Optimal. Leaf size=128 \[ -\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac{5 b^{2/3} \log (a+b x)}{6 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3}}-\frac{5}{2 a^2 x^{2/3}}+\frac{1}{a x^{2/3} (a+b x)} \]
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Rubi [A] time = 0.115408, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac{5 b^{2/3} \log (a+b x)}{6 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3}}-\frac{5}{2 a^2 x^{2/3}}+\frac{1}{a x^{2/3} (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(5/3)*(a + b*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 15.4528, size = 126, normalized size = 0.98 \[ \frac{1}{a x^{\frac{2}{3}} \left (a + b x\right )} - \frac{5}{2 a^{2} x^{\frac{2}{3}}} - \frac{5 b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 a^{\frac{8}{3}}} + \frac{5 b^{\frac{2}{3}} \log{\left (a + b x \right )}}{6 a^{\frac{8}{3}}} + \frac{5 \sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{8}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(5/3)/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.174068, size = 147, normalized size = 1.15 \[ \frac{5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )-\frac{6 a^{2/3} b \sqrt [3]{x}}{a+b x}-\frac{9 a^{2/3}}{x^{2/3}}-10 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+10 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 a^{8/3}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(5/3)*(a + b*x)^2),x]
[Out]
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Maple [A] time = 0.02, size = 121, normalized size = 1. \[ -{\frac{3}{2\,{a}^{2}}{x}^{-{\frac{2}{3}}}}-{\frac{b}{{a}^{2} \left ( bx+a \right ) }\sqrt [3]{x}}-{\frac{5}{3\,{a}^{2}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5}{6\,{a}^{2}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,\sqrt{3}}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(5/3)/(b*x+a)^2,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(1/((b*x + a)^2*x^(5/3)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232116, size = 270, normalized size = 2.11 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (b x + a\right )} x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{\frac{2}{3}} + a b x^{\frac{1}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 10 \, \sqrt{3}{\left (b x + a\right )} x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{\frac{1}{3}} - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 30 \,{\left (b x + a\right )} x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x^{\frac{1}{3}} + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (5 \, b x + 3 \, a\right )}\right )}}{18 \,{\left (a^{2} b x + a^{3}\right )} x^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*x^(5/3)),x, algorithm="fricas")
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Sympy [A] time = 6.08757, size = 615, normalized size = 4.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(5/3)/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.218071, size = 185, normalized size = 1.45 \[ \frac{5 \, b \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3}} - \frac{5 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3}} - \frac{b x^{\frac{1}{3}}}{{\left (b x + a\right )} a^{2}} - \frac{5 \, \left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{3}} - \frac{3}{2 \, a^{2} x^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*x^(5/3)),x, algorithm="giac")
[Out]