Optimal. Leaf size=79 \[ \frac{2 (e x)^{3/2} (a B+2 A b)}{9 a^2 b e \sqrt{a+b x^3}}+\frac{2 (e x)^{3/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.0311775, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {457, 264} \[ \frac{2 (e x)^{3/2} (a B+2 A b)}{9 a^2 b e \sqrt{a+b x^3}}+\frac{2 (e x)^{3/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 457
Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt{e x} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac{2 (A b-a B) (e x)^{3/2}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac{\left (2 \left (3 A b+\frac{3 a B}{2}\right )\right ) \int \frac{\sqrt{e x}}{\left (a+b x^3\right )^{3/2}} \, dx}{9 a b}\\ &=\frac{2 (A b-a B) (e x)^{3/2}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac{2 (2 A b+a B) (e x)^{3/2}}{9 a^2 b e \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0371331, size = 44, normalized size = 0.56 \[ \frac{2 x \sqrt{e x} \left (3 a A+a B x^3+2 A b x^3\right )}{9 a^2 \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 39, normalized size = 0.5 \begin{align*}{\frac{2\,x \left ( 2\,A{x}^{3}b+Ba{x}^{3}+3\,Aa \right ) }{9\,{a}^{2}}\sqrt{ex} \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{3} + A\right )} \sqrt{e x}}{{\left (b x^{3} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36708, size = 128, normalized size = 1.62 \begin{align*} \frac{2 \,{\left ({\left (B a + 2 \, A b\right )} x^{4} + 3 \, A a x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{9 \,{\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.47911, size = 86, normalized size = 1.09 \begin{align*} \frac{2 \, x^{\frac{3}{2}}{\left (\frac{3 \, A e^{5}}{a} + \frac{{\left (B a^{5} b^{5} e^{21} + 2 \, A a^{4} b^{6} e^{21}\right )} x^{3} e^{\left (-16\right )}}{a^{6} b^{5}}\right )} e^{\frac{3}{2}}}{9 \,{\left (b x^{3} e^{4} + a e^{4}\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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