3.742 \(\int \frac{\sqrt [3]{a+b x^2}}{(c x)^{11/3}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{8/3}} \]

[Out]

(-3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(8/3))

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Rubi [A]  time = 0.0056703, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {264} \[ -\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{8/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(1/3)/(c*x)^(11/3),x]

[Out]

(-3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(8/3))

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{\sqrt [3]{a+b x^2}}{(c x)^{11/3}} \, dx &=-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{8/3}}\\ \end{align*}

Mathematica [A]  time = 0.0086013, size = 26, normalized size = 0.93 \[ -\frac{3 x \left (a+b x^2\right )^{4/3}}{8 a (c x)^{11/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(1/3)/(c*x)^(11/3),x]

[Out]

(-3*x*(a + b*x^2)^(4/3))/(8*a*(c*x)^(11/3))

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Maple [A]  time = 0.003, size = 21, normalized size = 0.8 \begin{align*} -{\frac{3\,x}{8\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}} \left ( cx \right ) ^{-{\frac{11}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(1/3)/(c*x)^(11/3),x)

[Out]

-3/8*x*(b*x^2+a)^(4/3)/a/(c*x)^(11/3)

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Maxima [A]  time = 1.92746, size = 47, normalized size = 1.68 \begin{align*} -\frac{3 \,{\left (b c^{\frac{1}{3}} x^{3} + a c^{\frac{1}{3}} x\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{8 \, a c^{4} x^{\frac{11}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/3)/(c*x)^(11/3),x, algorithm="maxima")

[Out]

-3/8*(b*c^(1/3)*x^3 + a*c^(1/3)*x)*(b*x^2 + a)^(1/3)/(a*c^4*x^(11/3))

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Fricas [A]  time = 2.50888, size = 65, normalized size = 2.32 \begin{align*} -\frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} \left (c x\right )^{\frac{1}{3}}}{8 \, a c^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/3)/(c*x)^(11/3),x, algorithm="fricas")

[Out]

-3/8*(b*x^2 + a)^(4/3)*(c*x)^(1/3)/(a*c^4*x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(1/3)/(c*x)**(11/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{\left (c x\right )^{\frac{11}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/3)/(c*x)^(11/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/3)/(c*x)^(11/3), x)