3.317 \(\int \frac{1}{x^3 (a+b x^2)^2 (c+d x^2)^3} \, dx\)

Optimal. Leaf size=215 \[ \frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}-\frac{b^4}{2 a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac{b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{1}{2 a^2 c^3 x^2}-\frac{d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

-1/(2*a^2*c^3*x^2) - b^4/(2*a^2*(b*c - a*d)^3*(a + b*x^2)) - d^3/(4*c^2*(b*c - a*d)^2*(c + d*x^2)^2) - (d^3*(2
*b*c - a*d))/(c^3*(b*c - a*d)^3*(c + d*x^2)) - ((2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[a
+ b*x^2])/(2*a^3*(b*c - a*d)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x^2])/(2*c^4*(b*c - a*
d)^4)

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Rubi [A]  time = 0.294329, antiderivative size = 215, normalized size of antiderivative = 1., 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 = {446, 88} \[ \frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}-\frac{b^4}{2 a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac{b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{1}{2 a^2 c^3 x^2}-\frac{d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*a^2*c^3*x^2) - b^4/(2*a^2*(b*c - a*d)^3*(a + b*x^2)) - d^3/(4*c^2*(b*c - a*d)^2*(c + d*x^2)^2) - (d^3*(2
*b*c - a*d))/(c^3*(b*c - a*d)^3*(c + d*x^2)) - ((2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[a
+ b*x^2])/(2*a^3*(b*c - a*d)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x^2])/(2*c^4*(b*c - a*
d)^4)

Rule 446

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]]

Rule 88

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]))

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^2 (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 c^3 x^2}+\frac{-2 b c-3 a d}{a^3 c^4 x}-\frac{b^5}{a^2 (-b c+a d)^3 (a+b x)^2}-\frac{b^5 (-2 b c+5 a d)}{a^3 (-b c+a d)^4 (a+b x)}+\frac{d^4}{c^2 (b c-a d)^2 (c+d x)^3}+\frac{2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)^2}+\frac{d^4 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )}{c^4 (b c-a d)^4 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 a^2 c^3 x^2}-\frac{b^4}{2 a^2 (b c-a d)^3 \left (a+b x^2\right )}-\frac{d^3}{4 c^2 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{d^3 (2 b c-a d)}{c^3 (b c-a d)^3 \left (c+d x^2\right )}-\frac{(2 b c+3 a d) \log (x)}{a^3 c^4}+\frac{b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 0.321434, size = 208, normalized size = 0.97 \[ \frac{1}{4} \left (\frac{2 d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{c^4 (b c-a d)^4}+\frac{2 b^4}{a^2 \left (a+b x^2\right ) (a d-b c)^3}+\frac{2 b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^4}-\frac{4 \log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{2}{a^2 c^3 x^2}+\frac{4 d^3 (a d-2 b c)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3}{c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2/(a^2*c^3*x^2) + (2*b^4)/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - d^3/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (4*d^
3*(-2*b*c + a*d))/(c^3*(b*c - a*d)^3*(c + d*x^2)) - (4*(2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (2*b^4*(2*b*c - 5*a
*d)*Log[a + b*x^2])/(a^3*(b*c - a*d)^4) + (2*d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x^2])/(c^4*(b
*c - a*d)^4))/4

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Maple [A]  time = 0.028, size = 405, normalized size = 1.9 \begin{align*}{\frac{3\,{d}^{5}\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{c}^{4} \left ( ad-bc \right ) ^{4}}}-5\,{\frac{{d}^{4}\ln \left ( d{x}^{2}+c \right ) ab}{{c}^{3} \left ( ad-bc \right ) ^{4}}}+5\,{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ){b}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{4}}}-{\frac{{d}^{5}{a}^{2}}{4\,{c}^{2} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{d}^{4}ab}{2\,c \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{b}^{2}{d}^{3}}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{d}^{5}{a}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}+3\,{\frac{{d}^{4}ab}{{c}^{2} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}-2\,{\frac{{b}^{2}{d}^{3}}{c \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}-{\frac{1}{2\,{a}^{2}{c}^{3}{x}^{2}}}-3\,{\frac{\ln \left ( x \right ) d}{{a}^{2}{c}^{4}}}-2\,{\frac{\ln \left ( x \right ) b}{{a}^{3}{c}^{3}}}-{\frac{5\,{b}^{4}\ln \left ( b{x}^{2}+a \right ) d}{2\,{a}^{2} \left ( ad-bc \right ) ^{4}}}+{\frac{{b}^{5}\ln \left ( b{x}^{2}+a \right ) c}{{a}^{3} \left ( ad-bc \right ) ^{4}}}+{\frac{{b}^{4}d}{2\,a \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{5}c}{2\,{a}^{2} \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.18909, size = 879, normalized size = 4.09 \begin{align*} \frac{{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )}} + \frac{{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )}} - \frac{2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \,{\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{6} +{\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{4} +{\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x^{2}}{4 \,{\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{8} +{\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{6} +{\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{4} +{\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x^{2}\right )}} - \frac{{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^5*c - 5*a*b^4*d)*log(b*x^2 + a)/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 +
a^7*d^4) + 1/2*(10*b^2*c^2*d^3 - 10*a*b*c*d^4 + 3*a^2*d^5)*log(d*x^2 + c)/(b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2
*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4) - 1/4*(2*a*b^3*c^5 - 6*a^2*b^2*c^4*d + 6*a^3*b*c^3*d^2 - 2*a^4*c^2*d
^3 + 2*(2*b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 7*a^2*b^2*c*d^4 - 3*a^3*b*d^5)*x^6 + (8*b^4*c^4*d - 10*a*b^3*c^3*d^2
+ 15*a^2*b^2*c^2*d^3 + 5*a^3*b*c*d^4 - 6*a^4*d^5)*x^4 + (4*b^4*c^5 - 2*a*b^3*c^4*d - 6*a^2*b^2*c^3*d^2 + 19*a
^3*b*c^2*d^3 - 9*a^4*c*d^4)*x^2)/((a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^
8 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)*x^6 + (a^2*b^4*c^8
- a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^4 + (a^3*b^3*c^8 - 3*a^4*b^2*c^7*d +
3*a^5*b*c^6*d^2 - a^6*c^5*d^3)*x^2) - 1/2*(2*b*c + 3*a*d)*log(x^2)/(a^3*c^4)

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Fricas [B]  time = 86.5762, size = 2446, normalized size = 11.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a^2*b^4*c^7 - 8*a^3*b^3*c^6*d + 12*a^4*b^2*c^5*d^2 - 8*a^5*b*c^4*d^3 + 2*a^6*c^3*d^4 + 2*(2*a*b^5*c^5*
d^2 - 5*a^2*b^4*c^4*d^3 + 10*a^3*b^3*c^3*d^4 - 10*a^4*b^2*c^2*d^5 + 3*a^5*b*c*d^6)*x^6 + (8*a*b^5*c^6*d - 18*a
^2*b^4*c^5*d^2 + 25*a^3*b^3*c^4*d^3 - 10*a^4*b^2*c^3*d^4 - 11*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^4 + (4*a*b^5*c^7
- 6*a^2*b^4*c^6*d - 4*a^3*b^3*c^5*d^2 + 25*a^4*b^2*c^4*d^3 - 28*a^5*b*c^3*d^4 + 9*a^6*c^2*d^5)*x^2 - 2*((2*b^6
*c^5*d^2 - 5*a*b^5*c^4*d^3)*x^8 + (4*b^6*c^6*d - 8*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3)*x^6 + (2*b^6*c^7 - a*b^5
*c^6*d - 10*a^2*b^4*c^5*d^2)*x^4 + (2*a*b^5*c^7 - 5*a^2*b^4*c^6*d)*x^2)*log(b*x^2 + a) - 2*((10*a^3*b^3*c^2*d^
5 - 10*a^4*b^2*c*d^6 + 3*a^5*b*d^7)*x^8 + (20*a^3*b^3*c^3*d^4 - 10*a^4*b^2*c^2*d^5 - 4*a^5*b*c*d^6 + 3*a^6*d^7
)*x^6 + (10*a^3*b^3*c^4*d^3 + 10*a^4*b^2*c^3*d^4 - 17*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^4 + (10*a^4*b^2*c^4*d^3 -
10*a^5*b*c^3*d^4 + 3*a^6*c^2*d^5)*x^2)*log(d*x^2 + c) + 4*((2*b^6*c^5*d^2 - 5*a*b^5*c^4*d^3 + 10*a^3*b^3*c^2*
d^5 - 10*a^4*b^2*c*d^6 + 3*a^5*b*d^7)*x^8 + (4*b^6*c^6*d - 8*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3 + 20*a^3*b^3*c^
3*d^4 - 10*a^4*b^2*c^2*d^5 - 4*a^5*b*c*d^6 + 3*a^6*d^7)*x^6 + (2*b^6*c^7 - a*b^5*c^6*d - 10*a^2*b^4*c^5*d^2 +
10*a^3*b^3*c^4*d^3 + 10*a^4*b^2*c^3*d^4 - 17*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^4 + (2*a*b^5*c^7 - 5*a^2*b^4*c^6*d
+ 10*a^4*b^2*c^4*d^3 - 10*a^5*b*c^3*d^4 + 3*a^6*c^2*d^5)*x^2)*log(x))/((a^3*b^5*c^8*d^2 - 4*a^4*b^4*c^7*d^3 +
6*a^5*b^3*c^6*d^4 - 4*a^6*b^2*c^5*d^5 + a^7*b*c^4*d^6)*x^8 + (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d^2 + 8*a^5*b^3
*c^7*d^3 - 2*a^6*b^2*c^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x^6 + (a^3*b^5*c^10 - 2*a^4*b^4*c^9*d - 2*a^5*b^
3*c^8*d^2 + 8*a^6*b^2*c^7*d^3 - 7*a^7*b*c^6*d^4 + 2*a^8*c^5*d^5)*x^4 + (a^4*b^4*c^10 - 4*a^5*b^3*c^9*d + 6*a^6
*b^2*c^8*d^2 - 4*a^7*b*c^7*d^3 + a^8*c^6*d^4)*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.17303, size = 861, normalized size = 4. \begin{align*} \frac{{\left (2 \, b^{6} c - 5 \, a b^{5} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )}} + \frac{{\left (10 \, b^{2} c^{2} d^{4} - 10 \, a b c d^{5} + 3 \, a^{2} d^{6}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )}} + \frac{10 \, a^{2} b^{3} c^{2} d^{3} x^{4} - 10 \, a^{3} b^{2} c d^{4} x^{4} + 3 \, a^{4} b d^{5} x^{4} - 4 \, b^{5} c^{5} x^{2} + 10 \, a b^{4} c^{4} d x^{2} - 12 \, a^{2} b^{3} c^{3} d^{2} x^{2} + 18 \, a^{3} b^{2} c^{2} d^{3} x^{2} - 12 \, a^{4} b c d^{4} x^{2} + 3 \, a^{5} d^{5} x^{2} - 2 \, a b^{4} c^{5} + 8 \, a^{2} b^{3} c^{4} d - 12 \, a^{3} b^{2} c^{3} d^{2} + 8 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}}{4 \,{\left (a^{2} b^{4} c^{8} - 4 \, a^{3} b^{3} c^{7} d + 6 \, a^{4} b^{2} c^{6} d^{2} - 4 \, a^{5} b c^{5} d^{3} + a^{6} c^{4} d^{4}\right )}{\left (b x^{4} + a x^{2}\right )}} - \frac{30 \, b^{2} c^{2} d^{5} x^{4} - 30 \, a b c d^{6} x^{4} + 9 \, a^{2} d^{7} x^{4} + 68 \, b^{2} c^{3} d^{4} x^{2} - 72 \, a b c^{2} d^{5} x^{2} + 22 \, a^{2} c d^{6} x^{2} + 39 \, b^{2} c^{4} d^{3} - 44 \, a b c^{3} d^{4} + 14 \, a^{2} c^{2} d^{5}}{4 \,{\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )}{\left (d x^{2} + c\right )}^{2}} - \frac{{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^6*c - 5*a*b^5*d)*log(abs(b*x^2 + a))/(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c
*d^3 + a^7*b*d^4) + 1/2*(10*b^2*c^2*d^4 - 10*a*b*c*d^5 + 3*a^2*d^6)*log(abs(d*x^2 + c))/(b^4*c^8*d - 4*a*b^3*c
^7*d^2 + 6*a^2*b^2*c^6*d^3 - 4*a^3*b*c^5*d^4 + a^4*c^4*d^5) + 1/4*(10*a^2*b^3*c^2*d^3*x^4 - 10*a^3*b^2*c*d^4*x
^4 + 3*a^4*b*d^5*x^4 - 4*b^5*c^5*x^2 + 10*a*b^4*c^4*d*x^2 - 12*a^2*b^3*c^3*d^2*x^2 + 18*a^3*b^2*c^2*d^3*x^2 -
12*a^4*b*c*d^4*x^2 + 3*a^5*d^5*x^2 - 2*a*b^4*c^5 + 8*a^2*b^3*c^4*d - 12*a^3*b^2*c^3*d^2 + 8*a^4*b*c^2*d^3 - 2*
a^5*c*d^4)/((a^2*b^4*c^8 - 4*a^3*b^3*c^7*d + 6*a^4*b^2*c^6*d^2 - 4*a^5*b*c^5*d^3 + a^6*c^4*d^4)*(b*x^4 + a*x^2
)) - 1/4*(30*b^2*c^2*d^5*x^4 - 30*a*b*c*d^6*x^4 + 9*a^2*d^7*x^4 + 68*b^2*c^3*d^4*x^2 - 72*a*b*c^2*d^5*x^2 + 22
*a^2*c*d^6*x^2 + 39*b^2*c^4*d^3 - 44*a*b*c^3*d^4 + 14*a^2*c^2*d^5)/((b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d
^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4)*(d*x^2 + c)^2) - 1/2*(2*b*c + 3*a*d)*log(x^2)/(a^3*c^4)