Optimal. Leaf size=134 \[ \frac{2 B \left (a+b x^3\right )^{7/2} (e x)^{m+1}}{b e (2 m+23)}-\frac{a^2 \sqrt{a+b x^3} (e x)^{m+1} (2 a B (m+1)-A b (2 m+23)) \, _2F_1\left (-\frac{5}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{b e (m+1) (2 m+23) \sqrt{\frac{b x^3}{a}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0791409, antiderivative size = 134, 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 = {459, 365, 364} \[ \frac{2 B \left (a+b x^3\right )^{7/2} (e x)^{m+1}}{b e (2 m+23)}-\frac{a^2 \sqrt{a+b x^3} (e x)^{m+1} (2 a B (m+1)-A b (2 m+23)) \, _2F_1\left (-\frac{5}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{b e (m+1) (2 m+23) \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 459
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (e x)^m \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx &=\frac{2 B (e x)^{1+m} \left (a+b x^3\right )^{7/2}}{b e (23+2 m)}-\frac{\left (a B (1+m)-A b \left (\frac{23}{2}+m\right )\right ) \int (e x)^m \left (a+b x^3\right )^{5/2} \, dx}{b \left (\frac{23}{2}+m\right )}\\ &=\frac{2 B (e x)^{1+m} \left (a+b x^3\right )^{7/2}}{b e (23+2 m)}-\frac{\left (a^2 \left (a B (1+m)-A b \left (\frac{23}{2}+m\right )\right ) \sqrt{a+b x^3}\right ) \int (e x)^m \left (1+\frac{b x^3}{a}\right )^{5/2} \, dx}{b \left (\frac{23}{2}+m\right ) \sqrt{1+\frac{b x^3}{a}}}\\ &=\frac{2 B (e x)^{1+m} \left (a+b x^3\right )^{7/2}}{b e (23+2 m)}-\frac{a^2 (2 a B (1+m)-A b (23+2 m)) (e x)^{1+m} \sqrt{a+b x^3} \, _2F_1\left (-\frac{5}{2},\frac{1+m}{3};\frac{4+m}{3};-\frac{b x^3}{a}\right )}{b e (1+m) (23+2 m) \sqrt{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [A] time = 0.111362, size = 113, normalized size = 0.84 \[ \frac{a^2 x \sqrt{a+b x^3} (e x)^m \left (A (m+4) \, _2F_1\left (-\frac{5}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )+B (m+1) x^3 \, _2F_1\left (-\frac{5}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right )\right )}{(m+1) (m+4) \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}} \left ( B{x}^{3}+A \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{2} x^{9} +{\left (2 \, B a b + A b^{2}\right )} x^{6} +{\left (B a^{2} + 2 \, A a b\right )} x^{3} + A a^{2}\right )} \sqrt{b x^{3} + a} \left (e x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]