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

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

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Rubi [A]  time = 0.09, antiderivative size = 87, 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 = {446, 72} \begin {gather*} \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} \end {gather*}

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 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]

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]]

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*}

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Mathematica [A]  time = 0.04, size = 88, normalized size = 1.01 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*1/(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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^5 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 8.27, size = 99, normalized size = 1.14 \begin {gather*} \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 \relax (x) - a b c^{2} + a^{2} c d}{4 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{4}} \end {gather*}

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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giac [A]  time = 0.19, size = 112, normalized size = 1.29 \begin {gather*} \frac {b^{3} \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, {\left (a^{2} b^{2} c - a^{3} b d\right )}} - \frac {d^{3} \log \left ({\left | d x^{4} + c \right |}\right )}{4 \, {\left (b c^{3} d - a c^{2} d^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{4}\right )}{4 \, a^{2} c^{2}} + \frac {b c x^{4} + a d x^{4} - a c}{4 \, a^{2} c^{2} x^{4}} \end {gather*}

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]

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

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maple [A]  time = 0.06, size = 87, normalized size = 1.00 \begin {gather*} -\frac {b^{2} \ln \left (b \,x^{4}+a \right )}{4 \left (a d -b c \right ) a^{2}}+\frac {d^{2} \ln \left (d \,x^{4}+c \right )}{4 \left (a d -b c \right ) c^{2}}-\frac {d \ln \relax (x )}{a \,c^{2}}-\frac {b \ln \relax (x )}{a^{2} c}-\frac {1}{4 a c \,x^{4}} \end {gather*}

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 = 0.55, size = 87, normalized size = 1.00 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 6.21, size = 87, normalized size = 1.00 \begin {gather*} -\frac {b^2\,\ln \left (b\,x^4+a\right )}{4\,\left (a^3\,d-a^2\,b\,c\right )}-\frac {d^2\,\ln \left (d\,x^4+c\right )}{4\,\left (b\,c^3-a\,c^2\,d\right )}-\frac {1}{4\,a\,c\,x^4}-\frac {\ln \relax (x)\,\left (a\,d+b\,c\right )}{a^2\,c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  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**5/(b*x**4+a)/(d*x**4+c),x)

[Out]

Timed out

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