Optimal. Leaf size=341 \[ \frac{4 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{35 \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{24 a^{5/4} A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{4 \sqrt{e x} \sqrt{a+c x^2} (5 a B+21 A c x)}{35 e^2}-\frac{2 \left (a+c x^2\right )^{3/2} (7 A-B x)}{7 e \sqrt{e x}}+\frac{24 a A \sqrt{c} x \sqrt{a+c x^2}}{5 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]
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Rubi [A] time = 0.34522, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {813, 815, 842, 840, 1198, 220, 1196} \[ \frac{4 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{35 \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{24 a^{5/4} A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{4 \sqrt{e x} \sqrt{a+c x^2} (5 a B+21 A c x)}{35 e^2}-\frac{2 \left (a+c x^2\right )^{3/2} (7 A-B x)}{7 e \sqrt{e x}}+\frac{24 a A \sqrt{c} x \sqrt{a+c x^2}}{5 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]
Antiderivative was successfully verified.
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Rule 813
Rule 815
Rule 842
Rule 840
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{(e x)^{3/2}} \, dx &=-\frac{2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt{e x}}-\frac{6 \int \frac{(-a B e-7 A c e x) \sqrt{a+c x^2}}{\sqrt{e x}} \, dx}{7 e^2}\\ &=\frac{4 \sqrt{e x} (5 a B+21 A c x) \sqrt{a+c x^2}}{35 e^2}-\frac{2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt{e x}}-\frac{8 \int \frac{-\frac{5}{2} a^2 B c e^3-\frac{21}{2} a A c^2 e^3 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{35 c e^4}\\ &=\frac{4 \sqrt{e x} (5 a B+21 A c x) \sqrt{a+c x^2}}{35 e^2}-\frac{2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt{e x}}-\frac{\left (8 \sqrt{x}\right ) \int \frac{-\frac{5}{2} a^2 B c e^3-\frac{21}{2} a A c^2 e^3 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{35 c e^4 \sqrt{e x}}\\ &=\frac{4 \sqrt{e x} (5 a B+21 A c x) \sqrt{a+c x^2}}{35 e^2}-\frac{2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt{e x}}-\frac{\left (16 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{5}{2} a^2 B c e^3-\frac{21}{2} a A c^2 e^3 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{35 c e^4 \sqrt{e x}}\\ &=\frac{4 \sqrt{e x} (5 a B+21 A c x) \sqrt{a+c x^2}}{35 e^2}-\frac{2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt{e x}}+\frac{\left (8 a^{3/2} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{35 e \sqrt{e x}}-\frac{\left (24 a^{3/2} A \sqrt{c} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{5 e \sqrt{e x}}\\ &=\frac{24 a A \sqrt{c} x \sqrt{a+c x^2}}{5 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{4 \sqrt{e x} (5 a B+21 A c x) \sqrt{a+c x^2}}{35 e^2}-\frac{2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt{e x}}-\frac{24 a^{5/4} A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{4 a^{5/4} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{35 \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0306419, size = 81, normalized size = 0.24 \[ \frac{2 a x \sqrt{a+c x^2} \left (B x \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{a}\right )-A \, _2F_1\left (-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{c x^2}{a}\right )\right )}{(e x)^{3/2} \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 340, normalized size = 1. \begin{align*} -{\frac{2}{35\,ce} \left ( 42\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c-84\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c-10\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{a}^{2}-5\,B{c}^{3}{x}^{5}-7\,A{c}^{3}{x}^{4}-20\,aB{c}^{2}{x}^{3}+28\,aA{c}^{2}{x}^{2}-15\,{a}^{2}Bcx+35\,A{a}^{2}c \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \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 (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{\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 c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt{c x^{2} + 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 = 12.8568, size = 202, normalized size = 0.59 \begin{align*} \frac{A a^{\frac{3}{2}} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} + \frac{A \sqrt{a} c x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} + \frac{B a^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac{3}{2}} \Gamma \left (\frac{5}{4}\right )} + \frac{B \sqrt{a} c x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\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 (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{\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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