3.1978 \(\int \frac{x^5}{(a+\frac{b}{x^3})^2} \, dx\)

Optimal. Leaf size=56 \[ \frac{b^3}{3 a^4 \left (a x^3+b\right )}+\frac{b^2 \log \left (a x^3+b\right )}{a^4}-\frac{2 b x^3}{3 a^3}+\frac{x^6}{6 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0401671, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac{b^3}{3 a^4 \left (a x^3+b\right )}+\frac{b^2 \log \left (a x^3+b\right )}{a^4}-\frac{2 b x^3}{3 a^3}+\frac{x^6}{6 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b/x^3)^2,x]

[Out]

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

Rule 263

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

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^5}{\left (a+\frac{b}{x^3}\right )^2} \, dx &=\int \frac{x^{11}}{\left (b+a x^3\right )^2} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{(b+a x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{2 b}{a^3}+\frac{x}{a^2}-\frac{b^3}{a^3 (b+a x)^2}+\frac{3 b^2}{a^3 (b+a x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 b x^3}{3 a^3}+\frac{x^6}{6 a^2}+\frac{b^3}{3 a^4 \left (b+a x^3\right )}+\frac{b^2 \log \left (b+a x^3\right )}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.0178297, size = 49, normalized size = 0.88 \[ \frac{a^2 x^6+\frac{2 b^3}{a x^3+b}+6 b^2 \log \left (a x^3+b\right )-4 a b x^3}{6 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b/x^3)^2,x]

[Out]

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

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Maple [A]  time = 0.005, size = 51, normalized size = 0.9 \begin{align*} -{\frac{2\,b{x}^{3}}{3\,{a}^{3}}}+{\frac{{x}^{6}}{6\,{a}^{2}}}+{\frac{{b}^{3}}{3\,{a}^{4} \left ( a{x}^{3}+b \right ) }}+{\frac{{b}^{2}\ln \left ( a{x}^{3}+b \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+b/x^3)^2,x)

[Out]

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

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Maxima [A]  time = 1.00343, size = 72, normalized size = 1.29 \begin{align*} \frac{b^{3}}{3 \,{\left (a^{5} x^{3} + a^{4} b\right )}} + \frac{b^{2} \log \left (a x^{3} + b\right )}{a^{4}} + \frac{a x^{6} - 4 \, b x^{3}}{6 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/x^3)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.42172, size = 143, normalized size = 2.55 \begin{align*} \frac{a^{3} x^{9} - 3 \, a^{2} b x^{6} - 4 \, a b^{2} x^{3} + 2 \, b^{3} + 6 \,{\left (a b^{2} x^{3} + b^{3}\right )} \log \left (a x^{3} + b\right )}{6 \,{\left (a^{5} x^{3} + a^{4} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/x^3)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.604334, size = 53, normalized size = 0.95 \begin{align*} \frac{b^{3}}{3 a^{5} x^{3} + 3 a^{4} b} + \frac{x^{6}}{6 a^{2}} - \frac{2 b x^{3}}{3 a^{3}} + \frac{b^{2} \log{\left (a x^{3} + b \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a+b/x**3)**2,x)

[Out]

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

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Giac [A]  time = 1.20849, size = 73, normalized size = 1.3 \begin{align*} \frac{b^{2} \log \left ({\left | a x^{3} + b \right |}\right )}{a^{4}} + \frac{b^{3}}{3 \,{\left (a x^{3} + b\right )} a^{4}} + \frac{a^{2} x^{6} - 4 \, a b x^{3}}{6 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/x^3)^2,x, algorithm="giac")

[Out]

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