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

Optimal. Leaf size=52 \[ -\frac{a^2}{6 b^3 \left (a+b x^3\right )^2}+\frac{2 a}{3 b^3 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right )}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0373049, antiderivative size = 52, 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} \[ -\frac{a^2}{6 b^3 \left (a+b x^3\right )^2}+\frac{2 a}{3 b^3 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right )}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^3)^3,x]

[Out]

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

Mathematica [A]  time = 0.0151773, size = 39, normalized size = 0.75 \[ \frac{\frac{a \left (3 a+4 b x^3\right )}{\left (a+b x^3\right )^2}+2 \log \left (a+b x^3\right )}{6 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^3)^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 47, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{2\,a}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{\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)^3,x)

[Out]

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

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Maxima [A]  time = 0.957314, size = 74, normalized size = 1.42 \begin{align*} \frac{4 \, a b x^{3} + 3 \, a^{2}}{6 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac{\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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47579, size = 143, normalized size = 2.75 \begin{align*} \frac{4 \, a b x^{3} + 3 \, a^{2} + 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.83584, size = 53, normalized size = 1.02 \begin{align*} \frac{3 a^{2} + 4 a b x^{3}}{6 a^{2} b^{3} + 12 a b^{4} x^{3} + 6 b^{5} x^{6}} + \frac{\log{\left (a + b x^{3} \right )}}{3 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17542, size = 57, normalized size = 1.1 \begin{align*} \frac{\log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac{3 \, b x^{6} + 2 \, a x^{3}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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