Optimal. Leaf size=133 \[ \frac{3 \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^2 x^4}-\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{256 a^{5/2}}-\frac{\left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 a x^8} \]
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Rubi [A] time = 0.116517, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1114, 720, 724, 206} \[ \frac{3 \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^2 x^4}-\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{256 a^{5/2}}-\frac{\left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 a x^8} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 720
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac{\left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 a x^8}-\frac{\left (3 \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^3} \, dx,x,x^2\right )}{32 a}\\ &=\frac{3 \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^2 x^4}-\frac{\left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 a x^8}+\frac{\left (3 \left (b^2-4 a c\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{256 a^2}\\ &=\frac{3 \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^2 x^4}-\frac{\left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 a x^8}-\frac{\left (3 \left (b^2-4 a c\right )^2\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 )}{128 a^2}\\ &=\frac{3 \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^2 x^4}-\frac{\left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 a x^8}-\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{256 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.162051, size = 138, normalized size = 1.04 \[ -\frac{\frac{3 \left (b^2-4 a c\right ) \left (x^4 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )-2 \sqrt{a} \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}\right )}{8 a^{3/2} x^4}+\frac{2 \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{x^8}}{32 a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.18, size = 260, normalized size = 2. \begin{align*} -{\frac{{b}^{2}}{64\,a{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,{b}^{3}}{128\,{a}^{2}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{a}{8\,{x}^{8}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,c}{16\,{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,{b}^{2}c}{32}\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{5\,bc}{32\,a{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,b}{16\,{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{4}}{256}\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}}}}-{\frac{3\,{c}^{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 ){\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 = 2.39964, size = 726, normalized size = 5.46 \begin{align*} \left [\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{a} x^{8} \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^{3} - 20 \, a^{2} b c\right )} x^{6} - 24 \, a^{3} b x^{2} - 2 \,{\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{4} - 16 \, a^{4}\right )} \sqrt{c x^{4} + b x^{2} + a}}{512 \, a^{3} x^{8}}, \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-a} x^{8} \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^{3} - 20 \, a^{2} b c\right )} x^{6} - 24 \, a^{3} b x^{2} - 2 \,{\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{4} - 16 \, a^{4}\right )} \sqrt{c x^{4} + b x^{2} + a}}{256 \, a^{3} x^{8}}\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^{9}}\, 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^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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