Optimal. Leaf size=581 \[ \frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{2 \sqrt [4]{a} d \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )}+\frac{e^2 x \sqrt{a+c x^4}}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}-\frac{\sqrt{c} e x \sqrt{a+c x^4}}{2 d \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2+c d^2\right )}+\frac{\sqrt{e} \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{4 d^{3/2} \left (a e^2+c d^2\right )^{3/2}}+\frac{\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 d \sqrt{a+c x^4} \left (a e^2+c d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (a e^2+3 c d^2\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.762326, antiderivative size = 581, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1224, 1715, 1196, 1709, 220, 1707} \[ \frac{e^2 x \sqrt{a+c x^4}}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}-\frac{\sqrt{c} e x \sqrt{a+c x^4}}{2 d \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2+c d^2\right )}+\frac{\sqrt{e} \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{4 d^{3/2} \left (a e^2+c d^2\right )^{3/2}}+\frac{\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 d \sqrt{a+c x^4} \left (a e^2+c d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (a e^2+3 c d^2\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right )}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} d \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )} \]
Antiderivative was successfully verified.
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Rule 1224
Rule 1715
Rule 1196
Rule 1709
Rule 220
Rule 1707
Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^2\right )^2 \sqrt{a+c x^4}} \, dx &=\frac{e^2 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 c d^2-a e^2+2 c d e x^2+c e^2 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^2 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^2+c e \left (-2 c d^2-a e^2\right )+\left (2 c^2 d e^2-c e^2 \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} 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} 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^2 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} 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{\sqrt{c} \int \frac{1}{\sqrt{a+c x^4}} \, dx}{d \left (\sqrt{c} d-\sqrt{a} e\right )}-\frac{\left (\sqrt{a} e \left (3 c d^2+a e^2\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} 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^2 x \sqrt{a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac{\sqrt{e} \left (3 c d^2+a e^2\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} \left (c d^2+a e^2\right )^{3/2}}+\frac{\sqrt [4]{a} \sqrt [4]{c} 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{\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{\sqrt [4]{a} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (3 c d^2+a 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}} \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]{c} d^2 \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 = 0.77866, size = 522, normalized size = 0.9 \[ \frac{\sqrt{c} d \sqrt{\frac{c x^4}{a}+1} \left (d+e x^2\right ) \left (\sqrt{a} e+i \sqrt{c} d\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\right ),-1\right )-3 i c d^2 e x^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-3 i c d^3 \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+c d e^2 x^5 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}-i a e^3 x^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-i a d e^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+a d e^2 x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}-\sqrt{a} \sqrt{c} d e \sqrt{\frac{c x^4}{a}+1} \left (d+e x^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{2 d^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} \left (d+e x^2\right ) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.197, size = 556, normalized size = 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{1}{\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{1}{\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{1}{\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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