Optimal. Leaf size=442 \[ -\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt{c+d x^2}}-\frac{e^2 \sqrt{e x} \sqrt{c+d x^2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{10 c d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{e (e x)^{3/2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{30 c d^3 \sqrt{c+d x^2}}+\frac{(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.383712, 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 = {463, 459, 288, 329, 305, 220, 1196} \[ -\frac{e^2 \sqrt{e x} \sqrt{c+d x^2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{10 c d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{e (e x)^{3/2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{30 c d^3 \sqrt{c+d x^2}}+\frac{(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 463
Rule 459
Rule 288
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac{(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac{\int \frac{(e x)^{5/2} \left (\frac{1}{2} \left (-6 a^2 d^2+7 (b c-a d)^2\right )-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2}\\ &=\frac{(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}}-\frac{\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) \int \frac{(e x)^{5/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{30 c d^2}\\ &=\frac{(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}}-\frac{\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{20 c d^3}\\ &=\frac{(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}}-\frac{\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{10 c d^3}\\ &=\frac{(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}}-\frac{\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{10 \sqrt{c} d^{7/2}}+\frac{\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2\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 )}{10 \sqrt{c} d^{7/2}}\\ &=\frac{(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}}-\frac{\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2 \sqrt{e x} \sqrt{c+d x^2}}{10 c d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \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 )}{10 c^{3/4} d^{15/4} \sqrt{c+d x^2}}-\frac{\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \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 )}{20 c^{3/4} d^{15/4} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.184941, size = 153, normalized size = 0.35 \[ \frac{e (e x)^{3/2} \left (-3 \sqrt{\frac{c}{d x^2}+1} \left (c+d x^2\right ) \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right )+5 a^2 d^2 \left (c+3 d x^2\right )-10 a b c d \left (7 c+9 d x^2\right )+b^2 c \left (77 c^2+99 c d x^2+12 d^2 x^4\right )\right )}{30 c d^3 \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 1191, 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 (e x\right )^{\frac{5}{2}}}{{\left (d x^{2} + c\right )}^{\frac{5}{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} e^{2} x^{6} + 2 \, a b e^{2} x^{4} + a^{2} e^{2} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{e x}}{d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}}, 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 (e x\right )^{\frac{5}{2}}}{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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