Optimal. Leaf size=353 \[ \frac{\left (8 \sqrt{a} b \sqrt{c}+4 a c+3 b^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{6 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a+b x^2+c x^4}}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}-\frac{\left (3 b-2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{3 x}+\frac{8 b \sqrt{c} x \sqrt{a+b x^2+c x^4}}{3 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{8 \sqrt [4]{a} b \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 \sqrt{a+b x^2+c x^4}} \]
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Rubi [A] time = 0.152889, antiderivative size = 353, 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 = {1117, 1271, 1197, 1103, 1195} \[ \frac{\left (8 \sqrt{a} b \sqrt{c}+4 a c+3 b^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{6 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a+b x^2+c x^4}}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}-\frac{\left (3 b-2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{3 x}+\frac{8 b \sqrt{c} x \sqrt{a+b x^2+c x^4}}{3 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{8 \sqrt [4]{a} b \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1117
Rule 1271
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x^4} \, dx &=-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}+\int \frac{\left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{x^2} \, dx\\ &=-\frac{\left (3 b-2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{3 x}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}-\frac{1}{3} \int \frac{-3 b^2-4 a c-8 b c x^2}{\sqrt{a+b x^2+c x^4}} \, dx\\ &=-\frac{\left (3 b-2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{3 x}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}-\frac{1}{3} \left (8 \sqrt{a} b \sqrt{c}\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx-\frac{1}{3} \left (-3 b^2-8 \sqrt{a} b \sqrt{c}-4 a c\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx\\ &=\frac{8 b \sqrt{c} x \sqrt{a+b x^2+c x^4}}{3 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\left (3 b-2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{3 x}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}-\frac{8 \sqrt [4]{a} b \sqrt [4]{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 \sqrt{a+b x^2+c x^4}}+\frac{\left (3 b^2+8 \sqrt{a} b \sqrt{c}+4 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 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.903069, size = 473, normalized size = 1.34 \[ \frac{-i x^3 \left (4 b \sqrt{b^2-4 a c}+4 a c-b^2\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )+2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (-a^2-5 a b x^2-4 b^2 x^4-3 b c x^6+c^2 x^8\right )+4 i b x^3 \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{6 x^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.226, size = 428, normalized size = 1.2 \begin{align*} -{\frac{a}{3\,{x}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{4\,b}{3\,x}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{cx}{3}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{\sqrt{2}}{4} \left ({\frac{4\,ac}{3}}+{b}^{2} \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{4\,abc\sqrt{2}}{3}\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 \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{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 (\frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{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 \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{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 (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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