Optimal. Leaf size=875 \[ \frac{\sqrt{c} \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) \sqrt{c x^4+a} x}{8 d^2 \left (c d^2+a e^2\right )^2 \left (\sqrt{c} x^2+\sqrt{a}\right )}-\frac{e \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) \sqrt{c x^4+a} x}{8 d^2 \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}-\frac{e (B d-A e) \sqrt{c x^4+a} x}{4 d \left (c d^2+a e^2\right ) \left (e x^2+d\right )^2}+\frac{\left (3 A e \left (5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c e^2 d^3-a^2 e^4 d\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right )}{16 d^{5/2} \sqrt{e} \left (c d^2+a e^2\right )^{5/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 d^2 \left (c d^2+a e^2\right )^2 \sqrt{c x^4+a}}+\frac{\sqrt [4]{c} \left (4 A c d^2+\sqrt{a} \sqrt{c} (B d-A e) d+a e (B d+3 A e)\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{8 \sqrt [4]{a} d^2 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \sqrt{c x^4+a}}-\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \left (3 A e \left (5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c e^2 d^3-a^2 e^4 d\right )\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt{c x^4+a}} \]
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Rubi [A] time = 1.78415, antiderivative size = 875, 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 = {1697, 1715, 1196, 1709, 220, 1707} \[ \frac{\sqrt{c} \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) \sqrt{c x^4+a} x}{8 d^2 \left (c d^2+a e^2\right )^2 \left (\sqrt{c} x^2+\sqrt{a}\right )}-\frac{e \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) \sqrt{c x^4+a} x}{8 d^2 \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}-\frac{e (B d-A e) \sqrt{c x^4+a} x}{4 d \left (c d^2+a e^2\right ) \left (e x^2+d\right )^2}+\frac{\left (3 A e \left (5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c e^2 d^3-a^2 e^4 d\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right )}{16 d^{5/2} \sqrt{e} \left (c d^2+a e^2\right )^{5/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 d^2 \left (c d^2+a e^2\right )^2 \sqrt{c x^4+a}}+\frac{\sqrt [4]{c} \left (4 A c d^2+\sqrt{a} \sqrt{c} (B d-A e) d+a e (B d+3 A e)\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} d^2 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \sqrt{c x^4+a}}-\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \left (3 A e \left (5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c e^2 d^3-a^2 e^4 d\right )\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt{c x^4+a}} \]
Antiderivative was successfully verified.
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Rule 1697
Rule 1715
Rule 1196
Rule 1709
Rule 220
Rule 1707
Rubi steps
\begin{align*} \int \frac{A+B x^2}{\left (d+e x^2\right )^3 \sqrt{a+c x^4}} \, dx &=-\frac{e (B d-A e) x \sqrt{a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac{\int \frac{-4 A c d^2-a B d e-3 a A e^2-4 c d (B d-A e) x^2+c e (B d-A e) x^4}{\left (d+e x^2\right )^2 \sqrt{a+c x^4}} \, dx}{4 d \left (c d^2+a e^2\right )}\\ &=-\frac{e (B d-A e) x \sqrt{a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac{e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x \sqrt{a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{\int \frac{a B d e \left (7 c d^2+a e^2\right )+A \left (8 c^2 d^4+5 a c d^2 e^2+3 a^2 e^4\right )+4 c d \left (2 B c d^3-4 A c d^2 e-a B d e^2-a A e^3\right ) x^2+c e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x^4}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{8 d^2 \left (c d^2+a e^2\right )^2}\\ &=-\frac{e (B d-A e) x \sqrt{a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac{e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x \sqrt{a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{\int \frac{\sqrt{a} c^{3/2} d e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right )+c e \left (a B d e \left (7 c d^2+a e^2\right )+A \left (8 c^2 d^4+5 a c d^2 e^2+3 a^2 e^4\right )\right )+\left (-c e \left (c d-\sqrt{a} \sqrt{c} e\right ) \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right )+4 c^2 d e \left (2 B c d^3-4 A c d^2 e-a B d e^2-a A e^3\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{8 c d^2 e \left (c d^2+a e^2\right )^2}-\frac{\left (\sqrt{a} \sqrt{c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{8 d^2 \left (c d^2+a e^2\right )^2}\\ &=\frac{\sqrt{c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x \sqrt{a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{e (B d-A e) x \sqrt{a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac{e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x \sqrt{a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 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 )}{8 d^2 \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{\left (\sqrt{c} \left (4 A c d^2+\sqrt{a} \sqrt{c} d (B d-A e)+a e (B d+3 A e)\right )\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{4 d^2 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )}-\frac{\left (\sqrt{a} \left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c d^3 e^2-a^2 d e^4\right )\right )\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{8 d^2 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2}\\ &=\frac{\sqrt{c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x \sqrt{a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{e (B d-A e) x \sqrt{a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac{e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x \sqrt{a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{\left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c d^3 e^2-a^2 d e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{16 d^{5/2} \sqrt{e} \left (c d^2+a e^2\right )^{5/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 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 )}{8 d^2 \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}+\frac{\sqrt [4]{c} \left (4 A c d^2+\sqrt{a} \sqrt{c} d (B d-A e)+a e (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 )}{8 \sqrt [4]{a} d^2 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}-\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c d^3 e^2-a^2 d e^4\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 2.29221, size = 453, normalized size = 0.52 \[ \frac{-\frac{d e^2 x \left (a+c x^4\right ) \left (\left (d+e x^2\right ) \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )+2 d \left (a e^2+c d^2\right ) (B d-A e)\right )}{\left (d+e x^2\right )^2}-\frac{i \sqrt{\frac{c x^4}{a}+1} \left (\sqrt{c} d \left (\sqrt{c} d-i \sqrt{a} e\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\right ),-1\right ) \left (A e \left (2 i \sqrt{a} \sqrt{c} d e-3 a e^2-7 c d^2\right )+B d \left (-2 i \sqrt{a} \sqrt{c} d e-a e^2+3 c d^2\right )\right )+\left (3 A e \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right )+B \left (a^2 d e^4+10 a c d^3 e^2-3 c^2 d^5\right )\right ) \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-i \sqrt{a} \sqrt{c} d e E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right ) \left (3 a A e^3+a B d e^2+9 A c d^2 e-5 B c d^3\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}}{8 d^3 e \sqrt{a+c x^4} \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 1591, normalized size = 1.8 \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{B x^{2} + A}{\sqrt{c x^{4} + a}{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x^{2}}{\sqrt{a + c x^{4}} \left (d + e x^{2}\right )^{3}}\, 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{B x^{2} + A}{\sqrt{c x^{4} + a}{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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