Optimal. Leaf size=73 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt{c x^4-a}} \]
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Rubi [A] time = 0.041897, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1219, 1218} \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt{c x^4-a}} \]
Antiderivative was successfully verified.
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Rule 1219
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^2\right ) \sqrt{-a+c x^4}} \, dx &=\frac{\sqrt{1-\frac{c x^4}{a}} \int \frac{1}{\left (d+e x^2\right ) \sqrt{1-\frac{c x^4}{a}}} \, dx}{\sqrt{-a+c x^4}}\\ &=\frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt{-a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.149835, size = 92, normalized size = 1.26 \[ -\frac{i \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{d \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \sqrt{c x^4-a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.18, size = 99, normalized size = 1.4 \begin{align*}{\frac{1}{d}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}},{\frac{e}{d}\sqrt{a}{\frac{1}{\sqrt{c}}}},{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{-{\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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