Optimal. Leaf size=213 \[ \frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left (\frac{5 \sqrt{c} d \left (a e^2+c d^2\right )}{\sqrt{a}}-3 e \left (a e^2+5 c d^2\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 c^{7/4} \sqrt{a-c x^4}}+\frac{3 a^{3/4} e \sqrt{1-\frac{c x^4}{a}} \left (a e^2+5 c d^2\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt{a-c x^4}}-\frac{d e^2 x \sqrt{a-c x^4}}{c}-\frac{e^3 x^3 \sqrt{a-c x^4}}{5 c} \]
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Rubi [A] time = 0.281801, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {1207, 1888, 1201, 224, 221, 1200, 1199, 424} \[ \frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left (\frac{5 \sqrt{c} d \left (a e^2+c d^2\right )}{\sqrt{a}}-3 e \left (a e^2+5 c d^2\right )\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt{a-c x^4}}+\frac{3 a^{3/4} e \sqrt{1-\frac{c x^4}{a}} \left (a e^2+5 c d^2\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt{a-c x^4}}-\frac{d e^2 x \sqrt{a-c x^4}}{c}-\frac{e^3 x^3 \sqrt{a-c x^4}}{5 c} \]
Antiderivative was successfully verified.
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Rule 1207
Rule 1888
Rule 1201
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^3}{\sqrt{a-c x^4}} \, dx &=-\frac{e^3 x^3 \sqrt{a-c x^4}}{5 c}-\frac{\int \frac{-5 c d^3-3 e \left (5 c d^2+a e^2\right ) x^2-15 c d e^2 x^4}{\sqrt{a-c x^4}} \, dx}{5 c}\\ &=-\frac{d e^2 x \sqrt{a-c x^4}}{c}-\frac{e^3 x^3 \sqrt{a-c x^4}}{5 c}+\frac{\int \frac{15 c d \left (c d^2+a e^2\right )+9 c e \left (5 c d^2+a e^2\right ) x^2}{\sqrt{a-c x^4}} \, dx}{15 c^2}\\ &=-\frac{d e^2 x \sqrt{a-c x^4}}{c}-\frac{e^3 x^3 \sqrt{a-c x^4}}{5 c}+\frac{\left (3 \sqrt{a} e \left (5 c d^2+a e^2\right )\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a-c x^4}} \, dx}{5 c^{3/2}}+\frac{\left (5 \sqrt{c} d \left (c d^2+a e^2\right )-3 \sqrt{a} e \left (5 c d^2+a e^2\right )\right ) \int \frac{1}{\sqrt{a-c x^4}} \, dx}{5 c^{3/2}}\\ &=-\frac{d e^2 x \sqrt{a-c x^4}}{c}-\frac{e^3 x^3 \sqrt{a-c x^4}}{5 c}+\frac{\left (3 \sqrt{a} e \left (5 c d^2+a e^2\right ) \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{1-\frac{c x^4}{a}}} \, dx}{5 c^{3/2} \sqrt{a-c x^4}}+\frac{\left (\left (5 \sqrt{c} d \left (c d^2+a e^2\right )-3 \sqrt{a} e \left (5 c d^2+a e^2\right )\right ) \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{c x^4}{a}}} \, dx}{5 c^{3/2} \sqrt{a-c x^4}}\\ &=-\frac{d e^2 x \sqrt{a-c x^4}}{c}-\frac{e^3 x^3 \sqrt{a-c x^4}}{5 c}+\frac{\sqrt [4]{a} \left (5 \sqrt{c} d \left (c d^2+a e^2\right )-3 \sqrt{a} e \left (5 c d^2+a e^2\right )\right ) \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt{a-c x^4}}+\frac{\left (3 \sqrt{a} e \left (5 c d^2+a e^2\right ) \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{\sqrt{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}}{\sqrt{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}} \, dx}{5 c^{3/2} \sqrt{a-c x^4}}\\ &=-\frac{d e^2 x \sqrt{a-c x^4}}{c}-\frac{e^3 x^3 \sqrt{a-c x^4}}{5 c}+\frac{3 a^{3/4} e \left (5 c d^2+a e^2\right ) \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt{a-c x^4}}+\frac{\sqrt [4]{a} \left (5 \sqrt{c} d \left (c d^2+a e^2\right )-3 \sqrt{a} e \left (5 c d^2+a e^2\right )\right ) \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt{a-c x^4}}\\ \end{align*}
Mathematica [C] time = 0.167146, size = 141, normalized size = 0.66 \[ \frac{5 d x \sqrt{1-\frac{c x^4}{a}} \left (a e^2+c d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{c x^4}{a}\right )+e x \left (x^2 \sqrt{1-\frac{c x^4}{a}} \left (a e^2+5 c d^2\right ) \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{c x^4}{a}\right )+e \left (c x^4-a\right ) \left (5 d+e x^2\right )\right )}{5 c \sqrt{a-c x^4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.291, size = 360, normalized size = 1.7 \begin{align*}{e}^{3} \left ( -{\frac{{x}^{3}}{5\,c}\sqrt{-c{x}^{4}+a}}-{\frac{3}{5}{a}^{{\frac{3}{2}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){c}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}} \right ) +3\,d{e}^{2} \left ( -1/3\,{\frac{x\sqrt{-c{x}^{4}+a}}{c}}+1/3\,{\frac{a}{c\sqrt{-c{x}^{4}+a}}\sqrt{1-{\frac{{x}^{2}\sqrt{c}}{\sqrt{a}}}}\sqrt{1+{\frac{{x}^{2}\sqrt{c}}{\sqrt{a}}}}{\it EllipticF} \left ( x\sqrt{{\frac{\sqrt{c}}{\sqrt{a}}}},i \right ){\frac{1}{\sqrt{{\frac{\sqrt{c}}{\sqrt{a}}}}}}} \right ) -3\,{\frac{{d}^{2}e\sqrt{a}}{\sqrt{-c{x}^{4}+a}\sqrt{c}}\sqrt{1-{\frac{{x}^{2}\sqrt{c}}{\sqrt{a}}}}\sqrt{1+{\frac{{x}^{2}\sqrt{c}}{\sqrt{a}}}} \left ({\it EllipticF} \left ( x\sqrt{{\frac{\sqrt{c}}{\sqrt{a}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{\frac{\sqrt{c}}{\sqrt{a}}}},i \right ) \right ){\frac{1}{\sqrt{{\frac{\sqrt{c}}{\sqrt{a}}}}}}}+{{d}^{3}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}} \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 (e x^{2} + d\right )}^{3}}{\sqrt{-c x^{4} + a}}\,{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 (e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}\right )} \sqrt{-c x^{4} + a}}{c x^{4} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.68116, size = 180, normalized size = 0.85 \begin{align*} \frac{d^{3} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} + \frac{3 d^{2} e x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} + \frac{3 d e^{2} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} + \frac{e^{3} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{11}{4}\right )} \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 (e x^{2} + d\right )}^{3}}{\sqrt{-c x^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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