Optimal. Leaf size=425 \[ \frac{2 c^{5/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{195 d^{11/4} \sqrt{c+d x^2}}-\frac{4 c^{5/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{11/4} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (39 a^2 d^2+b c (7 b c-26 a d)\right )}{195 d^2 e}+\frac{4 c \sqrt{e x} \sqrt{c+d x^2} \left (39 a^2 d^2+b c (7 b c-26 a d)\right )}{195 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{117 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3} \]
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Rubi [A] time = 0.414861, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {464, 459, 279, 329, 305, 220, 1196} \[ \frac{2 c^{5/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{11/4} \sqrt{c+d x^2}}-\frac{4 c^{5/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{11/4} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (39 a^2 d^2+b c (7 b c-26 a d)\right )}{195 d^2 e}+\frac{4 c \sqrt{e x} \sqrt{c+d x^2} \left (39 a^2 d^2+b c (7 b c-26 a d)\right )}{195 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{117 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3} \]
Antiderivative was successfully verified.
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Rule 464
Rule 459
Rule 279
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \sqrt{e x} \left (a+b x^2\right )^2 \sqrt{c+d x^2} \, dx &=\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}+\frac{2 \int \sqrt{e x} \sqrt{c+d x^2} \left (\frac{13 a^2 d}{2}-\frac{1}{2} b (7 b c-26 a d) x^2\right ) \, dx}{13 d}\\ &=-\frac{2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}+\frac{1}{39} \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right ) \int \sqrt{e x} \sqrt{c+d x^2} \, dx\\ &=\frac{2 \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right ) (e x)^{3/2} \sqrt{c+d x^2}}{195 e}-\frac{2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}+\frac{1}{195} \left (2 c \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right )\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx\\ &=\frac{2 \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right ) (e x)^{3/2} \sqrt{c+d x^2}}{195 e}-\frac{2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}+\frac{\left (4 c \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{195 e}\\ &=\frac{2 \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right ) (e x)^{3/2} \sqrt{c+d x^2}}{195 e}-\frac{2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}+\frac{\left (4 c^{3/2} \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{195 \sqrt{d}}-\frac{\left (4 c^{3/2} \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{d} x^2}{\sqrt{c} e}}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{195 \sqrt{d}}\\ &=\frac{2 \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right ) (e x)^{3/2} \sqrt{c+d x^2}}{195 e}+\frac{4 c \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right ) \sqrt{e x} \sqrt{c+d x^2}}{195 \sqrt{d} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}-\frac{4 c^{5/4} \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right ) \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{3/4} \sqrt{c+d x^2}}+\frac{2 c^{5/4} \left (39 a^2+\frac{b c (7 b c-26 a d)}{d^2}\right ) \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{3/4} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.132363, size = 145, normalized size = 0.34 \[ \frac{2 \sqrt{e x} \left (6 c x \sqrt{\frac{c}{d x^2}+1} \left (39 a^2 d^2-26 a b c d+7 b^2 c^2\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right )-x \left (c+d x^2\right ) \left (-117 a^2 d^2-26 a b d \left (2 c+5 d x^2\right )+b^2 \left (14 c^2-10 c d x^2-45 d^2 x^4\right )\right )\right )}{585 d^2 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 658, normalized size = 1.6 \begin{align*}{\frac{2}{585\,{d}^{3}x}\sqrt{ex} \left ( 45\,{x}^{8}{b}^{2}{d}^{4}+130\,{x}^{6}ab{d}^{4}+55\,{x}^{6}{b}^{2}c{d}^{3}+234\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{c}^{2}{d}^{2}-156\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{3}d+42\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{4}-117\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{c}^{2}{d}^{2}+78\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{3}d-21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{4}+117\,{x}^{4}{a}^{2}{d}^{4}+182\,{x}^{4}abc{d}^{3}-4\,{x}^{4}{b}^{2}{c}^{2}{d}^{2}+117\,{x}^{2}{a}^{2}c{d}^{3}+52\,{x}^{2}ab{c}^{2}{d}^{2}-14\,{x}^{2}{b}^{2}{c}^{3}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \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 (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c} \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 ({\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c} \sqrt{e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.68144, size = 148, normalized size = 0.35 \begin{align*} \frac{a^{2} \sqrt{c} \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{d x^{2} e^{i \pi }}{c}} \right )}}{2 e \Gamma \left (\frac{7}{4}\right )} + \frac{a b \sqrt{c} \left (e x\right )^{\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{d x^{2} e^{i \pi }}{c}} \right )}}{e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{b^{2} \sqrt{c} \left (e x\right )^{\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{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{5} \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 (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c} \sqrt{e x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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