Optimal. Leaf size=53 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{a e+b d}}{\sqrt{d} \sqrt{a-b x^2+c x^4}}\right )}{\sqrt{d} \sqrt{a e+b d}} \]
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Rubi [A] time = 0.260311, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {2112, 205} \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{a e+b d}}{\sqrt{d} \sqrt{a-b x^2+c x^4}}\right )}{\sqrt{d} \sqrt{a e+b d}} \]
Antiderivative was successfully verified.
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Rule 2112
Rule 205
Rubi steps
\begin{align*} \int \frac{a-c x^4}{\sqrt{a-b x^2+c x^4} \left (a d+a e x^2+c d x^4\right )} \, dx &=a \operatorname{Subst}\left (\int \frac{1}{a d-\left (-a b d-a^2 e\right ) x^2} \, dx,x,\frac{x}{\sqrt{a-b x^2+c x^4}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b d+a e} x}{\sqrt{d} \sqrt{a-b x^2+c x^4}}\right )}{\sqrt{d} \sqrt{b d+a e}}\\ \end{align*}
Mathematica [C] time = 1.42708, size = 416, normalized size = 7.85 \[ \frac{i \sqrt{\frac{4 c x^2}{\sqrt{b^2-4 a c}-b}+2} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}} \left (-\Pi \left (\frac{\left (b-\sqrt{b^2-4 a c}\right ) d}{\sqrt{a} \sqrt{a e^2-4 c d^2}-a e};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )-\Pi \left (\frac{\left (\sqrt{b^2-4 a c}-b\right ) d}{a e+\sqrt{a} \sqrt{a e^2-4 c d^2}};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )+F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )\right )}{2 d \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} \sqrt{a-b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.053, size = 517, normalized size = 9.8 \begin{align*} -{\frac{\sqrt{2}}{4\,d}\sqrt{4-2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \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 ( b+\sqrt{-4\,ac+{b}^{2}} \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 ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}-b{x}^{2}+a}}}}-{\frac{a}{4\,d}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}cd+{{\it \_Z}}^{2}ae+ad \right ) }{\frac{-{{\it \_alpha}}^{2}e-2\,d}{{\it \_alpha}\, \left ( 2\,{{\it \_alpha}}^{2}cd+ae \right ) } \left ( -{{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}c{x}^{2}-b{{\it \_alpha}}^{2}-b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{-{\frac{{{\it \_alpha}}^{2} \left ( ae+bd \right ) }{d}}}}}{\frac{1}{\sqrt{c{x}^{4}-b{x}^{2}+a}}}} \right ){\frac{1}{\sqrt{-{\frac{{{\it \_alpha}}^{2} \left ( ae+bd \right ) }{d}}}}}}+{\frac{\sqrt{2}{\it \_alpha}\, \left ({{\it \_alpha}}^{2}cd+ae \right ) }{ad}\sqrt{2-{\frac{b{x}^{2}}{a}}-{\frac{{x}^{2}}{a}\sqrt{-4\,ac+{b}^{2}}}}\sqrt{2-{\frac{b{x}^{2}}{a}}+{\frac{{x}^{2}}{a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},-{\frac{1}{2\,acd} \left ( -{{\it \_alpha}}^{2}\sqrt{-4\,ac+{b}^{2}}cd+{{\it \_alpha}}^{2}bcd-\sqrt{-4\,ac+{b}^{2}}ae+abe \right ) },{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \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{c x^{4} - a}{{\left (c d x^{4} + a e x^{2} + a d\right )} \sqrt{c x^{4} - b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 118.877, size = 647, normalized size = 12.21 \begin{align*} \left [-\frac{\sqrt{-b d^{2} - a d e} \log \left (-\frac{c^{2} d^{2} x^{8} - 2 \,{\left (4 \, b c d^{2} + 3 \, a c d e\right )} x^{6} +{\left (8 \, a b d e + a^{2} e^{2} + 2 \,{\left (4 \, b^{2} + a c\right )} d^{2}\right )} x^{4} + a^{2} d^{2} - 2 \,{\left (4 \, a b d^{2} + 3 \, a^{2} d e\right )} x^{2} + 4 \,{\left (c d x^{5} -{\left (2 \, b d + a e\right )} x^{3} + a d x\right )} \sqrt{c x^{4} - b x^{2} + a} \sqrt{-b d^{2} - a d e}}{c^{2} d^{2} x^{8} + 2 \, a c d e x^{6} + 2 \, a^{2} d e x^{2} +{\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{4} + a^{2} d^{2}}\right )}{4 \,{\left (b d^{2} + a d e\right )}}, \frac{\arctan \left (\frac{2 \, \sqrt{c x^{4} - b x^{2} + a} \sqrt{b d^{2} + a d e} x}{c d x^{4} -{\left (2 \, b d + a e\right )} x^{2} + a d}\right )}{2 \, \sqrt{b d^{2} + a d e}}\right ] \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{a}{a d \sqrt{a - b x^{2} + c x^{4}} + a e x^{2} \sqrt{a - b x^{2} + c x^{4}} + c d x^{4} \sqrt{a - b x^{2} + c x^{4}}}\, dx - \int \frac{c x^{4}}{a d \sqrt{a - b x^{2} + c x^{4}} + a e x^{2} \sqrt{a - b x^{2} + c x^{4}} + c d x^{4} \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{c x^{4} - a}{{\left (c d x^{4} + a e x^{2} + a d\right )} \sqrt{c x^{4} - b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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