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

Optimal. Leaf size=211 \[ -\frac{a \sqrt [3]{c} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{6 b^{2/3}}+\frac{a \sqrt [3]{c} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{12 b^{2/3}}-\frac{a \sqrt [3]{c} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{2 \sqrt{3} b^{2/3}}+\frac{(c x)^{4/3} \sqrt [3]{a+b x^2}}{2 c} \]

[Out]

((c*x)^(4/3)*(a + b*x^2)^(1/3))/(2*c) - (a*c^(1/3)*ArcTan[(c^(2/3) + (2*b^(1/3)*
(c*x)^(2/3))/(a + b*x^2)^(1/3))/(Sqrt[3]*c^(2/3))])/(2*Sqrt[3]*b^(2/3)) - (a*c^(
1/3)*Log[c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)])/(6*b^(2/3)) + (a*c^
(1/3)*Log[c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(
c*x)^(2/3))/(a + b*x^2)^(1/3)])/(12*b^(2/3))

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Rubi [A]  time = 0.590294, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{a \sqrt [3]{c} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{6 b^{2/3}}+\frac{a \sqrt [3]{c} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{12 b^{2/3}}-\frac{a \sqrt [3]{c} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{2 \sqrt{3} b^{2/3}}+\frac{(c x)^{4/3} \sqrt [3]{a+b x^2}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

((c*x)^(4/3)*(a + b*x^2)^(1/3))/(2*c) - (a*c^(1/3)*ArcTan[(c^(2/3) + (2*b^(1/3)*
(c*x)^(2/3))/(a + b*x^2)^(1/3))/(Sqrt[3]*c^(2/3))])/(2*Sqrt[3]*b^(2/3)) - (a*c^(
1/3)*Log[c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)])/(6*b^(2/3)) + (a*c^
(1/3)*Log[c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(
c*x)^(2/3))/(a + b*x^2)^(1/3)])/(12*b^(2/3))

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Rubi in Sympy [A]  time = 56.9619, size = 196, normalized size = 0.93 \[ - \frac{a \sqrt [3]{c} \log{\left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}} \right )}}{6 b^{\frac{2}{3}}} + \frac{a \sqrt [3]{c} \log{\left (\frac{b^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}{c^{\frac{4}{3}} \left (a + b x^{2}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{c^{\frac{2}{3}} \sqrt [3]{a + b x^{2}}} + 1 \right )}}{12 b^{\frac{2}{3}}} - \frac{\sqrt{3} a \sqrt [3]{c} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{2 \sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{3 \sqrt [3]{a + b x^{2}}} + \frac{c^{\frac{2}{3}}}{3}\right )}{c^{\frac{2}{3}}} \right )}}{6 b^{\frac{2}{3}}} + \frac{\left (c x\right )^{\frac{4}{3}} \sqrt [3]{a + b x^{2}}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*c**(1/3)*log(-b**(1/3)*(c*x)**(2/3)/(a + b*x**2)**(1/3) + c**(2/3))/(6*b**(2/
3)) + a*c**(1/3)*log(b**(2/3)*(c*x)**(4/3)/(c**(4/3)*(a + b*x**2)**(2/3)) + b**(
1/3)*(c*x)**(2/3)/(c**(2/3)*(a + b*x**2)**(1/3)) + 1)/(12*b**(2/3)) - sqrt(3)*a*
c**(1/3)*atan(sqrt(3)*(2*b**(1/3)*(c*x)**(2/3)/(3*(a + b*x**2)**(1/3)) + c**(2/3
)/3)/c**(2/3))/(6*b**(2/3)) + (c*x)**(4/3)*(a + b*x**2)**(1/3)/(2*c)

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Mathematica [C]  time = 0.049813, size = 68, normalized size = 0.32 \[ \frac{x \sqrt [3]{c x} \left (a \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right )+2 \left (a+b x^2\right )\right )}{4 \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int \sqrt [3]{cx}\sqrt [3]{b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [A]  time = 6.71784, size = 46, normalized size = 0.22 \[ \frac{\sqrt [3]{a} \sqrt [3]{c} x^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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