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

Optimal. Leaf size=59 \[ \frac{3 a (c x)^{2/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right )}{2 c \sqrt [3]{\frac{b x^2}{a}+1}} \]

[Out]

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

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Rubi [A]  time = 0.0675049, antiderivative size = 59, 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 \[ \frac{3 a (c x)^{2/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right )}{2 c \sqrt [3]{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.56447, size = 51, normalized size = 0.86 \[ \frac{3 a \left (c x\right )^{\frac{2}{3}} \sqrt [3]{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{2 c \sqrt [3]{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0591085, size = 81, normalized size = 1.37 \[ \frac{3 x \left (2 a^2 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^2}{a}\right )+3 a^2+4 a b x^2+b^2 x^4\right )}{10 \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*x*(3*a^2 + 4*a*b*x^2 + b^2*x^4 + 2*a^2*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F
1[1/3, 2/3, 4/3, -((b*x^2)/a)]))/(10*(c*x)^(1/3)*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{1 \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}{\frac{1}{\sqrt [3]{cx}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)/(c*x)^(1/3),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 68.4535, size = 46, normalized size = 0.78 \[ \frac{a^{\frac{4}{3}} x^{\frac{2}{3}} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [3]{c} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)/(c*x)^(1/3),x, algorithm="giac")

[Out]

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