Optimal. Leaf size=387 \[ \frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right )}{5 c^2 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{5 c e (e x)^{5/2}}-\frac{2 a \sqrt{c+d x^2} (10 b c-3 a d)}{5 c^2 e^3 \sqrt{e x}} \]
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Rubi [A] time = 0.345841, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {462, 453, 329, 305, 220, 1196} \[ \frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right )}{5 c^2 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{5 c e (e x)^{5/2}}-\frac{2 a \sqrt{c+d x^2} (10 b c-3 a d)}{5 c^2 e^3 \sqrt{e x}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 453
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{(e x)^{7/2} \sqrt{c+d x^2}} \, dx &=-\frac{2 a^2 \sqrt{c+d x^2}}{5 c e (e x)^{5/2}}+\frac{2 \int \frac{\frac{1}{2} a (10 b c-3 a d)+\frac{5}{2} b^2 c x^2}{(e x)^{3/2} \sqrt{c+d x^2}} \, dx}{5 c e^2}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{5 c e (e x)^{5/2}}-\frac{2 a (10 b c-3 a d) \sqrt{c+d x^2}}{5 c^2 e^3 \sqrt{e x}}+\frac{\left (5 b^2 c^2+10 a b c d-3 a^2 d^2\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{5 c^2 e^4}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{5 c e (e x)^{5/2}}-\frac{2 a (10 b c-3 a d) \sqrt{c+d x^2}}{5 c^2 e^3 \sqrt{e x}}+\frac{\left (2 \left (5 b^2 c^2+10 a b c d-3 a^2 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 )}{5 c^2 e^5}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{5 c e (e x)^{5/2}}-\frac{2 a (10 b c-3 a d) \sqrt{c+d x^2}}{5 c^2 e^3 \sqrt{e x}}+\frac{\left (2 \left (5 b^2 c^2+10 a b c d-3 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 c^{3/2} \sqrt{d} e^4}-\frac{\left (2 \left (5 b^2 c^2+10 a b c d-3 a^2 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 )}{5 c^{3/2} \sqrt{d} e^4}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{5 c e (e x)^{5/2}}-\frac{2 a (10 b c-3 a d) \sqrt{c+d x^2}}{5 c^2 e^3 \sqrt{e x}}+\frac{2 \left (5 b^2 c^2+10 a b c d-3 a^2 d^2\right ) \sqrt{e x} \sqrt{c+d x^2}}{5 c^2 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 \left (5 b^2 c^2+10 a b c d-3 a^2 d^2\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^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{\left (5 b^2 c^2+10 a b c d-3 a^2 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 )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.124356, size = 116, normalized size = 0.3 \[ \frac{x \left (2 x^4 \sqrt{\frac{c}{d x^2}+1} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right )-2 a \left (c+d x^2\right ) \left (a \left (c-3 d x^2\right )+10 b c x^2\right )\right )}{5 c^2 (e x)^{7/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 626, normalized size = 1.6 \begin{align*} -{\frac{1}{5\,d{x}^{2}{e}^{3}{c}^{2}} \left ( 6\,\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-10\,\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}-3\,\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}{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+5\,\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}{b}^{2}{c}^{3}-6\,{x}^{4}{a}^{2}{d}^{3}+20\,{x}^{4}abc{d}^{2}-4\,{x}^{2}{a}^{2}c{d}^{2}+20\,{x}^{2}ab{c}^{2}d+2\,{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}}{d e^{4} x^{6} + c e^{4} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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