Optimal. Leaf size=395 \[ \frac{\sqrt [4]{a} \left (2 \sqrt{a} \sqrt{c} \left (2 b^2-5 a c\right )-29 a b c+8 b^3\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{210 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{2 x \left (2 b^2-5 a c\right ) \sqrt{a+b x^2+c x^4}}{105 c^2}+\frac{b x \left (8 b^2-29 a c\right ) \sqrt{a+b x^2+c x^4}}{105 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} b \left (8 b^2-29 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{105 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{x^3 \left (b+5 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c} \]
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Rubi [A] time = 0.251072, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1116, 1279, 1197, 1103, 1195} \[ -\frac{2 x \left (2 b^2-5 a c\right ) \sqrt{a+b x^2+c x^4}}{105 c^2}+\frac{b x \left (8 b^2-29 a c\right ) \sqrt{a+b x^2+c x^4}}{105 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} \left (2 \sqrt{a} \sqrt{c} \left (2 b^2-5 a c\right )-29 a b c+8 b^3\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{210 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} b \left (8 b^2-29 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{105 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{x^3 \left (b+5 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c} \]
Antiderivative was successfully verified.
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Rule 1116
Rule 1279
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int x^4 \sqrt{a+b x^2+c x^4} \, dx &=\frac{x^3 \left (b+5 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c}-\frac{\int \frac{x^2 \left (3 a b+2 \left (2 b^2-5 a c\right ) x^2\right )}{\sqrt{a+b x^2+c x^4}} \, dx}{35 c}\\ &=-\frac{2 \left (2 b^2-5 a c\right ) x \sqrt{a+b x^2+c x^4}}{105 c^2}+\frac{x^3 \left (b+5 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c}+\frac{\int \frac{2 a \left (2 b^2-5 a c\right )+b \left (8 b^2-29 a c\right ) x^2}{\sqrt{a+b x^2+c x^4}} \, dx}{105 c^2}\\ &=-\frac{2 \left (2 b^2-5 a c\right ) x \sqrt{a+b x^2+c x^4}}{105 c^2}+\frac{x^3 \left (b+5 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c}-\frac{\left (\sqrt{a} b \left (8 b^2-29 a c\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{105 c^{5/2}}+\frac{\left (\sqrt{a} \left (b \left (8 b^2-29 a c\right )+2 \sqrt{a} \sqrt{c} \left (2 b^2-5 a c\right )\right )\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{105 c^{5/2}}\\ &=-\frac{2 \left (2 b^2-5 a c\right ) x \sqrt{a+b x^2+c x^4}}{105 c^2}+\frac{b \left (8 b^2-29 a c\right ) x \sqrt{a+b x^2+c x^4}}{105 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{x^3 \left (b+5 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c}-\frac{\sqrt [4]{a} b \left (8 b^2-29 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}} 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 )}{105 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\sqrt [4]{a} \left (8 b^3-29 a b c+2 \sqrt{a} \sqrt{c} \left (2 b^2-5 a c\right )\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 )}{210 c^{11/4} \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 1.55288, size = 538, normalized size = 1.36 \[ \frac{-i \left (-20 a^2 c^2+8 b^3 \sqrt{b^2-4 a c}+37 a b^2 c-29 a b c \sqrt{b^2-4 a c}-8 b^4\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )+4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (10 a^2 c+a \left (-4 b^2+13 b c x^2+25 c^2 x^4\right )-b^2 c x^4-4 b^3 x^2+18 b c^2 x^6+15 c^3 x^8\right )+i b \left (8 b^2-29 a c\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{420 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.273, size = 476, normalized size = 1.2 \begin{align*}{\frac{{x}^{5}}{7}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{b{x}^{3}}{35\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{x}{3\,c} \left ({\frac{2\,a}{7}}-{\frac{4\,{b}^{2}}{35\,c}} \right ) \sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{a\sqrt{2}}{12\,c} \left ({\frac{2\,a}{7}}-{\frac{4\,{b}^{2}}{35\,c}} \right ) \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\sqrt{2}}{2} \left ( -{\frac{3\,ab}{35\,c}}-{\frac{2\,b}{3\,c} \left ({\frac{2\,a}{7}}-{\frac{4\,{b}^{2}}{35\,c}} \right ) } \right ) \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}}} \left ({\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 ) -{\it EllipticE} \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 ) \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}}} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \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 \sqrt{c x^{4} + b x^{2} + a} x^{4}\,{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 (\sqrt{c x^{4} + b x^{2} + a} x^{4}, 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 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 \sqrt{c x^{4} + b x^{2} + a} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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