Optimal. Leaf size=453 \[ \frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right ) \left (-63 a^{3/2} \sqrt{c} e^2 (A e+3 B d)+25 a^2 B e^3+105 \sqrt{a} c^{3/2} d^2 (3 A e+B d)-105 a c d e (A e+B d)+105 A c^2 d^3\right )}{210 \sqrt [4]{a} c^{9/4} \sqrt{a+c x^4}}+\frac{e x \sqrt{a+c x^4} \left (-5 a B e^2+21 A c d e+21 B c d^2\right )}{21 c^2}+\frac{x \sqrt{a+c x^4} \left (-3 a A e^3-9 a B d e^2+15 A c d^2 e+5 B c d^3\right )}{5 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) \left (-3 a A e^3-9 a B d e^2+15 A c d^2 e+5 B c d^3\right )}{5 c^{7/4} \sqrt{a+c x^4}}+\frac{e^2 x^3 \sqrt{a+c x^4} (A e+3 B d)}{5 c}+\frac{B e^3 x^5 \sqrt{a+c x^4}}{7 c} \]
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Rubi [A] time = 0.509473, antiderivative size = 878, normalized size of antiderivative = 1.94, number of steps used = 15, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {1721, 220, 305, 1196, 321} \[ \frac{B e^3 \sqrt{c x^4+a} x^5}{7 c}+\frac{e^2 (3 B d+A e) \sqrt{c x^4+a} x^3}{5 c}-\frac{5 a B e^3 \sqrt{c x^4+a} x}{21 c^2}+\frac{d e (B d+A e) \sqrt{c x^4+a} x}{c}-\frac{3 a e^2 (3 B d+A e) \sqrt{c x^4+a} x}{5 c^{3/2} \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{d^2 (B d+3 A e) \sqrt{c x^4+a} x}{\sqrt{c} \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{3 a^{5/4} e^2 (3 B d+A e) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{7/4} \sqrt{c x^4+a}}-\frac{\sqrt [4]{a} d^2 (B d+3 A e) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{c x^4+a}}+\frac{A d^3 \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt{c x^4+a}}+\frac{5 a^{7/4} B e^3 \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{42 c^{9/4} \sqrt{c x^4+a}}-\frac{a^{3/4} d e (B d+A e) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 c^{5/4} \sqrt{c x^4+a}}-\frac{3 a^{5/4} e^2 (3 B d+A e) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 c^{7/4} \sqrt{c x^4+a}}+\frac{\sqrt [4]{a} d^2 (B d+3 A e) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 c^{3/4} \sqrt{c x^4+a}} \]
Antiderivative was successfully verified.
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Rule 1721
Rule 220
Rule 305
Rule 1196
Rule 321
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt{a+c x^4}} \, dx &=\int \left (\frac{A d^3}{\sqrt{a+c x^4}}+\frac{d^2 (B d+3 A e) x^2}{\sqrt{a+c x^4}}+\frac{3 d e (B d+A e) x^4}{\sqrt{a+c x^4}}+\frac{e^2 (3 B d+A e) x^6}{\sqrt{a+c x^4}}+\frac{B e^3 x^8}{\sqrt{a+c x^4}}\right ) \, dx\\ &=\left (A d^3\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx+\left (B e^3\right ) \int \frac{x^8}{\sqrt{a+c x^4}} \, dx+(3 d e (B d+A e)) \int \frac{x^4}{\sqrt{a+c x^4}} \, dx+\left (e^2 (3 B d+A e)\right ) \int \frac{x^6}{\sqrt{a+c x^4}} \, dx+\left (d^2 (B d+3 A e)\right ) \int \frac{x^2}{\sqrt{a+c x^4}} \, dx\\ &=\frac{d e (B d+A e) x \sqrt{a+c x^4}}{c}+\frac{e^2 (3 B d+A e) x^3 \sqrt{a+c x^4}}{5 c}+\frac{B e^3 x^5 \sqrt{a+c x^4}}{7 c}+\frac{A d^3 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a+c x^4}}-\frac{\left (5 a B e^3\right ) \int \frac{x^4}{\sqrt{a+c x^4}} \, dx}{7 c}-\frac{(a d e (B d+A e)) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{c}-\frac{\left (3 a e^2 (3 B d+A e)\right ) \int \frac{x^2}{\sqrt{a+c x^4}} \, dx}{5 c}+\frac{\left (\sqrt{a} d^2 (B d+3 A e)\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{\sqrt{c}}-\frac{\left (\sqrt{a} d^2 (B d+3 A e)\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{\sqrt{c}}\\ &=-\frac{5 a B e^3 x \sqrt{a+c x^4}}{21 c^2}+\frac{d e (B d+A e) x \sqrt{a+c x^4}}{c}+\frac{e^2 (3 B d+A e) x^3 \sqrt{a+c x^4}}{5 c}+\frac{B e^3 x^5 \sqrt{a+c x^4}}{7 c}+\frac{d^2 (B d+3 A e) x \sqrt{a+c x^4}}{\sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} d^2 (B d+3 A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{a+c x^4}}+\frac{A d^3 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a+c x^4}}-\frac{a^{3/4} d e (B d+A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 c^{5/4} \sqrt{a+c x^4}}+\frac{\sqrt [4]{a} d^2 (B d+3 A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 c^{3/4} \sqrt{a+c x^4}}+\frac{\left (5 a^2 B e^3\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{21 c^2}-\frac{\left (3 a^{3/2} e^2 (3 B d+A e)\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{5 c^{3/2}}+\frac{\left (3 a^{3/2} e^2 (3 B d+A e)\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{5 c^{3/2}}\\ &=-\frac{5 a B e^3 x \sqrt{a+c x^4}}{21 c^2}+\frac{d e (B d+A e) x \sqrt{a+c x^4}}{c}+\frac{e^2 (3 B d+A e) x^3 \sqrt{a+c x^4}}{5 c}+\frac{B e^3 x^5 \sqrt{a+c x^4}}{7 c}-\frac{3 a e^2 (3 B d+A e) x \sqrt{a+c x^4}}{5 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{d^2 (B d+3 A e) x \sqrt{a+c x^4}}{\sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{3 a^{5/4} e^2 (3 B d+A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{7/4} \sqrt{a+c x^4}}-\frac{\sqrt [4]{a} d^2 (B d+3 A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{a+c x^4}}+\frac{A d^3 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{5 a^{7/4} B e^3 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{42 c^{9/4} \sqrt{a+c x^4}}-\frac{a^{3/4} d e (B d+A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 c^{5/4} \sqrt{a+c x^4}}-\frac{3 a^{5/4} e^2 (3 B d+A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 c^{7/4} \sqrt{a+c x^4}}+\frac{\sqrt [4]{a} d^2 (B d+3 A e) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 c^{3/4} \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.281571, size = 217, normalized size = 0.48 \[ \frac{7 c x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right ) \left (-3 a A e^3-9 a B d e^2+15 A c d^2 e+5 B c d^3\right )+5 x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right ) \left (21 A c d \left (c d^2-a e^2\right )+a B e \left (5 a e^2-21 c d^2\right )\right )+e x \left (a+c x^4\right ) \left (-25 a B e^2+21 A c e \left (5 d+e x^2\right )+3 B c \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )}{105 c^2 \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.054, size = 533, normalized size = 1.2 \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 )}{\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{B e^{3} x^{8} +{\left (3 \, B d e^{2} + A e^{3}\right )} x^{6} + 3 \,{\left (B d^{2} e + A d e^{2}\right )} x^{4} + A d^{3} +{\left (B d^{3} + 3 \, A d^{2} e\right )} x^{2}}{\sqrt{c x^{4} + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.84416, size = 364, normalized size = 0.8 \begin{align*} \frac{A 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^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} + \frac{3 A 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^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} + \frac{3 A 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^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} + \frac{A 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^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{11}{4}\right )} + \frac{B d^{3} 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^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} + \frac{3 B d^{2} e 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^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} + \frac{3 B d e^{2} 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^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{11}{4}\right )} + \frac{B e^{3} x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{13}{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 (B x^{2} + A\right )}{\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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