3.298 \(\int \frac{1}{x^5 (a+b x^2)^2 (c+d x^2)} \, dx\)

Optimal. Leaf size=160 \[ \frac{\log (x) \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{a^4 c^3}+\frac{b^3}{2 a^3 \left (a+b x^2\right ) (b c-a d)}-\frac{b^3 (3 b c-4 a d) \log \left (a+b x^2\right )}{2 a^4 (b c-a d)^2}+\frac{a d+2 b c}{2 a^3 c^2 x^2}-\frac{1}{4 a^2 c x^4}-\frac{d^4 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.193524, antiderivative size = 160, 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} \[ \frac{\log (x) \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{a^4 c^3}+\frac{b^3}{2 a^3 \left (a+b x^2\right ) (b c-a d)}-\frac{b^3 (3 b c-4 a d) \log \left (a+b x^2\right )}{2 a^4 (b c-a d)^2}+\frac{a d+2 b c}{2 a^3 c^2 x^2}-\frac{1}{4 a^2 c x^4}-\frac{d^4 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.192717, size = 155, normalized size = 0.97 \[ \frac{1}{4} \left (\frac{4 \log (x) \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{a^4 c^3}-\frac{2 b^3}{a^3 \left (a+b x^2\right ) (a d-b c)}+\frac{2 b^3 (4 a d-3 b c) \log \left (a+b x^2\right )}{a^4 (b c-a d)^2}+\frac{2 a d+4 b c}{a^3 c^2 x^2}-\frac{1}{a^2 c x^4}-\frac{2 d^4 \log \left (c+d x^2\right )}{c^3 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.018, size = 209, normalized size = 1.3 \begin{align*} -{\frac{{d}^{4}\ln \left ( d{x}^{2}+c \right ) }{2\,{c}^{3} \left ( ad-bc \right ) ^{2}}}-{\frac{1}{4\,{a}^{2}c{x}^{4}}}+{\frac{d}{2\,{a}^{2}{c}^{2}{x}^{2}}}+{\frac{b}{{a}^{3}c{x}^{2}}}+{\frac{\ln \left ( x \right ){d}^{2}}{{a}^{2}{c}^{3}}}+2\,{\frac{\ln \left ( x \right ) bd}{{a}^{3}{c}^{2}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{4}c}}+2\,{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) d}{{a}^{3} \left ( ad-bc \right ) ^{2}}}-{\frac{3\,{b}^{4}\ln \left ( b{x}^{2}+a \right ) c}{2\,{a}^{4} \left ( ad-bc \right ) ^{2}}}-{\frac{{b}^{3}d}{2\,{a}^{2} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{4}c}{2\,{a}^{3} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02167, size = 348, normalized size = 2.17 \begin{align*} -\frac{d^{4} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}} - \frac{{\left (3 \, b^{4} c - 4 \, a b^{3} d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )}} - \frac{a^{2} b c^{2} - a^{3} c d - 2 \,{\left (3 \, b^{3} c^{2} - a b^{2} c d - a^{2} b d^{2}\right )} x^{4} -{\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{2}}{4 \,{\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{6} +{\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{4}\right )}} + \frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 35.3027, size = 706, normalized size = 4.41 \begin{align*} -\frac{a^{3} b^{2} c^{4} - 2 \, a^{4} b c^{3} d + a^{5} c^{2} d^{2} - 2 \,{\left (3 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + a^{4} b c d^{3}\right )} x^{4} -{\left (3 \, a^{2} b^{3} c^{4} - 4 \, a^{3} b^{2} c^{3} d - a^{4} b c^{2} d^{2} + 2 \, a^{5} c d^{3}\right )} x^{2} + 2 \,{\left ({\left (3 \, b^{5} c^{4} - 4 \, a b^{4} c^{3} d\right )} x^{6} +{\left (3 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d\right )} x^{4}\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left (a^{4} b d^{4} x^{6} + a^{5} d^{4} x^{4}\right )} \log \left (d x^{2} + c\right ) - 4 \,{\left ({\left (3 \, b^{5} c^{4} - 4 \, a b^{4} c^{3} d + a^{4} b d^{4}\right )} x^{6} +{\left (3 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + a^{5} d^{4}\right )} x^{4}\right )} \log \left (x\right )}{4 \,{\left ({\left (a^{4} b^{3} c^{5} - 2 \, a^{5} b^{2} c^{4} d + a^{6} b c^{3} d^{2}\right )} x^{6} +{\left (a^{5} b^{2} c^{5} - 2 \, a^{6} b c^{4} d + a^{7} c^{3} d^{2}\right )} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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**5/(b*x**2+a)**2/(d*x**2+c),x)

[Out]

Timed out

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Giac [A]  time = 1.18213, size = 379, normalized size = 2.37 \begin{align*} -\frac{d^{5} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )}} - \frac{{\left (3 \, b^{5} c - 4 \, a b^{4} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\right )}} + \frac{3 \, b^{5} c x^{2} - 4 \, a b^{4} d x^{2} + 4 \, a b^{4} c - 5 \, a^{2} b^{3} d}{2 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )}{\left (b x^{2} + a\right )}} + \frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4} c^{3}} - \frac{9 \, b^{2} c^{2} x^{4} + 6 \, a b c d x^{4} + 3 \, a^{2} d^{2} x^{4} - 4 \, a b c^{2} x^{2} - 2 \, a^{2} c d x^{2} + a^{2} c^{2}}{4 \, a^{4} c^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d^5*log(abs(d*x^2 + c))/(b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3) - 1/2*(3*b^5*c - 4*a*b^4*d)*log(abs(b*x
^2 + a))/(a^4*b^3*c^2 - 2*a^5*b^2*c*d + a^6*b*d^2) + 1/2*(3*b^5*c*x^2 - 4*a*b^4*d*x^2 + 4*a*b^4*c - 5*a^2*b^3*
d)/((a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)*(b*x^2 + a)) + 1/2*(3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*log(x^2)/(a^4*c
^3) - 1/4*(9*b^2*c^2*x^4 + 6*a*b*c*d*x^4 + 3*a^2*d^2*x^4 - 4*a*b*c^2*x^2 - 2*a^2*c*d*x^2 + a^2*c^2)/(a^4*c^3*x
^4)