3.753 \(\int \frac{\sqrt [3]{a+b x^2}}{(c x)^{4/3}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{3 \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{1}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \sqrt [3]{\frac{b x^2}{a}+1}} \]

[Out]

(-3*(a + b*x^2)^(1/3)*Hypergeometric2F1[-1/3, -1/6, 5/6, -((b*x^2)/a)])/(c*(c*x)^(1/3)*(1 + (b*x^2)/a)^(1/3))

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Rubi [A]  time = 0.0182894, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {365, 364} \[ -\frac{3 \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{1}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \sqrt [3]{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(1/3)/(c*x)^(4/3),x]

[Out]

(-3*(a + b*x^2)^(1/3)*Hypergeometric2F1[-1/3, -1/6, 5/6, -((b*x^2)/a)])/(c*(c*x)^(1/3)*(1 + (b*x^2)/a)^(1/3))

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{a+b x^2}}{(c x)^{4/3}} \, dx &=\frac{\sqrt [3]{a+b x^2} \int \frac{\sqrt [3]{1+\frac{b x^2}{a}}}{(c x)^{4/3}} \, dx}{\sqrt [3]{1+\frac{b x^2}{a}}}\\ &=-\frac{3 \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{1}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \sqrt [3]{1+\frac{b x^2}{a}}}\\ \end{align*}

Mathematica [A]  time = 0.0113679, size = 54, normalized size = 0.96 \[ -\frac{3 x \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{1}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right )}{(c x)^{4/3} \sqrt [3]{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(1/3)/(c*x)^(4/3),x]

[Out]

(-3*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-1/3, -1/6, 5/6, -((b*x^2)/a)])/((c*x)^(4/3)*(1 + (b*x^2)/a)^(1/3))

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(1/3)/(c*x)^(4/3),x)

[Out]

int((b*x^2+a)^(1/3)/(c*x)^(4/3),x)

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Maxima [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{4}{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)^(4/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/3)/(c*x)^(4/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/3)/(c*x)^(4/3),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 3.09779, size = 49, normalized size = 0.88 \begin{align*} \frac{\sqrt [3]{a} \Gamma \left (- \frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{1}{6} \\ \frac{5}{6} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac{4}{3}} \sqrt [3]{x} \Gamma \left (\frac{5}{6}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(1/3)/(c*x)**(4/3),x)

[Out]

a**(1/3)*gamma(-1/6)*hyper((-1/3, -1/6), (5/6,), b*x**2*exp_polar(I*pi)/a)/(2*c**(4/3)*x**(1/3)*gamma(5/6))

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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{4}{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)^(4/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/3)/(c*x)^(4/3), x)