Optimal. Leaf size=610 \[ -\frac{c^{3/4} d^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \left (a e^4+c d^4\right )}-\frac{e^3 \sqrt{a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}+\frac{\sqrt{c} e^2 x \sqrt{a+c x^4}}{\left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^4+c d^4\right )}-\frac{c d^3 e \tan ^{-1}\left (\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right )}{\left (-a e^4-c d^4\right )^{3/2}}-\frac{c d^3 e \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt{a+c x^4} \left (a e^4+c d^4\right )} \]
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Rubi [A] time = 0.760191, antiderivative size = 610, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.579, Rules used = {1727, 1742, 12, 1248, 725, 206, 1715, 1196, 1709, 220, 1707} \[ -\frac{c^{3/4} d^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \left (a e^4+c d^4\right )}-\frac{e^3 \sqrt{a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}+\frac{\sqrt{c} e^2 x \sqrt{a+c x^4}}{\left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^4+c d^4\right )}-\frac{c d^3 e \tan ^{-1}\left (\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right )}{\left (-a e^4-c d^4\right )^{3/2}}-\frac{c d^3 e \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt{a+c x^4} \left (a e^4+c d^4\right )} \]
Antiderivative was successfully verified.
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Rule 1727
Rule 1742
Rule 12
Rule 1248
Rule 725
Rule 206
Rule 1715
Rule 1196
Rule 1709
Rule 220
Rule 1707
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \sqrt{a+c x^4}} \, dx &=-\frac{e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}-\frac{c \int \frac{-d^3+d^2 e x-d e^2 x^2-e^3 x^3}{(d+e x) \sqrt{a+c x^4}} \, dx}{c d^4+a e^4}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}-\frac{c \int \frac{2 d^3 e x}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx}{c d^4+a e^4}-\frac{c \int \frac{-d^4-2 d^2 e^2 x^2+e^4 x^4}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx}{c d^4+a e^4}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}+\frac{\int \frac{c d^4 e^2+\sqrt{a} \sqrt{c} d^2 e^4+\left (2 c d^2 e^4-e^4 \left (c d^2+\sqrt{a} \sqrt{c} e^2\right )\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx}{e^2 \left (c d^4+a e^4\right )}-\frac{\left (2 c d^3 e\right ) \int \frac{x}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx}{c d^4+a e^4}-\frac{\left (\sqrt{a} \sqrt{c} e^2\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{c d^4+a e^4}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}+\frac{\sqrt{c} e^2 x \sqrt{a+c x^4}}{\left (c d^4+a e^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} e^2 \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 )}{\left (c d^4+a e^4\right ) \sqrt{a+c x^4}}+\frac{\sqrt{c} \int \frac{1}{\sqrt{a+c x^4}} \, dx}{\sqrt{c} d^2+\sqrt{a} e^2}-\frac{\left (c d^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d^2-e^2 x\right ) \sqrt{a+c x^2}} \, dx,x,x^2\right )}{c d^4+a e^4}+\frac{\left (2 \sqrt{a} c d^4 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}{\left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \left (c d^4+a e^4\right )}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}+\frac{\sqrt{c} e^2 x \sqrt{a+c x^4}}{\left (c d^4+a e^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{c d^3 e \tan ^{-1}\left (\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{a+c x^4}}\right )}{\left (-c d^4-a e^4\right )^{3/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} e^2 \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 )}{\left (c d^4+a e^4\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} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{a+c x^4}}-\frac{c^{3/4} d^2 \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}} \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 )}{2 \sqrt [4]{a} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt{a+c x^4}}+\frac{\left (c d^3 e\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 )}{c d^4+a e^4}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}+\frac{\sqrt{c} e^2 x \sqrt{a+c x^4}}{\left (c d^4+a e^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{c d^3 e \tan ^{-1}\left (\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{a+c x^4}}\right )}{\left (-c d^4-a e^4\right )^{3/2}}-\frac{c d^3 e \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 )}{\left (c d^4+a e^4\right )^{3/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} e^2 \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 )}{\left (c d^4+a e^4\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} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{a+c x^4}}-\frac{c^{3/4} d^2 \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}} \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 )}{2 \sqrt [4]{a} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 1.10013, size = 425, normalized size = 0.7 \[ \frac{-\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (2 \sqrt [4]{-1} \sqrt [4]{a} c^{3/4} d^2 \sqrt{\frac{c x^4}{a}+1} (d+e x) \sqrt{a e^4+c d^4} \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )+e^3 \left (a+c x^4\right ) \sqrt{a e^4+c d^4}+c d^3 e \sqrt{a+c x^4} (d+e x) \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )\right )+i \sqrt{c} \sqrt{\frac{c x^4}{a}+1} (d+e x) \left (\sqrt{c} d^2+i \sqrt{a} e^2\right ) \sqrt{a e^4+c d^4} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{a} \sqrt{c} e^2 \sqrt{\frac{c x^4}{a}+1} (d+e x) \sqrt{a e^4+c d^4} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} (d+e x) \left (a e^4+c d^4\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.014, size = 421, normalized size = 0.7 \begin{align*} -{\frac{{e}^{3}}{ \left ( a{e}^{4}+c{d}^{4} \right ) \left ( ex+d \right ) }\sqrt{c{x}^{4}+a}}-{\frac{c{d}^{2}}{a{e}^{4}+c{d}^{4}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{i{e}^{2}}{a{e}^{4}+c{d}^{4}}\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}}}}+2\,{\frac{c{d}^{3}}{ \left ( a{e}^{4}+c{d}^{4} \right ) e} \left ( -1/2\,{{\it Artanh} \left ( 1/2\,{\frac{1}{\sqrt{c{x}^{4}+a}} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}+2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}}+{\frac{e}{d\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}}}}}}} \right ) } \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 + 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\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 + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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