Optimal. Leaf size=912 \[ \text{result too large to display} \]
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Rubi [A] time = 0.72556, antiderivative size = 912, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1721, 1179, 1198, 220, 1196, 305, 321} \[ -\frac{a^{3/4} B \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 ) e^3}{6 c^{9/4} \sqrt{c x^4+a}}+\frac{B x \sqrt{c x^4+a} e^3}{3 c^2}-\frac{\sqrt [4]{a} (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 ) e^2}{c^{7/4} \sqrt{c x^4+a}}+\frac{\sqrt [4]{a} (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 ) e^2}{2 c^{7/4} \sqrt{c x^4+a}}+\frac{(3 B d+A e) x \sqrt{c x^4+a} e^2}{c^{3/2} \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{\left (3 B c d^2+3 A c e d-a B e^2\right ) \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 ) e}{2 \sqrt [4]{a} c^{9/4} \sqrt{c x^4+a}}+\frac{\left (B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right ) \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 )}{2 a^{3/4} c^{7/4} \sqrt{c x^4+a}}+\frac{\left (A c^2 d^3-\sqrt{a} c^{3/2} (B d+3 A e) d^2-3 a c e (B d+A e) d+a^2 B e^3+a^{3/2} \sqrt{c} e^2 (3 B d+A e)\right ) \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 )}{4 a^{5/4} c^{9/4} \sqrt{c x^4+a}}-\frac{\left (B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right ) x \sqrt{c x^4+a}}{2 a c^{3/2} \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{x \left (c \left (B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right ) x^2+A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )\right )}{2 a c^2 \sqrt{c x^4+a}} \]
Antiderivative was successfully verified.
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Rule 1721
Rule 1179
Rule 1198
Rule 220
Rule 1196
Rule 305
Rule 321
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx &=\int \left (\frac{A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2}{c^2 \left (a+c x^4\right )^{3/2}}+\frac{e \left (3 B c d^2+3 A c d e-a B e^2\right )}{c^2 \sqrt{a+c x^4}}+\frac{e^2 (3 B d+A e) x^2}{c \sqrt{a+c x^4}}+\frac{B e^3 x^4}{c \sqrt{a+c x^4}}\right ) \, dx\\ &=\frac{\int \frac{A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2}{\left (a+c x^4\right )^{3/2}} \, dx}{c^2}+\frac{\left (B e^3\right ) \int \frac{x^4}{\sqrt{a+c x^4}} \, dx}{c}+\frac{\left (e^2 (3 B d+A e)\right ) \int \frac{x^2}{\sqrt{a+c x^4}} \, dx}{c}+\frac{\left (e \left (3 B c d^2+3 A c d e-a B e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{c^2}\\ &=\frac{x \left (A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2\right )}{2 a c^2 \sqrt{a+c x^4}}+\frac{B e^3 x \sqrt{a+c x^4}}{3 c^2}+\frac{e \left (3 B c d^2+3 A c d e-a B e^2\right ) \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} c^{9/4} \sqrt{a+c x^4}}-\frac{\int \frac{-A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2}{\sqrt{a+c x^4}} \, dx}{2 a c^2}-\frac{\left (a B e^3\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{3 c^2}+\frac{\left (\sqrt{a} e^2 (3 B d+A e)\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{c^{3/2}}-\frac{\left (\sqrt{a} e^2 (3 B d+A e)\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{c^{3/2}}\\ &=\frac{x \left (A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2\right )}{2 a c^2 \sqrt{a+c x^4}}+\frac{B e^3 x \sqrt{a+c x^4}}{3 c^2}+\frac{e^2 (3 B d+A e) x \sqrt{a+c x^4}}{c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} 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 )}{c^{7/4} \sqrt{a+c x^4}}-\frac{a^{3/4} 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 )}{6 c^{9/4} \sqrt{a+c x^4}}+\frac{\sqrt [4]{a} 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 )}{2 c^{7/4} \sqrt{a+c x^4}}+\frac{e \left (3 B c d^2+3 A c d e-a B e^2\right ) \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} c^{9/4} \sqrt{a+c x^4}}+\frac{\left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{2 \sqrt{a} c^{3/2}}+\frac{\left (A c^2 d^3+a^2 B e^3-3 a c d e (B d+A e)+a^{3/2} \sqrt{c} e^2 (3 B d+A e)-\sqrt{a} c^{3/2} d^2 (B d+3 A e)\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{2 a c^2}\\ &=\frac{x \left (A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2\right )}{2 a c^2 \sqrt{a+c x^4}}+\frac{B e^3 x \sqrt{a+c x^4}}{3 c^2}+\frac{e^2 (3 B d+A e) x \sqrt{a+c x^4}}{c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x \sqrt{a+c x^4}}{2 a c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} 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 )}{c^{7/4} \sqrt{a+c x^4}}+\frac{\left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) \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 )}{2 a^{3/4} c^{7/4} \sqrt{a+c x^4}}-\frac{a^{3/4} 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 )}{6 c^{9/4} \sqrt{a+c x^4}}+\frac{\sqrt [4]{a} 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 )}{2 c^{7/4} \sqrt{a+c x^4}}+\frac{e \left (3 B c d^2+3 A c d e-a B e^2\right ) \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} c^{9/4} \sqrt{a+c x^4}}+\frac{\left (A c^2 d^3+a^2 B e^3-3 a c d e (B d+A e)+a^{3/2} \sqrt{c} e^2 (3 B d+A e)-\sqrt{a} c^{3/2} d^2 (B d+3 A e)\right ) \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 )}{4 a^{5/4} c^{9/4} \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.286017, size = 222, normalized size = 0.24 \[ \frac{2 c x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^4}{a}\right ) \left (-3 a A e^3-9 a B d e^2+3 A c d^2 e+B c d^3\right )+x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right ) \left (3 A c d \left (3 a e^2+c d^2\right )+a B e \left (9 c d^2-5 a e^2\right )\right )+3 A c x \left (a e^2 \left (2 e x^2-3 d\right )+c d^3\right )+a B e x \left (5 a e^2+c \left (-9 d^2+18 d e x^2+2 e^2 x^4\right )\right )}{6 a c^2 \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.036, size = 588, normalized size = 0.6 \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}}{{\left (c x^{4} + a\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 (\frac{{\left (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}\right )} \sqrt{c x^{4} + a}}{c^{2} x^{8} + 2 \, a c x^{4} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{3}}{\left (a + c x^{4}\right )^{\frac{3}{2}}}\, dx \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}}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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