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

Optimal. Leaf size=58 \[ \frac{3 (c x)^{5/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{1}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right )}{5 c \sqrt [3]{\frac{b x^2}{a}+1}} \]

[Out]

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

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Rubi [A]  time = 0.0655194, antiderivative size = 58, 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 \[ \frac{3 (c x)^{5/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{1}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right )}{5 c \sqrt [3]{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.49475, size = 49, normalized size = 0.84 \[ \frac{3 \left (c x\right )^{\frac{5}{3}} \sqrt [3]{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{5 c \sqrt [3]{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(c*x)**(5/3)*(a + b*x**2)**(1/3)*hyper((-1/3, 5/6), (11/6,), -b*x**2/a)/(5*c*(
1 + b*x**2/a)**(1/3))

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Mathematica [A]  time = 0.0519905, size = 69, normalized size = 1.19 \[ \frac{3 x (c x)^{2/3} \left (2 a \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right )+5 \left (a+b x^2\right )\right )}{35 \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*x*(c*x)^(2/3)*(5*(a + b*x^2) + 2*a*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[2/
3, 5/6, 11/6, -((b*x^2)/a)]))/(35*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{{\frac{2}{3}}}\sqrt [3]{b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.7127, size = 46, normalized size = 0.79 \[ \frac{\sqrt [3]{a} c^{\frac{2}{3}} x^{\frac{5}{3}} \Gamma \left (\frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{11}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/3)*c**(2/3)*x**(5/3)*gamma(5/6)*hyper((-1/3, 5/6), (11/6,), b*x**2*exp_pol
ar(I*pi)/a)/(2*gamma(11/6))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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