Optimal. Leaf size=64 \[ \frac {2}{3} A \sqrt {a+b x^3}-\frac {2}{3} \sqrt {a} A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {2 B \left (a+b x^3\right )^{3/2}}{9 b} \]
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Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 80, 50, 63, 208} \begin {gather*} \frac {2}{3} A \sqrt {a+b x^3}-\frac {2}{3} \sqrt {a} A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {2 B \left (a+b x^3\right )^{3/2}}{9 b} \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 {a+b x^3} \left (A+B x^3\right )}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x} (A+B x)}{x} \, dx,x,x^3\right )\\ &=\frac {2 B \left (a+b x^3\right )^{3/2}}{9 b}+\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 B \left (a+b x^3\right )^{3/2}}{9 b}+\frac {1}{3} (a A) \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 B \left (a+b x^3\right )^{3/2}}{9 b}+\frac {(2 a A) \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 B \left (a+b x^3\right )^{3/2}}{9 b}-\frac {2}{3} \sqrt {a} A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 0.94 \begin {gather*} \frac {2}{9} \left (\frac {\sqrt {a+b x^3} \left (B \left (a+b x^3\right )+3 A b\right )}{b}-3 \sqrt {a} A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 61, normalized size = 0.95 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (a B+3 A b+b B x^3\right )}{9 b}-\frac {2}{3} \sqrt {a} A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 125, normalized size = 1.95 \begin {gather*} \left [\frac {3 \, A \sqrt {a} b \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (B b x^{3} + B a + 3 \, A b\right )} \sqrt {b x^{3} + a}}{9 \, b}, \frac {2 \, {\left (3 \, A \sqrt {-a} b \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left (B b x^{3} + B a + 3 \, A b\right )} \sqrt {b x^{3} + a}\right )}}{9 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 61, normalized size = 0.95 \begin {gather*} \frac {2 \, A a \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} + \frac {2 \, {\left ({\left (b x^{3} + a\right )}^{\frac {3}{2}} B b^{2} + 3 \, \sqrt {b x^{3} + a} A b^{3}\right )}}{9 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 50, normalized size = 0.78 \begin {gather*} \left (-\frac {2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3}+\frac {2 \sqrt {b \,x^{3}+a}}{3}\right ) A +\frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} B}{9 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 67, normalized size = 1.05 \begin {gather*} \frac {1}{3} \, {\left (\sqrt {a} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right ) + 2 \, \sqrt {b x^{3} + a}\right )} A + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} B}{9 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.71, size = 80, normalized size = 1.25 \begin {gather*} \frac {2\,B\,x^3\,\sqrt {b\,x^3+a}}{9}+\frac {\sqrt {b\,x^3+a}\,\left (2\,A\,b+\frac {2\,B\,a}{3}\right )}{3\,b}+\frac {A\,\sqrt {a}\,\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 25.83, size = 76, normalized size = 1.19 \begin {gather*} - \frac {A \left (- \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + b x^{3}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} - 2 \sqrt {a + b x^{3}}\right )}{3} - \frac {B \left (\begin {cases} - \sqrt {a} x^{3} & \text {for}\: b = 0 \\- \frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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