Optimal. Leaf size=296 \[ \frac{e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (3 \sqrt{a} B+A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{2 \sqrt [4]{a} c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e \sqrt{e x} (A+B x)}{c \sqrt{a+c x^2}}+\frac{3 B e^2 x \sqrt{a+c x^2}}{c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{3 \sqrt [4]{a} B e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{7/4} \sqrt{e x} \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.266385, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {819, 842, 840, 1198, 220, 1196} \[ \frac{e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (3 \sqrt{a} B+A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e \sqrt{e x} (A+B x)}{c \sqrt{a+c x^2}}+\frac{3 B e^2 x \sqrt{a+c x^2}}{c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{3 \sqrt [4]{a} B e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{7/4} \sqrt{e x} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 819
Rule 842
Rule 840
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{e \sqrt{e x} (A+B x)}{c \sqrt{a+c x^2}}+\frac{\int \frac{\frac{1}{2} a A e^2+\frac{3}{2} a B e^2 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{a c}\\ &=-\frac{e \sqrt{e x} (A+B x)}{c \sqrt{a+c x^2}}+\frac{\sqrt{x} \int \frac{\frac{1}{2} a A e^2+\frac{3}{2} a B e^2 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{a c \sqrt{e x}}\\ &=-\frac{e \sqrt{e x} (A+B x)}{c \sqrt{a+c x^2}}+\frac{\left (2 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} a A e^2+\frac{3}{2} a B e^2 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{a c \sqrt{e x}}\\ &=-\frac{e \sqrt{e x} (A+B x)}{c \sqrt{a+c x^2}}-\frac{\left (3 \sqrt{a} B e^2 \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 )}{c^{3/2} \sqrt{e x}}+\frac{\left (2 \left (\frac{3}{2} a B e^2+\frac{1}{2} \sqrt{a} A \sqrt{c} e^2\right ) \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{\sqrt{a} c^{3/2} \sqrt{e x}}\\ &=-\frac{e \sqrt{e x} (A+B x)}{c \sqrt{a+c x^2}}+\frac{3 B e^2 x \sqrt{a+c x^2}}{c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{3 \sqrt [4]{a} B e^2 \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 )}{c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{\left (3 \sqrt{a} B+A \sqrt{c}\right ) e^2 \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 )}{2 \sqrt [4]{a} c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.051173, size = 102, normalized size = 0.34 \[ \frac{e \sqrt{e x} \left (A \sqrt{\frac{c x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{a}\right )+B x \sqrt{\frac{c x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{a}\right )-A-B x\right )}{c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 290, normalized size = 1. \begin{align*}{\frac{e}{2\,x{c}^{2}}\sqrt{ex} \left ( A\sqrt{{ \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{ \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{ \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ac}-3\,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 ) a+6\,B\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\,Bc{x}^{2}-2\,Acx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \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 + A\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (c x^{2} + a\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 e x^{2} + A e x\right )} \sqrt{c x^{2} + a} \sqrt{e x}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 52.8504, size = 94, normalized size = 0.32 \begin{align*} \frac{A e^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} + \frac{B e^{\frac{3}{2}} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{11}{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 (B x + A\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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