Optimal. Leaf size=60 \[ \frac {6 \sqrt [3]{x}}{b \sqrt {a x+b x^{2/3}}}-\frac {6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2006, 2029, 206} \[ \frac {6 \sqrt [3]{x}}{b \sqrt {a x+b x^{2/3}}}-\frac {6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2006
Rule 2029
Rubi steps
\begin {align*} \int \frac {1}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx &=\frac {6 \sqrt [3]{x}}{b \sqrt {b x^{2/3}+a x}}+\frac {\int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{b}\\ &=\frac {6 \sqrt [3]{x}}{b \sqrt {b x^{2/3}+a x}}-\frac {6 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{b}\\ &=\frac {6 \sqrt [3]{x}}{b \sqrt {b x^{2/3}+a x}}-\frac {6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 45, normalized size = 0.75 \[ \frac {6 \sqrt [3]{x} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{b \sqrt {a x+b x^{2/3}}} \]
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 = 0.21, size = 71, normalized size = 1.18 \[ \frac {6 \, \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {6 \, {\left (\sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + \sqrt {-b}\right )}}{\sqrt {-b} b^{\frac {3}{2}}} + \frac {6}{\sqrt {a x^{\frac {1}{3}} + b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 56, normalized size = 0.93 \[ -\frac {6 \left (a \,x^{\frac {1}{3}}+b \right ) \left (\sqrt {a \,x^{\frac {1}{3}}+b}\, b \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-b^{\frac {3}{2}}\right ) x}{\left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} b^{\frac {5}{2}}} \]
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 \[ \int \frac {1}{{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.36, size = 40, normalized size = 0.67 \[ -\frac {2\,x\,{\left (\frac {b}{a\,x^{1/3}}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {3}{2};\ \frac {5}{2};\ -\frac {b}{a\,x^{1/3}}\right )}{{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \]
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 \[ \int \frac {1}{\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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