Optimal. Leaf size=614 \[ -\frac{9\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{4/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+14 A b) \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{224 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} a^{4/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+14 A b) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{112 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{27 \left (1+\sqrt{3}\right ) a \sqrt{e x} \sqrt{a+b x^3} (a B+14 A b)}{112 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{3/2} (a B+14 A b)}{7 a e^4}+\frac{9 (e x)^{5/2} \sqrt{a+b x^3} (a B+14 A b)}{56 e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{a e \sqrt{e x}} \]
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Rubi [A] time = 0.644303, antiderivative size = 614, 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 = {453, 279, 329, 308, 225, 1881} \[ -\frac{9\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{4/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+14 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{224 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} a^{4/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+14 A b) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{112 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{27 \left (1+\sqrt{3}\right ) a \sqrt{e x} \sqrt{a+b x^3} (a B+14 A b)}{112 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{3/2} (a B+14 A b)}{7 a e^4}+\frac{9 (e x)^{5/2} \sqrt{a+b x^3} (a B+14 A b)}{56 e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{a e \sqrt{e x}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 279
Rule 329
Rule 308
Rule 225
Rule 1881
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx &=-\frac{2 A \left (a+b x^3\right )^{5/2}}{a e \sqrt{e x}}+\frac{(14 A b+a B) \int (e x)^{3/2} \left (a+b x^3\right )^{3/2} \, dx}{a e^3}\\ &=\frac{(14 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 a e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{a e \sqrt{e x}}+\frac{(9 (14 A b+a B)) \int (e x)^{3/2} \sqrt{a+b x^3} \, dx}{14 e^3}\\ &=\frac{9 (14 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{56 e^4}+\frac{(14 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 a e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{a e \sqrt{e x}}+\frac{(27 a (14 A b+a B)) \int \frac{(e x)^{3/2}}{\sqrt{a+b x^3}} \, dx}{112 e^3}\\ &=\frac{9 (14 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{56 e^4}+\frac{(14 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 a e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{a e \sqrt{e x}}+\frac{(27 a (14 A b+a B)) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{56 e^4}\\ &=\frac{9 (14 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{56 e^4}+\frac{(14 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 a e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{a e \sqrt{e x}}-\frac{(27 a (14 A b+a B)) \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 )}{112 b^{2/3} e^4}-\frac{\left (27 \left (1-\sqrt{3}\right ) a^{5/3} (14 A b+a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{112 b^{2/3} e^2}\\ &=\frac{9 (14 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{56 e^4}+\frac{27 \left (1+\sqrt{3}\right ) a (14 A b+a B) \sqrt{e x} \sqrt{a+b x^3}}{112 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{(14 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 a e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{a e \sqrt{e x}}-\frac{27 \sqrt [4]{3} a^{4/3} (14 A b+a B) \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 )}{112 b^{2/3} e^2 \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^{4/3} (14 A b+a B) \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 )}{224 b^{2/3} e^2 \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 = 0.0736531, size = 84, normalized size = 0.14 \[ \frac{2 x \sqrt{a+b x^3} \left (\frac{x^3 (a B+14 A b) \, _2F_1\left (-\frac{3}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\frac{5 A \left (a+b x^3\right )^2}{a}\right )}{5 (e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.052, size = 6142, normalized size = 10. \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{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 (\frac{{\left (B b x^{6} +{\left (B a + A b\right )} x^{3} + A a\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{e^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 31.626, size = 202, normalized size = 0.33 \begin{align*} \frac{A a^{\frac{3}{2}} \Gamma \left (- \frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{6} \\ \frac{5}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{5}{6}\right )} + \frac{A \sqrt{a} b x^{\frac{5}{2}} \Gamma \left (\frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \Gamma \left (\frac{11}{6}\right )} + \frac{B a^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \Gamma \left (\frac{11}{6}\right )} + \frac{B \sqrt{a} b x^{\frac{11}{2}} \Gamma \left (\frac{11}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{6} \\ \frac{17}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \Gamma \left (\frac{17}{6}\right )} \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{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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