Optimal. Leaf size=139 \[ -\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{5/2}}+\frac{-2 a c+b^2+b c x^2}{a x^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]
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Rubi [A] time = 0.125908, antiderivative size = 139, 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, 740, 806, 724, 206} \[ -\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{5/2}}+\frac{-2 a c+b^2+b c x^2}{a x^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 740
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^2 \sqrt{a+b x^2+c x^4}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-3 b^2+8 a c\right )-b c x}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{a \left (b^2-4 a c\right )}\\ &=\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^2 \sqrt{a+b x^2+c x^4}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^2}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{4 a^2}\\ &=\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^2 \sqrt{a+b x^2+c x^4}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac{(3 b) \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 )}{2 a^2}\\ &=\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^2 \sqrt{a+b x^2+c x^4}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0853652, size = 137, normalized size = 0.99 \[ \frac{\frac{2 \sqrt{a} \left (-4 a^2 c+a \left (b^2-10 b c x^2-8 c^2 x^4\right )+3 b^2 x^2 \left (b+c x^2\right )\right )}{x^2 \sqrt{a+b x^2+c x^4}}-3 b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{5/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.17, size = 195, normalized size = 1.4 \begin{align*} -{\frac{1}{2\,a{x}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,b}{4\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{3\,{b}^{2}c{x}^{2}}{2\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{3\,{b}^{3}}{4\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{3\,b}{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 ){a}^{-{\frac{5}{2}}}}-2\,{\frac{c \left ( 2\,c{x}^{2}+b \right ) }{a \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{4}+b{x}^{2}+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 = 2.17469, size = 1025, normalized size = 7.37 \begin{align*} \left [\frac{3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{4} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt{a} \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 \,{\left ({\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c +{\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{8 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{6} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{4} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2}\right )}}, -\frac{3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{4} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt{-a} \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 \,{\left ({\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c +{\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{4 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{6} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{4} +{\left (a^{4} b^{2} - 4 \, a^{5} c\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{1}{x^{3} \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 \frac{1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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