Optimal. Leaf size=334 \[ \frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )}-\frac{a^{3/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right )^2 \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{c} d \sqrt{a+c x^4} \left (c d^2-a e^2\right )}+\frac{\sqrt{e} \tan ^{-1}\left (\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{2 \sqrt{d} \sqrt{a e^2+c d^2}} \]
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Rubi [A] time = 0.265683, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1217, 220, 1707} \[ -\frac{a^{3/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right )^2 \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{c} d \sqrt{a+c x^4} \left (c d^2-a e^2\right )}+\frac{\sqrt{e} \tan ^{-1}\left (\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{2 \sqrt{d} \sqrt{a e^2+c d^2}}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )} \]
Antiderivative was successfully verified.
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Rule 1217
Rule 220
Rule 1707
Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx &=\frac{\sqrt{c} \int \frac{1}{\sqrt{a+c x^4}} \, dx}{\sqrt{c} d-\sqrt{a} e}-\frac{\left (\sqrt{a} e\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{\sqrt{c} d-\sqrt{a} e}\\ &=\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{2 \sqrt{d} \sqrt{c d^2+a e^2}}+\frac{\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} \left (\sqrt{c} d-\sqrt{a} e\right ) \sqrt{a+c x^4}}-\frac{\sqrt [4]{a} \left (\frac{\sqrt{c} d}{\sqrt{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}} \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 )}{4 \sqrt [4]{c} d \left (\sqrt{c} d-\sqrt{a} e\right ) \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.15367, size = 95, normalized size = 0.28 \[ -\frac{i \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{d \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.191, size = 107, normalized size = 0.3 \begin{align*}{\frac{1}{d}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},{\frac{ie}{d}\sqrt{a}{\frac{1}{\sqrt{c}}}},{\sqrt{{-i\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{{i\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{1}{\sqrt{c x^{4} + a}{\left (e x^{2} + d\right )}}\,{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{\sqrt{c x^{4} + a}}{c e x^{6} + c d x^{4} + a e x^{2} + a d}, 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{1}{\sqrt{a + c x^{4}} \left (d + e x^{2}\right )}\, 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{1}{\sqrt{c x^{4} + a}{\left (e x^{2} + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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