3.3.100 \(\int \frac {1}{x^7 (a+b x^2)^2 (c+d x^2)} \, dx\) [300]

Optimal. Leaf size=210 \[ -\frac {1}{6 a^2 c x^6}+\frac {2 b c+a d}{4 a^3 c^2 x^4}-\frac {3 b^2 c^2+2 a b c d+a^2 d^2}{2 a^4 c^3 x^2}-\frac {b^4}{2 a^4 (b c-a d) \left (a+b x^2\right )}-\frac {\left (4 b^3 c^3+3 a b^2 c^2 d+2 a^2 b c d^2+a^3 d^3\right ) \log (x)}{a^5 c^4}+\frac {b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{2 a^5 (b c-a d)^2}+\frac {d^5 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^2} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 90} \begin {gather*} \frac {b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{2 a^5 (b c-a d)^2}-\frac {b^4}{2 a^4 \left (a+b x^2\right ) (b c-a d)}+\frac {a d+2 b c}{4 a^3 c^2 x^4}-\frac {1}{6 a^2 c x^6}-\frac {a^2 d^2+2 a b c d+3 b^2 c^2}{2 a^4 c^3 x^2}-\frac {\log (x) \left (a^3 d^3+2 a^2 b c d^2+3 a b^2 c^2 d+4 b^3 c^3\right )}{a^5 c^4}+\frac {d^5 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

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

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]
time = 0.20, size = 202, normalized size = 0.96 \begin {gather*} \frac {1}{12} \left (-\frac {2}{a^2 c x^6}+\frac {6 b c+3 a d}{a^3 c^2 x^4}-\frac {6 \left (3 b^2 c^2+2 a b c d+a^2 d^2\right )}{a^4 c^3 x^2}+\frac {6 b^4}{a^4 (-b c+a d) \left (a+b x^2\right )}-\frac {12 \left (4 b^3 c^3+3 a b^2 c^2 d+2 a^2 b c d^2+a^3 d^3\right ) \log (x)}{a^5 c^4}+\frac {6 b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{a^5 (b c-a d)^2}+\frac {6 d^5 \log \left (c+d x^2\right )}{c^4 (b c-a d)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

(-2/(a^2*c*x^6) + (6*b*c + 3*a*d)/(a^3*c^2*x^4) - (6*(3*b^2*c^2 + 2*a*b*c*d + a^2*d^2))/(a^4*c^3*x^2) + (6*b^4
)/(a^4*(-(b*c) + a*d)*(a + b*x^2)) - (12*(4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*Log[x])/(a^5*c^
4) + (6*b^4*(4*b*c - 5*a*d)*Log[a + b*x^2])/(a^5*(b*c - a*d)^2) + (6*d^5*Log[c + d*x^2])/(c^4*(b*c - a*d)^2))/
12

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Maple [A]
time = 0.16, size = 201, normalized size = 0.96

method	result	size
default	\(-\frac {b^{5} \left (\frac {\left (5 a d -4 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a \left (a d -b c \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{5} \left (a d -b c \right )^{2}}+\frac {d^{5} \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a d -b c \right )^{2}}-\frac {1}{6 a^{2} c \,x^{6}}-\frac {-a d -2 b c}{4 a^{3} c^{2} x^{4}}-\frac {a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}}{2 a^{4} c^{3} x^{2}}+\frac {\left (-a^{3} d^{3}-2 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{5} c^{4}}\)	\(201\)
norman	\(\frac {-\frac {1}{6 a c}+\frac {\left (3 a d +4 b c \right ) x^{2}}{12 a^{2} c^{2}}-\frac {\left (2 a^{2} d^{2}+3 a b c d +4 b^{2} c^{2}\right ) x^{4}}{4 a^{3} c^{3}}+\frac {\left (a^{3} b \,d^{3}+a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d -4 b^{4} c^{3}\right ) b \,x^{8}}{2 c^{3} \left (a d -b c \right ) a^{5}}}{x^{6} \left (b \,x^{2}+a \right )}+\frac {d^{5} \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{5} c^{4}}-\frac {b^{4} \left (5 a d -4 b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(270\)
risch	\(\frac {-\frac {b \left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{6}}{2 a^{4} c^{3} \left (a d -b c \right )}-\frac {\left (2 a^{2} d^{2}+3 a b c d +4 b^{2} c^{2}\right ) x^{4}}{4 a^{3} c^{3}}+\frac {\left (3 a d +4 b c \right ) x^{2}}{12 a^{2} c^{2}}-\frac {1}{6 a c}}{x^{6} \left (b \,x^{2}+a \right )}-\frac {\ln \left (x \right ) d^{3}}{a^{2} c^{4}}-\frac {2 \ln \left (x \right ) b \,d^{2}}{a^{3} c^{3}}-\frac {3 \ln \left (x \right ) b^{2} d}{a^{4} c^{2}}-\frac {4 \ln \left (x \right ) b^{3}}{a^{5} c}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right ) d}{2 a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 b^{5} \ln \left (b \,x^{2}+a \right ) c}{a^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {d^{5} \ln \left (-d \,x^{2}-c \right )}{2 c^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(310\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^2+a)^2/(d*x^2+c),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 339, normalized size = 1.61 \begin {gather*} \frac {d^{5} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}\right )}} + \frac {{\left (4 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2}\right )}} - \frac {2 \, a^{3} b c^{3} - 2 \, a^{4} c^{2} d + 6 \, {\left (4 \, b^{4} c^{3} - a b^{3} c^{2} d - a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{6} + 3 \, {\left (4 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{4} - {\left (4 \, a^{2} b^{2} c^{3} - a^{3} b c^{2} d - 3 \, a^{4} c d^{2}\right )} x^{2}}{12 \, {\left ({\left (a^{4} b^{2} c^{4} - a^{5} b c^{3} d\right )} x^{8} + {\left (a^{5} b c^{4} - a^{6} c^{3} d\right )} x^{6}\right )}} - \frac {{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")

[Out]

1/2*d^5*log(d*x^2 + c)/(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2) + 1/2*(4*b^5*c - 5*a*b^4*d)*log(b*x^2 + a)/(a^5*b
^2*c^2 - 2*a^6*b*c*d + a^7*d^2) - 1/12*(2*a^3*b*c^3 - 2*a^4*c^2*d + 6*(4*b^4*c^3 - a*b^3*c^2*d - a^2*b^2*c*d^2
- a^3*b*d^3)*x^6 + 3*(4*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c*d^2 - 2*a^4*d^3)*x^4 - (4*a^2*b^2*c^3 - a^3*b*c^2
*d - 3*a^4*c*d^2)*x^2)/((a^4*b^2*c^4 - a^5*b*c^3*d)*x^8 + (a^5*b*c^4 - a^6*c^3*d)*x^6) - 1/2*(4*b^3*c^3 + 3*a*
b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*log(x^2)/(a^5*c^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (198) = 396\).
time = 8.81, size = 410, normalized size = 1.95 \begin {gather*} -\frac {2 \, a^{4} b^{2} c^{5} - 4 \, a^{5} b c^{4} d + 2 \, a^{6} c^{3} d^{2} + 6 \, {\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d + a^{5} b c d^{4}\right )} x^{6} + 3 \, {\left (4 \, a^{2} b^{4} c^{5} - 5 \, a^{3} b^{3} c^{4} d - a^{5} b c^{2} d^{3} + 2 \, a^{6} c d^{4}\right )} x^{4} - {\left (4 \, a^{3} b^{3} c^{5} - 5 \, a^{4} b^{2} c^{4} d - 2 \, a^{5} b c^{3} d^{2} + 3 \, a^{6} c^{2} d^{3}\right )} x^{2} - 6 \, {\left ({\left (4 \, b^{6} c^{5} - 5 \, a b^{5} c^{4} d\right )} x^{8} + {\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (a^{5} b d^{5} x^{8} + a^{6} d^{5} x^{6}\right )} \log \left (d x^{2} + c\right ) + 12 \, {\left ({\left (4 \, b^{6} c^{5} - 5 \, a b^{5} c^{4} d + a^{5} b d^{5}\right )} x^{8} + {\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d + a^{6} d^{5}\right )} x^{6}\right )} \log \left (x\right )}{12 \, {\left ({\left (a^{5} b^{3} c^{6} - 2 \, a^{6} b^{2} c^{5} d + a^{7} b c^{4} d^{2}\right )} x^{8} + {\left (a^{6} b^{2} c^{6} - 2 \, a^{7} b c^{5} d + a^{8} c^{4} d^{2}\right )} x^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")

[Out]

-1/12*(2*a^4*b^2*c^5 - 4*a^5*b*c^4*d + 2*a^6*c^3*d^2 + 6*(4*a*b^5*c^5 - 5*a^2*b^4*c^4*d + a^5*b*c*d^4)*x^6 + 3
*(4*a^2*b^4*c^5 - 5*a^3*b^3*c^4*d - a^5*b*c^2*d^3 + 2*a^6*c*d^4)*x^4 - (4*a^3*b^3*c^5 - 5*a^4*b^2*c^4*d - 2*a^
5*b*c^3*d^2 + 3*a^6*c^2*d^3)*x^2 - 6*((4*b^6*c^5 - 5*a*b^5*c^4*d)*x^8 + (4*a*b^5*c^5 - 5*a^2*b^4*c^4*d)*x^6)*l
og(b*x^2 + a) - 6*(a^5*b*d^5*x^8 + a^6*d^5*x^6)*log(d*x^2 + c) + 12*((4*b^6*c^5 - 5*a*b^5*c^4*d + a^5*b*d^5)*x
^8 + (4*a*b^5*c^5 - 5*a^2*b^4*c^4*d + a^6*d^5)*x^6)*log(x))/((a^5*b^3*c^6 - 2*a^6*b^2*c^5*d + a^7*b*c^4*d^2)*x
^8 + (a^6*b^2*c^6 - 2*a^7*b*c^5*d + a^8*c^4*d^2)*x^6)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(b*x**2+a)**2/(d*x**2+c),x)

[Out]

Timed out

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Giac [A]
time = 0.88, size = 354, normalized size = 1.69 \begin {gather*} \frac {d^{6} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )}} + \frac {{\left (4 \, b^{6} c - 5 \, a b^{5} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{5} b^{3} c^{2} - 2 \, a^{6} b^{2} c d + a^{7} b d^{2}\right )}} - \frac {4 \, b^{6} c x^{2} - 5 \, a b^{5} d x^{2} + 5 \, a b^{5} c - 6 \, a^{2} b^{4} d}{2 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2}\right )} {\left (b x^{2} + a\right )}} - \frac {{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5} c^{4}} + \frac {44 \, b^{3} c^{3} x^{6} + 33 \, a b^{2} c^{2} d x^{6} + 22 \, a^{2} b c d^{2} x^{6} + 11 \, a^{3} d^{3} x^{6} - 18 \, a b^{2} c^{3} x^{4} - 12 \, a^{2} b c^{2} d x^{4} - 6 \, a^{3} c d^{2} x^{4} + 6 \, a^{2} b c^{3} x^{2} + 3 \, a^{3} c^{2} d x^{2} - 2 \, a^{3} c^{3}}{12 \, a^{5} c^{4} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")

[Out]

1/2*d^6*log(abs(d*x^2 + c))/(b^2*c^6*d - 2*a*b*c^5*d^2 + a^2*c^4*d^3) + 1/2*(4*b^6*c - 5*a*b^5*d)*log(abs(b*x^
2 + a))/(a^5*b^3*c^2 - 2*a^6*b^2*c*d + a^7*b*d^2) - 1/2*(4*b^6*c*x^2 - 5*a*b^5*d*x^2 + 5*a*b^5*c - 6*a^2*b^4*d
)/((a^5*b^2*c^2 - 2*a^6*b*c*d + a^7*d^2)*(b*x^2 + a)) - 1/2*(4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d
^3)*log(x^2)/(a^5*c^4) + 1/12*(44*b^3*c^3*x^6 + 33*a*b^2*c^2*d*x^6 + 22*a^2*b*c*d^2*x^6 + 11*a^3*d^3*x^6 - 18*
a*b^2*c^3*x^4 - 12*a^2*b*c^2*d*x^4 - 6*a^3*c*d^2*x^4 + 6*a^2*b*c^3*x^2 + 3*a^3*c^2*d*x^2 - 2*a^3*c^3)/(a^5*c^4
*x^6)

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Mupad [B]
time = 0.66, size = 278, normalized size = 1.32 \begin {gather*} \frac {\ln \left (b\,x^2+a\right )\,\left (4\,b^5\,c-5\,a\,b^4\,d\right )}{2\,a^7\,d^2-4\,a^6\,b\,c\,d+2\,a^5\,b^2\,c^2}-\frac {\frac {1}{6\,a\,c}-\frac {x^2\,\left (3\,a\,d+4\,b\,c\right )}{12\,a^2\,c^2}+\frac {x^4\,\left (2\,a^2\,d^2+3\,a\,b\,c\,d+4\,b^2\,c^2\right )}{4\,a^3\,c^3}+\frac {x^6\,\left (a^3\,b\,d^3+a^2\,b^2\,c\,d^2+a\,b^3\,c^2\,d-4\,b^4\,c^3\right )}{2\,a^4\,c^3\,\left (a\,d-b\,c\right )}}{b\,x^8+a\,x^6}+\frac {d^5\,\ln \left (d\,x^2+c\right )}{2\,\left (a^2\,c^4\,d^2-2\,a\,b\,c^5\,d+b^2\,c^6\right )}-\frac {\ln \left (x\right )\,\left (a^3\,d^3+2\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+4\,b^3\,c^3\right )}{a^5\,c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(a + b*x^2)^2*(c + d*x^2)),x)

[Out]

(log(a + b*x^2)*(4*b^5*c - 5*a*b^4*d))/(2*a^7*d^2 + 2*a^5*b^2*c^2 - 4*a^6*b*c*d) - (1/(6*a*c) - (x^2*(3*a*d +
4*b*c))/(12*a^2*c^2) + (x^4*(2*a^2*d^2 + 4*b^2*c^2 + 3*a*b*c*d))/(4*a^3*c^3) + (x^6*(a^3*b*d^3 - 4*b^4*c^3 + a
^2*b^2*c*d^2 + a*b^3*c^2*d))/(2*a^4*c^3*(a*d - b*c)))/(a*x^6 + b*x^8) + (d^5*log(c + d*x^2))/(2*(b^2*c^6 + a^2
*c^4*d^2 - 2*a*b*c^5*d)) - (log(x)*(a^3*d^3 + 4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2))/(a^5*c^4)

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