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 = 0.0637845, 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 \[ -\frac{3 \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{1}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \sqrt [3]{\frac{b x^2}{a}+1}} \]

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))

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Rubi in Sympy [A]  time = 7.47105, size = 51, normalized size = 0.91 \[ - \frac{3 \sqrt [3]{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{1}{6} \\ \frac{5}{6} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{c \sqrt [3]{c x} \sqrt [3]{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(a + b*x**2)**(1/3)*hyper((-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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Mathematica [A]  time = 0.0510562, size = 72, normalized size = 1.29 \[ \frac{x \left (6 b x^2 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right )-15 \left (a+b x^2\right )\right )}{5 (c x)^{4/3} \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-15*(a + b*x^2) + 6*b*x^2*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[2/3, 5/6,
11/6, -((b*x^2)/a)]))/(5*(c*x)^(4/3)*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{1\sqrt [3]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{4}{3}}}}\, dx \]

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. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{\left (c x\right )^{\frac{4}{3}}}\,{d x} \]

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. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{\left (c x\right )^{\frac{1}{3}} c x}, x\right ) \]

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)^(1/3)*c*x), x)

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Sympy [A]  time = 12.9224, size = 49, normalized size = 0.88 \[ \frac{\sqrt [3]{a} \Gamma \left (- \frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{1}{6} \\ \frac{5}{6} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac{4}{3}} \sqrt [3]{x} \Gamma \left (\frac{5}{6}\right )} \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{\left (c x\right )^{\frac{4}{3}}}\,{d x} \]

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)