Optimal. Leaf size=121 \[ \frac{a \sqrt{e} (4 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{12 b^{3/2}}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (4 A b-a B)}{12 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e} \]
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Rubi [A] time = 0.0893504, antiderivative size = 121, 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 = {459, 279, 329, 275, 217, 206} \[ \frac{a \sqrt{e} (4 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{12 b^{3/2}}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (4 A b-a B)}{12 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e} \]
Antiderivative was successfully verified.
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Rule 459
Rule 279
Rule 329
Rule 275
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \sqrt{e x} \sqrt{a+b x^3} \left (A+B x^3\right ) \, dx &=\frac{B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}-\frac{\left (-6 A b+\frac{3 a B}{2}\right ) \int \sqrt{e x} \sqrt{a+b x^3} \, dx}{6 b}\\ &=\frac{(4 A b-a B) (e x)^{3/2} \sqrt{a+b x^3}}{12 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}+\frac{(a (4 A b-a B)) \int \frac{\sqrt{e x}}{\sqrt{a+b x^3}} \, dx}{8 b}\\ &=\frac{(4 A b-a B) (e x)^{3/2} \sqrt{a+b x^3}}{12 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}+\frac{(a (4 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 )}{4 b e}\\ &=\frac{(4 A b-a B) (e x)^{3/2} \sqrt{a+b x^3}}{12 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}+\frac{(a (4 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 )}{12 b e}\\ &=\frac{(4 A b-a B) (e x)^{3/2} \sqrt{a+b x^3}}{12 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}+\frac{(a (4 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 )}{12 b e}\\ &=\frac{(4 A b-a B) (e x)^{3/2} \sqrt{a+b x^3}}{12 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}+\frac{a (4 A b-a B) \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{12 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.124822, size = 119, normalized size = 0.98 \[ \frac{\sqrt{e x} \sqrt{a+b x^3} \left (\sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1} \left (B \left (a+2 b x^3\right )+4 A b\right )-\sqrt{a} (a B-4 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )\right )}{12 b^{3/2} \sqrt{x} \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.056, size = 6858, normalized size = 56.7 \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 )} \sqrt{b x^{3} + a} \sqrt{e x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.40025, size = 501, normalized size = 4.14 \begin{align*} \left [-\frac{{\left (B a^{2} - 4 \, A a b\right )} \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 (2 \, B b x^{4} +{\left (B a + 4 \, A b\right )} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{48 \, b}, \frac{{\left (B a^{2} - 4 \, A a b\right )} \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 (2 \, B b x^{4} +{\left (B a + 4 \, A b\right )} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{24 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.3376, size = 201, normalized size = 1.66 \begin{align*} \frac{A \sqrt{a} \left (e x\right )^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e} + \frac{A a \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{3 \sqrt{b}} + \frac{B a^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}}}{12 b e \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{B \sqrt{a} \left (e x\right )^{\frac{9}{2}}}{4 e^{4} \sqrt{1 + \frac{b x^{3}}{a}}} - \frac{B a^{2} \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{12 b^{\frac{3}{2}}} + \frac{B b \left (e x\right )^{\frac{15}{2}}}{6 \sqrt{a} e^{7} \sqrt{1 + \frac{b x^{3}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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