\(\int \frac {1}{(d+e x)^3 (a+c x^4)^2} \, dx\) [407]

Optimal result

Integrand size = 17, antiderivative size = 1384 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^2} \, dx=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (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 )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\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 )}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}-\frac {\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 )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c d^4 e^4+9 a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\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 )^3}-\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 )+3 \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 )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c d^4 e^4+9 a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\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 )^3}+\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 )+3 \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 )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c d^4 e^4+9 a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 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 )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}+\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 )-3 \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 )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c d^4 e^4+9 a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 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 )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}-\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 )-3 \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 )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4} \]

[Out]

-1/2*e^7/(a*e^4+c*d^4)^2/(e*x+d)^2-8*c*d^3*e^7/(a*e^4+c*d^4)^3/(e*x+d)+1/4*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)+12*c*d^2*e^7*(-a*e^4+3*c*d^4)*ln(e*x+d)/(a*e^4+c*d^4)^4-3*c*d^2*e^7*(-a*e^4+3
*c*d^4)*ln(c*x^4+a)/(a*e^4+c*d^4)^4-1/4*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^(3/2)/(a*e^4+c*d^4)^3-1/2*e^5*(a^2*e^8-26*a*c*d^4*e^4+21*c^2*d^8)*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/(a*
e^4+c*d^4)^4/a^(1/2)-1/32*c^(3/4)*d*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(3*c^2*d^8-36*a*c*d^4*e
^4+9*a^2*e^8-2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/32*c^(3/4)*d*ln(a
^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(3*c^2*d^8-36*a*c*d^4*e^4+9*a^2*e^8-2*d^2*e^2*(-5*a*e^4+3*c*d^4)
*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/16*c^(3/4)*d*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(3*c^2*d
^8-36*a*c*d^4*e^4+9*a^2*e^8+2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/16
*c^(3/4)*d*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(3*c^2*d^8-36*a*c*d^4*e^4+9*a^2*e^8+2*d^2*e^2*(-5*a*e^4+3*c*d^4
)*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/8*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))*(-3*a^2*e^8+30*a*c*d^4*e^4-15*c^2*d^8+4*d^2*e^2*(-5*a*e^4+7*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c
*d^4)^4*2^(1/2)-1/8*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))*(-3*a^2*e^8+30*a*c*d^4*e^4
-15*c^2*d^8+4*d^2*e^2*(-5*a*e^4+7*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^4*2^(1/2)+1/4*c^(3/4)*d*e^4*ar
ctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(3*a^2*e^8-30*a*c*d^4*e^4+15*c^2*d^8+4*d^2*e^2*(-5*a*e^4+7*c*d^4)*a^(1/2)*c
^(1/2))/a^(3/4)/(a*e^4+c*d^4)^4*2^(1/2)+1/4*c^(3/4)*d*e^4*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(3*a^2*e^8-30*a*
c*d^4*e^4+15*c^2*d^8+4*d^2*e^2*(-5*a*e^4+7*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^4*2^(1/2)

Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 1384, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {6874, 1868, 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 )^2} \, dx=\frac {12 c d^2 \left (3 c d^4-a e^4\right ) \log (d+e x) e^7}{\left (c d^4+a e^4\right )^4}-\frac {3 c d^2 \left (3 c d^4-a e^4\right ) \log \left (c x^4+a\right ) e^7}{\left (c d^4+a e^4\right )^4}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\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}{2 \sqrt {a} \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+3 \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}{2 \sqrt {2} a^{3/4} \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+3 \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}{2 \sqrt {2} a^{3/4} \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 )-3 \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}{4 \sqrt {2} a^{3/4} \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 )-3 \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}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {\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}{4 a^{3/2} \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 )}{4 a \left (c d^4+a e^4\right )^3 \left (c x^4+a\right )}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\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 )^3}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3} \]

[In]

Int[1/((d + e*x)^3*(a + c*x^4)^2),x]

[Out]

-1/2*e^7/((c*d^4 + a*e^4)^2*(d + e*x)^2) - (8*c*d^3*e^7)/((c*d^4 + a*e^4)^3*(d + e*x)) + (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)))/(4*a*(c*d^4 + a*e^4)^3*(a + c*x^4)) - (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]])/(2*Sqrt[a]*(c*d^4 + a*e^4)^4) - (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]])/(4*a^(3/2)*(c*d^4 + a*e^4)^3) - (c^(3/4)*d*(3*c^2*d^8 - 36
*a*c*d^4*e^4 + 9*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*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^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) + 3
*(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)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 +
a*e^4)^4) + (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^4 + 9*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*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^(3/4)*d*e^4*(4*Sqrt[a]*S
qrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) + 3*(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)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) + (12*c*d^2*e^7*(3*c*d^4 - a*e^4)*Log[d + e*x])/(c*d^4 + a*e^4
)^4 - (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^4 + 9*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*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)^3) + (c^(3/4)*d*e^4*(4
*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) - 3*(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])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) + (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*
e^4 + 9*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*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)^3) - (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e
^4) - 3*(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])/(4*Sqr
t[2]*a^(3/4)*(c*d^4 + a*e^4)^4) - (3*c*d^2*e^7*(3*c*d^4 - a*e^4)*Log[a + c*x^4])/(c*d^4 + a*e^4)^4

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 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^8}{\left (c d^4+a e^4\right )^2 (d+e x)^3}+\frac {8 c d^3 e^8}{\left (c d^4+a e^4\right )^3 (d+e x)^2}+\frac {12 c d^2 e^8 \left (3 c d^4-a e^4\right )}{\left (c d^4+a e^4\right )^4 (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 )^2}+\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 )}\right ) \, dx \\ & = -\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\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}{a+c x^4} \, 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 )^2} \, dx}{\left (c d^4+a e^4\right )^3} \\ & = -\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (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 )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \int \left (\frac {3 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}+\frac {x \left (-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^4}-\frac {c \int \frac {-3 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+2 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}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^3} \\ & = -\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (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 )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\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 )+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \int \frac {x \left (-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}-\frac {c \int \left (\frac {2 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 {-3 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-2 c d^3 e^2 \left (3 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 )^3} \\ & = -\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (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 )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \text {Subst}\left (\int \frac {-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^4}-\frac {c \int \frac {-3 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-2 c d^3 e^2 \left (3 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 )^3}-\frac {\left (c e \left (3 c^2 d^8-12 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 )^3}-\frac {\left (\sqrt {c} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}+\frac {\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\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 )^4} \\ & = -\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (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 )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}-\frac {\left (6 c^2 d^2 e^7 \left (3 c d^4-a e^4\right )\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^4}-\frac {\left (c e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^4}-\frac {\left (c e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^4+a e^4\right )^3}+\frac {\left (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 )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {\left (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 )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^4}+\frac {\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^4}-\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^3} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 996, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^2} \, dx=\frac {-\frac {16 e^7 \left (c d^4+a e^4\right )^2}{(d+e x)^2}-\frac {256 c d^3 e^7 \left (c d^4+a e^4\right )}{d+e x}+\frac {8 c \left (c d^4+a e^4\right ) \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 )}-\frac {6 \sqrt {c} \left (\sqrt {2} c^{13/4} d^{13}-4 \sqrt [4]{a} c^3 d^{12} e+2 \sqrt {2} \sqrt {a} c^{11/4} d^{11} e^2+9 \sqrt {2} a c^{9/4} d^9 e^4-44 a^{5/4} c^2 d^8 e^5+36 \sqrt {2} a^{3/2} c^{7/4} d^7 e^6-49 \sqrt {2} a^2 c^{5/4} d^5 e^8+84 a^{9/4} c d^4 e^9-30 \sqrt {2} a^{5/2} c^{3/4} d^3 e^{10}+7 \sqrt {2} a^3 \sqrt [4]{c} d e^{12}-4 a^{13/4} e^{13}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {6 \sqrt {c} \left (\sqrt {2} c^{13/4} d^{13}+4 \sqrt [4]{a} c^3 d^{12} e+2 \sqrt {2} \sqrt {a} c^{11/4} d^{11} e^2+9 \sqrt {2} a c^{9/4} d^9 e^4+44 a^{5/4} c^2 d^8 e^5+36 \sqrt {2} a^{3/2} c^{7/4} d^7 e^6-49 \sqrt {2} a^2 c^{5/4} d^5 e^8-84 a^{9/4} c d^4 e^9-30 \sqrt {2} a^{5/2} c^{3/4} d^3 e^{10}+7 \sqrt {2} a^3 \sqrt [4]{c} d e^{12}+4 a^{13/4} e^{13}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+384 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)-\frac {3 \sqrt {2} c^{3/4} \left (c^3 d^{13}-2 \sqrt {a} c^{5/2} d^{11} e^2+9 a c^2 d^9 e^4-36 a^{3/2} c^{3/2} d^7 e^6-49 a^2 c d^5 e^8+30 a^{5/2} \sqrt {c} d^3 e^{10}+7 a^3 d e^{12}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {3 \sqrt {2} c^{3/4} \left (c^3 d^{13}-2 \sqrt {a} c^{5/2} d^{11} e^2+9 a c^2 d^9 e^4-36 a^{3/2} c^{3/2} d^7 e^6-49 a^2 c d^5 e^8+30 a^{5/2} \sqrt {c} d^3 e^{10}+7 a^3 d e^{12}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}-96 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{32 \left (c d^4+a e^4\right )^4} \]

[In]

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

[Out]

((-16*e^7*(c*d^4 + a*e^4)^2)/(d + e*x)^2 - (256*c*d^3*e^7*(c*d^4 + a*e^4))/(d + e*x) + (8*c*(c*d^4 + a*e^4)*(-
(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)) - (6*Sqrt[c]*(Sqrt[2]*c^(13/4)*d^13 - 4*a^(1/4)*c^3*d^12*e + 2
*Sqrt[2]*Sqrt[a]*c^(11/4)*d^11*e^2 + 9*Sqrt[2]*a*c^(9/4)*d^9*e^4 - 44*a^(5/4)*c^2*d^8*e^5 + 36*Sqrt[2]*a^(3/2)
*c^(7/4)*d^7*e^6 - 49*Sqrt[2]*a^2*c^(5/4)*d^5*e^8 + 84*a^(9/4)*c*d^4*e^9 - 30*Sqrt[2]*a^(5/2)*c^(3/4)*d^3*e^10
+ 7*Sqrt[2]*a^3*c^(1/4)*d*e^12 - 4*a^(13/4)*e^13)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) + (6*Sqrt[
c]*(Sqrt[2]*c^(13/4)*d^13 + 4*a^(1/4)*c^3*d^12*e + 2*Sqrt[2]*Sqrt[a]*c^(11/4)*d^11*e^2 + 9*Sqrt[2]*a*c^(9/4)*d
^9*e^4 + 44*a^(5/4)*c^2*d^8*e^5 + 36*Sqrt[2]*a^(3/2)*c^(7/4)*d^7*e^6 - 49*Sqrt[2]*a^2*c^(5/4)*d^5*e^8 - 84*a^(
9/4)*c*d^4*e^9 - 30*Sqrt[2]*a^(5/2)*c^(3/4)*d^3*e^10 + 7*Sqrt[2]*a^3*c^(1/4)*d*e^12 + 4*a^(13/4)*e^13)*ArcTan[
1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) + 384*c*d^2*e^7*(3*c*d^4 - a*e^4)*Log[d + e*x] - (3*Sqrt[2]*c^(3/4)*
(c^3*d^13 - 2*Sqrt[a]*c^(5/2)*d^11*e^2 + 9*a*c^2*d^9*e^4 - 36*a^(3/2)*c^(3/2)*d^7*e^6 - 49*a^2*c*d^5*e^8 + 30*
a^(5/2)*Sqrt[c]*d^3*e^10 + 7*a^3*d*e^12)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (3*
Sqrt[2]*c^(3/4)*(c^3*d^13 - 2*Sqrt[a]*c^(5/2)*d^11*e^2 + 9*a*c^2*d^9*e^4 - 36*a^(3/2)*c^(3/2)*d^7*e^6 - 49*a^2
*c*d^5*e^8 + 30*a^(5/2)*Sqrt[c]*d^3*e^10 + 7*a^3*d*e^12)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2
])/a^(7/4) - 96*c*d^2*e^7*(3*c*d^4 - a*e^4)*Log[a + c*x^4])/(32*(c*d^4 + a*e^4)^4)

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 680, normalized size of antiderivative = 0.49

method	result	size
default	\(\frac {c \left (\frac {-\frac {c \,d^{3} e^{2} \left (5 a^{2} e^{8}+2 a c \,d^{4} e^{4}-3 c^{2} d^{8}\right ) x^{3}}{2 a}-\frac {e \left (a^{3} e^{12}-11 a^{2} c \,d^{4} e^{8}-9 a \,c^{2} d^{8} e^{4}+3 c^{3} d^{12}\right ) x^{2}}{4 a}+\frac {d \left (3 a^{3} e^{12}-9 a^{2} c \,d^{4} e^{8}-11 a \,c^{2} d^{8} e^{4}+c^{3} d^{12}\right ) x}{4 a}-\frac {d^{2} e^{3} \left (3 a^{2} e^{8}-2 a c \,d^{4} e^{4}-5 c^{2} d^{8}\right )}{2}}{c \,x^{4}+a}+\frac {\frac {3 \left (7 a^{3} d \,e^{12}-49 a^{2} c \,d^{5} e^{8}+9 a \,c^{2} d^{9} e^{4}+c^{3} d^{13}\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 )}{32 a}+\frac {3 \left (-2 a^{3} e^{13}+42 a^{2} c \,d^{4} e^{9}-22 a \,c^{2} d^{8} e^{5}-2 c^{3} d^{12} e \right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{8 \sqrt {a c}}+\frac {3 \left (-30 a^{2} c \,d^{3} e^{10}+36 a \,c^{2} d^{7} e^{6}+2 c^{3} d^{11} 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 )}{32 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}+\frac {3 \left (16 a^{2} c \,d^{2} e^{11}-48 a \,c^{2} d^{6} e^{7}\right ) \ln \left (c \,x^{4}+a \right )}{16 c}}{a}\right )}{\left (e^{4} a +d^{4} c \right )^{4}}-\frac {e^{7}}{2 \left (e^{4} a +d^{4} c \right )^{2} \left (e x +d \right )^{2}}-\frac {8 c \,d^{3} e^{7}}{\left (e^{4} a +d^{4} c \right )^{3} \left (e x +d \right )}-\frac {12 e^{7} c \,d^{2} \left (e^{4} a -3 d^{4} c \right ) \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{4}}\)	\(680\)
risch	\(\text {Expression too large to display}\)	\(1541\)

[In]

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

[Out]

c/(a*e^4+c*d^4)^4*((-1/2*c*d^3*e^2*(5*a^2*e^8+2*a*c*d^4*e^4-3*c^2*d^8)/a*x^3-1/4*e*(a^3*e^12-11*a^2*c*d^4*e^8-
9*a*c^2*d^8*e^4+3*c^3*d^12)/a*x^2+1/4*d*(3*a^3*e^12-9*a^2*c*d^4*e^8-11*a*c^2*d^8*e^4+c^3*d^12)/a*x-1/2*d^2*e^3
*(3*a^2*e^8-2*a*c*d^4*e^4-5*c^2*d^8))/(c*x^4+a)+3/4/a*(1/8*(7*a^3*d*e^12-49*a^2*c*d^5*e^8+9*a*c^2*d^9*e^4+c^3*
d^13)*(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*(-2*a^3*e^13+42*a^2*c*d^4*e^9-22*a
*c^2*d^8*e^5-2*c^3*d^12*e)/(a*c)^(1/2)*arctan(x^2*(c/a)^(1/2))+1/8*(-30*a^2*c*d^3*e^10+36*a*c^2*d^7*e^6+2*c^3*
d^11*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*(16*a^2*c*d^2*e^11-48*a*c^2*d^
6*e^7)/c*ln(c*x^4+a)))-1/2*e^7/(a*e^4+c*d^4)^2/(e*x+d)^2-8*c*d^3*e^7/(a*e^4+c*d^4)^3/(e*x+d)-12*e^7*c*d^2*(a*e
^4-3*c*d^4)/(a*e^4+c*d^4)^4*ln(e*x+d)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 1394, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-3/32*c*(sqrt(2)*(48*sqrt(2)*a^(7/4)*c^(9/4)*d^6*e^7 - 16*sqrt(2)*a^(11/4)*c^(5/4)*d^2*e^11 - c^4*d^13 + 2*sqr
t(a)*c^(7/2)*d^11*e^2 - 9*a*c^3*d^9*e^4 + 36*a^(3/2)*c^(5/2)*d^7*e^6 + 49*a^2*c^2*d^5*e^8 - 30*a^(5/2)*c^(3/2)
*d^3*e^10 - 7*a^3*c*d*e^12)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + sqrt(2)
*(48*sqrt(2)*a^(7/4)*c^(9/4)*d^6*e^7 - 16*sqrt(2)*a^(11/4)*c^(5/4)*d^2*e^11 + c^4*d^13 - 2*sqrt(a)*c^(7/2)*d^1
1*e^2 + 9*a*c^3*d^9*e^4 - 36*a^(3/2)*c^(5/2)*d^7*e^6 - 49*a^2*c^2*d^5*e^8 + 30*a^(5/2)*c^(3/2)*d^3*e^10 + 7*a^
3*c*d*e^12)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) - 2*(sqrt(2)*a^(1/4)*c^(1
7/4)*d^13 + 2*sqrt(2)*a^(3/4)*c^(15/4)*d^11*e^2 + 9*sqrt(2)*a^(5/4)*c^(13/4)*d^9*e^4 + 36*sqrt(2)*a^(7/4)*c^(1
1/4)*d^7*e^6 - 49*sqrt(2)*a^(9/4)*c^(9/4)*d^5*e^8 - 30*sqrt(2)*a^(11/4)*c^(7/4)*d^3*e^10 + 7*sqrt(2)*a^(13/4)*
c^(5/4)*d*e^12 + 4*sqrt(a)*c^4*d^12*e + 44*a^(3/2)*c^3*d^8*e^5 - 84*a^(5/2)*c^2*d^4*e^9 + 4*a^(7/2)*c*e^13)*ar
ctan(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*(sqrt(2)*a^(1/4)*c^(17/4)*d^13 + 2*sqrt(2)*a^(3/4)*c^(15/4)*d^11*e^2 + 9*sqrt(2)*a^(5/4)*c^(13/4
)*d^9*e^4 + 36*sqrt(2)*a^(7/4)*c^(11/4)*d^7*e^6 - 49*sqrt(2)*a^(9/4)*c^(9/4)*d^5*e^8 - 30*sqrt(2)*a^(11/4)*c^(
7/4)*d^3*e^10 + 7*sqrt(2)*a^(13/4)*c^(5/4)*d*e^12 - 4*sqrt(a)*c^4*d^12*e - 44*a^(3/2)*c^3*d^8*e^5 + 84*a^(5/2)
*c^2*d^4*e^9 - 4*a^(7/2)*c*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)))/(a*c^4*d^16 + 4*a^2*c^3*d^12*e^4 + 6*a^3*c^2*d^8*e^8 + 4*a^4*c*d
^4*e^12 + a^5*e^16) + 12*(3*c^2*d^6*e^7 - a*c*d^2*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) + 1/4*(10*a*c^2*d^8*e^3 - 40*a^2*c*d^4*e^7 - 2*a^3*e^11 + 6*(c^3*d^7*e^
4 - 7*a*c^2*d^3*e^8)*x^5 + 3*(3*c^3*d^8*e^3 - 14*a*c^2*d^4*e^7 - a^2*c*e^11)*x^4 + (c^3*d^9*e^2 + 2*a*c^2*d^5*
e^6 + a^2*c*d*e^10)*x^3 - (c^3*d^10*e + 2*a*c^2*d^6*e^5 + a^2*c*d^2*e^9)*x^2 + (c^3*d^11 + 8*a*c^2*d^7*e^4 - 4
1*a^2*c*d^3*e^8)*x)/(a^2*c^3*d^14 + 3*a^3*c^2*d^10*e^4 + 3*a^4*c*d^6*e^8 + a^5*d^2*e^12 + (a*c^4*d^12*e^2 + 3*
a^2*c^3*d^8*e^6 + 3*a^3*c^2*d^4*e^10 + a^4*c*e^14)*x^6 + 2*(a*c^4*d^13*e + 3*a^2*c^3*d^9*e^5 + 3*a^3*c^2*d^5*e
^9 + a^4*c*d*e^13)*x^5 + (a*c^4*d^14 + 3*a^2*c^3*d^10*e^4 + 3*a^3*c^2*d^6*e^8 + a^4*c*d^2*e^12)*x^4 + (a^2*c^3
*d^12*e^2 + 3*a^3*c^2*d^8*e^6 + 3*a^4*c*d^4*e^10 + a^5*e^14)*x^2 + 2*(a^2*c^3*d^13*e + 3*a^3*c^2*d^9*e^5 + 3*a
^4*c*d^5*e^9 + a^5*d*e^13)*x)

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 1557, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

3/8*(4*sqrt(2)*a*c^3*d^2*e^3 + 2*sqrt(2)*sqrt(a*c)*c^3*d^4*e + 2*sqrt(2)*sqrt(a*c)*a*c^2*e^5 + (a*c^3)^(1/4)*c
^3*d^5 - 9*(a*c^3)^(1/4)*a*c^2*d*e^4 + 2*(a*c^3)^(3/4)*c*d^3*e^2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4
))/(a/c)^(1/4))/(sqrt(2)*a^2*c^4*d^8 + 34*sqrt(2)*a^3*c^3*d^4*e^4 + sqrt(2)*a^4*c^2*e^8 + 16*sqrt(2)*sqrt(a*c)
*a^2*c^3*d^6*e^2 + 16*sqrt(2)*sqrt(a*c)*a^3*c^2*d^2*e^6 - 8*(a*c^3)^(1/4)*a^2*c^3*d^7*e - 40*(a*c^3)^(1/4)*a^3
*c^2*d^3*e^5 - 40*(a*c^3)^(3/4)*a^2*c*d^5*e^3 - 8*(a*c^3)^(3/4)*a^3*d*e^7) - 3/8*(4*sqrt(2)*a*c^3*d^2*e^3 - 2*
sqrt(2)*sqrt(a*c)*c^3*d^4*e - 2*sqrt(2)*sqrt(a*c)*a*c^2*e^5 - (a*c^3)^(1/4)*c^3*d^5 + 9*(a*c^3)^(1/4)*a*c^2*d*
e^4 - 2*(a*c^3)^(3/4)*c*d^3*e^2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^4*
d^8 + 34*sqrt(2)*a^3*c^3*d^4*e^4 + sqrt(2)*a^4*c^2*e^8 + 16*sqrt(2)*sqrt(a*c)*a^2*c^3*d^6*e^2 + 16*sqrt(2)*sqr
t(a*c)*a^3*c^2*d^2*e^6 + 8*(a*c^3)^(1/4)*a^2*c^3*d^7*e + 40*(a*c^3)^(1/4)*a^3*c^2*d^3*e^5 + 40*(a*c^3)^(3/4)*a
^2*c*d^5*e^3 + 8*(a*c^3)^(3/4)*a^3*d*e^7) + 3/32*(sqrt(2)*(a*c^3)^(1/4)*c^4*d^13 + 9*sqrt(2)*(a*c^3)^(1/4)*a*c
^3*d^9*e^4 - 49*sqrt(2)*(a*c^3)^(1/4)*a^2*c^2*d^5*e^8 + 7*sqrt(2)*(a*c^3)^(1/4)*a^3*c*d*e^12 - 2*sqrt(2)*(a*c^
3)^(3/4)*c^2*d^11*e^2 - 36*sqrt(2)*(a*c^3)^(3/4)*a*c*d^7*e^6 + 30*sqrt(2)*(a*c^3)^(3/4)*a^2*d^3*e^10)*log(x^2
+ sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^5*d^16 + 4*a^3*c^4*d^12*e^4 + 6*a^4*c^3*d^8*e^8 + 4*a^5*c^2*d^4*e^
12 + a^6*c*e^16) - 3/32*(sqrt(2)*(a*c^3)^(1/4)*c^4*d^13 + 9*sqrt(2)*(a*c^3)^(1/4)*a*c^3*d^9*e^4 - 49*sqrt(2)*(
a*c^3)^(1/4)*a^2*c^2*d^5*e^8 + 7*sqrt(2)*(a*c^3)^(1/4)*a^3*c*d*e^12 - 2*sqrt(2)*(a*c^3)^(3/4)*c^2*d^11*e^2 - 3
6*sqrt(2)*(a*c^3)^(3/4)*a*c*d^7*e^6 + 30*sqrt(2)*(a*c^3)^(3/4)*a^2*d^3*e^10)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) +
sqrt(a/c))/(a^2*c^5*d^16 + 4*a^3*c^4*d^12*e^4 + 6*a^4*c^3*d^8*e^8 + 4*a^5*c^2*d^4*e^12 + a^6*c*e^16) - 3*(3*c
^2*d^6*e^7 - a*c*d^2*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) + 12*(3*c^2*d^6*e^8 - a*c*d^2*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) + 1/4*(10*a*c^3*d^12*e^3 - 30*a^2*c^2*d^8*e^7 - 42*a^3*c*d^4*e^11 -
2*a^4*e^15 + 6*(c^4*d^11*e^4 - 6*a*c^3*d^7*e^8 - 7*a^2*c^2*d^3*e^12)*x^5 + 3*(3*c^4*d^12*e^3 - 11*a*c^3*d^8*e
^7 - 15*a^2*c^2*d^4*e^11 - a^3*c*e^15)*x^4 + (c^4*d^13*e^2 + 3*a*c^3*d^9*e^6 + 3*a^2*c^2*d^5*e^10 + a^3*c*d*e^
14)*x^3 - (c^4*d^14*e + 3*a*c^3*d^10*e^5 + 3*a^2*c^2*d^6*e^9 + a^3*c*d^2*e^13)*x^2 + (c^4*d^15 + 9*a*c^3*d^11*
e^4 - 33*a^2*c^2*d^7*e^8 - 41*a^3*c*d^3*e^12)*x)/((c*d^4 + a*e^4)^4*(c*x^4 + a)*(e*x + d)^2*a)

Mupad [B] (verification not implemented)

Time = 10.85 (sec) , antiderivative size = 3256, normalized size of antiderivative = 2.35 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/((a + c*x^4)^2*(d + e*x)^3),x)

[Out]

symsum(log(root(262144*a^10*c*d^4*e^12*z^4 + 393216*a^9*c^2*d^8*e^8*z^4 + 262144*a^8*c^3*d^12*e^4*z^4 + 65536*
a^7*c^4*d^16*z^4 + 65536*a^11*e^16*z^4 - 786432*a^8*c*d^2*e^11*z^3 + 2359296*a^7*c^2*d^6*e^7*z^3 + 755712*a^5*
c^2*d^4*e^6*z^2 + 36864*a^4*c^3*d^8*e^2*z^2 + 18432*a^6*c*e^10*z^2 + 58752*a^3*c^2*d^2*e^5*z + 3456*a^2*c^3*d^
6*e*z + 1296*a*c^2*e^4 + 81*c^3*d^4, z, k)*((108*a*c^10*d^19*e^3 + 3888*a^2*c^9*d^15*e^7 - 99576*a^3*c^8*d^11*
e^11 + 591408*a^4*c^7*d^7*e^15 - 79380*a^5*c^6*d^3*e^19)/(256*(a^10*e^24 + a^4*c^6*d^24 + 6*a^9*c*d^4*e^20 + 6
*a^5*c^5*d^20*e^4 + 15*a^6*c^4*d^16*e^8 + 20*a^7*c^3*d^12*e^12 + 15*a^8*c^2*d^8*e^16)) + root(262144*a^10*c*d^
4*e^12*z^4 + 393216*a^9*c^2*d^8*e^8*z^4 + 262144*a^8*c^3*d^12*e^4*z^4 + 65536*a^7*c^4*d^16*z^4 + 65536*a^11*e^
16*z^4 - 786432*a^8*c*d^2*e^11*z^3 + 2359296*a^7*c^2*d^6*e^7*z^3 + 755712*a^5*c^2*d^4*e^6*z^2 + 36864*a^4*c^3*
d^8*e^2*z^2 + 18432*a^6*c*e^10*z^2 + 58752*a^3*c^2*d^2*e^5*z + 3456*a^2*c^3*d^6*e*z + 1296*a*c^2*e^4 + 81*c^3*
d^4, z, k)*((6912*a^8*c^5*d*e^24 + 4608*a^3*c^10*d^21*e^4 + 154368*a^4*c^9*d^17*e^8 - 331776*a^5*c^8*d^13*e^12
+ 5976576*a^6*c^7*d^9*e^16 - 612864*a^7*c^6*d^5*e^20)/(256*(a^10*e^24 + a^4*c^6*d^24 + 6*a^9*c*d^4*e^20 + 6*a
^5*c^5*d^20*e^4 + 15*a^6*c^4*d^16*e^8 + 20*a^7*c^3*d^12*e^12 + 15*a^8*c^2*d^8*e^16)) + root(262144*a^10*c*d^4*
e^12*z^4 + 393216*a^9*c^2*d^8*e^8*z^4 + 262144*a^8*c^3*d^12*e^4*z^4 + 65536*a^7*c^4*d^16*z^4 + 65536*a^11*e^16
*z^4 - 786432*a^8*c*d^2*e^11*z^3 + 2359296*a^7*c^2*d^6*e^7*z^3 + 755712*a^5*c^2*d^4*e^6*z^2 + 36864*a^4*c^3*d^
8*e^2*z^2 + 18432*a^6*c*e^10*z^2 + 58752*a^3*c^2*d^2*e^5*z + 3456*a^2*c^3*d^6*e*z + 1296*a*c^2*e^4 + 81*c^3*d^
4, z, k)*((18432*a^5*c^10*d^23*e^5 - 3072*a^4*c^11*d^27*e + 1170432*a^6*c^9*d^19*e^9 + 2863104*a^7*c^8*d^15*e^
13 + 1797120*a^8*c^7*d^11*e^17 - 423936*a^9*c^6*d^7*e^21 - 506880*a^10*c^5*d^3*e^25)/(256*(a^10*e^24 + a^4*c^6
*d^24 + 6*a^9*c*d^4*e^20 + 6*a^5*c^5*d^20*e^4 + 15*a^6*c^4*d^16*e^8 + 20*a^7*c^3*d^12*e^12 + 15*a^8*c^2*d^8*e^
16)) + root(262144*a^10*c*d^4*e^12*z^4 + 393216*a^9*c^2*d^8*e^8*z^4 + 262144*a^8*c^3*d^12*e^4*z^4 + 65536*a^7*
c^4*d^16*z^4 + 65536*a^11*e^16*z^4 - 786432*a^8*c*d^2*e^11*z^3 + 2359296*a^7*c^2*d^6*e^7*z^3 + 755712*a^5*c^2*
d^4*e^6*z^2 + 36864*a^4*c^3*d^8*e^2*z^2 + 18432*a^6*c*e^10*z^2 + 58752*a^3*c^2*d^2*e^5*z + 3456*a^2*c^3*d^6*e*
z + 1296*a*c^2*e^4 + 81*c^3*d^4, z, k)*((98304*a^13*c^4*d*e^30 - 32768*a^6*c^11*d^29*e^2 - 98304*a^7*c^10*d^25
*e^6 + 98304*a^8*c^9*d^21*e^10 + 819200*a^9*c^8*d^17*e^14 + 1474560*a^10*c^7*d^13*e^18 + 1277952*a^11*c^6*d^9*
e^22 + 557056*a^12*c^5*d^5*e^26)/(256*(a^10*e^24 + a^4*c^6*d^24 + 6*a^9*c*d^4*e^20 + 6*a^5*c^5*d^20*e^4 + 15*a
^6*c^4*d^16*e^8 + 20*a^7*c^3*d^12*e^12 + 15*a^8*c^2*d^8*e^16)) + (x*(81920*a^13*c^4*e^31 - 49152*a^6*c^11*d^28
*e^3 - 212992*a^7*c^10*d^24*e^7 - 245760*a^8*c^9*d^20*e^11 + 245760*a^9*c^8*d^16*e^15 + 901120*a^10*c^7*d^12*e
^19 + 933888*a^11*c^6*d^8*e^23 + 442368*a^12*c^5*d^4*e^27))/(256*(a^10*e^24 + a^4*c^6*d^24 + 6*a^9*c*d^4*e^20
+ 6*a^5*c^5*d^20*e^4 + 15*a^6*c^4*d^16*e^8 + 20*a^7*c^3*d^12*e^12 + 15*a^8*c^2*d^8*e^16))) - (x*(12288*a^4*c^1
1*d^26*e^2 + 98304*a^5*c^10*d^22*e^6 - 1413120*a^6*c^9*d^18*e^10 - 4030464*a^7*c^8*d^14*e^14 - 2813952*a^8*c^7
*d^10*e^18 + 393216*a^9*c^6*d^6*e^22 + 675840*a^10*c^5*d^2*e^26))/(256*(a^10*e^24 + a^4*c^6*d^24 + 6*a^9*c*d^4
*e^20 + 6*a^5*c^5*d^20*e^4 + 15*a^6*c^4*d^16*e^8 + 20*a^7*c^3*d^12*e^12 + 15*a^8*c^2*d^8*e^16))) + (x*(20736*a
^8*c^5*e^25 - 576*a^2*c^11*d^24*e - 576*a^3*c^10*d^20*e^5 + 484992*a^4*c^9*d^16*e^9 + 2468736*a^5*c^8*d^12*e^1
3 + 4093632*a^6*c^7*d^8*e^17 - 228672*a^7*c^6*d^4*e^21))/(256*(a^10*e^24 + a^4*c^6*d^24 + 6*a^9*c*d^4*e^20 + 6
*a^5*c^5*d^20*e^4 + 15*a^6*c^4*d^16*e^8 + 20*a^7*c^3*d^12*e^12 + 15*a^8*c^2*d^8*e^16))) + (x*(216*a*c^10*d^18*
e^4 + 25056*a^2*c^9*d^14*e^8 - 2160*a^3*c^8*d^10*e^12 + 59616*a^4*c^7*d^6*e^16 + 86616*a^5*c^6*d^2*e^20))/(256
*(a^10*e^24 + a^4*c^6*d^24 + 6*a^9*c*d^4*e^20 + 6*a^5*c^5*d^20*e^4 + 15*a^6*c^4*d^16*e^8 + 20*a^7*c^3*d^12*e^1
2 + 15*a^8*c^2*d^8*e^16))) + (81*c^9*d^13*e^6 - 2430*a*c^8*d^9*e^10 + 1296*a^3*c^6*d*e^18 + 3969*a^2*c^7*d^5*e
^14)/(256*(a^10*e^24 + a^4*c^6*d^24 + 6*a^9*c*d^4*e^20 + 6*a^5*c^5*d^20*e^4 + 15*a^6*c^4*d^16*e^8 + 20*a^7*c^3
*d^12*e^12 + 15*a^8*c^2*d^8*e^16)) + (x*(1296*a^3*c^6*e^19 + 81*c^9*d^12*e^7 - 6318*a*c^8*d^8*e^11 + 5265*a^2*
c^7*d^4*e^15))/(256*(a^10*e^24 + a^4*c^6*d^24 + 6*a^9*c*d^4*e^20 + 6*a^5*c^5*d^20*e^4 + 15*a^6*c^4*d^16*e^8 +
20*a^7*c^3*d^12*e^12 + 15*a^8*c^2*d^8*e^16)))*root(262144*a^10*c*d^4*e^12*z^4 + 393216*a^9*c^2*d^8*e^8*z^4 + 2
62144*a^8*c^3*d^12*e^4*z^4 + 65536*a^7*c^4*d^16*z^4 + 65536*a^11*e^16*z^4 - 786432*a^8*c*d^2*e^11*z^3 + 235929
6*a^7*c^2*d^6*e^7*z^3 + 755712*a^5*c^2*d^4*e^6*z^2 + 36864*a^4*c^3*d^8*e^2*z^2 + 18432*a^6*c*e^10*z^2 + 58752*
a^3*c^2*d^2*e^5*z + 3456*a^2*c^3*d^6*e*z + 1296*a*c^2*e^4 + 81*c^3*d^4, z, k), k, 1, 4) - ((a^2*e^11 - 5*c^2*d
^8*e^3 + 20*a*c*d^4*e^7)/(2*(a*e^4 + c*d^4)*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) - (3*x^5*(c^3*d^7*e^4 - 7*a*c
^2*d^3*e^8))/(2*a*(a^3*e^12 + c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8)) + (3*x^4*(a^2*c*e^11 - 3*c^3*d^8*
e^3 + 14*a*c^2*d^4*e^7))/(4*a*(a^3*e^12 + c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8)) + (c*d^2*e*x^2)/(4*a*
(a*e^4 + c*d^4)) - (c*d*e^2*x^3)/(4*a*(a*e^4 + c*d^4)) - (d*x*(c^3*d^10 + 8*a*c^2*d^6*e^4 - 41*a^2*c*d^2*e^8))
/(4*a*(a*e^4 + c*d^4)*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)))/(a*d^2 + a*e^2*x^2 + c*d^2*x^4 + c*e^2*x^6 + 2*a*d
*e*x + 2*c*d*e*x^5) + (log(d + e*x)*(36*c^2*d^6*e^7 - 12*a*c*d^2*e^11))/(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)