Optimal. Leaf size=650 \[ -\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{81 \left (1+\sqrt{3}\right ) a^2 \sqrt{e x} \sqrt{a+b x^3} (a B+20 A b)}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{81 \sqrt [4]{3} a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{5/2} (a B+20 A b)}{10 a e^4}+\frac{3 (e x)^{5/2} \left (a+b x^3\right )^{3/2} (a B+20 A b)}{28 e^4}+\frac{27 a (e x)^{5/2} \sqrt{a+b x^3} (a B+20 A b)}{224 e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}} \]
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Rubi [A] time = 0.698379, antiderivative size = 650, 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, 308, 225, 1881} \[ \frac{81 \left (1+\sqrt{3}\right ) a^2 \sqrt{e x} \sqrt{a+b x^3} (a B+20 A b)}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{81 \sqrt [4]{3} a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{5/2} (a B+20 A b)}{10 a e^4}+\frac{3 (e x)^{5/2} \left (a+b x^3\right )^{3/2} (a B+20 A b)}{28 e^4}+\frac{27 a (e x)^{5/2} \sqrt{a+b x^3} (a B+20 A b)}{224 e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}} \]
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 )^{5/2} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx &=-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}+\frac{(20 A b+a B) \int (e x)^{3/2} \left (a+b x^3\right )^{5/2} \, dx}{a e^3}\\ &=\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}+\frac{(3 (20 A b+a B)) \int (e x)^{3/2} \left (a+b x^3\right )^{3/2} \, dx}{4 e^3}\\ &=\frac{3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}+\frac{(27 a (20 A b+a B)) \int (e x)^{3/2} \sqrt{a+b x^3} \, dx}{56 e^3}\\ &=\frac{27 a (20 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 e^4}+\frac{3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}+\frac{\left (81 a^2 (20 A b+a B)\right ) \int \frac{(e x)^{3/2}}{\sqrt{a+b x^3}} \, dx}{448 e^3}\\ &=\frac{27 a (20 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 e^4}+\frac{3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}+\frac{\left (81 a^2 (20 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 e^4}\\ &=\frac{27 a (20 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 e^4}+\frac{3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}-\frac{\left (81 a^2 (20 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^{2/3} e^4}-\frac{\left (81 \left (1-\sqrt{3}\right ) a^{8/3} (20 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 )}{448 b^{2/3} e^2}\\ &=\frac{27 a (20 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 e^4}+\frac{81 \left (1+\sqrt{3}\right ) a^2 (20 A b+a B) \sqrt{e x} \sqrt{a+b x^3}}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}-\frac{81 \sqrt [4]{3} a^{7/3} (20 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 )}{448 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\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} (20 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 )}{896 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 = 0.0439877, size = 87, normalized size = 0.13 \[ \frac{2 x \sqrt{a+b x^3} \left (\frac{a^2 x^3 (a B+20 A b) \, _2F_1\left (-\frac{5}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right )}{\sqrt{\frac{b x^3}{a}+1}}-5 A \left (a+b x^3\right )^3\right )}{5 a (e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.056, size = 6530, normalized size = 10.1 \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{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^{2} x^{9} +{\left (2 \, B a b + A b^{2}\right )} x^{6} +{\left (B a^{2} + 2 \, A a b\right )} x^{3} + A a^{2}\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 = 159.126, size = 311, normalized size = 0.48 \begin{align*} \frac{A a^{\frac{5}{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{2 A a^{\frac{3}{2}} 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{A \sqrt{a} b^{2} 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 )} + \frac{B a^{\frac{5}{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{2 B a^{\frac{3}{2}} 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 )} + \frac{B \sqrt{a} b^{2} x^{\frac{17}{2}} \Gamma \left (\frac{17}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{17}{6} \\ \frac{23}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \Gamma \left (\frac{23}{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{5}{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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