Optimal. Leaf size=78 \[ -6 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )+\frac {6 b \sqrt {a x+b x^{2/3}}}{\sqrt [3]{x}}+\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{x} \]
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Rubi [A] time = 0.14, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2021, 2029, 206} \[ -6 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )+\frac {6 b \sqrt {a x+b x^{2/3}}}{\sqrt [3]{x}}+\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{x} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2021
Rule 2029
Rubi steps
\begin {align*} \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx &=\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{x}+b \int \frac {\sqrt {b x^{2/3}+a x}}{x^{4/3}} \, dx\\ &=\frac {6 b \sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}}+\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{x}+b^2 \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx\\ &=\frac {6 b \sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}}+\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{x}-\left (6 b^2\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 )\\ &=\frac {6 b \sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}}+\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{x}-6 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 88, normalized size = 1.13 \[ \frac {2 \sqrt {a x+b x^{2/3}} \left (\sqrt {a \sqrt [3]{x}+b} \left (a \sqrt [3]{x}+4 b\right )-3 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sqrt [3]{x}+b}}{\sqrt {b}}\right )\right )}{\sqrt [3]{x} \sqrt {a \sqrt [3]{x}+b}} \]
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 = 0.23, size = 83, normalized size = 1.06 \[ \frac {6 \, b^{2} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + 2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} + 6 \, \sqrt {a x^{\frac {1}{3}} + b} b - \frac {2 \, {\left (3 \, b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b} b^{\frac {3}{2}}\right )}}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 69, normalized size = 0.88 \[ -\frac {2 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} \left (3 b^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-3 \sqrt {a \,x^{\frac {1}{3}}+b}\, b -\left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}}\right )}{\left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}} x} \]
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 \[ \int \frac {{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^2} \,d x \]
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 \[ \int \frac {\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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