3.283 \(\int \frac{x^{11}}{(a x^2+b x^5)^3} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^6}{6 a \left (a+b x^3\right )^2} \]

[Out]

x^6/(6*a*(a + b*x^3)^2)

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Rubi [A]  time = 0.0100698, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 264} \[ \frac{x^6}{6 a \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a*x^2 + b*x^5)^3,x]

[Out]

x^6/(6*a*(a + b*x^3)^2)

Rule 1584

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

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^{11}}{\left (a x^2+b x^5\right )^3} \, dx &=\int \frac{x^5}{\left (a+b x^3\right )^3} \, dx\\ &=\frac{x^6}{6 a \left (a+b x^3\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.00836, size = 24, normalized size = 1.26 \[ -\frac{a+2 b x^3}{6 b^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a*x^2 + b*x^5)^3,x]

[Out]

-(a + 2*b*x^3)/(6*b^2*(a + b*x^3)^2)

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Maple [A]  time = 0.007, size = 31, normalized size = 1.6 \begin{align*}{\frac{a}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{1}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x^5+a*x^2)^3,x)

[Out]

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

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Maxima [B]  time = 1.17217, size = 49, normalized size = 2.58 \begin{align*} -\frac{2 \, b x^{3} + a}{6 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^5+a*x^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.71859, size = 73, normalized size = 3.84 \begin{align*} -\frac{2 \, b x^{3} + a}{6 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^5+a*x^2)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.50915, size = 36, normalized size = 1.89 \begin{align*} - \frac{a + 2 b x^{3}}{6 a^{2} b^{2} + 12 a b^{3} x^{3} + 6 b^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x**5+a*x**2)**3,x)

[Out]

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

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Giac [A]  time = 1.16428, size = 30, normalized size = 1.58 \begin{align*} -\frac{2 \, b x^{3} + a}{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^11/(b*x^5+a*x^2)^3,x, algorithm="giac")

[Out]

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