Optimal. Leaf size=52 \[ \frac {2}{9} \sqrt {a x^3+b} \left (a x^3+7 b\right )-\frac {4}{3} b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 80, 50, 63, 208} \begin {gather*} -\frac {4}{3} b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )+\frac {4}{3} b \sqrt {a x^3+b}+\frac {2}{9} \left (a x^3+b\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {b+a x} (2 b+a x)}{x} \, dx,x,x^3\right )\\ &=\frac {2}{9} \left (b+a x^3\right )^{3/2}+\frac {1}{3} (2 b) \operatorname {Subst}\left (\int \frac {\sqrt {b+a x}}{x} \, dx,x,x^3\right )\\ &=\frac {4}{3} b \sqrt {b+a x^3}+\frac {2}{9} \left (b+a x^3\right )^{3/2}+\frac {1}{3} \left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b+a x}} \, dx,x,x^3\right )\\ &=\frac {4}{3} b \sqrt {b+a x^3}+\frac {2}{9} \left (b+a x^3\right )^{3/2}+\frac {\left (4 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b+a x^3}\right )}{3 a}\\ &=\frac {4}{3} b \sqrt {b+a x^3}+\frac {2}{9} \left (b+a x^3\right )^{3/2}-\frac {4}{3} b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.98 \begin {gather*} \frac {2}{9} \left (\sqrt {a x^3+b} \left (a x^3+7 b\right )-6 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 52, normalized size = 1.00 \begin {gather*} \frac {2}{9} \sqrt {b+a x^3} \left (7 b+a x^3\right )-\frac {4}{3} b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 103, normalized size = 1.98 \begin {gather*} \left [\frac {2}{3} \, b^{\frac {3}{2}} \log \left (\frac {a x^{3} - 2 \, \sqrt {a x^{3} + b} \sqrt {b} + 2 \, b}{x^{3}}\right ) + \frac {2}{9} \, {\left (a x^{3} + 7 \, b\right )} \sqrt {a x^{3} + b}, \frac {4}{3} \, \sqrt {-b} b \arctan \left (\frac {\sqrt {a x^{3} + b} \sqrt {-b}}{b}\right ) + \frac {2}{9} \, {\left (a x^{3} + 7 \, b\right )} \sqrt {a x^{3} + b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 50, normalized size = 0.96 \begin {gather*} \frac {4 \, b^{2} \arctan \left (\frac {\sqrt {a x^{3} + b}}{\sqrt {-b}}\right )}{3 \, \sqrt {-b}} + \frac {2}{9} \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} + \frac {4}{3} \, \sqrt {a x^{3} + b} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 47, normalized size = 0.90
method | result | size |
default | \(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}}}{9}+2 b \left (\frac {2 \sqrt {a \,x^{3}+b}}{3}-\frac {2 \sqrt {b}\, \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3}\right )\) | \(47\) |
elliptic | \(\frac {2 a \,x^{3} \sqrt {a \,x^{3}+b}}{9}+\frac {14 b \sqrt {a \,x^{3}+b}}{9}-\frac {4 b^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 63, normalized size = 1.21 \begin {gather*} \frac {2}{3} \, {\left (\sqrt {b} \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right ) + 2 \, \sqrt {a x^{3} + b}\right )} b + \frac {2}{9} \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 68, normalized size = 1.31 \begin {gather*} \frac {2\,b^{3/2}\,\ln \left (\frac {{\left (\sqrt {a\,x^3+b}-\sqrt {b}\right )}^3\,\left (\sqrt {a\,x^3+b}+\sqrt {b}\right )}{x^6}\right )}{3}+\frac {14\,b\,\sqrt {a\,x^3+b}}{9}+\frac {2\,a\,x^3\,\sqrt {a\,x^3+b}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.74, size = 78, normalized size = 1.50 \begin {gather*} - \frac {a \left (\begin {cases} - \sqrt {b} x^{3} & \text {for}\: a = 0 \\- \frac {2 \left (a x^{3} + b\right )^{\frac {3}{2}}}{3 a} & \text {otherwise} \end {cases}\right )}{3} - \frac {2 b \left (- \frac {2 b \operatorname {atan}{\left (\frac {\sqrt {a x^{3} + b}}{\sqrt {- b}} \right )}}{\sqrt {- b}} - 2 \sqrt {a x^{3} + b}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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