Optimal. Leaf size=89 \[ \frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}-\frac{b x}{c^2}+\frac{x^2}{2 c} \]
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Rubi [A] time = 0.093805, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1585, 701, 634, 618, 206, 628} \[ \frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}-\frac{b x}{c^2}+\frac{x^2}{2 c} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 701
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^5}{a x^2+b x^3+c x^4} \, dx &=\int \frac{x^3}{a+b x+c x^2} \, dx\\ &=\int \left (-\frac{b}{c^2}+\frac{x}{c}+\frac{a b+\left (b^2-a c\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac{b x}{c^2}+\frac{x^2}{2 c}+\frac{\int \frac{a b+\left (b^2-a c\right ) x}{a+b x+c x^2} \, dx}{c^2}\\ &=-\frac{b x}{c^2}+\frac{x^2}{2 c}-\frac{\left (b \left (b^2-3 a c\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^3}+\frac{\left (b^2-a c\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^3}\\ &=-\frac{b x}{c^2}+\frac{x^2}{2 c}+\frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac{\left (b \left (b^2-3 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^3}\\ &=-\frac{b x}{c^2}+\frac{x^2}{2 c}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}+\frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.109711, size = 84, normalized size = 0.94 \[ \frac{\left (b^2-a c\right ) \log (a+x (b+c x))-\frac{2 b \left (b^2-3 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+c x (c x-2 b)}{2 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 132, normalized size = 1.5 \begin{align*}{\frac{{x}^{2}}{2\,c}}-{\frac{bx}{{c}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) a}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}}{2\,{c}^{3}}}+3\,{\frac{ab}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}}{{c}^{3}}\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 = 1.59039, size = 653, normalized size = 7.34 \begin{align*} \left [\frac{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} -{\left (b^{3} - 3 \, a b c\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 ) - 2 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x +{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 2 \,{\left (b^{3} - 3 \, a b c\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 ) - 2 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x +{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.783942, size = 381, normalized size = 4.28 \begin{align*} - \frac{b x}{c^{2}} + \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{2 c^{3}}\right ) \log{\left (x + \frac{2 a^{2} c - a b^{2} + 4 a c^{3} \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{2 c^{3}}\right ) - b^{2} c^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{2 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{2 c^{3}}\right ) \log{\left (x + \frac{2 a^{2} c - a b^{2} + 4 a c^{3} \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{2 c^{3}}\right ) - b^{2} c^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{2 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \frac{x^{2}}{2 c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08281, size = 116, normalized size = 1.3 \begin{align*} \frac{c x^{2} - 2 \, b x}{2 \, c^{2}} + \frac{{\left (b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac{{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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