Optimal. Leaf size=188 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{24 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{5/2} (a B+6 A b)}{9 a e^4}+\frac{5 (e x)^{3/2} \left (a+b x^3\right )^{3/2} (a B+6 A b)}{36 e^4}+\frac{5 a (e x)^{3/2} \sqrt{a+b x^3} (a B+6 A b)}{24 e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}} \]
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Rubi [A] time = 0.128335, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{24 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{5/2} (a B+6 A b)}{9 a e^4}+\frac{5 (e x)^{3/2} \left (a+b x^3\right )^{3/2} (a B+6 A b)}{36 e^4}+\frac{5 a (e x)^{3/2} \sqrt{a+b x^3} (a B+6 A b)}{24 e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}} \]
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{\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{(e x)^{5/2}} \, dx &=-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac{(6 A b+a B) \int \sqrt{e x} \left (a+b x^3\right )^{5/2} \, dx}{a e^3}\\ &=\frac{(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac{(5 (6 A b+a B)) \int \sqrt{e x} \left (a+b x^3\right )^{3/2} \, dx}{6 e^3}\\ &=\frac{5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac{(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac{(5 a (6 A b+a B)) \int \sqrt{e x} \sqrt{a+b x^3} \, dx}{8 e^3}\\ &=\frac{5 a (6 A b+a B) (e x)^{3/2} \sqrt{a+b x^3}}{24 e^4}+\frac{5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac{(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac{\left (5 a^2 (6 A b+a B)\right ) \int \frac{\sqrt{e x}}{\sqrt{a+b x^3}} \, dx}{16 e^3}\\ &=\frac{5 a (6 A b+a B) (e x)^{3/2} \sqrt{a+b x^3}}{24 e^4}+\frac{5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac{(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac{\left (5 a^2 (6 A b+a B)\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{8 e^4}\\ &=\frac{5 a (6 A b+a B) (e x)^{3/2} \sqrt{a+b x^3}}{24 e^4}+\frac{5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac{(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac{\left (5 a^2 (6 A b+a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{24 e^4}\\ &=\frac{5 a (6 A b+a B) (e x)^{3/2} \sqrt{a+b x^3}}{24 e^4}+\frac{5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac{(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac{\left (5 a^2 (6 A b+a B)\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 )}{24 e^4}\\ &=\frac{5 a (6 A b+a B) (e x)^{3/2} \sqrt{a+b x^3}}{24 e^4}+\frac{5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac{(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac{5 a^2 (6 A b+a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{24 \sqrt{b} e^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.193192, size = 150, normalized size = 0.8 \[ \frac{x \sqrt{a+b x^3} \left (\sqrt{b} \sqrt{\frac{b x^3}{a}+1} \left (a^2 \left (33 B x^3-48 A\right )+a \left (54 A b x^3+26 b B x^6\right )+4 b^2 x^6 \left (3 A+2 B x^3\right )\right )+15 a^{3/2} x^{3/2} (a B+6 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )\right )}{72 \sqrt{b} (e x)^{5/2} \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.053, size = 7544, normalized size = 40.1 \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 )}{\left (b x^{3} + a\right )}^{\frac{5}{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 = 4.33935, size = 714, normalized size = 3.8 \begin{align*} \left [\frac{15 \,{\left (B a^{3} + 6 \, A a^{2} 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 (8 \, B b^{3} x^{9} + 2 \,{\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} x^{6} - 48 \, A a^{2} b + 3 \,{\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} x^{3}\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{288 \, b e^{3} x^{2}}, -\frac{15 \,{\left (B a^{3} + 6 \, A a^{2} 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 (8 \, B b^{3} x^{9} + 2 \,{\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} x^{6} - 48 \, A a^{2} b + 3 \,{\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} x^{3}\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{144 \, b 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{{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{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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