Optimal. Leaf size=146 \[ \frac {105 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{8 b^{9/2}}-\frac {105 a^2 \sqrt {a x+b x^{2/3}}}{8 b^4 x^{2/3}}+\frac {35 a \sqrt {a x+b x^{2/3}}}{4 b^3 x}-\frac {7 \sqrt {a x+b x^{2/3}}}{b^2 x^{4/3}}+\frac {6}{b x^{2/3} \sqrt {a x+b x^{2/3}}} \]
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Rubi [A] time = 0.24, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2023, 2025, 2029, 206} \[ -\frac {105 a^2 \sqrt {a x+b x^{2/3}}}{8 b^4 x^{2/3}}+\frac {105 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{8 b^{9/2}}+\frac {35 a \sqrt {a x+b x^{2/3}}}{4 b^3 x}-\frac {7 \sqrt {a x+b x^{2/3}}}{b^2 x^{4/3}}+\frac {6}{b x^{2/3} \sqrt {a x+b x^{2/3}}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2023
Rule 2025
Rule 2029
Rubi steps
\begin {align*} \int \frac {1}{x \left (b x^{2/3}+a x\right )^{3/2}} \, dx &=\frac {6}{b x^{2/3} \sqrt {b x^{2/3}+a x}}+\frac {7 \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{b}\\ &=\frac {6}{b x^{2/3} \sqrt {b x^{2/3}+a x}}-\frac {7 \sqrt {b x^{2/3}+a x}}{b^2 x^{4/3}}-\frac {(35 a) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{6 b^2}\\ &=\frac {6}{b x^{2/3} \sqrt {b x^{2/3}+a x}}-\frac {7 \sqrt {b x^{2/3}+a x}}{b^2 x^{4/3}}+\frac {35 a \sqrt {b x^{2/3}+a x}}{4 b^3 x}+\frac {\left (35 a^2\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{8 b^3}\\ &=\frac {6}{b x^{2/3} \sqrt {b x^{2/3}+a x}}-\frac {7 \sqrt {b x^{2/3}+a x}}{b^2 x^{4/3}}+\frac {35 a \sqrt {b x^{2/3}+a x}}{4 b^3 x}-\frac {105 a^2 \sqrt {b x^{2/3}+a x}}{8 b^4 x^{2/3}}-\frac {\left (35 a^3\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{16 b^4}\\ &=\frac {6}{b x^{2/3} \sqrt {b x^{2/3}+a x}}-\frac {7 \sqrt {b x^{2/3}+a x}}{b^2 x^{4/3}}+\frac {35 a \sqrt {b x^{2/3}+a x}}{4 b^3 x}-\frac {105 a^2 \sqrt {b x^{2/3}+a x}}{8 b^4 x^{2/3}}+\frac {\left (105 a^3\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 )}{8 b^4}\\ &=\frac {6}{b x^{2/3} \sqrt {b x^{2/3}+a x}}-\frac {7 \sqrt {b x^{2/3}+a x}}{b^2 x^{4/3}}+\frac {35 a \sqrt {b x^{2/3}+a x}}{4 b^3 x}-\frac {105 a^2 \sqrt {b x^{2/3}+a x}}{8 b^4 x^{2/3}}+\frac {105 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{8 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 48, normalized size = 0.33 \[ -\frac {6 a^3 \sqrt [3]{x} \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{b^4 \sqrt {a x+b x^{2/3}}} \]
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 = 0.26, size = 105, normalized size = 0.72 \[ -\frac {105 \, a^{3} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{8 \, \sqrt {-b} b^{4}} - \frac {6 \, a^{3}}{\sqrt {a x^{\frac {1}{3}} + b} b^{4}} - \frac {57 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{3} - 136 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{3} b + 87 \, \sqrt {a x^{\frac {1}{3}} + b} a^{3} b^{2}}{8 \, a^{3} b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 88, normalized size = 0.60 \[ -\frac {\left (a \,x^{\frac {1}{3}}+b \right ) \left (-105 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{3} x \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )+105 a^{3} \sqrt {b}\, x +35 a^{2} b^{\frac {3}{2}} x^{\frac {2}{3}}-14 a \,b^{\frac {5}{2}} x^{\frac {1}{3}}+8 b^{\frac {7}{2}}\right )}{8 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} b^{\frac {9}{2}}} \]
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 \[ \int \frac {1}{{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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