + 276824064*a^8*c^12*d^30*e^6 + 968884224*a^9*c^11*d^26*e^10 - 1
3010731008*a^10*c^10*d^22*e^14 - 53433335808*a^11*c^9*d^18*e^18 - 71647100928*a^12*c^8*d^14*e^22 - 34313601024
*a^13*c^7*d^10*e^26 + 1837105152*a^14*c^6*d^6*e^30 + 4193255424*a^15*c^5*d^2*e^34))/(1048576*(a^16*e^32 + a^8*
c^8*d^32 + 8*a^15*c*d^4*e^28 + 8*a^9*c^7*d^28*e^4 + 28*a^10*c^6*d^24*e^8 + 56*a^11*c^5*d^20*e^12 + 70*a^12*c^4
*d^16*e^16 + 56*a^13*c^3*d^12*e^20 + 28*a^14*c^2*d^8*e^24))) + (33914880*a^12*c^5*d*e^32 + 13713408*a^5*c^12*d
^29*e^4 + 225902592*a^6*c^11*d^25*e^8 + 2352070656*a^7*c^10*d^21*e^12 + 2474606592*a^8*c^9*d^17*e^16 - 2136180
3264*a^9*c^8*d^13*e^20 + 88707170304*a^10*c^7*d^9*e^24 - 5526503424*a^11*c^6*d^5*e^28)/(1048576*(a^16*e^32 + a
^8*c^8*d^32 + 8*a^15*c*d^4*e^28 + 8*a^9*c^7*d^28*e^4 + 28*a^10*c^6*d^24*e^8 + 56*a^11*c^5*d^20*e^12 + 70*a^12*
c^4*d^16*e^16 + 56*a^13*c^3*d^12*e^20 + 28*a^14*c^2*d^8*e^24)) + (x*(132710400*a^12*c^5*e^33 - 1806336*a^4*c^1
3*d^32*e - 2027520*a^5*c^12*d^28*e^5 + 162017280*a^6*c^11*d^24*e^9 + 4635316224*a^7*c^10*d^20*e^13 + 156042731
52*a^8*c^9*d^16*e^17 + 39318663168*a^9*c^8*d^12*e^21 + 64184389632*a^10*c^7*d^8*e^25 - 2525073408*a^11*c^6*d^4
*e^29))/(1048576*(a^16*e^32 + a^8*c^8*d^32 + 8*a^15*c*d^4*e^28 + 8*a^9*c^7*d^28*e^4 + 28*a^10*c^6*d^24*e^8 + 5
6*a^11*c^5*d^20*e^12 + 70*a^12*c^4*d^16*e^16 + 56*a^13*c^3*d^12*e^20 + 28*a^14*c^2*d^8*e^24))) + (320544*a^2*c
^12*d^27*e^3 + 11448000*a^3*c^11*d^23*e^7 + 114031584*a^4*c^10*d^19*e^11 - 213750144*a^5*c^9*d^15*e^15 - 34992
71712*a^6*c^8*d^11*e^19 + 9699804864*a^7*c^7*d^7*e^23 - 933615072*a^8*c^6*d^3*e^27)/(1048576*(a^16*e^32 + a^8*
c^8*d^32 + 8*a^15*c*d^4*e^28 + 8*a^9*c^7*d^28*e^4 + 28*a^10*c^6*d^24*e^8 + 56*a^11*c^5*d^20*e^12 + 70*a^12*c^4
*d^16*e^16 + 56*a^13*c^3*d^12*e^20 + 28*a^14*c^2*d^8*e^24)) + (x*(514944*a^2*c^12*d^26*e^4 + 14314752*a^3*c^11
*d^22*e^8 + 266343552*a^4*c^10*d^18*e^12 + 297948672*a^5*c^9*d^14*e^16 - 2642613120*a^6*c^8*d^10*e^20 + 178245
9648*a^7*c^7*d^6*e^24 + 846599040*a^8*c^6*d^2*e^28))/(1048576*(a^16*e^32 + a^8*c^8*d^32 + 8*a^15*c*d^4*e^28 +
8*a^9*c^7*d^28*e^4 + 28*a^10*c^6*d^24*e^8 + 56*a^11*c^5*d^20*e^12 + 70*a^12*c^4*d^16*e^16 + 56*a^13*c^3*d^12*e
^20 + 28*a^14*c^2*d^8*e^24))) + (194481*c^11*d^21*e^6 + 2430324*a*c^10*d^17*e^10 + 12960000*a^5*c^6*d*e^26 - 5
918346*a^2*c^9*d^13*e^14 - 83522988*a^3*c^8*d^9*e^18 + 71628705*a^4*c^7*d^5*e^22)/(1048576*(a^16*e^32 + a^8*c^
8*d^32 + 8*a^15*c*d^4*e^28 + 8*a^9*c^7*d^28*e^4 + 28*a^10*c^6*d^24*e^8 + 56*a^11*c^5*d^20*e^12 + 70*a^12*c^4*d
^16*e^16 + 56*a^13*c^3*d^12*e^20 + 28*a^14*c^2*d^8*e^24)) + (x*(12960000*a^5*c^6*e^27 + 194481*c^11*d^20*e^7 +
2430324*a*c^10*d^16*e^11 - 21081546*a^2*c^9*d^12*e^15 - 227814444*a^3*c^8*d^8*e^19 + 105734241*a^4*c^7*d^4*e^
23))/(1048576*(a^16*e^32 + a^8*c^8*d^32 + 8*a^15*c*d^4*e^28 + 8*a^9*c^7*d^28*e^4 + 28*a^10*c^6*d^24*e^8 + 56*a
^11*c^5*d^20*e^12 + 70*a^12*c^4*d^16*e^16 + 56*a^13*c^3*d^12*e^20 + 28*a^14*c^2*d^8*e^24)))*root(2684354560*a^
12*c^9*d^36*e^4*z^5 + 32212254720*a^18*c^3*d^12*e^28*z^5 + 32212254720*a^14*c^7*d^28*e^12*z^5 + 2684354560*a^2
0*c*d^4*e^36*z^5 + 56371445760*a^17*c^4*d^16*e^24*z^5 + 56371445760*a^15*c^6*d^24*e^16*z^5 + 12079595520*a^19*
c^2*d^8*e^32*z^5 + 12079595520*a^13*c^8*d^32*e^8*z^5 + 67645734912*a^16*c^5*d^20*e^20*z^5 + 268435456*a^11*c^1
0*d^40*z^5 + 268435456*a^21*e^40*z^5 + 45339770880*a^9*c^6*d^20*e^14*z^3 - 79148482560*a^13*c^2*d^4*e^30*z^3 +
791941349376*a^12*c^3*d^8*e^26*z^3 + 1239810048*a^7*c^8*d^28*e^6*z^3 - 1555444924416*a^11*c^4*d^12*e^22*z^3 +
83755008*a^6*c^9*d^32*e^2*z^3 + 81566760960*a^10*c^5*d^16*e^18*z^3 + 12177506304*a^8*c^7*d^24*e^10*z^3 + 1179
64800*a^14*c*e^34*z^3 - 2785204224*a^6*c^6*d^18*e^13*z^2 + 8128512*a^3*c^9*d^30*e*z^2 + 2700933120*a^10*c^2*d^
2*e^29*z^2 - 543361222656*a^8*c^4*d^10*e^21*z^2 + 1048135680*a^5*c^7*d^22*e^9*z^2 + 118499328*a^4*c^8*d^26*e^5
*z^2 - 55938263040*a^7*c^5*d^14*e^17*z^2 + 123990497280*a^9*c^3*d^6*e^25*z^2 + 24139215*a^2*c^7*d^20*e^8*z + 2
819286*a*c^8*d^24*e^4*z + 10462847841*a^6*c^3*d^4*e^24*z - 5777473473*a^4*c^5*d^12*e^16*z - 43509753450*a^5*c^
4*d^8*e^20*z - 548810316*a^3*c^6*d^16*e^12*z + 12960000*a^7*c^2*e^28*z + 194481*c^9*d^28*z - 977636142*a^2*c^4
*d^6*e^19 + 233280000*a^3*c^3*d^2*e^23 - 140556060*a*c^5*d^10*e^15 - 15169518*c^6*d^14*e^11, z, k), k, 1, 5) -
((2*a^3*e^15 - 5*c^3*d^12*e^3 - 38*a*c^2*d^8*e^7 + 65*a^2*c*d^4*e^11)/(4*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)^
2) + (3*x^8*(5*a^3*c^2*e^15 - 7*c^5*d^12*e^3 - 49*a*c^4*d^8*e^7 + 91*a^2*c^3*d^4*e^11))/(16*a^2*(a^4*e^16 + c^
4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8)) - (x^5*(7*c^5*d^15 + 97*a*c^4*d^11*e^4 + 46
1*a^2*c^3*d^7*e^8 - 1165*a^3*c^2*d^3*e^12))/(32*a^2*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)^2) - (3*x^9*(5*c^5*d^1
1*e^4 + 34*a*c^4*d^7*e^8 - 99*a^2*c^3*d^3*e^12))/(16*a^2*(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4
*e^12 + 6*a^2*c^2*d^8*e^8)) + (x^2*(c^2*d^6*e + 3*a*c*d^2*e^5))/(4*a*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) - (x
^3*(5*c^2*d^5*e^2 + 21*a*c*d*e^6))/(32*a*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) - (x*(11*c^4*d^15 + 79*a*c^3*d^1
1*e^4 - 567*a^3*c*d^3*e^12 + 269*a^2*c^2*d^7*e^8))/(32*a*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)^2) + (x^6*(c^3*d^
6*e + 5*a*c^2*d^2*e^5))/(8*a^2*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) + (x^4*(25*a^3*c*e^15 - 39*c^4*d^12*e^3 -
293*a*c^3*d^8*e^7 + 539*a^2*c^2*d^4*e^11))/(16*a*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)^2) - (e^2*x^7*(c^3*d^5 +
17*a*c^2*d*e^4))/(32*a^2*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)))/(a^2*d^2 + a^2*e^2*x^2 + c^2*d^2*x^8 + c^2*e^2*
x^10 + 2*a^2*d*e*x + 2*a*c*d^2*x^4 + 2*a*c*e^2*x^6 + 2*c^2*d*e*x^9 + 4*a*c*d*e*x^5)