Optimal. Leaf size=236 \[ -\frac {9009 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{512 b^{15/2}}+\frac {9009 a^5 \sqrt {a x+b x^{2/3}}}{512 b^7 x^{2/3}}-\frac {3003 a^4 \sqrt {a x+b x^{2/3}}}{256 b^6 x}+\frac {3003 a^3 \sqrt {a x+b x^{2/3}}}{320 b^5 x^{4/3}}-\frac {1287 a^2 \sqrt {a x+b x^{2/3}}}{160 b^4 x^{5/3}}+\frac {143 a \sqrt {a x+b x^{2/3}}}{20 b^3 x^2}-\frac {13 \sqrt {a x+b x^{2/3}}}{2 b^2 x^{7/3}}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}} \]
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Rubi [A] time = 0.41, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 206
Rule 2023
Rule 2025
Rule 2029
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*}
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Mathematica [C] time = 0.08, size = 48, normalized size = 0.20 \[ \frac {6 a^6 \sqrt [3]{x} \, _2F_1\left (-\frac {1}{2},7;\frac {1}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{b^7 \sqrt {a x+b x^{2/3}}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 156, normalized size = 0.66 \[ \frac {9009 \, a^{6} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{512 \, \sqrt {-b} b^{7}} + \frac {6 \, a^{6}}{\sqrt {a x^{\frac {1}{3}} + b} b^{7}} + \frac {29685 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{6} - 163095 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{6} b + 364194 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{6} b^{2} - 416094 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{6} b^{3} + 246505 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{6} b^{4} - 62475 \, \sqrt {a x^{\frac {1}{3}} + b} a^{6} b^{5}}{2560 \, a^{6} b^{7} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 126, normalized size = 0.53 \[ -\frac {\left (a \,x^{\frac {1}{3}}+b \right ) \left (45045 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{6} x^{2} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-45045 a^{6} \sqrt {b}\, x^{2}-15015 a^{5} b^{\frac {3}{2}} x^{\frac {5}{3}}+6006 a^{4} b^{\frac {5}{2}} x^{\frac {4}{3}}-3432 a^{3} b^{\frac {7}{2}} x +2288 a^{2} b^{\frac {9}{2}} x^{\frac {2}{3}}-1664 a \,b^{\frac {11}{2}} x^{\frac {1}{3}}+1280 b^{\frac {13}{2}}\right )}{2560 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} b^{\frac {15}{2}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^2\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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