Optimal. Leaf size=115 \[ \frac{x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{b \sqrt{a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 c^{3/2}} \]
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Rubi [A] time = 0.0922341, antiderivative size = 115, 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 = {1114, 738, 640, 621, 206} \[ \frac{x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{b \sqrt{a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 738
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\operatorname{Subst}\left (\int \frac{2 a+b x}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{b^2-4 a c}\\ &=\frac{x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{b \sqrt{a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 c}\\ &=\frac{x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{b \sqrt{a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{c}\\ &=\frac{x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{b \sqrt{a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0996468, size = 107, normalized size = 0.93 \[ \frac{\frac{2 \sqrt{c} \left (a \left (b-2 c x^2\right )+b^2 x^2\right )}{\sqrt{a+b x^2+c x^4}}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 c^{3/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.171, size = 149, normalized size = 1.3 \begin{align*} -{\frac{{x}^{2}}{2\,c}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{b}{4\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{{b}^{2}{x}^{2}}{2\,c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{{b}^{3}}{4\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{1}{2}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \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 = 1.83386, size = 833, normalized size = 7.24 \begin{align*} \left [\frac{{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 4 \, a c\right ) - 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b c +{\left (b^{2} c - 2 \, a c^{2}\right )} x^{2}\right )}}{4 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )}}, -\frac{{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b c +{\left (b^{2} c - 2 \, a c^{2}\right )} x^{2}\right )}}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )}}\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^{5}}{\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 [A] time = 1.28782, size = 255, normalized size = 2.22 \begin{align*} -\frac{\frac{{\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x^{2}}{b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}} + \frac{a b^{3} - 4 \, a^{2} b c}{b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}}}{\sqrt{c x^{4} + b x^{2} + a}} - \frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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