Optimal. Leaf size=641 \[ \frac{A \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{2 \sqrt [4]{a} d \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )}+\frac{\sqrt{c} x \sqrt{a+c x^4} (B d-A e)}{2 d \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2+c d^2\right )}-\frac{e x \sqrt{a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}-\frac{\left (-a A e^3-a B d e^2-3 A c d^2 e+B c d^3\right ) \tan ^{-1}\left (\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{4 d^{3/2} \sqrt{e} \left (a e^2+c d^2\right )^{3/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} (B d-A e) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 d \sqrt{a+c x^4} \left (a e^2+c d^2\right )}+\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (-a A e^3-a B d e^2-3 A c d^2 e+B c d^3\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 e \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 1.18937, antiderivative size = 641, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{\sqrt{c} x \sqrt{a+c x^4} (B d-A e)}{2 d \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2+c d^2\right )}-\frac{e x \sqrt{a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}-\frac{\left (-a A e^3-a B d e^2-3 A c d^2 e+B c d^3\right ) \tan ^{-1}\left (\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{4 d^{3/2} \sqrt{e} \left (a e^2+c d^2\right )^{3/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} (B d-A e) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 d \sqrt{a+c x^4} \left (a e^2+c d^2\right )}+\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (-a A e^3-a B d e^2-3 A c d^2 e+B c d^3\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 e \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right )}+\frac{A \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} d \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )} \]
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 )^2 \sqrt{a+c x^4}} \, dx &=-\frac{e (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}-\frac{\int \frac{-2 A c d^2-a B d e-a A e^2-2 c d (B d-A e) x^2-c e (B d-A e) x^4}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )}\\ &=-\frac{e (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}-\frac{\int \frac{-\sqrt{a} c^{3/2} d e (B d-A e)+c e \left (-2 A c d^2-a B d e-a A e^2\right )+\left (-2 c^2 d e (B d-A e)+c e (B d-A e) \left (c d-\sqrt{a} \sqrt{c} e\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{2 c d e \left (c d^2+a e^2\right )}-\frac{\left (\sqrt{a} \sqrt{c} (B d-A e)\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )}\\ &=\frac{\sqrt{c} (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{e (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} (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 )}{2 d \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}+\frac{\left (A \sqrt{c}\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{d \left (\sqrt{c} d-\sqrt{a} e\right )}+\frac{\left (\sqrt{a} \left (B c d^3-3 A c d^2 e-a B d e^2-a A e^3\right )\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{2 d \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )}\\ &=\frac{\sqrt{c} (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{e (B d-A e) x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}-\frac{\left (B c d^3-3 A c d^2 e-a B d e^2-a A e^3\right ) \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{4 d^{3/2} \sqrt{e} \left (c d^2+a e^2\right )^{3/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} (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 )}{2 d \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}+\frac{A \sqrt [4]{c} \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} d \left (\sqrt{c} d-\sqrt{a} e\right ) \sqrt{a+c x^4}}+\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \left (B c d^3-3 A c d^2 e-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}} \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 )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 e \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 1.23753, size = 297, normalized size = 0.46 \[ \frac{\frac{d e x \left (a+c x^4\right ) (A e-B d)}{\left (d+e x^2\right ) \left (a e^2+c d^2\right )}-\frac{i \sqrt{\frac{c x^4}{a}+1} \left (\sqrt{c} d \left (\sqrt{c} d-i \sqrt{a} e\right ) (B d-A e) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\right ),-1\right )+\left (a A e^3+a B d e^2+3 A c d^2 e-B c d^3\right ) \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+i \sqrt{a} \sqrt{c} d e (B d-A e) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (a e^3+c d^2 e\right )}}{2 d^2 \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.025, size = 679, normalized size = 1.1 \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 )}^{2}}\,{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 )^{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{B x^{2} + A}{\sqrt{c x^{4} + a}{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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