\(\int \frac {1}{(d+e x)^2 (a+c x^4)^3} \, dx\) [190]

Optimal result

Integrand size = 17, antiderivative size = 1533 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Output:

-e^11/(a*e^4+c*d^4)^3/(e*x+d)+1/2*c*d^3*e^3/(a*e^4+c*d^4)^2/(c*x^4+a)^2+1/ 
8*c*x*(d^2*(-3*a*e^4+c*d^4)-2*d*e*(-a*e^4+c*d^4)*x+e^2*(-a*e^4+3*c*d^4)*x^ 
2)/a/(a*e^4+c*d^4)^2/(c*x^4+a)^2+2*c*d^3*e^7/(a*e^4+c*d^4)^3/(c*x^4+a)+1/3 
2*c*x*(7*d^2*(-3*a*e^4+c*d^4)-12*d*e*(-a*e^4+c*d^4)*x+5*e^2*(-a*e^4+3*c*d^ 
4)*x^2)/a^2/(a*e^4+c*d^4)^2/(c*x^4+a)+1/4*c*e^4*x*(d^2*(-3*a*e^4+5*c*d^4)- 
2*d*e*(-a*e^4+3*c*d^4)*x+e^2*(-a*e^4+7*c*d^4)*x^2)/a/(a*e^4+c*d^4)^3/(c*x^ 
4+a)-c^(1/2)*d*e^9*(-a*e^4+5*c*d^4)*arctan(c^(1/2)*x^2/a^(1/2))/a^(1/2)/(a 
*e^4+c*d^4)^4-1/2*c^(1/2)*d*e^5*(-a*e^4+3*c*d^4)*arctan(c^(1/2)*x^2/a^(1/2 
))/a^(3/2)/(a*e^4+c*d^4)^3-3/8*c^(1/2)*d*e*(-a*e^4+c*d^4)*arctan(c^(1/2)*x 
^2/a^(1/2))/a^(5/2)/(a*e^4+c*d^4)^2+1/128*c^(1/4)*(21*c^(1/2)*d^2*(-3*a*e^ 
4+c*d^4)+5*a^(1/2)*e^2*(-a*e^4+3*c*d^4))*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/ 
4))*2^(1/2)/a^(11/4)/(a*e^4+c*d^4)^2+1/16*c^(1/4)*e^4*(3*c^(1/2)*d^2*(-3*a 
*e^4+5*c*d^4)+a^(1/2)*e^2*(-a*e^4+7*c*d^4))*arctan(-1+2^(1/2)*c^(1/4)*x/a^ 
(1/4))*2^(1/2)/a^(7/4)/(a*e^4+c*d^4)^3+1/4*c^(1/4)*e^8*(3*c^(1/2)*d^2*(-a* 
e^4+3*c*d^4)+a^(1/2)*e^2*(-a*e^4+11*c*d^4))*arctan(-1+2^(1/2)*c^(1/4)*x/a^ 
(1/4))*2^(1/2)/a^(3/4)/(a*e^4+c*d^4)^4+1/128*c^(1/4)*(21*c^(1/2)*d^2*(-3*a 
*e^4+c*d^4)+5*a^(1/2)*e^2*(-a*e^4+3*c*d^4))*arctan(1+2^(1/2)*c^(1/4)*x/a^( 
1/4))*2^(1/2)/a^(11/4)/(a*e^4+c*d^4)^2+1/16*c^(1/4)*e^4*(3*c^(1/2)*d^2*(-3 
*a*e^4+5*c*d^4)+a^(1/2)*e^2*(-a*e^4+7*c*d^4))*arctan(1+2^(1/2)*c^(1/4)*x/a 
^(1/4))*2^(1/2)/a^(7/4)/(a*e^4+c*d^4)^3+1/4*c^(1/4)*e^8*(3*c^(1/2)*d^2*...
 

Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 1115, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[1/((d + e*x)^2*(a + c*x^4)^3),x]
 

Output:

((-256*e^11*(c*d^4 + a*e^4))/(d + e*x) + (8*c*(c*d^4 + a*e^4)*(c^2*d^8*x*( 
7*d^2 - 12*d*e*x + 15*e^2*x^2) + 2*a*c*d^4*e^4*x*(13*d^2 - 24*d*e*x + 33*e 
^2*x^2) + a^2*e^7*(64*d^3 - 45*d^2*e*x + 28*d*e^2*x^2 - 13*e^3*x^3)))/(a^2 
*(a + c*x^4)) + (32*c*(c*d^4 + a*e^4)^2*(c*d^4*x*(d^2 - 2*d*e*x + 3*e^2*x^ 
2) + a*e^3*(4*d^3 - 3*d^2*e*x + 2*d*e^2*x^2 - e^3*x^3)))/(a*(a + c*x^4)^2) 
 - (6*c^(1/4)*(7*Sqrt[2]*c^(7/2)*d^14 - 16*a^(1/4)*c^(13/4)*d^13*e + 5*Sqr 
t[2]*Sqrt[a]*c^3*d^12*e^2 + 33*Sqrt[2]*a*c^(5/2)*d^10*e^4 - 80*a^(5/4)*c^( 
9/4)*d^9*e^5 + 27*Sqrt[2]*a^(3/2)*c^2*d^8*e^6 + 77*Sqrt[2]*a^2*c^(3/2)*d^6 
*e^8 - 240*a^(9/4)*c^(5/4)*d^5*e^9 + 135*Sqrt[2]*a^(5/2)*c*d^4*e^10 - 77*S 
qrt[2]*a^3*Sqrt[c]*d^2*e^12 + 80*a^(13/4)*c^(1/4)*d*e^13 - 15*Sqrt[2]*a^(7 
/2)*e^14)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(11/4) + (6*c^(1/4)*( 
7*Sqrt[2]*c^(7/2)*d^14 + 16*a^(1/4)*c^(13/4)*d^13*e + 5*Sqrt[2]*Sqrt[a]*c^ 
3*d^12*e^2 + 33*Sqrt[2]*a*c^(5/2)*d^10*e^4 + 80*a^(5/4)*c^(9/4)*d^9*e^5 + 
27*Sqrt[2]*a^(3/2)*c^2*d^8*e^6 + 77*Sqrt[2]*a^2*c^(3/2)*d^6*e^8 + 240*a^(9 
/4)*c^(5/4)*d^5*e^9 + 135*Sqrt[2]*a^(5/2)*c*d^4*e^10 - 77*Sqrt[2]*a^3*Sqrt 
[c]*d^2*e^12 - 80*a^(13/4)*c^(1/4)*d*e^13 - 15*Sqrt[2]*a^(7/2)*e^14)*ArcTa 
n[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(11/4) + 3072*c*d^3*e^11*Log[d + e*x 
] - (3*Sqrt[2]*c^(1/4)*(7*c^(7/2)*d^14 - 5*Sqrt[a]*c^3*d^12*e^2 + 33*a*c^( 
5/2)*d^10*e^4 - 27*a^(3/2)*c^2*d^8*e^6 + 77*a^2*c^(3/2)*d^6*e^8 - 135*a^(5 
/2)*c*d^4*e^10 - 77*a^3*Sqrt[c]*d^2*e^12 + 15*a^(7/2)*e^14)*Log[Sqrt[a]...
 

Rubi [A] (verified)

Time = 4.90 (sec) , antiderivative size = 1830, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\left (a+c x^4\right )^3 (d+e x)^2} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{12}}{(d+e x)^2 \left (a e^4+c d^4\right )^3}+\frac {12 c d^3 e^{12}}{(d+e x) \left (a e^4+c d^4\right )^4}+\frac {c e^4 \left (-2 d e x \left (3 c d^4-a e^4\right )+e^2 x^2 \left (7 c d^4-a e^4\right )+d^2 \left (5 c d^4-3 a e^4\right )-8 c d^3 e^3 x^3\right )}{\left (a+c x^4\right )^2 \left (a e^4+c d^4\right )^3}+\frac {c \left (-2 d e x \left (c d^4-a e^4\right )+e^2 x^2 \left (3 c d^4-a e^4\right )+d^2 \left (c d^4-3 a e^4\right )-4 c d^3 e^3 x^3\right )}{\left (a+c x^4\right )^3 \left (a e^4+c d^4\right )^2}+\frac {c e^8 \left (-2 d e x \left (5 c d^4-a e^4\right )+e^2 x^2 \left (11 c d^4-a e^4\right )+3 d^2 \left (3 c d^4-a e^4\right )-12 c d^3 e^3 x^3\right )}{\left (a+c x^4\right ) \left (a e^4+c d^4\right )^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {12 c d^3 \log (d+e x) e^{11}}{\left (c d^4+a e^4\right )^4}-\frac {3 c d^3 \log \left (c x^4+a\right ) e^{11}}{\left (c d^4+a e^4\right )^4}-\frac {e^{11}}{\left (c d^4+a e^4\right )^3 (d+e x)}-\frac {\sqrt {c} d \left (5 c d^4-a e^4\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^9}{\sqrt {a} \left (c d^4+a e^4\right )^4}-\frac {\sqrt [4]{c} \left (3 \sqrt {c} \left (3 c d^4-a e^4\right ) d^2+\sqrt {a} e^2 \left (11 c d^4-a e^4\right )\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 )^4}+\frac {\sqrt [4]{c} \left (3 \sqrt {c} \left (3 c d^4-a e^4\right ) d^2+\sqrt {a} e^2 \left (11 c d^4-a e^4\right )\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 )^4}-\frac {\sqrt [4]{c} \left (9 c^{3/2} d^6-11 \sqrt {a} c e^2 d^4-3 a \sqrt {c} e^4 d^2+a^{3/2} e^6\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 )^4}+\frac {\sqrt [4]{c} \left (9 c^{3/2} d^6-11 \sqrt {a} c e^2 d^4-3 a \sqrt {c} e^4 d^2+a^{3/2} e^6\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 )^4}-\frac {\sqrt {c} d \left (3 c d^4-a e^4\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{2 a^{3/2} \left (c d^4+a e^4\right )^3}+\frac {c \left (8 a d^3 e^3+x \left (\left (5 c d^4-3 a e^4\right ) d^2-2 e \left (3 c d^4-a e^4\right ) x d+e^2 \left (7 c d^4-a e^4\right ) x^2\right )\right ) e^4}{4 a \left (c d^4+a e^4\right )^3 \left (c x^4+a\right )}-\frac {\sqrt [4]{c} \left (3 \sqrt {c} \left (5 c d^4-3 a e^4\right ) d^2+\sqrt {a} e^2 \left (7 c d^4-a e^4\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 )^3}+\frac {\sqrt [4]{c} \left (3 \sqrt {c} \left (5 c d^4-3 a e^4\right ) d^2+\sqrt {a} e^2 \left (7 c d^4-a e^4\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 )^3}-\frac {\sqrt [4]{c} \left (3 \sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (7 c d^4-a e^4\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 )^3}+\frac {\sqrt [4]{c} \left (3 \sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (7 c d^4-a e^4\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 )^3}-\frac {3 \sqrt {c} d \left (c d^4-a e^4\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{8 a^{5/2} \left (c d^4+a e^4\right )^2}+\frac {c \left (4 a d^3 e^3+x \left (\left (c d^4-3 a e^4\right ) d^2-2 e \left (c d^4-a e^4\right ) x d+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^2 \left (c x^4+a\right )^2}-\frac {\sqrt [4]{c} \left (21 \sqrt {c} \left (c d^4-3 a e^4\right ) d^2+5 \sqrt {a} e^2 \left (3 c d^4-a e^4\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 )^2}+\frac {\sqrt [4]{c} \left (21 \sqrt {c} \left (c d^4-3 a e^4\right ) d^2+5 \sqrt {a} e^2 \left (3 c d^4-a e^4\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 )^2}-\frac {\sqrt [4]{c} \left (21 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )-5 \sqrt {a} e^2 \left (3 c d^4-a e^4\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 )^2}+\frac {\sqrt [4]{c} \left (21 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )-5 \sqrt {a} e^2 \left (3 c d^4-a e^4\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 )^2}+\frac {c x \left (7 \left (c d^4-3 a e^4\right ) d^2-12 e \left (c d^4-a e^4\right ) x d+5 e^2 \left (3 c d^4-a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^2 \left (c x^4+a\right )}\)

Input:

Int[1/((d + e*x)^2*(a + c*x^4)^3),x]
 

Output:

-(e^11/((c*d^4 + a*e^4)^3*(d + e*x))) + (c*x*(7*d^2*(c*d^4 - 3*a*e^4) - 12 
*d*e*(c*d^4 - a*e^4)*x + 5*e^2*(3*c*d^4 - a*e^4)*x^2))/(32*a^2*(c*d^4 + a* 
e^4)^2*(a + c*x^4)) + (c*(4*a*d^3*e^3 + x*(d^2*(c*d^4 - 3*a*e^4) - 2*d*e*( 
c*d^4 - a*e^4)*x + e^2*(3*c*d^4 - a*e^4)*x^2)))/(8*a*(c*d^4 + a*e^4)^2*(a 
+ c*x^4)^2) + (c*e^4*(8*a*d^3*e^3 + x*(d^2*(5*c*d^4 - 3*a*e^4) - 2*d*e*(3* 
c*d^4 - a*e^4)*x + e^2*(7*c*d^4 - a*e^4)*x^2)))/(4*a*(c*d^4 + a*e^4)^3*(a 
+ c*x^4)) - (Sqrt[c]*d*e^9*(5*c*d^4 - a*e^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]] 
)/(Sqrt[a]*(c*d^4 + a*e^4)^4) - (Sqrt[c]*d*e^5*(3*c*d^4 - a*e^4)*ArcTan[(S 
qrt[c]*x^2)/Sqrt[a]])/(2*a^(3/2)*(c*d^4 + a*e^4)^3) - (3*Sqrt[c]*d*e*(c*d^ 
4 - a*e^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(8*a^(5/2)*(c*d^4 + a*e^4)^2) - 
(c^(1/4)*(21*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) + 5*Sqrt[a]*e^2*(3*c*d^4 - a*e^ 
4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + 
 a*e^4)^2) - (c^(1/4)*e^4*(3*Sqrt[c]*d^2*(5*c*d^4 - 3*a*e^4) + Sqrt[a]*e^2 
*(7*c*d^4 - a*e^4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^ 
(7/4)*(c*d^4 + a*e^4)^3) - (c^(1/4)*e^8*(3*Sqrt[c]*d^2*(3*c*d^4 - a*e^4) + 
 Sqrt[a]*e^2*(11*c*d^4 - 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)^4) + (c^(1/4)*(21*Sqrt[c]*d^2*(c*d^4 - 
3*a*e^4) + 5*Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x) 
/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^2) + (c^(1/4)*e^4*(3*Sqrt[ 
c]*d^2*(5*c*d^4 - 3*a*e^4) + Sqrt[a]*e^2*(7*c*d^4 - a*e^4))*ArcTan[1 + ...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 829, normalized size of antiderivative = 0.54

method	result	size
default	\(-\frac {c \left (\frac {\frac {c \,e^{2} \left (13 a^{3} e^{12}-53 a^{2} c \,d^{4} e^{8}-81 a \,c^{2} d^{8} e^{4}-15 c^{3} d^{12}\right ) x^{7}}{32 a^{2}}-\frac {c d e \left (7 a^{3} e^{12}-5 a^{2} c \,d^{4} e^{8}-15 a \,c^{2} d^{8} e^{4}-3 c^{3} d^{12}\right ) x^{6}}{8 a^{2}}+\frac {c \,d^{2} \left (45 a^{3} e^{12}+19 a^{2} c \,d^{4} e^{8}-33 a \,c^{2} d^{8} e^{4}-7 c^{3} d^{12}\right ) x^{5}}{32 a^{2}}+\left (-2 a \,d^{3} e^{11} c -2 d^{7} e^{7} c^{2}\right ) x^{4}+\frac {e^{2} \left (17 a^{3} e^{12}-57 a^{2} c \,d^{4} e^{8}-101 a \,c^{2} d^{8} e^{4}-27 c^{3} d^{12}\right ) x^{3}}{32 a}-\frac {d e \left (9 a^{3} e^{12}-3 a^{2} c \,d^{4} e^{8}-17 a \,c^{2} d^{8} e^{4}-5 c^{3} d^{12}\right ) x^{2}}{8 a}+\frac {d^{2} \left (57 a^{3} e^{12}+39 a^{2} c \,d^{4} e^{8}-29 a \,c^{2} d^{8} e^{4}-11 c^{3} d^{12}\right ) x}{32 a}-\frac {5 a^{2} d^{3} e^{11}}{2}-3 a \,d^{7} e^{7} c -\frac {d^{11} e^{3} c^{2}}{2}}{\left (c \,x^{4}+a \right )^{2}}+\frac {\frac {3 \left (77 a^{3} d^{2} e^{12}-77 a^{2} c \,d^{6} e^{8}-33 a \,c^{2} d^{10} e^{4}-7 c^{3} d^{14}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a}+\frac {3 \left (-40 a^{3} d \,e^{13}+120 a^{2} c \,d^{5} e^{9}+40 a \,c^{2} d^{9} e^{5}+8 c^{3} d^{13} e \right ) \arctan \left (\sqrt {\frac {c}{a}}\, x^{2}\right )}{64 \sqrt {a c}}+\frac {3 \left (15 a^{3} e^{14}-135 a^{2} c \,d^{4} e^{10}-27 a \,c^{2} d^{8} e^{6}-5 c^{3} d^{12} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}+3 a^{2} d^{3} e^{11} \ln \left (c \,x^{4}+a \right )}{a^{2}}\right )}{\left (e^{4} a +c \,d^{4}\right )^{4}}-\frac {e^{11}}{\left (e^{4} a +c \,d^{4}\right )^{3} \left (e x +d \right )}+\frac {12 c \,d^{3} e^{11} \ln \left (e x +d \right )}{\left (e^{4} a +c \,d^{4}\right )^{4}}\)	\(829\)
risch	\(\text {Expression too large to display}\)	\(1673\)

Input:

int(1/(e*x+d)^2/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-c/(a*e^4+c*d^4)^4*((1/32*c*e^2*(13*a^3*e^12-53*a^2*c*d^4*e^8-81*a*c^2*d^8 
*e^4-15*c^3*d^12)/a^2*x^7-1/8*c*d*e*(7*a^3*e^12-5*a^2*c*d^4*e^8-15*a*c^2*d 
^8*e^4-3*c^3*d^12)/a^2*x^6+1/32*c*d^2*(45*a^3*e^12+19*a^2*c*d^4*e^8-33*a*c 
^2*d^8*e^4-7*c^3*d^12)/a^2*x^5+(-2*a*c*d^3*e^11-2*c^2*d^7*e^7)*x^4+1/32*e^ 
2*(17*a^3*e^12-57*a^2*c*d^4*e^8-101*a*c^2*d^8*e^4-27*c^3*d^12)/a*x^3-1/8*d 
*e*(9*a^3*e^12-3*a^2*c*d^4*e^8-17*a*c^2*d^8*e^4-5*c^3*d^12)/a*x^2+1/32*d^2 
*(57*a^3*e^12+39*a^2*c*d^4*e^8-29*a*c^2*d^8*e^4-11*c^3*d^12)/a*x-5/2*a^2*d 
^3*e^11-3*a*d^7*e^7*c-1/2*d^11*e^3*c^2)/(c*x^4+a)^2+3/32/a^2*(1/8*(77*a^3* 
d^2*e^12-77*a^2*c*d^6*e^8-33*a*c^2*d^10*e^4-7*c^3*d^14)*(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*(-40*a^3*d*e^13+120*a^2*c*d^5*e^9+40*a*c^2*d^9*e^5+8*c^3*d^1 
3*e)/(a*c)^(1/2)*arctan((c/a)^(1/2)*x^2)+1/8*(15*a^3*e^14-135*a^2*c*d^4*e^ 
10-27*a*c^2*d^8*e^6-5*c^3*d^12*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*arc 
tan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+32*a^2*d^3 
*e^11*ln(c*x^4+a)))-e^11/(a*e^4+c*d^4)^3/(e*x+d)+12*c*d^3*e^11*ln(e*x+d)/( 
a*e^4+c*d^4)^4
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)^2/(c*x^4+a)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)**2/(c*x**4+a)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1564, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^2/(c*x^4+a)^3,x, algorithm="maxima")
 

Output:

12*c*d^3*e^11*log(e*x + d)/(c^4*d^16 + 4*a*c^3*d^12*e^4 + 6*a^2*c^2*d^8*e^ 
8 + 4*a^3*c*d^4*e^12 + a^4*e^16) - 3/256*c*(sqrt(2)*(128*sqrt(2)*a^(11/4)* 
c^(5/4)*d^3*e^11 - 7*c^4*d^14 + 5*sqrt(a)*c^(7/2)*d^12*e^2 - 33*a*c^3*d^10 
*e^4 + 27*a^(3/2)*c^(5/2)*d^8*e^6 - 77*a^2*c^2*d^6*e^8 + 135*a^(5/2)*c^(3/ 
2)*d^4*e^10 + 77*a^3*c*d^2*e^12 - 15*a^(7/2)*sqrt(c)*e^14)*log(sqrt(c)*x^2 
 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + sqrt(2)*(128*s 
qrt(2)*a^(11/4)*c^(5/4)*d^3*e^11 + 7*c^4*d^14 - 5*sqrt(a)*c^(7/2)*d^12*e^2 
 + 33*a*c^3*d^10*e^4 - 27*a^(3/2)*c^(5/2)*d^8*e^6 + 77*a^2*c^2*d^6*e^8 - 1 
35*a^(5/2)*c^(3/2)*d^4*e^10 - 77*a^3*c*d^2*e^12 + 15*a^(7/2)*sqrt(c)*e^14) 
*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^(17/4)*d^14 + 5*sqrt(2)*a^(3/4)*c^(15/4)*d^12*e^2 
 + 33*sqrt(2)*a^(5/4)*c^(13/4)*d^10*e^4 + 27*sqrt(2)*a^(7/4)*c^(11/4)*d^8* 
e^6 + 77*sqrt(2)*a^(9/4)*c^(9/4)*d^6*e^8 + 135*sqrt(2)*a^(11/4)*c^(7/4)*d^ 
4*e^10 - 77*sqrt(2)*a^(13/4)*c^(5/4)*d^2*e^12 - 15*sqrt(2)*a^(15/4)*c^(3/4 
)*e^14 + 16*sqrt(a)*c^4*d^13*e + 80*a^(3/2)*c^3*d^9*e^5 + 240*a^(5/2)*c^2* 
d^5*e^9 - 80*a^(7/2)*c*d*e^13)*arctan(1/2*sqrt(2)*(2*sqrt(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)) - 2*(7*sqrt(2)*a^(1/4)*c^(17/4)*d^14 + 5*sqrt(2)*a^(3/4)*c^(15/4)*d^1 
2*e^2 + 33*sqrt(2)*a^(5/4)*c^(13/4)*d^10*e^4 + 27*sqrt(2)*a^(7/4)*c^(11/4) 
*d^8*e^6 + 77*sqrt(2)*a^(9/4)*c^(9/4)*d^6*e^8 + 135*sqrt(2)*a^(11/4)*c^...
 

Giac [A] (verification not implemented)

Time = 28.69 (sec) , antiderivative size = 1809, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^2/(c*x^4+a)^3,x, algorithm="giac")
 

Output:

12*c*d^3*e^12*log(abs(e*x + d))/(c^4*d^16*e + 4*a*c^3*d^12*e^5 + 6*a^2*c^2 
*d^8*e^9 + 4*a^3*c*d^4*e^13 + a^4*e^17) - 3*c*d^3*e^11*log(abs(c*x^4 + a)) 
/(c^4*d^16 + 4*a*c^3*d^12*e^4 + 6*a^2*c^2*d^8*e^8 + 4*a^3*c*d^4*e^12 + a^4 
*e^16) - 3/64*(32*sqrt(2)*a*c^3*d^3*e^3 - 20*sqrt(2)*sqrt(a*c)*c^3*d^5*e - 
 20*sqrt(2)*sqrt(a*c)*a*c^2*d*e^5 - 7*(a*c^3)^(1/4)*c^3*d^6 - 3*(a*c^3)^(1 
/4)*a*c^2*d^2*e^4 - 53*(a*c^3)^(3/4)*c*d^4*e^2 + 15*(a*c^3)^(3/4)*a*e^6)*a 
rctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^3*c^ 
4*d^8 + 34*sqrt(2)*a^4*c^3*d^4*e^4 + sqrt(2)*a^5*c^2*e^8 + 16*sqrt(2)*sqrt 
(a*c)*a^3*c^3*d^6*e^2 + 16*sqrt(2)*sqrt(a*c)*a^4*c^2*d^2*e^6 - 8*(a*c^3)^( 
1/4)*a^3*c^3*d^7*e - 40*(a*c^3)^(1/4)*a^4*c^2*d^3*e^5 - 40*(a*c^3)^(3/4)*a 
^3*c*d^5*e^3 - 8*(a*c^3)^(3/4)*a^4*d*e^7) + 3/64*(32*sqrt(2)*a*c^3*d^3*e^3 
 + 20*sqrt(2)*sqrt(a*c)*c^3*d^5*e + 20*sqrt(2)*sqrt(a*c)*a*c^2*d*e^5 + 7*( 
a*c^3)^(1/4)*c^3*d^6 + 3*(a*c^3)^(1/4)*a*c^2*d^2*e^4 + 53*(a*c^3)^(3/4)*c* 
d^4*e^2 - 15*(a*c^3)^(3/4)*a*e^6)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^ 
(1/4))/(a/c)^(1/4))/(sqrt(2)*a^3*c^4*d^8 + 34*sqrt(2)*a^4*c^3*d^4*e^4 + sq 
rt(2)*a^5*c^2*e^8 + 16*sqrt(2)*sqrt(a*c)*a^3*c^3*d^6*e^2 + 16*sqrt(2)*sqrt 
(a*c)*a^4*c^2*d^2*e^6 + 8*(a*c^3)^(1/4)*a^3*c^3*d^7*e + 40*(a*c^3)^(1/4)*a 
^4*c^2*d^3*e^5 + 40*(a*c^3)^(3/4)*a^3*c*d^5*e^3 + 8*(a*c^3)^(3/4)*a^4*d*e^ 
7) + 3/256*(7*sqrt(2)*(a*c^3)^(1/4)*c^5*d^14 + 33*sqrt(2)*(a*c^3)^(1/4)*a* 
c^4*d^10*e^4 + 77*sqrt(2)*(a*c^3)^(1/4)*a^2*c^3*d^6*e^8 - 77*sqrt(2)*(a...
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 25.24 (sec) , antiderivative size = 3572, normalized size of antiderivative = 2.33 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/((a + c*x^4)^3*(d + e*x)^2),x)
 

Output:

symsum(log((194481*c^9*d^17*e^6 + 1527012*a*c^8*d^13*e^10 + 4100625*a^4*c^ 
5*d*e^22 + 1926342*a^2*c^7*d^9*e^14 - 3102300*a^3*c^6*d^5*e^18)/(1048576*( 
a^14*e^24 + a^8*c^6*d^24 + 6*a^13*c*d^4*e^20 + 6*a^9*c^5*d^20*e^4 + 15*a^1 
0*c^4*d^16*e^8 + 20*a^11*c^3*d^12*e^12 + 15*a^12*c^2*d^8*e^16)) + root(161 
0612736*a^13*c^2*d^8*e^8*z^4 + 1073741824*a^12*c^3*d^12*e^4*z^4 + 10737418 
24*a^14*c*d^4*e^12*z^4 + 268435456*a^11*c^4*d^16*z^4 + 268435456*a^15*e^16 
*z^4 + 3221225472*a^11*c*d^3*e^11*z^3 + 239468544*a^7*c^2*d^6*e^6*z^2 + 39 
518208*a^6*c^3*d^10*e^2*z^2 + 1153105920*a^8*c*d^2*e^10*z^2 + 32071680*a^4 
*c^2*d^5*e^5*z + 5419008*a^3*c^3*d^9*e*z + 124416000*a^5*c*d*e^9*z + 11380 
50*a*c^2*d^4*e^4 + 4100625*a^2*c*e^8 + 194481*c^3*d^8, z, k)*(root(1610612 
736*a^13*c^2*d^8*e^8*z^4 + 1073741824*a^12*c^3*d^12*e^4*z^4 + 1073741824*a 
^14*c*d^4*e^12*z^4 + 268435456*a^11*c^4*d^16*z^4 + 268435456*a^15*e^16*z^4 
 + 3221225472*a^11*c*d^3*e^11*z^3 + 239468544*a^7*c^2*d^6*e^6*z^2 + 395182 
08*a^6*c^3*d^10*e^2*z^2 + 1153105920*a^8*c*d^2*e^10*z^2 + 32071680*a^4*c^2 
*d^5*e^5*z + 5419008*a^3*c^3*d^9*e*z + 124416000*a^5*c*d*e^9*z + 1138050*a 
*c^2*d^4*e^4 + 4100625*a^2*c*e^8 + 194481*c^3*d^8, z, k)*(root(1610612736* 
a^13*c^2*d^8*e^8*z^4 + 1073741824*a^12*c^3*d^12*e^4*z^4 + 1073741824*a^14* 
c*d^4*e^12*z^4 + 268435456*a^11*c^4*d^16*z^4 + 268435456*a^15*e^16*z^4 + 3 
221225472*a^11*c*d^3*e^11*z^3 + 239468544*a^7*c^2*d^6*e^6*z^2 + 39518208*a 
^6*c^3*d^10*e^2*z^2 + 1153105920*a^8*c*d^2*e^10*z^2 + 32071680*a^4*c^2*...
 

Reduce [B] (verification not implemented)

Time = 2.50 (sec) , antiderivative size = 11727, normalized size of antiderivative = 7.65 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)^2/(c*x^4+a)^3,x)
 

Output:

(90*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)* 
x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**5*d**2*e**14 + 90*c**(1/4)*a**(3/4)*sqr 
t(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqr 
t(2)))*a**5*d*e**15*x - 810*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1 
/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**4*c*d**6*e**10 
- 810*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c 
)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**4*c*d**5*e**11*x + 180*c**(1/4)*a**(3 
/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1 
/4)*sqrt(2)))*a**4*c*d**2*e**14*x**4 + 180*c**(1/4)*a**(3/4)*sqrt(2)*atan( 
(c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a** 
4*c*d*e**15*x**5 - 162*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*s 
qrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**3*c**2*d**10*e**6 - 
162*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)* 
x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**3*c**2*d**9*e**7*x - 1620*c**(1/4)*a**( 
3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**( 
1/4)*sqrt(2)))*a**3*c**2*d**6*e**10*x**4 - 1620*c**(1/4)*a**(3/4)*sqrt(2)* 
atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)) 
)*a**3*c**2*d**5*e**11*x**5 + 90*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)* 
a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**3*c**2*d** 
2*e**14*x**8 + 90*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqr...