Optimal. Leaf size=621 \[ -\frac{9\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{27 \left (1+\sqrt{3}\right ) a^2 e \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{448 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{27 \sqrt [4]{3} a^{7/3} e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{3/2} (4 A b-a B)}{28 b e}+\frac{9 a (e x)^{5/2} \sqrt{a+b x^3} (4 A b-a B)}{224 b e}+\frac{B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e} \]
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Rubi [A] time = 0.660539, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {459, 279, 329, 308, 225, 1881} \[ \frac{27 \left (1+\sqrt{3}\right ) a^2 e \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{448 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{9\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} a^{7/3} e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{3/2} (4 A b-a B)}{28 b e}+\frac{9 a (e x)^{5/2} \sqrt{a+b x^3} (4 A b-a B)}{224 b e}+\frac{B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e} \]
Antiderivative was successfully verified.
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Rule 459
Rule 279
Rule 329
Rule 308
Rule 225
Rule 1881
Rubi steps
\begin{align*} \int (e x)^{3/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac{B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}-\frac{\left (-10 A b+\frac{5 a B}{2}\right ) \int (e x)^{3/2} \left (a+b x^3\right )^{3/2} \, dx}{10 b}\\ &=\frac{(4 A b-a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 b e}+\frac{B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}+\frac{(9 a (4 A b-a B)) \int (e x)^{3/2} \sqrt{a+b x^3} \, dx}{56 b}\\ &=\frac{9 a (4 A b-a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 b e}+\frac{(4 A b-a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 b e}+\frac{B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}+\frac{\left (27 a^2 (4 A b-a B)\right ) \int \frac{(e x)^{3/2}}{\sqrt{a+b x^3}} \, dx}{448 b}\\ &=\frac{9 a (4 A b-a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 b e}+\frac{(4 A b-a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 b e}+\frac{B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}+\frac{\left (27 a^2 (4 A b-a B)\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{224 b e}\\ &=\frac{9 a (4 A b-a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 b e}+\frac{(4 A b-a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 b e}+\frac{B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}-\frac{\left (27 a^2 (4 A b-a B)\right ) \operatorname{Subst}\left (\int \frac{\left (-1+\sqrt{3}\right ) a^{2/3} e^2-2 b^{2/3} x^4}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{448 b^{5/3} e}-\frac{\left (27 \left (1-\sqrt{3}\right ) a^{8/3} (4 A b-a B) e\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{448 b^{5/3}}\\ &=\frac{9 a (4 A b-a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 b e}+\frac{27 \left (1+\sqrt{3}\right ) a^2 (4 A b-a B) e \sqrt{e x} \sqrt{a+b x^3}}{448 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{(4 A b-a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 b e}+\frac{B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}-\frac{27 \sqrt [4]{3} a^{7/3} (4 A b-a B) e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{9\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} (4 A b-a B) e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.115566, size = 96, normalized size = 0.15 \[ \frac{x (e x)^{3/2} \sqrt{a+b x^3} \left (a (4 A b-a B) \, _2F_1\left (-\frac{3}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right )+B \sqrt{\frac{b x^3}{a}+1} \left (a+b x^3\right )^2\right )}{10 b \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.062, size = 5790, normalized size = 9.3 \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{3}{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 ({\left (B b e x^{7} +{\left (B a + A b\right )} e x^{4} + A a e x\right )} \sqrt{b x^{3} + a} \sqrt{e x}, 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{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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