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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]