Optimal. Leaf size=59 \[ -\frac {2}{3} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {2}{3} a \sqrt {a+b x^3}+\frac {2}{9} \left (a+b x^3\right )^{3/2} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ -\frac {2}{3} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {2}{3} a \sqrt {a+b x^3}+\frac {2}{9} \left (a+b x^3\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{3/2}}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^3\right )\\ &=\frac {2}{9} \left (a+b x^3\right )^{3/2}+\frac {1}{3} a \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^3\right )\\ &=\frac {2}{3} a \sqrt {a+b x^3}+\frac {2}{9} \left (a+b x^3\right )^{3/2}+\frac {1}{3} a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {2}{3} a \sqrt {a+b x^3}+\frac {2}{9} \left (a+b x^3\right )^{3/2}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 b}\\ &=\frac {2}{3} a \sqrt {a+b x^3}+\frac {2}{9} \left (a+b x^3\right )^{3/2}-\frac {2}{3} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.86 \[ \frac {2}{9} \left (\sqrt {a+b x^3} \left (4 a+b x^3\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 103, normalized size = 1.75 \[ \left [\frac {1}{3} \, a^{\frac {3}{2}} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + \frac {2}{9} \, {\left (b x^{3} + 4 \, a\right )} \sqrt {b x^{3} + a}, \frac {2}{3} \, \sqrt {-a} a \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + \frac {2}{9} \, {\left (b x^{3} + 4 \, a\right )} \sqrt {b x^{3} + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 50, normalized size = 0.85 \[ \frac {2 \, a^{2} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} + \frac {2}{9} \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {b x^{3} + a} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 48, normalized size = 0.81 \[ \frac {2 \sqrt {b \,x^{3}+a}\, b \,x^{3}}{9}-\frac {2 a^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3}+\frac {8 \sqrt {b \,x^{3}+a}\, a}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.89, size = 61, normalized size = 1.03 \[ \frac {1}{3} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right ) + \frac {2}{9} \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {b x^{3} + a} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 68, normalized size = 1.15 \[ \frac {a^{3/2}\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{3}+\frac {8\,a\,\sqrt {b\,x^3+a}}{9}+\frac {2\,b\,x^3\,\sqrt {b\,x^3+a}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.24, size = 83, normalized size = 1.41 \[ \frac {8 a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{9} + \frac {a^{\frac {3}{2}} \log {\left (\frac {b x^{3}}{a} \right )}}{3} - \frac {2 a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3} + \frac {2 \sqrt {a} b x^{3} \sqrt {1 + \frac {b x^{3}}{a}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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