Optimal. Leaf size=302 \[ \frac{5 e^{7/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt{c+d x^2}}-\frac{5 e^3 \sqrt{e x} \sqrt{c+d x^2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{42 c d^4}+\frac{e (e x)^{5/2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{14 c d^3 \sqrt{c+d x^2}}+\frac{(e x)^{9/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.233711, antiderivative size = 302, 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 = {463, 459, 288, 321, 329, 220} \[ -\frac{5 e^3 \sqrt{e x} \sqrt{c+d x^2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{42 c d^4}+\frac{5 e^{7/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt{c+d x^2}}+\frac{e (e x)^{5/2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{14 c d^3 \sqrt{c+d x^2}}+\frac{(e x)^{9/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 463
Rule 459
Rule 288
Rule 321
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac{(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac{\int \frac{(e x)^{7/2} \left (-\frac{3}{2} \left (2 a^2 d^2-3 (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)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt{c+d x^2}}-\frac{\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) \int \frac{(e x)^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{14 c d^2}\\ &=\frac{(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt{c+d x^2}}-\frac{\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^2\right ) \int \frac{(e x)^{3/2}}{\sqrt{c+d x^2}} \, dx}{28 c d^3}\\ &=\frac{(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt{c+d x^2}}-\frac{5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt{e x} \sqrt{c+d x^2}}{42 c d^4}+\frac{\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^4\right ) \int \frac{1}{\sqrt{e x} \sqrt{c+d x^2}} \, dx}{84 d^4}\\ &=\frac{(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt{c+d x^2}}-\frac{5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt{e x} \sqrt{c+d x^2}}{42 c d^4}+\frac{\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{42 d^4}\\ &=\frac{(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt{c+d x^2}}-\frac{5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt{e x} \sqrt{c+d x^2}}{42 c d^4}+\frac{5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^{7/2} \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 )}{84 \sqrt [4]{c} d^{17/4} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.280124, size = 222, normalized size = 0.74 \[ \frac{(e x)^{7/2} \left (\frac{\sqrt{x} \left (-7 a^2 d^2 \left (5 c+7 d x^2\right )+14 a b d \left (15 c^2+21 c d x^2+4 d^2 x^4\right )+b^2 \left (-\left (273 c^2 d x^2+195 c^3+52 c d^2 x^4-12 d^3 x^6\right )\right )\right )}{d^4 \left (c+d x^2\right )}+\frac{5 i x \sqrt{\frac{c}{d x^2}+1} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )}{d^4 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{42 x^{7/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 696, normalized size = 2.3 \begin{align*}{\frac{{e}^{3}}{84\,x{d}^{5}} \left ( 35\,\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 ) \sqrt{-cd}{x}^{2}{a}^{2}{d}^{3}-210\,\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 ) \sqrt{-cd}{x}^{2}abc{d}^{2}+195\,\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 ) \sqrt{-cd}{x}^{2}{b}^{2}{c}^{2}d+24\,{x}^{7}{b}^{2}{d}^{4}+35\,\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 ) \sqrt{-cd}{a}^{2}c{d}^{2}-210\,\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 ) \sqrt{-cd}ab{c}^{2}d+195\,\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 ) \sqrt{-cd}{b}^{2}{c}^{3}+112\,{x}^{5}ab{d}^{4}-104\,{x}^{5}{b}^{2}c{d}^{3}-98\,{x}^{3}{a}^{2}{d}^{4}+588\,{x}^{3}abc{d}^{3}-546\,{x}^{3}{b}^{2}{c}^{2}{d}^{2}-70\,x{a}^{2}c{d}^{3}+420\,xab{c}^{2}{d}^{2}-390\,x{b}^{2}{c}^{3}d \right ) \sqrt{ex} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \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{7}{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^{3} x^{7} + 2 \, a b e^{3} x^{5} + a^{2} e^{3} x^{3}\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{7}{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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