Optimal. Leaf size=209 \[ \frac{a c^{7/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}-\frac{a c^{7/3} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{6 b^{5/3}}+\frac{a c^{7/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{\sqrt{3} b^{5/3}}+\frac{c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b} \]
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Rubi [A] time = 0.589379, antiderivative size = 209, 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 \[ \frac{a c^{7/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}-\frac{a c^{7/3} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{6 b^{5/3}}+\frac{a c^{7/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{\sqrt{3} b^{5/3}}+\frac{c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^(7/3)/(a + b*x^2)^(2/3),x]
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Rubi in Sympy [A] time = 58.1946, size = 197, normalized size = 0.94 \[ \frac{a c^{\frac{7}{3}} \log{\left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}} \right )}}{3 b^{\frac{5}{3}}} - \frac{a c^{\frac{7}{3}} \log{\left (\frac{b^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}{c^{\frac{4}{3}} \left (a + b x^{2}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{c^{\frac{2}{3}} \sqrt [3]{a + b x^{2}}} + 1 \right )}}{6 b^{\frac{5}{3}}} + \frac{\sqrt{3} a c^{\frac{7}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{2 \sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{3 \sqrt [3]{a + b x^{2}}} + \frac{c^{\frac{2}{3}}}{3}\right )}{c^{\frac{2}{3}}} \right )}}{3 b^{\frac{5}{3}}} + \frac{c \left (c x\right )^{\frac{4}{3}} \sqrt [3]{a + b x^{2}}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(7/3)/(b*x**2+a)**(2/3),x)
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Mathematica [C] time = 0.0565378, size = 69, normalized size = 0.33 \[ \frac{c (c x)^{4/3} \left (-a \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right )+a+b x^2\right )}{2 b \left (a+b x^2\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^(7/3)/(a + b*x^2)^(2/3),x]
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Maple [F] time = 0.033, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{7}{3}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(7/3)/(b*x^2+a)^(2/3),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((c*x)^(7/3)/(b*x^2 + a)^(2/3),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((c*x)^(7/3)/(b*x^2 + a)^(2/3),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**(7/3)/(b*x**2+a)**(2/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{7}{3}}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(7/3)/(b*x^2 + a)^(2/3),x, algorithm="giac")
[Out]