3.251 \(\int \frac {1}{x^5 (d+e x^2) (a+c x^4)^2} \, dx\)

Optimal. Leaf size=265 \[ \frac {c^{3/2} e \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2} \left (a e^2+c d^2\right )^2}+\frac {c^{3/2} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{5/2} \left (a e^2+c d^2\right )}+\frac {c^2 d \left (3 a e^2+2 c d^2\right ) \log \left (a+c x^4\right )}{4 a^3 \left (a e^2+c d^2\right )^2}-\frac {\log (x) \left (2 c d^2-a e^2\right )}{a^3 d^3}-\frac {c^2 \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac {e}{2 a^2 d^2 x^2}-\frac {1}{4 a^2 d x^4}-\frac {e^6 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2+c d^2\right )^2} \]

[Out]

-1/4/a^2/d/x^4+1/2*e/a^2/d^2/x^2-1/4*c^2*(-e*x^2+d)/a^2/(a*e^2+c*d^2)/(c*x^4+a)+1/4*c^(3/2)*e*arctan(x^2*c^(1/
2)/a^(1/2))/a^(5/2)/(a*e^2+c*d^2)+1/2*c^(3/2)*e*(2*a*e^2+c*d^2)*arctan(x^2*c^(1/2)/a^(1/2))/a^(5/2)/(a*e^2+c*d
^2)^2-(-a*e^2+2*c*d^2)*ln(x)/a^3/d^3-1/2*e^6*ln(e*x^2+d)/d^3/(a*e^2+c*d^2)^2+1/4*c^2*d*(3*a*e^2+2*c*d^2)*ln(c*
x^4+a)/a^3/(a*e^2+c*d^2)^2

________________________________________________________________________________________

Rubi [A]  time = 0.33, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1252, 894, 639, 205, 635, 260} \[ -\frac {c^2 \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac {c^2 d \left (3 a e^2+2 c d^2\right ) \log \left (a+c x^4\right )}{4 a^3 \left (a e^2+c d^2\right )^2}+\frac {c^{3/2} e \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2} \left (a e^2+c d^2\right )^2}+\frac {c^{3/2} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{5/2} \left (a e^2+c d^2\right )}-\frac {\log (x) \left (2 c d^2-a e^2\right )}{a^3 d^3}+\frac {e}{2 a^2 d^2 x^2}-\frac {1}{4 a^2 d x^4}-\frac {e^6 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

-1/(4*a^2*d*x^4) + e/(2*a^2*d^2*x^2) - (c^2*(d - e*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (c^(3/2)*e*ArcT
an[(Sqrt[c]*x^2)/Sqrt[a]])/(4*a^(5/2)*(c*d^2 + a*e^2)) + (c^(3/2)*e*(c*d^2 + 2*a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqr
t[a]])/(2*a^(5/2)*(c*d^2 + a*e^2)^2) - ((2*c*d^2 - a*e^2)*Log[x])/(a^3*d^3) - (e^6*Log[d + e*x^2])/(2*d^3*(c*d
^2 + a*e^2)^2) + (c^2*d*(2*c*d^2 + 3*a*e^2)*Log[a + c*x^4])/(4*a^3*(c*d^2 + a*e^2)^2)

Rule 205

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

Rule 260

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 635

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 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1252

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

Rubi steps

\begin {align*} \int \frac {1}{x^5 \left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^3 (d+e x) \left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^2 d x^3}-\frac {e}{a^2 d^2 x^2}+\frac {-2 c d^2+a e^2}{a^3 d^3 x}-\frac {e^7}{d^3 \left (c d^2+a e^2\right )^2 (d+e x)}+\frac {c^2 (a e+c d x)}{a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {c^2 \left (a e \left (c d^2+2 a e^2\right )+c d \left (2 c d^2+3 a e^2\right ) x\right )}{a^3 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a^2 d x^4}+\frac {e}{2 a^2 d^2 x^2}-\frac {\left (2 c d^2-a e^2\right ) \log (x)}{a^3 d^3}-\frac {e^6 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2+a e^2\right )^2}+\frac {c^2 \operatorname {Subst}\left (\int \frac {a e \left (c d^2+2 a e^2\right )+c d \left (2 c d^2+3 a e^2\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 a^3 \left (c d^2+a e^2\right )^2}+\frac {c^2 \operatorname {Subst}\left (\int \frac {a e+c d x}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2+a e^2\right )}\\ &=-\frac {1}{4 a^2 d x^4}+\frac {e}{2 a^2 d^2 x^2}-\frac {c^2 \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {\left (2 c d^2-a e^2\right ) \log (x)}{a^3 d^3}-\frac {e^6 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2+a e^2\right )^2}+\frac {\left (c^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 a^2 \left (c d^2+a e^2\right )}+\frac {\left (c^2 e \left (c d^2+2 a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2+a e^2\right )^2}+\frac {\left (c^3 d \left (2 c d^2+3 a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{2 a^3 \left (c d^2+a e^2\right )^2}\\ &=-\frac {1}{4 a^2 d x^4}+\frac {e}{2 a^2 d^2 x^2}-\frac {c^2 \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {c^{3/2} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{5/2} \left (c d^2+a e^2\right )}+\frac {c^{3/2} e \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2} \left (c d^2+a e^2\right )^2}-\frac {\left (2 c d^2-a e^2\right ) \log (x)}{a^3 d^3}-\frac {e^6 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2+a e^2\right )^2}+\frac {c^2 d \left (2 c d^2+3 a e^2\right ) \log \left (a+c x^4\right )}{4 a^3 \left (c d^2+a e^2\right )^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.39, size = 278, normalized size = 1.05 \[ \frac {1}{4} \left (-\frac {c^{3/2} e \left (5 a e^2+3 c d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{5/2} \left (a e^2+c d^2\right )^2}-\frac {c^{3/2} e \left (5 a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{5/2} \left (a e^2+c d^2\right )^2}+\frac {c^2 \left (3 a d e^2+2 c d^3\right ) \log \left (a+c x^4\right )}{a^3 \left (a e^2+c d^2\right )^2}+\frac {4 \log (x) \left (a e^2-2 c d^2\right )}{a^3 d^3}+\frac {c^2 \left (e x^2-d\right )}{a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac {2 e}{a^2 d^2 x^2}-\frac {1}{a^2 d x^4}-\frac {2 e^6 \log \left (d+e x^2\right )}{d^3 \left (a e^2+c d^2\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

(-(1/(a^2*d*x^4)) + (2*e)/(a^2*d^2*x^2) + (c^2*(-d + e*x^2))/(a^2*(c*d^2 + a*e^2)*(a + c*x^4)) - (c^(3/2)*e*(3
*c*d^2 + 5*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/2)*(c*d^2 + a*e^2)^2) - (c^(3/2)*e*(3*c*d^2 +
5*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/2)*(c*d^2 + a*e^2)^2) + (4*(-2*c*d^2 + a*e^2)*Log[x])
/(a^3*d^3) - (2*e^6*Log[d + e*x^2])/(d^3*(c*d^2 + a*e^2)^2) + (c^2*(2*c*d^3 + 3*a*d*e^2)*Log[a + c*x^4])/(a^3*
(c*d^2 + a*e^2)^2))/4

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [A]  time = 0.37, size = 350, normalized size = 1.32 \[ \frac {{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \log \left (c x^{4} + a\right )}{4 \, {\left (a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}\right )}} - \frac {e^{7} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c^{2} d^{7} e + 2 \, a c d^{5} e^{3} + a^{2} d^{3} e^{5}\right )}} + \frac {{\left (3 \, c^{3} d^{2} e + 5 \, a c^{2} e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \sqrt {a c}} - \frac {2 \, c^{4} d^{3} x^{4} + 3 \, a c^{3} d x^{4} e^{2} - a c^{3} d^{2} x^{2} e + 3 \, a c^{3} d^{3} - a^{2} c^{2} x^{2} e^{3} + 4 \, a^{2} c^{2} d e^{2}}{4 \, {\left (a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}\right )} {\left (c x^{4} + a\right )}} - \frac {{\left (2 \, c d^{2} - a e^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{3} d^{3}} + \frac {6 \, c d^{2} x^{4} - 3 \, a x^{4} e^{2} + 2 \, a d x^{2} e - a d^{2}}{4 \, a^{3} d^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="giac")

[Out]

1/4*(2*c^3*d^3 + 3*a*c^2*d*e^2)*log(c*x^4 + a)/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4) - 1/2*e^7*log(abs(x^2
*e + d))/(c^2*d^7*e + 2*a*c*d^5*e^3 + a^2*d^3*e^5) + 1/4*(3*c^3*d^2*e + 5*a*c^2*e^3)*arctan(c*x^2/sqrt(a*c))/(
(a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4)*sqrt(a*c)) - 1/4*(2*c^4*d^3*x^4 + 3*a*c^3*d*x^4*e^2 - a*c^3*d^2*x^2*
e + 3*a*c^3*d^3 - a^2*c^2*x^2*e^3 + 4*a^2*c^2*d*e^2)/((a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4)*(c*x^4 + a)) -
1/2*(2*c*d^2 - a*e^2)*log(x^2)/(a^3*d^3) + 1/4*(6*c*d^2*x^4 - 3*a*x^4*e^2 + 2*a*d*x^2*e - a*d^2)/(a^3*d^3*x^4
)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 363, normalized size = 1.37 \[ \frac {c^{2} e^{3} x^{2}}{4 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{4}+a \right ) a}+\frac {c^{3} d^{2} e \,x^{2}}{4 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{4}+a \right ) a^{2}}+\frac {5 c^{2} e^{3} \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{4 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {a c}\, a}+\frac {3 c^{3} d^{2} e \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{4 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {a c}\, a^{2}}-\frac {c^{2} d \,e^{2}}{4 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{4}+a \right ) a}-\frac {c^{3} d^{3}}{4 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{4}+a \right ) a^{2}}+\frac {3 c^{2} d \,e^{2} \ln \left (c \,x^{4}+a \right )}{4 \left (a \,e^{2}+c \,d^{2}\right )^{2} a^{2}}+\frac {c^{3} d^{3} \ln \left (c \,x^{4}+a \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a^{3}}-\frac {e^{6} \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} d^{3}}+\frac {e^{2} \ln \relax (x )}{a^{2} d^{3}}-\frac {2 c \ln \relax (x )}{a^{3} d}+\frac {e}{2 a^{2} d^{2} x^{2}}-\frac {1}{4 a^{2} d \,x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(e*x^2+d)/(c*x^4+a)^2,x)

[Out]

-1/4/a^2/d/x^4+1/a^2/d^3*ln(x)*e^2-2/a^3/d*ln(x)*c+1/2*e/a^2/d^2/x^2+1/4*c^2/(a*e^2+c*d^2)^2/a/(c*x^4+a)*e^3*x
^2+1/4*c^3/(a*e^2+c*d^2)^2/a^2/(c*x^4+a)*x^2*e*d^2-1/4*c^2/(a*e^2+c*d^2)^2/a/(c*x^4+a)*d*e^2-1/4*c^3/(a*e^2+c*
d^2)^2/a^2/(c*x^4+a)*d^3+3/4*c^2/(a*e^2+c*d^2)^2/a^2*ln(c*x^4+a)*d*e^2+1/2*c^3/(a*e^2+c*d^2)^2/a^3*ln(c*x^4+a)
*d^3+5/4*c^2/(a*e^2+c*d^2)^2/a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x^2)*e^3+3/4*c^3/(a*e^2+c*d^2)^2/a^2/(a*c)^(
1/2)*arctan(1/(a*c)^(1/2)*c*x^2)*e*d^2-1/2*e^6*ln(e*x^2+d)/d^3/(a*e^2+c*d^2)^2

________________________________________________________________________________________

maxima [A]  time = 2.08, size = 332, normalized size = 1.25 \[ -\frac {e^{6} \log \left (e x^{2} + d\right )}{2 \, {\left (c^{2} d^{7} + 2 \, a c d^{5} e^{2} + a^{2} d^{3} e^{4}\right )}} + \frac {{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \log \left (c x^{4} + a\right )}{4 \, {\left (a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}\right )}} + \frac {{\left (3 \, c^{3} d^{2} e + 5 \, a c^{2} e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \sqrt {a c}} + \frac {{\left (3 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{6} - a c d^{3} - a^{2} d e^{2} - {\left (2 \, c^{2} d^{3} + a c d e^{2}\right )} x^{4} + 2 \, {\left (a c d^{2} e + a^{2} e^{3}\right )} x^{2}}{4 \, {\left ({\left (a^{2} c^{2} d^{4} + a^{3} c d^{2} e^{2}\right )} x^{8} + {\left (a^{3} c d^{4} + a^{4} d^{2} e^{2}\right )} x^{4}\right )}} - \frac {{\left (2 \, c d^{2} - a e^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

-1/2*e^6*log(e*x^2 + d)/(c^2*d^7 + 2*a*c*d^5*e^2 + a^2*d^3*e^4) + 1/4*(2*c^3*d^3 + 3*a*c^2*d*e^2)*log(c*x^4 +
a)/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4) + 1/4*(3*c^3*d^2*e + 5*a*c^2*e^3)*arctan(c*x^2/sqrt(a*c))/((a^2*c
^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4)*sqrt(a*c)) + 1/4*((3*c^2*d^2*e + 2*a*c*e^3)*x^6 - a*c*d^3 - a^2*d*e^2 - (2
*c^2*d^3 + a*c*d*e^2)*x^4 + 2*(a*c*d^2*e + a^2*e^3)*x^2)/((a^2*c^2*d^4 + a^3*c*d^2*e^2)*x^8 + (a^3*c*d^4 + a^4
*d^2*e^2)*x^4) - 1/2*(2*c*d^2 - a*e^2)*log(x^2)/(a^3*d^3)

________________________________________________________________________________________

mupad [B]  time = 3.48, size = 1545, normalized size = 5.83 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + c*x^4)^2*(d + e*x^2)),x)

[Out]

(log(6400*a^13*c^18*d^28*x^2 + 1024*a^27*c^4*e^28*x^2 - 6400*a^3*c^13*d^28*(-a^7*c^3)^(3/2) + 1024*a^24*c^2*e^
28*(-a^7*c^3)^(1/2) - 10688*a^6*d^8*e^20*(-a^7*c^3)^(5/2) - 2048*a^16*d^2*e^26*(-a^7*c^3)^(3/2) + 536959*c^6*d
^20*e^8*(-a^7*c^3)^(5/2) + 54944*a^14*c^17*d^26*e^2*x^2 + 200881*a^15*c^16*d^24*e^4*x^2 + 413414*a^16*c^15*d^2
2*e^6*x^2 + 536959*a^17*c^14*d^20*e^8*x^2 + 465092*a^18*c^13*d^18*e^10*x^2 + 256991*a^19*c^12*d^16*e^12*x^2 +
52822*a^20*c^11*d^14*e^14*x^2 - 37423*a^21*c^10*d^12*e^16*x^2 - 27472*a^22*c^9*d^10*e^18*x^2 - 10688*a^23*c^8*
d^8*e^20*x^2 - 10288*a^24*c^7*d^6*e^22*x^2 - 3584*a^25*c^6*d^4*e^24*x^2 + 2048*a^26*c^5*d^2*e^26*x^2 + 465092*
a*c^5*d^18*e^10*(-a^7*c^3)^(5/2) - 27472*a^5*c*d^10*e^18*(-a^7*c^3)^(5/2) + 3584*a^15*c*d^4*e^24*(-a^7*c^3)^(3
/2) + 256991*a^2*c^4*d^16*e^12*(-a^7*c^3)^(5/2) + 52822*a^3*c^3*d^14*e^14*(-a^7*c^3)^(5/2) - 37423*a^4*c^2*d^1
2*e^16*(-a^7*c^3)^(5/2) - 54944*a^4*c^12*d^26*e^2*(-a^7*c^3)^(3/2) - 200881*a^5*c^11*d^24*e^4*(-a^7*c^3)^(3/2)
- 413414*a^6*c^10*d^22*e^6*(-a^7*c^3)^(3/2) + 10288*a^14*c^2*d^6*e^22*(-a^7*c^3)^(3/2))*(4*a^3*c^3*d^3 + 5*a*
e^3*(-a^7*c^3)^(1/2) + 6*a^4*c^2*d*e^2 + 3*c*d^2*e*(-a^7*c^3)^(1/2)))/(8*(a^8*e^4 + a^6*c^2*d^4 + 2*a^7*c*d^2*
e^2)) - (e^6*log(d + e*x^2))/(2*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)) - (1/(4*a*d) - (e*x^2)/(2*a*d^2) + (x
^4*(2*c^2*d^2 + a*c*e^2))/(4*a^2*d*(a*e^2 + c*d^2)) - (c*e*x^6*(2*a*e^2 + 3*c*d^2))/(4*a^2*d^2*(a*e^2 + c*d^2)
))/(a*x^4 + c*x^8) + (log(6400*a^13*c^18*d^28*x^2 + 1024*a^27*c^4*e^28*x^2 + 6400*a^3*c^13*d^28*(-a^7*c^3)^(3/
2) - 1024*a^24*c^2*e^28*(-a^7*c^3)^(1/2) + 10688*a^6*d^8*e^20*(-a^7*c^3)^(5/2) + 2048*a^16*d^2*e^26*(-a^7*c^3)
^(3/2) - 536959*c^6*d^20*e^8*(-a^7*c^3)^(5/2) + 54944*a^14*c^17*d^26*e^2*x^2 + 200881*a^15*c^16*d^24*e^4*x^2 +
413414*a^16*c^15*d^22*e^6*x^2 + 536959*a^17*c^14*d^20*e^8*x^2 + 465092*a^18*c^13*d^18*e^10*x^2 + 256991*a^19*
c^12*d^16*e^12*x^2 + 52822*a^20*c^11*d^14*e^14*x^2 - 37423*a^21*c^10*d^12*e^16*x^2 - 27472*a^22*c^9*d^10*e^18*
x^2 - 10688*a^23*c^8*d^8*e^20*x^2 - 10288*a^24*c^7*d^6*e^22*x^2 - 3584*a^25*c^6*d^4*e^24*x^2 + 2048*a^26*c^5*d
^2*e^26*x^2 - 465092*a*c^5*d^18*e^10*(-a^7*c^3)^(5/2) + 27472*a^5*c*d^10*e^18*(-a^7*c^3)^(5/2) - 3584*a^15*c*d
^4*e^24*(-a^7*c^3)^(3/2) - 256991*a^2*c^4*d^16*e^12*(-a^7*c^3)^(5/2) - 52822*a^3*c^3*d^14*e^14*(-a^7*c^3)^(5/2
) + 37423*a^4*c^2*d^12*e^16*(-a^7*c^3)^(5/2) + 54944*a^4*c^12*d^26*e^2*(-a^7*c^3)^(3/2) + 200881*a^5*c^11*d^24
*e^4*(-a^7*c^3)^(3/2) + 413414*a^6*c^10*d^22*e^6*(-a^7*c^3)^(3/2) - 10288*a^14*c^2*d^6*e^22*(-a^7*c^3)^(3/2))*
(4*a^3*c^3*d^3 - 5*a*e^3*(-a^7*c^3)^(1/2) + 6*a^4*c^2*d*e^2 - 3*c*d^2*e*(-a^7*c^3)^(1/2)))/(8*(a^8*e^4 + a^6*c
^2*d^4 + 2*a^7*c*d^2*e^2)) + (log(x)*(a*e^2 - 2*c*d^2))/(a^3*d^3)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________