Optimal. Leaf size=88 \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 a^{3/2}}-\frac{\left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{8 a x^4} \]
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Rubi [A] time = 0.0709131, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1114, 720, 724, 206} \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 a^{3/2}}-\frac{\left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{8 a x^4} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 720
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2+c x^4}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{8 a x^4}-\frac{\left (b^2-4 a c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{16 a}\\ &=-\frac{\left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{8 a x^4}+\frac{\left (b^2-4 a c\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )}{8 a}\\ &=-\frac{\left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{8 a x^4}+\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0388379, size = 88, normalized size = 1. \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 a^{3/2}}-\frac{\left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{8 a x^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.16, size = 193, normalized size = 2.2 \begin{align*} -{\frac{1}{4\,a{x}^{4}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{b}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}}{8\,{a}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{b}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{bc{x}^{2}}{8\,{a}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{c}{4\,a}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{c}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67553, size = 502, normalized size = 5.7 \begin{align*} \left [-\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{a} x^{4} \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )}}{32 \, a^{2} x^{4}}, -\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{-a} x^{4} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )}}{16 \, a^{2} x^{4}}\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{\sqrt{a + b x^{2} + c x^{4}}}{x^{5}}\, 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{\sqrt{c x^{4} + b x^{2} + a}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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