Optimal. Leaf size=98 \[ \frac{b \log (x)}{3 a^{5/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{\sqrt [3]{a+b x}}{a x} \]
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Rubi [A] time = 0.0834813, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{b \log (x)}{3 a^{5/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{\sqrt [3]{a+b x}}{a x} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x)^(2/3)),x]
[Out]
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Rubi in Sympy [A] time = 7.62188, size = 90, normalized size = 0.92 \[ - \frac{\sqrt [3]{a + b x}}{a x} + \frac{b \log{\left (x \right )}}{3 a^{\frac{5}{3}}} - \frac{b \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{a^{\frac{5}{3}}} + \frac{2 \sqrt{3} b \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x+a)**(2/3),x)
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Mathematica [C] time = 0.0370604, size = 60, normalized size = 0.61 \[ \frac{b x \left (\frac{a}{b x}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x}\right )-a-b x}{a x (a+b x)^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x)^(2/3)),x]
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Maple [A] time = 0.012, size = 95, normalized size = 1. \[ -{\frac{1}{ax}\sqrt [3]{bx+a}}-{\frac{2\,b}{3}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{5}{3}}}}+{\frac{b}{3}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{5}{3}}}}+{\frac{2\,b\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{5}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x+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(1/((b*x + a)^(2/3)*x^2),x, algorithm="maxima")
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Fricas [A] time = 0.246412, size = 190, normalized size = 1.94 \[ -\frac{\sqrt{3}{\left (\sqrt{3} b x \log \left (a^{2} - \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}} a + \left (-a^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} b x \log \left (a + \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right ) - 6 \, b x \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}}{3 \, a}\right ) + 3 \, \sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right )}}{9 \, \left (-a^{2}\right )^{\frac{1}{3}} a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(2/3)*x^2),x, algorithm="fricas")
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Sympy [A] time = 6.67593, size = 610, normalized size = 6.22 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x+a)**(2/3),x)
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GIAC/XCAS [A] time = 0.554939, size = 146, normalized size = 1.49 \[ \frac{\frac{2 \, \sqrt{3} b^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{5}{3}}} + \frac{b^{2}{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{5}{3}}} - \frac{2 \, b^{2}{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{5}{3}}} - \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}} b}{a x}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(2/3)*x^2),x, algorithm="giac")
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