Optimal. Leaf size=518 \[ \frac{\sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \sqrt{a+b x^2+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}+\frac{e \tan ^{-1}\left (\frac{x \sqrt{-a e^4-b d^2 e^2-c d^4}}{d e \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{-a e^4-b d^2 e^2-c d^4}}-\frac{e \tanh ^{-1}\left (\frac{2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt{a e^4+b d^2 e^2+c d^4}}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \sqrt{a+b x^2+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )} \]
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Rubi [A] time = 0.532925, antiderivative size = 518, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1724, 1216, 1103, 1706, 1247, 724, 206} \[ \frac{e \tan ^{-1}\left (\frac{x \sqrt{-a e^4-b d^2 e^2-c d^4}}{d e \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{-a e^4-b d^2 e^2-c d^4}}-\frac{e \tanh ^{-1}\left (\frac{2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt{a e^4+b d^2 e^2+c d^4}}+\frac{\sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \sqrt{a+b x^2+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \sqrt{a+b x^2+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )} \]
Antiderivative was successfully verified.
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Rule 1724
Rule 1216
Rule 1103
Rule 1706
Rule 1247
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \sqrt{a+b x^2+c x^4}} \, dx &=d \int \frac{1}{\left (d^2-e^2 x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx-e \int \frac{x}{\left (d^2-e^2 x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx\\ &=-\left (\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (d^2-e^2 x\right ) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )\right )+\frac{\left (\sqrt{c} d\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{c} d^2+\sqrt{a} e^2}+\frac{\left (\sqrt{a} d e^2\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (d^2-e^2 x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{c} d^2+\sqrt{a} e^2}\\ &=\frac{e \tan ^{-1}\left (\frac{\sqrt{-c d^4-b d^2 e^2-a e^4} x}{d e \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{-c d^4-b d^2 e^2-a e^4}}+\frac{\sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+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}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+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}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{a+b x^2+c x^4}}+e \operatorname{Subst}\left (\int \frac{1}{4 c d^4+4 b d^2 e^2+4 a e^4-x^2} \, dx,x,\frac{-b d^2-2 a e^2-\left (2 c d^2+b e^2\right ) x^2}{\sqrt{a+b x^2+c x^4}}\right )\\ &=\frac{e \tan ^{-1}\left (\frac{\sqrt{-c d^4-b d^2 e^2-a e^4} x}{d e \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{-c d^4-b d^2 e^2-a e^4}}-\frac{e \tanh ^{-1}\left (\frac{b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt{c d^4+b d^2 e^2+a e^4} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{c d^4+b d^2 e^2+a e^4}}+\frac{\sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+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}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+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}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [B] time = 6.9177, size = 1725, normalized size = 3.33 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 281, normalized size = 0.5 \begin{align*}{\frac{1}{e} \left ( -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,{\frac{c{x}^{2}{d}^{2}}{{e}^{2}}}+b{x}^{2}+{\frac{b{d}^{2}}{{e}^{2}}}+2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+{\frac{b{d}^{2}}{{e}^{2}}}+a}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+{\frac{b{d}^{2}}{{e}^{2}}}+a}}}}+{\frac{\sqrt{2}e}{d}\sqrt{1-{\frac{{x}^{2}}{2\,a} \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) }}\sqrt{1+{\frac{{x}^{2}}{2\,a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) }}},2\,{\frac{a{e}^{2}}{ \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ){d}^{2}}},{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) }}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+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} + b x^{2} + a}{\left (e x + d\right )}}\,{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}{\left (d + e x\right ) \sqrt{a + b x^{2} + c x^{4}}}\, 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} + b x^{2} + a}{\left (e x + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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