Optimal. Leaf size=67 \[ -\frac{2 (e x)^{3/2} (2 A b-a B)}{3 a^2 e^4 \sqrt{a+b x^3}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.029186, antiderivative size = 67, 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 = {453, 264} \[ -\frac{2 (e x)^{3/2} (2 A b-a B)}{3 a^2 e^4 \sqrt{a+b x^3}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{A+B x^3}{(e x)^{5/2} \left (a+b x^3\right )^{3/2}} \, dx &=-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^3}}-\frac{(2 A b-a B) \int \frac{\sqrt{e x}}{\left (a+b x^3\right )^{3/2}} \, dx}{a e^3}\\ &=-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^3}}-\frac{2 (2 A b-a B) (e x)^{3/2}}{3 a^2 e^4 \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0206157, size = 45, normalized size = 0.67 \[ \frac{x \left (-2 a A+2 a B x^3-4 A b x^3\right )}{3 a^2 (e x)^{5/2} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 39, normalized size = 0.6 \begin{align*} -{\frac{2\,x \left ( 2\,A{x}^{3}b-Ba{x}^{3}+Aa \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{3}+a}}} \left ( ex \right ) ^{-{\frac{5}{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{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} \left (e x\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 = 1.31067, size = 117, normalized size = 1.75 \begin{align*} \frac{2 \,{\left ({\left (B a - 2 \, A b\right )} x^{3} - A a\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{3 \,{\left (a^{2} b e^{3} x^{5} + a^{3} e^{3} x^{2}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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