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

Optimal. Leaf size=119 \[ \frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}-\frac{b^3 \log \left (a+b x^2\right )}{2 a^3 (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{d^3 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)}-\frac{1}{4 a c x^4} \]

[Out]

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

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Rubi [A]  time = 0.127446, antiderivative size = 119, 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, 72} \[ \frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}-\frac{b^3 \log \left (a+b x^2\right )}{2 a^3 (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{d^3 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)}-\frac{1}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0536383, size = 119, normalized size = 1. \[ \frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}+\frac{b^3 \log \left (a+b x^2\right )}{2 a^3 (a d-b c)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{d^3 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)}-\frac{1}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 124, normalized size = 1. \begin{align*} -{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ) }{2\,{c}^{3} \left ( ad-bc \right ) }}-{\frac{1}{4\,ac{x}^{4}}}+{\frac{d}{2\,a{c}^{2}{x}^{2}}}+{\frac{b}{2\,{a}^{2}c{x}^{2}}}+{\frac{\ln \left ( x \right ){d}^{2}}{a{c}^{3}}}+{\frac{\ln \left ( x \right ) bd}{{a}^{2}{c}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}c}}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{3} \left ( ad-bc \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10021, size = 158, normalized size = 1.33 \begin{align*} -\frac{b^{3} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{3} b c - a^{4} d\right )}} + \frac{d^{3} \log \left (d x^{2} + c\right )}{2 \,{\left (b c^{4} - a c^{3} d\right )}} + \frac{{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{3}} + \frac{2 \,{\left (b c + a d\right )} x^{2} - a c}{4 \, a^{2} c^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 7.42589, size = 255, normalized size = 2.14 \begin{align*} -\frac{2 \, b^{3} c^{3} x^{4} \log \left (b x^{2} + a\right ) - 2 \, a^{3} d^{3} x^{4} \log \left (d x^{2} + c\right ) + a^{2} b c^{3} - a^{3} c^{2} d - 4 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{4} \log \left (x\right ) - 2 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{2}}{4 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError