Optimal. Leaf size=468 \[ \frac{4 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac{8 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{9 c^2 e^5}+\frac{4 (e x)^{3/2} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 c e^5}+\frac{8 \sqrt{e x} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a \left (c+d x^2\right )^{5/2} (a d+2 b c)}{c^2 e^3 \sqrt{e x}} \]
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Rubi [A] time = 0.444668, antiderivative size = 468, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}+\frac{4 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{8 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{9 c^2 e^5}+\frac{4 (e x)^{3/2} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 c e^5}+\frac{8 \sqrt{e x} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a \left (c+d x^2\right )^{5/2} (a d+2 b c)}{c^2 e^3 \sqrt{e x}} \]
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 \left (c+d x^2\right )^{3/2}}{(e x)^{7/2}} \, dx &=-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}+\frac{2 \int \frac{\left (\frac{5}{2} a (2 b c+a d)+\frac{5}{2} b^2 c x^2\right ) \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx}{5 c e^2}\\ &=-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac{2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt{e x}}+\frac{\left (b^2 c^2+9 a d (2 b c+a d)\right ) \int \sqrt{e x} \left (c+d x^2\right )^{3/2} \, dx}{c^2 e^4}\\ &=\frac{2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac{2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt{e x}}+\frac{\left (2 \left (b^2 c^2+9 a d (2 b c+a d)\right )\right ) \int \sqrt{e x} \sqrt{c+d x^2} \, dx}{3 c e^4}\\ &=\frac{4 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \sqrt{c+d x^2}}{15 c e^5}+\frac{2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac{2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt{e x}}+\frac{\left (4 \left (b^2 c^2+9 a d (2 b c+a d)\right )\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{15 e^4}\\ &=\frac{4 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \sqrt{c+d x^2}}{15 c e^5}+\frac{2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac{2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt{e x}}+\frac{\left (8 \left (b^2 c^2+9 a d (2 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 )}{15 e^5}\\ &=\frac{4 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \sqrt{c+d x^2}}{15 c e^5}+\frac{2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac{2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt{e x}}+\frac{\left (8 \sqrt{c} \left (b^2 c^2+9 a d (2 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 )}{15 \sqrt{d} e^4}-\frac{\left (8 \sqrt{c} \left (b^2 c^2+9 a d (2 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 )}{15 \sqrt{d} e^4}\\ &=\frac{4 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \sqrt{c+d x^2}}{15 c e^5}+\frac{8 \left (b^2 c^2+9 a d (2 b c+a d)\right ) \sqrt{e x} \sqrt{c+d x^2}}{15 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \left (b^2 c^2+9 a d (2 b c+a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 c^2 e^5}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}-\frac{2 a (2 b c+a d) \left (c+d x^2\right )^{5/2}}{c^2 e^3 \sqrt{e x}}-\frac{8 \sqrt [4]{c} \left (b^2 c^2+9 a d (2 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 )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{4 \sqrt [4]{c} \left (b^2 c^2+9 a d (2 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 )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.162997, size = 141, normalized size = 0.3 \[ \frac{x \left (24 x^4 \sqrt{\frac{c}{d x^2}+1} \left (9 a^2 d^2+18 a b c d+b^2 c^2\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right )-2 \left (c+d x^2\right ) \left (9 a^2 \left (c+7 d x^2\right )-18 a b x^2 \left (d x^2-5 c\right )-b^2 x^4 \left (11 c+5 d x^2\right )\right )\right )}{45 (e x)^{7/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 668, normalized size = 1.4 \begin{align*}{\frac{2}{45\,d{x}^{2}{e}^{3}} \left ( 5\,{b}^{2}{d}^{3}{x}^{8}+108\,\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}+216\,\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+12\,\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}-54\,\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}-108\,\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-6\,\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}+18\,{x}^{6}ab{d}^{3}+16\,{x}^{6}{b}^{2}c{d}^{2}-63\,{x}^{4}{a}^{2}{d}^{3}-72\,{x}^{4}abc{d}^{2}+11\,{x}^{4}{b}^{2}{c}^{2}d-72\,{x}^{2}{a}^{2}c{d}^{2}-90\,{x}^{2}ab{c}^{2}d-9\,{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}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\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} 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^{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}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\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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