Optimal. Leaf size=381 \[ -\frac{\sqrt [4]{a} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{70 c^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{2 b x \left (b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{35 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{2 \sqrt [4]{a} b \left (b^2-8 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{35 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{x \left (10 a c+b^2+3 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c}+\frac{1}{7} x \left (a+b x^2+c x^4\right )^{3/2} \]
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Rubi [A] time = 0.251015, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1091, 1176, 1197, 1103, 1195} \[ -\frac{2 b x \left (b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{35 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{70 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{2 \sqrt [4]{a} b \left (b^2-8 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{35 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{x \left (10 a c+b^2+3 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c}+\frac{1}{7} x \left (a+b x^2+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 1091
Rule 1176
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\frac{1}{7} x \left (a+b x^2+c x^4\right )^{3/2}+\frac{3}{7} \int \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4} \, dx\\ &=\frac{x \left (b^2+10 a c+3 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c}+\frac{1}{7} x \left (a+b x^2+c x^4\right )^{3/2}+\frac{\int \frac{-a \left (b^2-20 a c\right )-2 b \left (b^2-8 a c\right ) x^2}{\sqrt{a+b x^2+c x^4}} \, dx}{35 c}\\ &=\frac{x \left (b^2+10 a c+3 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c}+\frac{1}{7} x \left (a+b x^2+c x^4\right )^{3/2}+\frac{\left (2 \sqrt{a} b \left (b^2-8 a c\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{35 c^{3/2}}-\frac{\left (\sqrt{a} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right )\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{35 c^{3/2}}\\ &=-\frac{2 b \left (b^2-8 a c\right ) x \sqrt{a+b x^2+c x^4}}{35 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{x \left (b^2+10 a c+3 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c}+\frac{1}{7} x \left (a+b x^2+c x^4\right )^{3/2}+\frac{2 \sqrt [4]{a} b \left (b^2-8 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 )}{35 c^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 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 )}{70 c^{7/4} \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 1.51035, size = 533, normalized size = 1.4 \[ \frac{i \left (-20 a^2 c^2+b^3 \sqrt{b^2-4 a c}+9 a b^2 c-8 a b c \sqrt{b^2-4 a c}-b^4\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )+2 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (15 a^2 c+a \left (b^2+23 b c x^2+20 c^2 x^4\right )+x^2 \left (9 b^2 c x^2+b^3+13 b c^2 x^4+5 c^3 x^6\right )\right )-i b \left (b^2-8 a c\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{70 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.214, size = 471, normalized size = 1.2 \begin{align*}{\frac{c{x}^{5}}{7}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{8\,b{x}^{3}}{35}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{x}{3\,c} \left ({\frac{9\,ac}{7}}+{\frac{3\,{b}^{2}}{35}} \right ) \sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{\sqrt{2}}{4} \left ({a}^{2}-{\frac{a}{3\,c} \left ({\frac{9\,ac}{7}}+{\frac{3\,{b}^{2}}{35}} \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}}}{\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{46\,ab}{35}}-{\frac{2\,b}{3\,c} \left ({\frac{9\,ac}{7}}+{\frac{3\,{b}^{2}}{35}} \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{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}\,{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 ({\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}, 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 \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}\, 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{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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