Optimal. Leaf size=201 \[ \frac{a^2 e^2 (e x)^{3/2} \sqrt{a+b x^3} (8 A b-3 a B)}{192 b^2}-\frac{a^3 e^{7/2} (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{192 b^{5/2}}+\frac{(e x)^{9/2} \left (a+b x^3\right )^{3/2} (8 A b-3 a B)}{72 b e}+\frac{a (e x)^{9/2} \sqrt{a+b x^3} (8 A b-3 a B)}{96 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e} \]
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Rubi [A] time = 0.138526, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {459, 279, 321, 329, 275, 217, 206} \[ \frac{a^2 e^2 (e x)^{3/2} \sqrt{a+b x^3} (8 A b-3 a B)}{192 b^2}-\frac{a^3 e^{7/2} (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{192 b^{5/2}}+\frac{(e x)^{9/2} \left (a+b x^3\right )^{3/2} (8 A b-3 a B)}{72 b e}+\frac{a (e x)^{9/2} \sqrt{a+b x^3} (8 A b-3 a B)}{96 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e} \]
Antiderivative was successfully verified.
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Rule 459
Rule 279
Rule 321
Rule 329
Rule 275
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac{\left (-12 A b+\frac{9 a B}{2}\right ) \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \, dx}{12 b}\\ &=\frac{(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}+\frac{(a (8 A b-3 a B)) \int (e x)^{7/2} \sqrt{a+b x^3} \, dx}{16 b}\\ &=\frac{a (8 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{96 b e}+\frac{(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}+\frac{\left (a^2 (8 A b-3 a B)\right ) \int \frac{(e x)^{7/2}}{\sqrt{a+b x^3}} \, dx}{64 b}\\ &=\frac{a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt{a+b x^3}}{192 b^2}+\frac{a (8 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{96 b e}+\frac{(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac{\left (a^3 (8 A b-3 a B) e^3\right ) \int \frac{\sqrt{e x}}{\sqrt{a+b x^3}} \, dx}{128 b^2}\\ &=\frac{a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt{a+b x^3}}{192 b^2}+\frac{a (8 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{96 b e}+\frac{(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac{\left (a^3 (8 A b-3 a B) e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{64 b^2}\\ &=\frac{a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt{a+b x^3}}{192 b^2}+\frac{a (8 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{96 b e}+\frac{(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac{\left (a^3 (8 A b-3 a B) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{192 b^2}\\ &=\frac{a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt{a+b x^3}}{192 b^2}+\frac{a (8 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{96 b e}+\frac{(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac{\left (a^3 (8 A b-3 a B) e^2\right ) \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 )}{192 b^2}\\ &=\frac{a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt{a+b x^3}}{192 b^2}+\frac{a (8 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{96 b e}+\frac{(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac{a^3 (8 A b-3 a B) e^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{192 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.257227, size = 167, normalized size = 0.83 \[ \frac{e^3 \sqrt{e x} \sqrt{a+b x^3} \left (\sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1} \left (6 a^2 b \left (4 A+B x^3\right )-9 a^3 B+8 a b^2 x^3 \left (14 A+9 B x^3\right )+16 b^3 x^6 \left (4 A+3 B x^3\right )\right )+3 a^{5/2} (3 a B-8 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )\right )}{576 b^{5/2} \sqrt{x} \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.065, size = 7705, normalized size = 38.3 \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{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.37852, size = 791, normalized size = 3.94 \begin{align*} \left [-\frac{3 \,{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{3} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (48 \, B b^{3} e^{3} x^{10} + 8 \,{\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} e^{3} x^{7} + 2 \,{\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} e^{3} x^{4} - 3 \,{\left (3 \, B a^{3} - 8 \, A a^{2} b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{2304 \, b^{2}}, -\frac{3 \,{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{3} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} b x \sqrt{-\frac{e}{b}}}{2 \, b e x^{3} + a e}\right ) - 2 \,{\left (48 \, B b^{3} e^{3} x^{10} + 8 \,{\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} e^{3} x^{7} + 2 \,{\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} e^{3} x^{4} - 3 \,{\left (3 \, B a^{3} - 8 \, A a^{2} b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{1152 \, b^{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{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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