3.772 \(\int \frac{1}{x^5 (a+b x^4) (c+d x^4)} \, dx\)

Optimal. Leaf size=87 \[ \frac{b^2 \log \left (a+b x^4\right )}{4 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^4\right )}{4 c^2 (b c-a d)}-\frac{1}{4 a c x^4} \]

[Out]

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

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Rubi [A]  time = 0.0899582, antiderivative size = 87, 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} \[ \frac{b^2 \log \left (a+b x^4\right )}{4 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^4\right )}{4 c^2 (b c-a d)}-\frac{1}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0413605, size = 88, normalized size = 1.01 \[ -\frac{b^2 \log \left (a+b x^4\right )}{4 a^2 (a d-b c)}+\frac{\log (x) (-a d-b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^4\right )}{4 c^2 (b c-a d)}-\frac{1}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 87, normalized size = 1. \begin{align*}{\frac{{d}^{2}\ln \left ( d{x}^{4}+c \right ) }{4\,{c}^{2} \left ( ad-bc \right ) }}-{\frac{{b}^{2}\ln \left ( b{x}^{4}+a \right ) }{4\,{a}^{2} \left ( ad-bc \right ) }}-{\frac{1}{4\,ac{x}^{4}}}-{\frac{\ln \left ( x \right ) d}{{c}^{2}a}}-{\frac{\ln \left ( x \right ) b}{{a}^{2}c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.937868, size = 117, normalized size = 1.34 \begin{align*} \frac{b^{2} \log \left (b x^{4} + a\right )}{4 \,{\left (a^{2} b c - a^{3} d\right )}} - \frac{d^{2} \log \left (d x^{4} + c\right )}{4 \,{\left (b c^{3} - a c^{2} d\right )}} - \frac{{\left (b c + a d\right )} \log \left (x^{4}\right )}{4 \, a^{2} c^{2}} - \frac{1}{4 \, a c x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 38.2724, size = 200, normalized size = 2.3 \begin{align*} \frac{b^{2} c^{2} x^{4} \log \left (b x^{4} + a\right ) - a^{2} d^{2} x^{4} \log \left (d x^{4} + c\right ) - 4 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} \log \left (x\right ) - a b c^{2} + a^{2} c d}{4 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: NotImplementedError