Optimal. Leaf size=336 \[ -\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} B \sqrt{c}-3 A c+2 b B\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{6 c^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{x \sqrt{a+b x^2+c x^4} (2 b B-3 A c)}{3 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} (2 b B-3 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{B x \sqrt{a+b x^2+c x^4}}{3 c} \]
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Rubi [A] time = 0.149233, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1279, 1197, 1103, 1195} \[ -\frac{x \sqrt{a+b x^2+c x^4} (2 b B-3 A c)}{3 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} B \sqrt{c}-3 A c+2 b B\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{6 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} (2 b B-3 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{B x \sqrt{a+b x^2+c x^4}}{3 c} \]
Antiderivative was successfully verified.
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Rule 1279
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \frac{x^2 \left (A+B x^2\right )}{\sqrt{a+b x^2+c x^4}} \, dx &=\frac{B x \sqrt{a+b x^2+c x^4}}{3 c}-\frac{\int \frac{a B+(2 b B-3 A c) x^2}{\sqrt{a+b x^2+c x^4}} \, dx}{3 c}\\ &=\frac{B x \sqrt{a+b x^2+c x^4}}{3 c}+\frac{\left (\sqrt{a} (2 b B-3 A c)\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{3 c^{3/2}}-\frac{\left (\sqrt{a} \left (2 b B+\sqrt{a} B \sqrt{c}-3 A c\right )\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{3 c^{3/2}}\\ &=\frac{B x \sqrt{a+b x^2+c x^4}}{3 c}-\frac{(2 b B-3 A c) x \sqrt{a+b x^2+c x^4}}{3 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} (2 b B-3 A c) \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}} E\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 )}{3 c^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (2 b B+\sqrt{a} B \sqrt{c}-3 A c\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}} 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 )}{6 c^{7/4} \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 1.38167, size = 479, normalized size = 1.43 \[ \frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \left (-3 A c \sqrt{b^2-4 a c}+2 b B \sqrt{b^2-4 a c}+2 a B c+3 A b c-2 b^2 B\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )-i \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} (2 b B-3 A c) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+4 B c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (a+b x^2+c x^4\right )}{12 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 607, normalized size = 1.8 \begin{align*} \text{result too large to display} \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{{\left (B x^{2} + A\right )} x^{2}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B x^{4} + A x^{2}}{\sqrt{c x^{4} + b x^{2} + a}}, x\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{x^{2} \left (A + B 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{{\left (B x^{2} + A\right )} x^{2}}{\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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