Optimal. Leaf size=271 \[ \frac{\left (x^2 \sqrt{\frac{c}{a}}+1\right ) \sqrt{\frac{a+b x^2+c x^4}{a \left (x^2 \sqrt{\frac{c}{a}}+1\right )^2}} \left (d \sqrt{\frac{c}{a}}+e\right ) \Pi \left (-\frac{\left (\sqrt{\frac{c}{a}} d-e\right )^2}{4 \sqrt{\frac{c}{a}} d e};2 \tan ^{-1}\left (\sqrt [4]{\frac{c}{a}} x\right )|\frac{1}{4} \left (2-\frac{b \sqrt{\frac{c}{a}}}{c}\right )\right )}{4 d e \sqrt [4]{\frac{c}{a}} \sqrt{a+b x^2+c x^4}}-\frac{\left (d \sqrt{\frac{c}{a}}-e\right ) \tan ^{-1}\left (\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}} \]
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Rubi [A] time = 0.186994, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.024, Rules used = {1706} \[ \frac{\left (x^2 \sqrt{\frac{c}{a}}+1\right ) \sqrt{\frac{a+b x^2+c x^4}{a \left (x^2 \sqrt{\frac{c}{a}}+1\right )^2}} \left (d \sqrt{\frac{c}{a}}+e\right ) \Pi \left (-\frac{\left (\sqrt{\frac{c}{a}} d-e\right )^2}{4 \sqrt{\frac{c}{a}} d e};2 \tan ^{-1}\left (\sqrt [4]{\frac{c}{a}} x\right )|\frac{1}{4} \left (2-\frac{b \sqrt{\frac{c}{a}}}{c}\right )\right )}{4 d e \sqrt [4]{\frac{c}{a}} \sqrt{a+b x^2+c x^4}}-\frac{\left (d \sqrt{\frac{c}{a}}-e\right ) \tan ^{-1}\left (\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}} \]
Antiderivative was successfully verified.
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Rule 1706
Rubi steps
\begin{align*} \int \frac{1+\sqrt{\frac{c}{a}} x^2}{\left (d+e x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx &=-\frac{\left (\sqrt{\frac{c}{a}} d-e\right ) \tan ^{-1}\left (\frac{\sqrt{c d^2-b d e+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{d} \sqrt{e} \sqrt{c d^2-b d e+a e^2}}+\frac{\left (\sqrt{\frac{c}{a}} d+e\right ) \left (1+\sqrt{\frac{c}{a}} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{a \left (1+\sqrt{\frac{c}{a}} x^2\right )^2}} \Pi \left (-\frac{\left (\sqrt{\frac{c}{a}} d-e\right )^2}{4 \sqrt{\frac{c}{a}} d e};2 \tan ^{-1}\left (\sqrt [4]{\frac{c}{a}} x\right )|\frac{1}{4} \left (2-\frac{b \sqrt{\frac{c}{a}}}{c}\right )\right )}{4 \sqrt [4]{\frac{c}{a}} d e \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.443384, size = 312, normalized size = 1.15 \[ -\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (d \sqrt{\frac{c}{a}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )+\left (e-d \sqrt{\frac{c}{a}}\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{\sqrt{2} d e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 369, normalized size = 1.4 \begin{align*}{\frac{\sqrt{2}}{4\,e}\sqrt{{\frac{c}{a}}}\sqrt{4-2\,{\frac{ \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \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}}}}+{\frac{\sqrt{2}}{ed} \left ( -d\sqrt{{\frac{c}{a}}}+e \right ) \sqrt{1-{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}+{\frac{b{x}^{2}}{2\,a}}}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) }}},-2\,{\frac{ae}{ \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) d}},{\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}\sqrt{-4\,ac+{b}^{2}}}-{\frac{b}{a}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+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{x^{2} \sqrt{\frac{c}{a}} + 1}{\sqrt{c x^{4} + b x^{2} + 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(-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{x^{2} \sqrt{\frac{c}{a}} + 1}{\left (d + e x^{2}\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{x^{2} \sqrt{\frac{c}{a}} + 1}{\sqrt{c x^{4} + b x^{2} + 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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