3.340 \(\int \frac{x^{11}}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=61 \[ \frac{a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac{a^2}{b^4 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right )}{b^4}+\frac{x^3}{3 b^3} \]

[Out]

x^3/(3*b^3) + a^3/(6*b^4*(a + b*x^3)^2) - a^2/(b^4*(a + b*x^3)) - (a*Log[a + b*x^3])/b^4

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Rubi [A]  time = 0.048149, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac{a^2}{b^4 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right )}{b^4}+\frac{x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^3)^3,x]

[Out]

x^3/(3*b^3) + a^3/(6*b^4*(a + b*x^3)^2) - a^2/(b^4*(a + b*x^3)) - (a*Log[a + b*x^3])/b^4

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^{11}}{\left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{b^3}-\frac{a^3}{b^3 (a+b x)^3}+\frac{3 a^2}{b^3 (a+b x)^2}-\frac{3 a}{b^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{x^3}{3 b^3}+\frac{a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac{a^2}{b^4 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right )}{b^4}\\ \end{align*}

Mathematica [A]  time = 0.0550541, size = 48, normalized size = 0.79 \[ -\frac{\frac{a^2 \left (5 a+6 b x^3\right )}{\left (a+b x^3\right )^2}+6 a \log \left (a+b x^3\right )-2 b x^3}{6 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^3)^3,x]

[Out]

-(-2*b*x^3 + (a^2*(5*a + 6*b*x^3))/(a + b*x^3)^2 + 6*a*Log[a + b*x^3])/(6*b^4)

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Maple [A]  time = 0.01, size = 58, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{3\,{b}^{3}}}+{\frac{{a}^{3}}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}}{{b}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{a\ln \left ( b{x}^{3}+a \right ) }{{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x^3+a)^3,x)

[Out]

1/3*x^3/b^3+1/6*a^3/b^4/(b*x^3+a)^2-a^2/b^4/(b*x^3+a)-a*ln(b*x^3+a)/b^4

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Maxima [A]  time = 0.973354, size = 89, normalized size = 1.46 \begin{align*} -\frac{6 \, a^{2} b x^{3} + 5 \, a^{3}}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac{x^{3}}{3 \, b^{3}} - \frac{a \log \left (b x^{3} + a\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45777, size = 186, normalized size = 3.05 \begin{align*} \frac{2 \, b^{3} x^{9} + 4 \, a b^{2} x^{6} - 4 \, a^{2} b x^{3} - 5 \, a^{3} - 6 \,{\left (a b^{2} x^{6} + 2 \, a^{2} b x^{3} + a^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.95331, size = 63, normalized size = 1.03 \begin{align*} - \frac{a \log{\left (a + b x^{3} \right )}}{b^{4}} - \frac{5 a^{3} + 6 a^{2} b x^{3}}{6 a^{2} b^{4} + 12 a b^{5} x^{3} + 6 b^{6} x^{6}} + \frac{x^{3}}{3 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x**3+a)**3,x)

[Out]

-a*log(a + b*x**3)/b**4 - (5*a**3 + 6*a**2*b*x**3)/(6*a**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6) + x**3/(3*b**3
)

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Giac [A]  time = 1.09897, size = 84, normalized size = 1.38 \begin{align*} \frac{x^{3}}{3 \, b^{3}} - \frac{a \log \left ({\left | b x^{3} + a \right |}\right )}{b^{4}} + \frac{9 \, a b^{2} x^{6} + 12 \, a^{2} b x^{3} + 4 \, a^{3}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/3*x^3/b^3 - a*log(abs(b*x^3 + a))/b^4 + 1/6*(9*a*b^2*x^6 + 12*a^2*b*x^3 + 4*a^3)/((b*x^3 + a)^2*b^4)