Optimal. Leaf size=400 \[ -\frac{4 a^{11/4} e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (77 \sqrt{a} B+65 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{5005 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^2 e \sqrt{e x} \sqrt{a+c x^2} (65 A+77 B x)}{5005 c}-\frac{8 a^3 B e^2 x \sqrt{a+c x^2}}{65 c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{8 a^{13/4} B e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 a e \sqrt{e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{3003 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c} \]
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Rubi [A] time = 0.495499, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {833, 815, 842, 840, 1198, 220, 1196} \[ -\frac{4 a^{11/4} e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (77 \sqrt{a} B+65 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^2 e \sqrt{e x} \sqrt{a+c x^2} (65 A+77 B x)}{5005 c}-\frac{8 a^3 B e^2 x \sqrt{a+c x^2}}{65 c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{8 a^{13/4} B e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 a e \sqrt{e x} \left (a+c x^2\right )^{3/2} (39 A+77 B x)}{3003 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c} \]
Antiderivative was successfully verified.
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Rule 833
Rule 815
Rule 842
Rule 840
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx &=\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac{2 \int \sqrt{e x} \left (-\frac{3}{2} a B e+\frac{13}{2} A c e x\right ) \left (a+c x^2\right )^{3/2} \, dx}{13 c}\\ &=\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac{4 \int \frac{\left (-\frac{13}{4} a A c e^2-\frac{33}{4} a B c e^2 x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt{e x}} \, dx}{143 c^2}\\ &=-\frac{2 a e \sqrt{e x} (39 A+77 B x) \left (a+c x^2\right )^{3/2}}{3003 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac{16 \int \frac{\left (-\frac{117}{8} a^2 A c^2 e^4-\frac{231}{8} a^2 B c^2 e^4 x\right ) \sqrt{a+c x^2}}{\sqrt{e x}} \, dx}{3003 c^3 e^2}\\ &=-\frac{4 a^2 e \sqrt{e x} (65 A+77 B x) \sqrt{a+c x^2}}{5005 c}-\frac{2 a e \sqrt{e x} (39 A+77 B x) \left (a+c x^2\right )^{3/2}}{3003 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac{64 \int \frac{-\frac{585}{16} a^3 A c^3 e^6-\frac{693}{16} a^3 B c^3 e^6 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{45045 c^4 e^4}\\ &=-\frac{4 a^2 e \sqrt{e x} (65 A+77 B x) \sqrt{a+c x^2}}{5005 c}-\frac{2 a e \sqrt{e x} (39 A+77 B x) \left (a+c x^2\right )^{3/2}}{3003 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac{\left (64 \sqrt{x}\right ) \int \frac{-\frac{585}{16} a^3 A c^3 e^6-\frac{693}{16} a^3 B c^3 e^6 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{45045 c^4 e^4 \sqrt{e x}}\\ &=-\frac{4 a^2 e \sqrt{e x} (65 A+77 B x) \sqrt{a+c x^2}}{5005 c}-\frac{2 a e \sqrt{e x} (39 A+77 B x) \left (a+c x^2\right )^{3/2}}{3003 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac{\left (128 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{585}{16} a^3 A c^3 e^6-\frac{693}{16} a^3 B c^3 e^6 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{45045 c^4 e^4 \sqrt{e x}}\\ &=-\frac{4 a^2 e \sqrt{e x} (65 A+77 B x) \sqrt{a+c x^2}}{5005 c}-\frac{2 a e \sqrt{e x} (39 A+77 B x) \left (a+c x^2\right )^{3/2}}{3003 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac{\left (8 a^{7/2} 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 )}{65 c^{3/2} \sqrt{e x}}-\frac{\left (8 a^3 \left (77 \sqrt{a} B+65 A \sqrt{c}\right ) e^2 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{5005 c^{3/2} \sqrt{e x}}\\ &=-\frac{4 a^2 e \sqrt{e x} (65 A+77 B x) \sqrt{a+c x^2}}{5005 c}-\frac{8 a^3 B e^2 x \sqrt{a+c x^2}}{65 c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 a e \sqrt{e x} (39 A+77 B x) \left (a+c x^2\right )^{3/2}}{3003 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac{8 a^{13/4} 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 )}{65 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^{11/4} \left (77 \sqrt{a} B+65 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 )}{5005 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.120267, size = 124, normalized size = 0.31 \[ \frac{2 e \sqrt{e x} \sqrt{a+c x^2} \left (-13 a^2 A \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{a}\right )-11 a^2 B x \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{a}\right )+\left (a+c x^2\right )^2 \sqrt{\frac{c x^2}{a}+1} (13 A+11 B x)\right )}{143 c \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 366, normalized size = 0.9 \begin{align*} -{\frac{2\,e}{15015\,x{c}^{2}}\sqrt{ex} \left ( -1155\,B{c}^{4}{x}^{8}-1365\,A{c}^{4}{x}^{7}-3080\,aB{c}^{3}{x}^{6}+390\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\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{2}\sqrt{-ac}{a}^{3}-3900\,aA{c}^{3}{x}^{5}+924\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\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 ) \sqrt{2}{a}^{4}-462\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\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{2}{a}^{4}-2233\,{a}^{2}B{c}^{2}{x}^{4}-3315\,{a}^{2}A{c}^{2}{x}^{3}-308\,{a}^{3}Bc{x}^{2}-780\,{a}^{3}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{\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 ({\left (B c e x^{4} + A c e x^{3} + B a e x^{2} + A a e x\right )} \sqrt{c x^{2} + a} \sqrt{e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 85.6933, size = 199, normalized size = 0.5 \begin{align*} \frac{A a^{\frac{3}{2}} e^{\frac{3}{2}} 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 \Gamma \left (\frac{9}{4}\right )} + \frac{A \sqrt{a} c e^{\frac{3}{2}} x^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{13}{4}\right )} + \frac{B a^{\frac{3}{2}} e^{\frac{3}{2}} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{11}{4}\right )} + \frac{B \sqrt{a} c e^{\frac{3}{2}} x^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{15}{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{\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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