Optimal. Leaf size=421 \[ \frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}-\frac{4 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c^2 e^5}+\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a \left (c+d x^2\right )^{3/2} (a d+10 b c)}{5 c^2 e^3 \sqrt{e x}} \]
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Rubi [A] time = 0.395567, antiderivative size = 421, 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 = {462, 453, 279, 329, 305, 220, 1196} \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{4 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c^2 e^5}+\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a \left (c+d x^2\right )^{3/2} (a d+10 b c)}{5 c^2 e^3 \sqrt{e x}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 453
Rule 279
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{(e x)^{7/2}} \, dx &=-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}+\frac{2 \int \frac{\left (\frac{1}{2} a (10 b c+a d)+\frac{5}{2} b^2 c x^2\right ) \sqrt{c+d x^2}}{(e x)^{3/2}} \, dx}{5 c e^2}\\ &=-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}-\frac{2 a (10 b c+a d) \left (c+d x^2\right )^{3/2}}{5 c^2 e^3 \sqrt{e x}}+\frac{\left (b^2 c^2+a d (10 b c+a d)\right ) \int \sqrt{e x} \sqrt{c+d x^2} \, dx}{c^2 e^4}\\ &=\frac{2 \left (b^2 c^2+a d (10 b c+a d)\right ) (e x)^{3/2} \sqrt{c+d x^2}}{5 c^2 e^5}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}-\frac{2 a (10 b c+a d) \left (c+d x^2\right )^{3/2}}{5 c^2 e^3 \sqrt{e x}}+\frac{\left (2 \left (b^2 c^2+a d (10 b c+a d)\right )\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{5 c e^4}\\ &=\frac{2 \left (b^2 c^2+a d (10 b c+a d)\right ) (e x)^{3/2} \sqrt{c+d x^2}}{5 c^2 e^5}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}-\frac{2 a (10 b c+a d) \left (c+d x^2\right )^{3/2}}{5 c^2 e^3 \sqrt{e x}}+\frac{\left (4 \left (b^2 c^2+a d (10 b c+a d)\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 c e^5}\\ &=\frac{2 \left (b^2 c^2+a d (10 b c+a d)\right ) (e x)^{3/2} \sqrt{c+d x^2}}{5 c^2 e^5}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}-\frac{2 a (10 b c+a d) \left (c+d x^2\right )^{3/2}}{5 c^2 e^3 \sqrt{e x}}+\frac{\left (4 \left (b^2 c^2+a d (10 b c+a d)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 \sqrt{c} \sqrt{d} e^4}-\frac{\left (4 \left (b^2 c^2+a d (10 b c+a d)\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 )}{5 \sqrt{c} \sqrt{d} e^4}\\ &=\frac{2 \left (b^2 c^2+a d (10 b c+a d)\right ) (e x)^{3/2} \sqrt{c+d x^2}}{5 c^2 e^5}+\frac{4 \left (b^2 c^2+a d (10 b c+a d)\right ) \sqrt{e x} \sqrt{c+d x^2}}{5 c \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}-\frac{2 a (10 b c+a d) \left (c+d x^2\right )^{3/2}}{5 c^2 e^3 \sqrt{e x}}-\frac{4 \left (b^2 c^2+a d (10 b c+a d)\right ) \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 )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 \left (b^2 c^2+a d (10 b c+a d)\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 )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.126577, size = 125, normalized size = 0.3 \[ \frac{x \left (4 x^4 \sqrt{\frac{c}{d x^2}+1} \left (a^2 d^2+10 a b c d+b^2 c^2\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right )-2 \left (c+d x^2\right ) \left (a^2 \left (c+2 d x^2\right )+10 a b c x^2-b^2 c x^4\right )\right )}{5 c (e x)^{7/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 648, normalized size = 1.5 \begin{align*}{\frac{2}{5\,d{x}^{2}{e}^{3}c} \left ( 2\,\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 ){x}^{2}{a}^{2}c{d}^{2}+20\,\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 ){x}^{2}ab{c}^{2}d+2\,\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 ){x}^{2}{b}^{2}{c}^{3}-\sqrt{{ \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{ \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{ \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}{a}^{2}c{d}^{2}-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 ){x}^{2}ab{c}^{2}d-\sqrt{{ \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{ \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{ \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}{b}^{2}{c}^{3}+{x}^{6}{b}^{2}c{d}^{2}-2\,{x}^{4}{a}^{2}{d}^{3}-10\,{x}^{4}abc{d}^{2}+{x}^{4}{b}^{2}{c}^{2}d-3\,{x}^{2}{a}^{2}c{d}^{2}-10\,{x}^{2}ab{c}^{2}d-{a}^{2}{c}^{2}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} \sqrt{d x^{2} + c}}{\left (e x\right )^{\frac{7}{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^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c} \sqrt{e x}}{e^{4} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 147.089, size = 160, normalized size = 0.38 \begin{align*} \frac{a^{2} \sqrt{c} \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac{7}{2}} x^{\frac{5}{2}} \Gamma \left (- \frac{1}{4}\right )} + \frac{a b \sqrt{c} \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{d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac{7}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} + \frac{b^{2} \sqrt{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{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac{7}{2}} \Gamma \left (\frac{7}{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} \sqrt{d x^{2} + c}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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