Optimal. Leaf size=379 \[ \frac{8 a^{9/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (15 \sqrt{a} B+77 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{231 \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}+\frac{16 a^2 A \sqrt{c} x \sqrt{a+c x^2}}{3 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{16 a^{9/4} A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{20 \sqrt{e x} \left (a+c x^2\right )^{3/2} (9 a B+77 A c x)}{693 e^2}+\frac{8 a \sqrt{e x} \sqrt{a+c x^2} (15 a B+77 A c x)}{231 e^2}-\frac{2 \left (a+c x^2\right )^{5/2} (11 A-B x)}{11 e \sqrt{e x}} \]
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Rubi [A] time = 0.413712, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{8 a^{9/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (15 \sqrt{a} B+77 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{231 \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}+\frac{16 a^2 A \sqrt{c} x \sqrt{a+c x^2}}{3 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{16 a^{9/4} A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{20 \sqrt{e x} \left (a+c x^2\right )^{3/2} (9 a B+77 A c x)}{693 e^2}+\frac{8 a \sqrt{e x} \sqrt{a+c x^2} (15 a B+77 A c x)}{231 e^2}-\frac{2 \left (a+c x^2\right )^{5/2} (11 A-B x)}{11 e \sqrt{e x}} \]
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 )^{5/2}}{(e x)^{3/2}} \, dx &=-\frac{2 (11 A-B x) \left (a+c x^2\right )^{5/2}}{11 e \sqrt{e x}}-\frac{10 \int \frac{(-a B e-11 A c e x) \left (a+c x^2\right )^{3/2}}{\sqrt{e x}} \, dx}{11 e^2}\\ &=\frac{20 \sqrt{e x} (9 a B+77 A c x) \left (a+c x^2\right )^{3/2}}{693 e^2}-\frac{2 (11 A-B x) \left (a+c x^2\right )^{5/2}}{11 e \sqrt{e x}}-\frac{40 \int \frac{\left (-\frac{9}{2} a^2 B c e^3-\frac{77}{2} a A c^2 e^3 x\right ) \sqrt{a+c x^2}}{\sqrt{e x}} \, dx}{231 c e^4}\\ &=\frac{8 a \sqrt{e x} (15 a B+77 A c x) \sqrt{a+c x^2}}{231 e^2}+\frac{20 \sqrt{e x} (9 a B+77 A c x) \left (a+c x^2\right )^{3/2}}{693 e^2}-\frac{2 (11 A-B x) \left (a+c x^2\right )^{5/2}}{11 e \sqrt{e x}}-\frac{32 \int \frac{-\frac{45}{4} a^3 B c^2 e^5-\frac{231}{4} a^2 A c^3 e^5 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{693 c^2 e^6}\\ &=\frac{8 a \sqrt{e x} (15 a B+77 A c x) \sqrt{a+c x^2}}{231 e^2}+\frac{20 \sqrt{e x} (9 a B+77 A c x) \left (a+c x^2\right )^{3/2}}{693 e^2}-\frac{2 (11 A-B x) \left (a+c x^2\right )^{5/2}}{11 e \sqrt{e x}}-\frac{\left (32 \sqrt{x}\right ) \int \frac{-\frac{45}{4} a^3 B c^2 e^5-\frac{231}{4} a^2 A c^3 e^5 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{693 c^2 e^6 \sqrt{e x}}\\ &=\frac{8 a \sqrt{e x} (15 a B+77 A c x) \sqrt{a+c x^2}}{231 e^2}+\frac{20 \sqrt{e x} (9 a B+77 A c x) \left (a+c x^2\right )^{3/2}}{693 e^2}-\frac{2 (11 A-B x) \left (a+c x^2\right )^{5/2}}{11 e \sqrt{e x}}-\frac{\left (64 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{45}{4} a^3 B c^2 e^5-\frac{231}{4} a^2 A c^3 e^5 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{693 c^2 e^6 \sqrt{e x}}\\ &=\frac{8 a \sqrt{e x} (15 a B+77 A c x) \sqrt{a+c x^2}}{231 e^2}+\frac{20 \sqrt{e x} (9 a B+77 A c x) \left (a+c x^2\right )^{3/2}}{693 e^2}-\frac{2 (11 A-B x) \left (a+c x^2\right )^{5/2}}{11 e \sqrt{e x}}+\frac{\left (16 a^{5/2} \left (15 \sqrt{a} B+77 A \sqrt{c}\right ) \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{231 e \sqrt{e x}}-\frac{\left (16 a^{5/2} 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 )}{3 e \sqrt{e x}}\\ &=\frac{16 a^2 A \sqrt{c} x \sqrt{a+c x^2}}{3 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{8 a \sqrt{e x} (15 a B+77 A c x) \sqrt{a+c x^2}}{231 e^2}+\frac{20 \sqrt{e x} (9 a B+77 A c x) \left (a+c x^2\right )^{3/2}}{693 e^2}-\frac{2 (11 A-B x) \left (a+c x^2\right )^{5/2}}{11 e \sqrt{e x}}-\frac{16 a^{9/4} 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 )}{3 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{8 a^{9/4} \left (15 \sqrt{a} B+77 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 )}{231 \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0304898, size = 83, normalized size = 0.22 \[ \frac{2 a^2 x \sqrt{a+c x^2} \left (B x \, _2F_1\left (-\frac{5}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{a}\right )-A \, _2F_1\left (-\frac{5}{2},-\frac{1}{4};\frac{3}{4};-\frac{c x^2}{a}\right )\right )}{(e x)^{3/2} \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 364, normalized size = 1. \begin{align*} -{\frac{2}{693\,ce} \left ( -63\,B{c}^{4}{x}^{7}-77\,A{c}^{4}{x}^{6}+924\,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}^{3}c-1848\,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}^{3}c-180\,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}^{3}-279\,aB{c}^{3}{x}^{5}-385\,aA{c}^{3}{x}^{4}-549\,{a}^{2}B{c}^{2}{x}^{3}+385\,{a}^{2}A{c}^{2}{x}^{2}-333\,{a}^{3}Bcx+693\,A{a}^{3}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{5}{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^{2} x^{5} + A c^{2} x^{4} + 2 \, B a c x^{3} + 2 \, A a c x^{2} + B a^{2} x + A a^{2}\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 = 40.5484, size = 304, normalized size = 0.8 \begin{align*} \frac{A a^{\frac{5}{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 a^{\frac{3}{2}} 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 )}}{e^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} + \frac{A \sqrt{a} c^{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 e^{\frac{3}{2}} \Gamma \left (\frac{11}{4}\right )} + \frac{B a^{\frac{5}{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 a^{\frac{3}{2}} 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 )}}{e^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} + \frac{B \sqrt{a} c^{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 e^{\frac{3}{2}} \Gamma \left (\frac{13}{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{5}{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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