Optimal. Leaf size=118 \[ -\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}-\frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{x \left (b^2-a c\right )}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c} \]
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Rubi [A] time = 0.104779, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1354, 701, 634, 618, 206, 628} \[ -\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}-\frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{x \left (b^2-a c\right )}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c} \]
Antiderivative was successfully verified.
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Rule 1354
Rule 701
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2}{c+\frac{a}{x^2}+\frac{b}{x}} \, dx &=\int \frac{x^4}{a+b x+c x^2} \, dx\\ &=\int \left (\frac{b^2-a c}{c^3}-\frac{b x}{c^2}+\frac{x^2}{c}-\frac{a \left (b^2-a c\right )+b \left (b^2-2 a c\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{\left (b^2-a c\right ) x}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c}-\frac{\int \frac{a \left (b^2-a c\right )+b \left (b^2-2 a c\right ) x}{a+b x+c x^2} \, dx}{c^3}\\ &=\frac{\left (b^2-a c\right ) x}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c}-\frac{\left (b \left (b^2-2 a c\right )\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}+\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^4}\\ &=\frac{\left (b^2-a c\right ) x}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c}-\frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4}\\ &=\frac{\left (b^2-a c\right ) x}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c}-\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}-\frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end{align*}
Mathematica [A] time = 0.084465, size = 112, normalized size = 0.95 \[ \frac{\frac{6 \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+c x \left (-6 a c+6 b^2-3 b c x+2 c^2 x^2\right )-3 \left (b^3-2 a b c\right ) \log (a+x (b+c x))}{6 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 190, normalized size = 1.6 \begin{align*}{\frac{{x}^{3}}{3\,c}}-{\frac{b{x}^{2}}{2\,{c}^{2}}}-{\frac{ax}{{c}^{2}}}+{\frac{{b}^{2}x}{{c}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) ab}{{c}^{3}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}}{2\,{c}^{4}}}+2\,{\frac{{a}^{2}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{a{b}^{2}}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}}{{c}^{4}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{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 = 1.82093, size = 829, normalized size = 7.03 \begin{align*} \left [\frac{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} - 3 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 3 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 6 \,{\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} x - 3 \,{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{6 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, \frac{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} - 3 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} - 6 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 6 \,{\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} x - 3 \,{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{6 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.963526, size = 496, normalized size = 4.2 \begin{align*} - \frac{b x^{2}}{2 c^{2}} + \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 3 a^{2} b c + a b^{3} + 4 a c^{4} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) - b^{2} c^{3} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right )}{2 a^{2} c^{2} - 4 a b^{2} c + b^{4}} \right )} + \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 3 a^{2} b c + a b^{3} + 4 a c^{4} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) - b^{2} c^{3} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right )}{2 a^{2} c^{2} - 4 a b^{2} c + b^{4}} \right )} + \frac{x^{3}}{3 c} - \frac{x \left (a c - b^{2}\right )}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12624, size = 153, normalized size = 1.3 \begin{align*} \frac{2 \, c^{2} x^{3} - 3 \, b c x^{2} + 6 \, b^{2} x - 6 \, a c x}{6 \, c^{3}} - \frac{{\left (b^{3} - 2 \, a b c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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