Optimal. Leaf size=340 \[ -\frac{4 c^{11/4} e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (57 a^2 d^2+b c (9 b c-38 a d)\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{4389 d^{13/4} \sqrt{c+d x^2}}+\frac{8 c^2 e \sqrt{e x} \sqrt{c+d x^2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{4389 d^3}+\frac{2 (e x)^{5/2} \left (c+d x^2\right )^{3/2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{627 d^2 e}+\frac{4 c (e x)^{5/2} \sqrt{c+d x^2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{1463 d^2 e}-\frac{2 b (e x)^{5/2} \left (c+d x^2\right )^{5/2} (9 b c-38 a d)}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3} \]
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Rubi [A] time = 0.323289, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {464, 459, 279, 321, 329, 220} \[ -\frac{4 c^{11/4} e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (57 a^2 d^2+b c (9 b c-38 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{4389 d^{13/4} \sqrt{c+d x^2}}+\frac{8 c^2 e \sqrt{e x} \sqrt{c+d x^2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{4389 d^3}+\frac{2 (e x)^{5/2} \left (c+d x^2\right )^{3/2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{627 d^2 e}+\frac{4 c (e x)^{5/2} \sqrt{c+d x^2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{1463 d^2 e}-\frac{2 b (e x)^{5/2} \left (c+d x^2\right )^{5/2} (9 b c-38 a d)}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3} \]
Antiderivative was successfully verified.
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Rule 464
Rule 459
Rule 279
Rule 321
Rule 329
Rule 220
Rubi steps
\begin{align*} \int (e x)^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx &=\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3}+\frac{2 \int (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (\frac{19 a^2 d}{2}-\frac{1}{2} b (9 b c-38 a d) x^2\right ) \, dx}{19 d}\\ &=-\frac{2 b (9 b c-38 a d) (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3}+\frac{1}{57} \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) \int (e x)^{3/2} \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac{2 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{627 e}-\frac{2 b (9 b c-38 a d) (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3}+\frac{1}{209} \left (2 c \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right )\right ) \int (e x)^{3/2} \sqrt{c+d x^2} \, dx\\ &=\frac{4 c \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) (e x)^{5/2} \sqrt{c+d x^2}}{1463 e}+\frac{2 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{627 e}-\frac{2 b (9 b c-38 a d) (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3}+\frac{\left (4 c^2 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right )\right ) \int \frac{(e x)^{3/2}}{\sqrt{c+d x^2}} \, dx}{1463}\\ &=\frac{8 c^2 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) e \sqrt{e x} \sqrt{c+d x^2}}{4389 d}+\frac{4 c \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) (e x)^{5/2} \sqrt{c+d x^2}}{1463 e}+\frac{2 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{627 e}-\frac{2 b (9 b c-38 a d) (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3}-\frac{\left (4 c^3 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) e^2\right ) \int \frac{1}{\sqrt{e x} \sqrt{c+d x^2}} \, dx}{4389 d}\\ &=\frac{8 c^2 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) e \sqrt{e x} \sqrt{c+d x^2}}{4389 d}+\frac{4 c \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) (e x)^{5/2} \sqrt{c+d x^2}}{1463 e}+\frac{2 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{627 e}-\frac{2 b (9 b c-38 a d) (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3}-\frac{\left (8 c^3 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) e\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{4389 d}\\ &=\frac{8 c^2 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) e \sqrt{e x} \sqrt{c+d x^2}}{4389 d}+\frac{4 c \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) (e x)^{5/2} \sqrt{c+d x^2}}{1463 e}+\frac{2 \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{627 e}-\frac{2 b (9 b c-38 a d) (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3}-\frac{4 c^{11/4} \left (57 a^2+\frac{b c (9 b c-38 a d)}{d^2}\right ) e^{3/2} \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 )}{4389 d^{5/4} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.263317, size = 259, normalized size = 0.76 \[ \frac{(e x)^{3/2} \left (\frac{2 \sqrt{x} \left (c+d x^2\right ) \left (285 a^2 d^2 \left (4 c^2+13 c d x^2+7 d^2 x^4\right )+38 a b d \left (12 c^2 d x^2-20 c^3+119 c d^2 x^4+77 d^3 x^6\right )+3 b^2 \left (28 c^2 d^2 x^4-36 c^3 d x^2+60 c^4+539 c d^3 x^6+385 d^4 x^8\right )\right )}{5 d^3}-\frac{8 i c^3 x \sqrt{\frac{c}{d x^2}+1} \left (57 a^2 d^2-38 a b c d+9 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )}{d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{4389 x^{3/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 489, normalized size = 1.4 \begin{align*} -{\frac{2\,e}{21945\,x{d}^{4}}\sqrt{ex} \left ( -1155\,{x}^{11}{b}^{2}{d}^{6}-2926\,{x}^{9}ab{d}^{6}-2772\,{x}^{9}{b}^{2}c{d}^{5}-1995\,{x}^{7}{a}^{2}{d}^{6}-7448\,{x}^{7}abc{d}^{5}-1701\,{x}^{7}{b}^{2}{c}^{2}{d}^{4}+570\,\sqrt{-cd}\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 ){a}^{2}{c}^{3}{d}^{2}-380\,\sqrt{-cd}\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 ) ab{c}^{4}d+90\,\sqrt{-cd}\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 ){b}^{2}{c}^{5}-5700\,{x}^{5}{a}^{2}c{d}^{5}-4978\,{x}^{5}ab{c}^{2}{d}^{4}+24\,{x}^{5}{b}^{2}{c}^{3}{d}^{3}-4845\,{x}^{3}{a}^{2}{c}^{2}{d}^{4}+304\,{x}^{3}ab{c}^{3}{d}^{3}-72\,{x}^{3}{b}^{2}{c}^{4}{d}^{2}-1140\,x{a}^{2}{c}^{3}{d}^{3}+760\,xab{c}^{4}{d}^{2}-180\,x{b}^{2}{c}^{5}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \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{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{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 ({\left (b^{2} d e x^{7} +{\left (b^{2} c + 2 \, a b d\right )} e x^{5} + a^{2} c e x +{\left (2 \, a b c + a^{2} d\right )} e x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{e x}, 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{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{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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