Optimal. Leaf size=1221 \[ \text{result too large to display} \]
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Rubi [A] time = 1.79971, antiderivative size = 1221, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 15, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.79, Rules used = {2153, 1227, 1198, 220, 1196, 1217, 1707, 1248, 733, 844, 217, 206, 725, 1336, 1209} \[ \frac{c \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{c d^4+a e^4} \sqrt{c x^4+a}}\right ) d^3}{e^3 \sqrt{c d^4+a e^4}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{c x^4+a}}\right ) d}{e^3}-\frac{\sqrt{c x^4+a} d}{e \left (d^2-e^2 x^2\right )}-\frac{2 \sqrt [4]{a} \sqrt [4]{c} \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 )}{e^2 \sqrt{c x^4+a}}+\frac{3 \sqrt [4]{a} \sqrt [4]{c} \left (\frac{\sqrt{c} d^2}{\sqrt{a}}+e^2\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 )}{4 e^4 \sqrt{c x^4+a}}-\frac{\sqrt [4]{c} \left (\sqrt{c} d^2-\sqrt{a} e^2\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 )}{2 \sqrt [4]{a} e^4 \sqrt{c x^4+a}}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2+\sqrt{a} e^2\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 )}{4 \sqrt [4]{a} e^4 \sqrt{c x^4+a}}-\frac{\sqrt [4]{c} \left (c d^4+a e^4\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 )}{2 \sqrt [4]{a} e^4 \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{c x^4+a}}+\frac{2 \sqrt{c} x \sqrt{c x^4+a}}{e^2 \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{x \sqrt{c x^4+a}}{d^2-e^2 x^2}-\frac{\left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{c x^4+a}}\right )}{2 e^3 \sqrt{-c d^4-a e^4} d}+\frac{\sqrt{-c d^4-a e^4} \tan ^{-1}\left (\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{c x^4+a}}\right )}{2 e^3 d}+\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right )^2 \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^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} e^4 \sqrt{c x^4+a} d^2}+\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (c d^4+a e^4\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^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} e^4 \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{c x^4+a} d^2} \]
Antiderivative was successfully verified.
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Rule 2153
Rule 1227
Rule 1198
Rule 220
Rule 1196
Rule 1217
Rule 1707
Rule 1248
Rule 733
Rule 844
Rule 217
Rule 206
Rule 725
Rule 1336
Rule 1209
Rubi steps
\begin{align*} \int \frac{\sqrt{a+c x^4}}{(d+e x)^2} \, dx &=\int \left (\frac{d^2 \sqrt{a+c x^4}}{\left (d^2-e^2 x^2\right )^2}-\frac{2 d e x \sqrt{a+c x^4}}{\left (d^2-e^2 x^2\right )^2}+\frac{e^2 x^2 \sqrt{a+c x^4}}{\left (-d^2+e^2 x^2\right )^2}\right ) \, dx\\ &=d^2 \int \frac{\sqrt{a+c x^4}}{\left (d^2-e^2 x^2\right )^2} \, dx-(2 d e) \int \frac{x \sqrt{a+c x^4}}{\left (d^2-e^2 x^2\right )^2} \, dx+e^2 \int \frac{x^2 \sqrt{a+c x^4}}{\left (-d^2+e^2 x^2\right )^2} \, dx\\ &=\frac{x \sqrt{a+c x^4}}{2 \left (d^2-e^2 x^2\right )}+\frac{1}{2} \left (a-\frac{c d^4}{e^4}\right ) \int \frac{1}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx+\frac{c \int \frac{d^2+e^2 x^2}{\sqrt{a+c x^4}} \, dx}{2 e^4}-(d e) \operatorname{Subst}\left (\int \frac{\sqrt{a+c x^2}}{\left (d^2-e^2 x\right )^2} \, dx,x,x^2\right )+e^2 \int \left (\frac{d^2 \sqrt{a+c x^4}}{e^2 \left (-d^2+e^2 x^2\right )^2}+\frac{\sqrt{a+c x^4}}{e^2 \left (-d^2+e^2 x^2\right )}\right ) \, dx\\ &=-\frac{d \sqrt{a+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac{x \sqrt{a+c x^4}}{2 \left (d^2-e^2 x^2\right )}+d^2 \int \frac{\sqrt{a+c x^4}}{\left (-d^2+e^2 x^2\right )^2} \, dx+\frac{1}{2} \left (\sqrt{a} \left (\sqrt{a}-\frac{\sqrt{c} d^2}{e^2}\right )\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx-\frac{\left (\sqrt{a} \sqrt{c}\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{2 e^2}+\frac{(c d) \operatorname{Subst}\left (\int \frac{x}{\left (d^2-e^2 x\right ) \sqrt{a+c x^2}} \, dx,x,x^2\right )}{e}-\frac{\left (\sqrt{c} \left (\sqrt{c} d^2-\sqrt{a} e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{2 e^4}+\frac{\left (c \left (d^2+\frac{\sqrt{a} e^2}{\sqrt{c}}\right )\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{2 e^4}+\int \frac{\sqrt{a+c x^4}}{-d^2+e^2 x^2} \, dx\\ &=\frac{\sqrt{c} x \sqrt{a+c x^4}}{2 e^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{d \sqrt{a+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac{x \sqrt{a+c x^4}}{d^2-e^2 x^2}-\frac{\left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{a+c x^4}}\right )}{4 d e^3 \sqrt{-c d^4-a e^4}}-\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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 e^2 \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} \left (\sqrt{c} d^2-\sqrt{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}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} e^4 \sqrt{a+c x^4}}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2+\sqrt{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}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} e^4 \sqrt{a+c x^4}}+\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right )^2 \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^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};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^4 \sqrt{a+c x^4}}+\frac{1}{2} \left (-a+\frac{c d^4}{e^4}\right ) \int \frac{1}{\left (-d^2+e^2 x^2\right ) \sqrt{a+c x^4}} \, dx+\left (a+\frac{c d^4}{e^4}\right ) \int \frac{1}{\left (-d^2+e^2 x^2\right ) \sqrt{a+c x^4}} \, dx-\frac{\int \frac{-c d^2-c e^2 x^2}{\sqrt{a+c x^4}} \, dx}{e^4}-\frac{c \int \frac{-d^2-e^2 x^2}{\sqrt{a+c x^4}} \, dx}{2 e^4}-\frac{(c d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^2}} \, dx,x,x^2\right )}{e^3}+\frac{\left (c d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d^2-e^2 x\right ) \sqrt{a+c x^2}} \, dx,x,x^2\right )}{e^3}\\ &=\frac{\sqrt{c} x \sqrt{a+c x^4}}{2 e^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{d \sqrt{a+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac{x \sqrt{a+c x^4}}{d^2-e^2 x^2}-\frac{\left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{a+c x^4}}\right )}{4 d e^3 \sqrt{-c d^4-a e^4}}-\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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 e^2 \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} \left (\sqrt{c} d^2-\sqrt{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}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} e^4 \sqrt{a+c x^4}}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2+\sqrt{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}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} e^4 \sqrt{a+c x^4}}+\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right )^2 \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^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};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^4 \sqrt{a+c x^4}}-\frac{1}{2} \left (\sqrt{a} \left (\sqrt{a}-\frac{\sqrt{c} d^2}{e^2}\right )\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (-d^2+e^2 x^2\right ) \sqrt{a+c x^4}} \, dx-\frac{(c d) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{a+c x^4}}\right )}{e^3}-\frac{\left (c d^3\right ) \operatorname{Subst}\left (\int \frac{1}{c d^4+a e^4-x^2} \, dx,x,\frac{-a e^2-c d^2 x^2}{\sqrt{a+c x^4}}\right )}{e^3}-\frac{\left (\sqrt{a} \sqrt{c}\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{2 e^2}-\frac{\left (\sqrt{a} \sqrt{c}\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{e^2}-\frac{\left (\sqrt{c} \left (\sqrt{c} d^2-\sqrt{a} e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{2 e^4}-\frac{\left (\sqrt{c} \left (a+\frac{c d^4}{e^4}\right )\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{\sqrt{c} d^2+\sqrt{a} e^2}+\frac{\left (\sqrt{a} \left (a+\frac{c d^4}{e^4}\right ) e^2\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (-d^2+e^2 x^2\right ) \sqrt{a+c x^4}} \, dx}{\sqrt{c} d^2+\sqrt{a} e^2}+\frac{\left (\sqrt{c} \left (\sqrt{c} d^2+\sqrt{a} e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{2 e^4}+\frac{\left (\sqrt{c} \left (\sqrt{c} d^2+\sqrt{a} e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{e^4}\\ &=\frac{2 \sqrt{c} x \sqrt{a+c x^4}}{e^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{d \sqrt{a+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac{x \sqrt{a+c x^4}}{d^2-e^2 x^2}+\frac{\sqrt{-c d^4-a e^4} \tan ^{-1}\left (\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{a+c x^4}}\right )}{2 d e^3}-\frac{\left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{a+c x^4}}\right )}{2 d e^3 \sqrt{-c d^4-a e^4}}-\frac{\sqrt{c} d \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{e^3}+\frac{c d^3 \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{c d^4+a e^4} \sqrt{a+c x^4}}\right )}{e^3 \sqrt{c d^4+a e^4}}-\frac{2 \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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{e^2 \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} \left (\sqrt{c} d^2-\sqrt{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}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} e^4 \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} \left (a+\frac{c d^4}{e^4}\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 )}{2 \sqrt [4]{a} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{a+c x^4}}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2+\sqrt{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}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt [4]{a} e^4 \sqrt{a+c x^4}}+\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right )^2 \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^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d^2 e^4 \sqrt{a+c x^4}}+\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (c d^4+a e^4\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^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d^2 e^4 \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 1.89189, size = 382, normalized size = 0.31 \[ \frac{2 \sqrt [4]{-1} \sqrt [4]{a} c^{3/4} d^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )-\frac{c d^3 e \sqrt{a+c x^4} \tanh ^{-1}\left (\frac{-a e^2-c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{\sqrt{a e^4+c d^4}}-\frac{2 \sqrt{c} \sqrt{\frac{c x^4}{a}+1} \left (\sqrt{a} e^2+i \sqrt{c} d^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}-\frac{e^3 \left (a+c x^4\right )}{d+e x}-\sqrt{c} d e \sqrt{a+c x^4} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-2 i a e^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{\frac{c x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{e^4 \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 402, normalized size = 0.3 \begin{align*} -{\frac{1}{e \left ( ex+d \right ) }\sqrt{c{x}^{4}+a}}+2\,{\frac{c{d}^{2}}{{e}^{4}\sqrt{c{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}},i \right ){\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}}-{\frac{d}{{e}^{3}}\sqrt{c}\ln \left ( 2\,{x}^{2}\sqrt{c}+2\,\sqrt{c{x}^{4}+a} \right ) }+{\frac{2\,i}{{e}^{2}}\sqrt{a}\sqrt{c}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{c{d}^{3}}{{e}^{5}}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}+2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}}-2\,{\frac{c{d}^{2}}{{e}^{4}\sqrt{c{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}},{\frac{-i\sqrt{a}{e}^{2}}{{d}^{2}\sqrt{c}}},{\sqrt{{\frac{-i\sqrt{c}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{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{\sqrt{c x^{4} + a}}{{\left (e x + 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{\sqrt{a + c x^{4}}}{\left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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