3.329 \(\int \frac{x^8}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{a^2}{3 b^3 \left (a+b x^3\right )}-\frac{2 a \log \left (a+b x^3\right )}{3 b^3}+\frac{x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0303931, antiderivative size = 46, 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} \[ -\frac{a^2}{3 b^3 \left (a+b x^3\right )}-\frac{2 a \log \left (a+b x^3\right )}{3 b^3}+\frac{x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^3)^2,x]

[Out]

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

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^8}{\left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{b^2}+\frac{a^2}{b^2 (a+b x)^2}-\frac{2 a}{b^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{x^3}{3 b^2}-\frac{a^2}{3 b^3 \left (a+b x^3\right )}-\frac{2 a \log \left (a+b x^3\right )}{3 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0180864, size = 38, normalized size = 0.83 \[ \frac{-\frac{a^2}{a+b x^3}-2 a \log \left (a+b x^3\right )+b x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^3)^2,x]

[Out]

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

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Maple [A]  time = 0.011, size = 41, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{2}}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,a\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x^3+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.986291, size = 58, normalized size = 1.26 \begin{align*} -\frac{a^{2}}{3 \,{\left (b^{4} x^{3} + a b^{3}\right )}} + \frac{x^{3}}{3 \, b^{2}} - \frac{2 \, a \log \left (b x^{3} + a\right )}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.43615, size = 113, normalized size = 2.46 \begin{align*} \frac{b^{2} x^{6} + a b x^{3} - a^{2} - 2 \,{\left (a b x^{3} + a^{2}\right )} \log \left (b x^{3} + a\right )}{3 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.58039, size = 42, normalized size = 0.91 \begin{align*} - \frac{a^{2}}{3 a b^{3} + 3 b^{4} x^{3}} - \frac{2 a \log{\left (a + b x^{3} \right )}}{3 b^{3}} + \frac{x^{3}}{3 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x**3+a)**2,x)

[Out]

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

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Giac [A]  time = 1.08214, size = 66, normalized size = 1.43 \begin{align*} \frac{x^{3}}{3 \, b^{2}} - \frac{2 \, a \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{2 \, a b x^{3} + a^{2}}{3 \,{\left (b x^{3} + a\right )} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^3+a)^2,x, algorithm="giac")

[Out]

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