Optimal. Leaf size=118 \[ \frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+2 A b)}{3 a e^4}+\frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 \sqrt{b} e^{5/2}}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}} \]
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Rubi [A] time = 0.0844829, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {453, 279, 329, 275, 217, 206} \[ \frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+2 A b)}{3 a e^4}+\frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 \sqrt{b} e^{5/2}}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 279
Rule 329
Rule 275
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^3} \left (A+B x^3\right )}{(e x)^{5/2}} \, dx &=-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}}+\frac{(2 A b+a B) \int \sqrt{e x} \sqrt{a+b x^3} \, dx}{a e^3}\\ &=\frac{(2 A b+a B) (e x)^{3/2} \sqrt{a+b x^3}}{3 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}}+\frac{(2 A b+a B) \int \frac{\sqrt{e x}}{\sqrt{a+b x^3}} \, dx}{2 e^3}\\ &=\frac{(2 A b+a B) (e x)^{3/2} \sqrt{a+b x^3}}{3 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}}+\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 )}{e^4}\\ &=\frac{(2 A b+a B) (e x)^{3/2} \sqrt{a+b x^3}}{3 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}}+\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 e^4}\\ &=\frac{(2 A b+a B) (e x)^{3/2} \sqrt{a+b x^3}}{3 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}}+\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 e^4}\\ &=\frac{(2 A b+a B) (e x)^{3/2} \sqrt{a+b x^3}}{3 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}}+\frac{(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 \sqrt{b} e^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.187416, size = 87, normalized size = 0.74 \[ \frac{x \sqrt{a+b x^3} \left (\frac{x^{3/2} (a B+2 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{\frac{b x^3}{a}+1}}-2 A+B x^3\right )}{3 (e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.059, size = 6668, normalized size = 56.5 \begin{align*} \text{output too large to display} \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{b x^{3} + a}}{\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 = 4.32924, size = 486, normalized size = 4.12 \begin{align*} \left [\frac{{\left (B a + 2 \, A b\right )} \sqrt{b e} x^{2} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b x^{4} + a x\right )} \sqrt{b x^{3} + a} \sqrt{b e} \sqrt{e x}\right ) + 4 \,{\left (B b x^{3} - 2 \, A b\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{12 \, b e^{3} x^{2}}, -\frac{{\left (B a + 2 \, A b\right )} \sqrt{-b e} x^{2} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x} x}{2 \, b e x^{3} + a e}\right ) - 2 \,{\left (B b x^{3} - 2 \, A b\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{6 \, b e^{3} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.1409, size = 160, normalized size = 1.36 \begin{align*} - \frac{2 A \sqrt{a}}{3 e^{\frac{5}{2}} x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{2 A \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 e^{\frac{5}{2}}} - \frac{2 A b x^{\frac{3}{2}}}{3 \sqrt{a} e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{B \sqrt{a} x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e^{\frac{5}{2}}} + \frac{B a \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 \sqrt{b} e^{\frac{5}{2}}} \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{{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a}}{\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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