Optimal. Leaf size=236 \[ \frac{9009 a^5 \sqrt{a x+b x^{2/3}}}{512 b^7 x^{2/3}}-\frac{3003 a^4 \sqrt{a x+b x^{2/3}}}{256 b^6 x}+\frac{3003 a^3 \sqrt{a x+b x^{2/3}}}{320 b^5 x^{4/3}}-\frac{1287 a^2 \sqrt{a x+b x^{2/3}}}{160 b^4 x^{5/3}}-\frac{9009 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{512 b^{15/2}}+\frac{143 a \sqrt{a x+b x^{2/3}}}{20 b^3 x^2}-\frac{13 \sqrt{a x+b x^{2/3}}}{2 b^2 x^{7/3}}+\frac{6}{b x^{5/3} \sqrt{a x+b x^{2/3}}} \]
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Rubi [A] time = 0.410442, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2023, 2025, 2029, 206} \[ \frac{9009 a^5 \sqrt{a x+b x^{2/3}}}{512 b^7 x^{2/3}}-\frac{3003 a^4 \sqrt{a x+b x^{2/3}}}{256 b^6 x}+\frac{3003 a^3 \sqrt{a x+b x^{2/3}}}{320 b^5 x^{4/3}}-\frac{1287 a^2 \sqrt{a x+b x^{2/3}}}{160 b^4 x^{5/3}}-\frac{9009 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{512 b^{15/2}}+\frac{143 a \sqrt{a x+b x^{2/3}}}{20 b^3 x^2}-\frac{13 \sqrt{a x+b x^{2/3}}}{2 b^2 x^{7/3}}+\frac{6}{b x^{5/3} \sqrt{a x+b x^{2/3}}} \]
Antiderivative was successfully verified.
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Rule 2023
Rule 2025
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (b x^{2/3}+a x\right )^{3/2}} \, dx &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}+\frac{13 \int \frac{1}{x^{8/3} \sqrt{b x^{2/3}+a x}} \, dx}{b}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}-\frac{(143 a) \int \frac{1}{x^{7/3} \sqrt{b x^{2/3}+a x}} \, dx}{12 b^2}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}+\frac{\left (429 a^2\right ) \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx}{40 b^3}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}-\frac{\left (3003 a^3\right ) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{320 b^4}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac{3003 a^3 \sqrt{b x^{2/3}+a x}}{320 b^5 x^{4/3}}+\frac{\left (1001 a^4\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{128 b^5}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac{3003 a^3 \sqrt{b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac{3003 a^4 \sqrt{b x^{2/3}+a x}}{256 b^6 x}-\frac{\left (3003 a^5\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{512 b^6}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac{3003 a^3 \sqrt{b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac{3003 a^4 \sqrt{b x^{2/3}+a x}}{256 b^6 x}+\frac{9009 a^5 \sqrt{b x^{2/3}+a x}}{512 b^7 x^{2/3}}+\frac{\left (3003 a^6\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{1024 b^7}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac{3003 a^3 \sqrt{b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac{3003 a^4 \sqrt{b x^{2/3}+a x}}{256 b^6 x}+\frac{9009 a^5 \sqrt{b x^{2/3}+a x}}{512 b^7 x^{2/3}}-\frac{\left (9009 a^6\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{512 b^7}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac{3003 a^3 \sqrt{b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac{3003 a^4 \sqrt{b x^{2/3}+a x}}{256 b^6 x}+\frac{9009 a^5 \sqrt{b x^{2/3}+a x}}{512 b^7 x^{2/3}}-\frac{9009 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{512 b^{15/2}}\\ \end{align*}
Mathematica [C] time = 0.0578095, size = 48, normalized size = 0.2 \[ \frac{6 a^6 \sqrt [3]{x} \, _2F_1\left (-\frac{1}{2},7;\frac{1}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^7 \sqrt{a x+b x^{2/3}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 126, normalized size = 0.5 \begin{align*} -{\frac{1}{2560\,x} \left ( b+a\sqrt [3]{x} \right ) \left ( 45045\,\sqrt{b+a\sqrt [3]{x}}{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){x}^{2}{a}^{6}-1664\,{b}^{11/2}\sqrt [3]{x}a+2288\,{b}^{9/2}{x}^{2/3}{a}^{2}-3432\,{b}^{7/2}x{a}^{3}+6006\,{b}^{5/2}{x}^{4/3}{a}^{4}-15015\,{b}^{3/2}{x}^{5/3}{a}^{5}-45045\,{x}^{2}{a}^{6}\sqrt{b}+1280\,{b}^{13/2} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{15}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b x^{\frac{2}{3}}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29799, size = 211, normalized size = 0.89 \begin{align*} \frac{9009 \, a^{6} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{512 \, \sqrt{-b} b^{7}} + \frac{6 \, a^{6}}{\sqrt{a x^{\frac{1}{3}} + b} b^{7}} + \frac{29685 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{6} - 163095 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{6} b + 364194 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{6} b^{2} - 416094 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{6} b^{3} + 246505 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{6} b^{4} - 62475 \, \sqrt{a x^{\frac{1}{3}} + b} a^{6} b^{5}}{2560 \, a^{6} b^{7} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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