Optimal. Leaf size=442 \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{(e x)^{3/2} \left (7 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}-\frac{2 a^2}{c e \sqrt{e x} \left (c+d x^2\right )^{3/2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d e^3 \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.420859, antiderivative size = 442, 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, 457, 290, 329, 305, 220, 1196} \[ -\frac{(e x)^{3/2} \left (7 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}-\frac{2 a^2}{c e \sqrt{e x} \left (c+d x^2\right )^{3/2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d e^3 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 457
Rule 290
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{5/2}} \, dx &=-\frac{2 a^2}{c e \sqrt{e x} \left (c+d x^2\right )^{3/2}}+\frac{2 \int \frac{\sqrt{e x} \left (\frac{1}{2} a (2 b c-7 a d)+\frac{1}{2} b^2 c x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx}{c e^2}\\ &=-\frac{2 a^2}{c e \sqrt{e x} \left (c+d x^2\right )^{3/2}}-\frac{\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac{\left (b^2 c^2+a d (2 b c-7 a d)\right ) \int \frac{\sqrt{e x}}{\left (c+d x^2\right )^{3/2}} \, dx}{2 c^2 d e^2}\\ &=-\frac{2 a^2}{c e \sqrt{e x} \left (c+d x^2\right )^{3/2}}-\frac{\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac{\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt{c+d x^2}}-\frac{\left (b^2 c^2+a d (2 b c-7 a d)\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{4 c^3 d e^2}\\ &=-\frac{2 a^2}{c e \sqrt{e x} \left (c+d x^2\right )^{3/2}}-\frac{\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac{\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt{c+d x^2}}-\frac{\left (b^2 c^2+a d (2 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 c^3 d e^3}\\ &=-\frac{2 a^2}{c e \sqrt{e x} \left (c+d x^2\right )^{3/2}}-\frac{\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac{\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt{c+d x^2}}-\frac{\left (b^2 c^2+a d (2 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 c^{5/2} d^{3/2} e^2}+\frac{\left (b^2 c^2+a d (2 b c-7 a d)\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 )}{2 c^{5/2} d^{3/2} e^2}\\ &=-\frac{2 a^2}{c e \sqrt{e x} \left (c+d x^2\right )^{3/2}}-\frac{\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac{\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt{c+d x^2}}-\frac{\left (b^2 c^2+a d (2 b c-7 a d)\right ) \sqrt{e x} \sqrt{c+d x^2}}{2 c^3 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\left (b^2 c^2+a d (2 b c-7 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 )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{\left (b^2 c^2+a d (2 b c-7 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 )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.146539, size = 161, normalized size = 0.36 \[ \frac{x \left (-x^2 \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \left (-7 a^2 d^2+2 a b c d+b^2 c^2\right ) \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right )+a^2 (-d) \left (12 c^2+35 c d x^2+21 d^2 x^4\right )+2 a b c d x^2 \left (5 c+3 d x^2\right )+b^2 c^2 x^2 \left (c+3 d x^2\right )\right )}{6 c^3 d (e x)^{3/2} \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 1187, normalized size = 2.7 \begin{align*} \text{result too large to display} \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{5}{2}} \left (e x\right )^{\frac{3}{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^{3} e^{2} x^{8} + 3 \, c d^{2} e^{2} x^{6} + 3 \, c^{2} d e^{2} x^{4} + c^{3} e^{2} x^{2}}, 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}}{{\left (d x^{2} + c\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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