3.2.19 \(\int \frac {1}{x^4 (a+b x^3) (c+d x^3)} \, dx\)

Optimal. Leaf size=87 \[ \frac {b^2 \log \left (a+b x^3\right )}{3 a^2 (b c-a d)}-\frac {\log (x) (a d+b c)}{a^2 c^2}-\frac {d^2 \log \left (c+d x^3\right )}{3 c^2 (b c-a d)}-\frac {1}{3 a c x^3} \]

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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^3\right )}{3 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^3\right )}{3 c^2 (b c-a d)}-\frac {1}{3 a c x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^3)*(c + d*x^3)),x]

[Out]

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

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Mathematica [A]  time = 0.11, size = 88, normalized size = 1.01 \begin {gather*} -\frac {b^2 \log \left (a+b x^3\right )}{3 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^3\right )}{3 c^2 (b c-a d)}-\frac {1}{3 a c x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^3)*(c + d*x^3)),x]

[Out]

-1/3*1/(a*c*x^3) + ((-(b*c) - a*d)*Log[x])/(a^2*c^2) - (b^2*Log[a + b*x^3])/(3*a^2*(-(b*c) + a*d)) - (d^2*Log[
c + d*x^3])/(3*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^4 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^4*(a + b*x^3)*(c + d*x^3)),x]

[Out]

IntegrateAlgebraic[1/(x^4*(a + b*x^3)*(c + d*x^3)), x]

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fricas [A]  time = 4.79, size = 99, normalized size = 1.14 \begin {gather*} \frac {b^{2} c^{2} x^{3} \log \left (b x^{3} + a\right ) - a^{2} d^{2} x^{3} \log \left (d x^{3} + c\right ) - 3 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{3} \log \relax (x) - a b c^{2} + a^{2} c d}{3 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 111, normalized size = 1.28 \begin {gather*} \frac {b^{3} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, {\left (a^{2} b^{2} c - a^{3} b d\right )}} - \frac {d^{3} \log \left ({\left | d x^{3} + c \right |}\right )}{3 \, {\left (b c^{3} d - a c^{2} d^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left ({\left | x \right |}\right )}{a^{2} c^{2}} + \frac {b c x^{3} + a d x^{3} - a c}{3 \, a^{2} c^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 87, normalized size = 1.00 \begin {gather*} -\frac {b^{2} \ln \left (b \,x^{3}+a \right )}{3 \left (a d -b c \right ) a^{2}}+\frac {d^{2} \ln \left (d \,x^{3}+c \right )}{3 \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}{3 a c \,x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

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maxima [A]  time = 0.49, size = 87, normalized size = 1.00 \begin {gather*} \frac {b^{2} \log \left (b x^{3} + a\right )}{3 \, {\left (a^{2} b c - a^{3} d\right )}} - \frac {d^{2} \log \left (d x^{3} + c\right )}{3 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{3}\right )}{3 \, a^{2} c^{2}} - \frac {1}{3 \, a c x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.22, size = 87, normalized size = 1.00 \begin {gather*} -\frac {b^2\,\ln \left (b\,x^3+a\right )}{3\,\left (a^3\,d-a^2\,b\,c\right )}-\frac {d^2\,\ln \left (d\,x^3+c\right )}{3\,\left (b\,c^3-a\,c^2\,d\right )}-\frac {1}{3\,a\,c\,x^3}-\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^4*(a + b*x^3)*(c + d*x^3)),x)

[Out]

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

[Out]

Timed out

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