Optimal. Leaf size=286 \[ \frac{4 c^{7/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (33 a^2 d^2+b c (b c-6 a d)\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{231 d^{9/4} \sqrt{e} \sqrt{c+d x^2}}+\frac{2 \sqrt{e x} \left (c+d x^2\right )^{3/2} \left (33 a^2 d^2+b c (b c-6 a d)\right )}{231 d^2 e}+\frac{4 c \sqrt{e x} \sqrt{c+d x^2} \left (33 a^2 d^2+b c (b c-6 a d)\right )}{231 d^2 e}-\frac{2 b \sqrt{e x} \left (c+d x^2\right )^{5/2} (b c-6 a d)}{33 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3} \]
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Rubi [A] time = 0.266818, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {464, 459, 279, 329, 220} \[ \frac{4 c^{7/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (33 a^2 d^2+b c (b c-6 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{9/4} \sqrt{e} \sqrt{c+d x^2}}+\frac{2 \sqrt{e x} \left (c+d x^2\right )^{3/2} \left (33 a^2 d^2+b c (b c-6 a d)\right )}{231 d^2 e}+\frac{4 c \sqrt{e x} \sqrt{c+d x^2} \left (33 a^2 d^2+b c (b c-6 a d)\right )}{231 d^2 e}-\frac{2 b \sqrt{e x} \left (c+d x^2\right )^{5/2} (b c-6 a d)}{33 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3} \]
Antiderivative was successfully verified.
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Rule 464
Rule 459
Rule 279
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{\sqrt{e x}} \, dx &=\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac{2 \int \frac{\left (c+d x^2\right )^{3/2} \left (\frac{15 a^2 d}{2}-\frac{5}{2} b (b c-6 a d) x^2\right )}{\sqrt{e x}} \, dx}{15 d}\\ &=-\frac{2 b (b c-6 a d) \sqrt{e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac{1}{33} \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right ) \int \frac{\left (c+d x^2\right )^{3/2}}{\sqrt{e x}} \, dx\\ &=\frac{2 \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right ) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{231 e}-\frac{2 b (b c-6 a d) \sqrt{e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac{1}{77} \left (2 c \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right )\right ) \int \frac{\sqrt{c+d x^2}}{\sqrt{e x}} \, dx\\ &=\frac{4 c \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right ) \sqrt{e x} \sqrt{c+d x^2}}{231 e}+\frac{2 \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right ) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{231 e}-\frac{2 b (b c-6 a d) \sqrt{e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac{1}{231} \left (4 c^2 \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right )\right ) \int \frac{1}{\sqrt{e x} \sqrt{c+d x^2}} \, dx\\ &=\frac{4 c \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right ) \sqrt{e x} \sqrt{c+d x^2}}{231 e}+\frac{2 \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right ) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{231 e}-\frac{2 b (b c-6 a d) \sqrt{e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac{\left (8 c^2 \left (33 a^2+\frac{b c (b c-6 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 )}{231 e}\\ &=\frac{4 c \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right ) \sqrt{e x} \sqrt{c+d x^2}}{231 e}+\frac{2 \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right ) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{231 e}-\frac{2 b (b c-6 a d) \sqrt{e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac{4 c^{7/4} \left (33 a^2+\frac{b c (b c-6 a d)}{d^2}\right ) \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 )}{231 \sqrt [4]{d} \sqrt{e} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.229998, size = 223, normalized size = 0.78 \[ \frac{\sqrt{x} \left (\frac{2 \sqrt{x} \left (c+d x^2\right ) \left (165 a^2 d^2 \left (3 c+d x^2\right )+30 a b d \left (4 c^2+13 c d x^2+7 d^2 x^4\right )+b^2 \left (12 c^2 d x^2-20 c^3+119 c d^2 x^4+77 d^3 x^6\right )\right )}{5 d^2}+\frac{8 i c^2 x \sqrt{\frac{c}{d x^2}+1} \left (33 a^2 d^2-6 a b c d+b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )}{d^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{231 \sqrt{e x} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 444, normalized size = 1.6 \begin{align*}{\frac{2}{1155\,{d}^{3}} \left ( 77\,{x}^{9}{b}^{2}{d}^{5}+210\,{x}^{7}ab{d}^{5}+196\,{x}^{7}{b}^{2}c{d}^{4}+330\,\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 ) \sqrt{-cd}{a}^{2}{c}^{2}{d}^{2}-60\,\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 ) \sqrt{-cd}ab{c}^{3}d+10\,\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 ) \sqrt{-cd}{b}^{2}{c}^{4}+165\,{x}^{5}{a}^{2}{d}^{5}+600\,{x}^{5}abc{d}^{4}+131\,{x}^{5}{b}^{2}{c}^{2}{d}^{3}+660\,{x}^{3}{a}^{2}c{d}^{4}+510\,{x}^{3}ab{c}^{2}{d}^{3}-8\,{x}^{3}{b}^{2}{c}^{3}{d}^{2}+495\,x{a}^{2}{c}^{2}{d}^{3}+120\,xab{c}^{3}{d}^{2}-20\,x{b}^{2}{c}^{4}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\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 (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\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 (\frac{{\left (b^{2} d x^{6} +{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c +{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{e x}}{e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 30.0793, size = 306, normalized size = 1.07 \begin{align*} \frac{a^{2} c^{\frac{3}{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{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{a^{2} \sqrt{c} d 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{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{9}{4}\right )} + \frac{a b c^{\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{d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt{e} \Gamma \left (\frac{9}{4}\right )} + \frac{a b \sqrt{c} d 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{d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt{e} \Gamma \left (\frac{13}{4}\right )} + \frac{b^{2} c^{\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{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{13}{4}\right )} + \frac{b^{2} \sqrt{c} d x^{\frac{13}{2}} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{17}{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^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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