Optimal. Leaf size=133 \[ \frac{\sqrt{\frac{b x^3}{a}+1} (e x)^{m+1} (2 a B (m+1)+A b (7-2 m)) \, _2F_1\left (\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{9 a^2 b e (m+1) \sqrt{a+b x^3}}+\frac{2 (e x)^{m+1} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.0789417, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {457, 365, 364} \[ \frac{\sqrt{\frac{b x^3}{a}+1} (e x)^{m+1} (2 a B (m+1)+A b (7-2 m)) \, _2F_1\left (\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{9 a^2 b e (m+1) \sqrt{a+b x^3}}+\frac{2 (e x)^{m+1} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 457
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac{2 (A b-a B) (e x)^{1+m}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac{\left (2 \left (-A b \left (-\frac{7}{2}+m\right )+a B (1+m)\right )\right ) \int \frac{(e x)^m}{\left (a+b x^3\right )^{3/2}} \, dx}{9 a b}\\ &=\frac{2 (A b-a B) (e x)^{1+m}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac{\left (2 \left (-A b \left (-\frac{7}{2}+m\right )+a B (1+m)\right ) \sqrt{1+\frac{b x^3}{a}}\right ) \int \frac{(e x)^m}{\left (1+\frac{b x^3}{a}\right )^{3/2}} \, dx}{9 a^2 b \sqrt{a+b x^3}}\\ &=\frac{2 (A b-a B) (e x)^{1+m}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac{(A b (7-2 m)+2 a B (1+m)) (e x)^{1+m} \sqrt{1+\frac{b x^3}{a}} \, _2F_1\left (\frac{3}{2},\frac{1+m}{3};\frac{4+m}{3};-\frac{b x^3}{a}\right )}{9 a^2 b e (1+m) \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0890926, size = 113, normalized size = 0.85 \[ \frac{x \sqrt{\frac{b x^3}{a}+1} (e x)^m \left (A (m+4) \, _2F_1\left (\frac{5}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )+B (m+1) x^3 \, _2F_1\left (\frac{5}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right )\right )}{a^2 (m+1) (m+4) \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex \right ) ^{m} \left ( B{x}^{3}+A \right ) \left ( b{x}^{3}+a \right ) ^{-{\frac{5}{2}}}}\, dx \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 )} \left (e x\right )^{m}}{{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a} \left (e x\right )^{m}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\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{{\left (B x^{3} + A\right )} \left (e x\right )^{m}}{{\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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