Optimal. Leaf size=328 \[ -\frac{2 a^{5/4} e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B-21 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{105 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^{5/4} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt{e x} \sqrt{a+c x^2} (5 a B-21 A c x)}{105 c}+\frac{4 a A e x \sqrt{a+c x^2}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{3/2}}{7 c} \]
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Rubi [A] time = 0.334809, antiderivative size = 328, 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 = {833, 815, 842, 840, 1198, 220, 1196} \[ -\frac{2 a^{5/4} e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B-21 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^{5/4} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt{e x} \sqrt{a+c x^2} (5 a B-21 A c x)}{105 c}+\frac{4 a A e x \sqrt{a+c x^2}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{3/2}}{7 c} \]
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 \sqrt{e x} (A+B x) \sqrt{a+c x^2} \, dx &=\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac{2 \int \frac{\left (-\frac{1}{2} a B e+\frac{7}{2} A c e x\right ) \sqrt{a+c x^2}}{\sqrt{e x}} \, dx}{7 c}\\ &=-\frac{2 \sqrt{e x} (5 a B-21 A c x) \sqrt{a+c x^2}}{105 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac{8 \int \frac{-\frac{5}{4} a^2 B c e^3+\frac{21}{4} a A c^2 e^3 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{105 c^2 e^2}\\ &=-\frac{2 \sqrt{e x} (5 a B-21 A c x) \sqrt{a+c x^2}}{105 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac{\left (8 \sqrt{x}\right ) \int \frac{-\frac{5}{4} a^2 B c e^3+\frac{21}{4} a A c^2 e^3 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{105 c^2 e^2 \sqrt{e x}}\\ &=-\frac{2 \sqrt{e x} (5 a B-21 A c x) \sqrt{a+c x^2}}{105 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac{\left (16 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{5}{4} a^2 B c e^3+\frac{21}{4} a A c^2 e^3 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{105 c^2 e^2 \sqrt{e x}}\\ &=-\frac{2 \sqrt{e x} (5 a B-21 A c x) \sqrt{a+c x^2}}{105 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac{\left (4 a^{3/2} \left (5 \sqrt{a} B-21 A \sqrt{c}\right ) e \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{105 c \sqrt{e x}}-\frac{\left (4 a^{3/2} A e \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 \sqrt{c} \sqrt{e x}}\\ &=\frac{4 a A e x \sqrt{a+c x^2}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 \sqrt{e x} (5 a B-21 A c x) \sqrt{a+c x^2}}{105 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac{4 a^{5/4} A e \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 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 a^{5/4} \left (5 \sqrt{a} B-21 A \sqrt{c}\right ) e \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 )}{105 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0647884, size = 111, normalized size = 0.34 \[ \frac{2 \sqrt{e x} \sqrt{a+c x^2} \left (7 A c x \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{a}\right )-3 a B \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{a}\right )+3 B \sqrt{\frac{c x^2}{a}+1} \left (a+c x^2\right )\right )}{21 c \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 333, normalized size = 1. \begin{align*} -{\frac{2}{105\,x{c}^{2}}\sqrt{ex} \left ( 21\,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-42\,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+5\,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}-15\,B{c}^{3}{x}^{5}-21\,A{c}^{3}{x}^{4}-25\,aB{c}^{2}{x}^{3}-21\,aA{c}^{2}{x}^{2}-10\,{a}^{2}Bcx \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 \sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{e x}\,{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 (\sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.54407, size = 95, normalized size = 0.29 \begin{align*} \frac{A \sqrt{a} \left (e x\right )^{\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 \Gamma \left (\frac{7}{4}\right )} + \frac{B \sqrt{a} \left (e x\right )^{\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^{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 \sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{e x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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