Optimal. Leaf size=83 \[ \frac {\sqrt {e} (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{3 b^{3/2}}+\frac {B (e x)^{3/2} \sqrt {a+b x^3}}{3 b e} \]
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Rubi [A] time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {459, 329, 275, 217, 206} \begin {gather*} \frac {\sqrt {e} (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{3 b^{3/2}}+\frac {B (e x)^{3/2} \sqrt {a+b x^3}}{3 b e} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 329
Rule 459
Rubi steps
\begin {align*} \int \frac {\sqrt {e x} \left (A+B x^3\right )}{\sqrt {a+b x^3}} \, dx &=\frac {B (e x)^{3/2} \sqrt {a+b x^3}}{3 b e}-\frac {\left (-3 A b+\frac {3 a B}{2}\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{3 b}\\ &=\frac {B (e x)^{3/2} \sqrt {a+b x^3}}{3 b e}+\frac {(2 A b-a B) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{b e}\\ &=\frac {B (e x)^{3/2} \sqrt {a+b x^3}}{3 b e}+\frac {(2 A b-a B) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{3 b e}\\ &=\frac {B (e x)^{3/2} \sqrt {a+b x^3}}{3 b e}+\frac {(2 A b-a B) \operatorname {Subst}\left (\int \frac {1}{1-\frac {b x^2}{e^3}} \, dx,x,\frac {(e x)^{3/2}}{\sqrt {a+b x^3}}\right )}{3 b e}\\ &=\frac {B (e x)^{3/2} \sqrt {a+b x^3}}{3 b e}+\frac {(2 A b-a B) \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{3 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 78, normalized size = 0.94 \begin {gather*} \frac {\sqrt {e x} \left ((2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a+b x^3}}\right )+\sqrt {b} B x^{3/2} \sqrt {a+b x^3}\right )}{3 b^{3/2} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.53, size = 89, normalized size = 1.07 \begin {gather*} \frac {B (e x)^{3/2} \sqrt {a+b x^3}}{3 b e}-\frac {e^2 \sqrt {\frac {b}{e^3}} (2 A b-a B) \log \left (\sqrt {a+b x^3}-\sqrt {\frac {b}{e^3}} (e x)^{3/2}\right )}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.28, size = 184, normalized size = 2.22 \begin {gather*} \left [\frac {4 \, \sqrt {b x^{3} + a} \sqrt {e x} B x - {\left (B a - 2 \, A b\right )} \sqrt {\frac {e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \, {\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt {b x^{3} + a} \sqrt {e x} \sqrt {\frac {e}{b}}\right )}{12 \, b}, \frac {2 \, \sqrt {b x^{3} + a} \sqrt {e x} B x + {\left (B a - 2 \, A b\right )} \sqrt {-\frac {e}{b}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {e x} b x \sqrt {-\frac {e}{b}}}{2 \, b e x^{3} + a e}\right )}{6 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 72, normalized size = 0.87 \begin {gather*} \frac {\sqrt {b x^{3} e^{4} + a e^{4}} B x^{\frac {3}{2}} e^{\left (-\frac {3}{2}\right )}}{3 \, b} + \frac {{\left (B a e^{5} - 2 \, A b e^{5}\right )} e^{\left (-\frac {9}{2}\right )} \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} e^{2} + \sqrt {b x^{3} e^{4} + a e^{4}} \right |}\right )}{3 \, b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.15, size = 6424, normalized size = 77.40 \begin {gather*} \text {output too large to display} \end {gather*}
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 \begin {gather*} \int \frac {{\left (B x^{3} + A\right )} \sqrt {e x}}{\sqrt {b x^{3} + a}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\left (B\,x^3+A\right )\,\sqrt {e\,x}}{\sqrt {b\,x^3+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.81, size = 107, normalized size = 1.29 \begin {gather*} \frac {2 A \sqrt {e} \operatorname {asinh}{\left (\frac {\sqrt {b} \left (e x\right )^{\frac {3}{2}}}{\sqrt {a} e^{\frac {3}{2}}} \right )}}{3 \sqrt {b}} + \frac {B \sqrt {a} \left (e x\right )^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{3 b e} - \frac {B a \sqrt {e} \operatorname {asinh}{\left (\frac {\sqrt {b} \left (e x\right )^{\frac {3}{2}}}{\sqrt {a} e^{\frac {3}{2}}} \right )}}{3 b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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