Optimal. Leaf size=169 \[ \frac{\left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}+\frac{\left (2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac{x \left (a c e+b^2 (-e)+b c d\right )}{c^3}+\frac{x^2 (c d-b e)}{2 c^2}+\frac{e x^3}{3 c} \]
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Rubi [A] time = 0.23571, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {800, 634, 618, 206, 628} \[ \frac{\left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}+\frac{\left (2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac{x \left (a c e+b^2 (-e)+b c d\right )}{c^3}+\frac{x^2 (c d-b e)}{2 c^2}+\frac{e x^3}{3 c} \]
Antiderivative was successfully verified.
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Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3 (d+e x)}{a+b x+c x^2} \, dx &=\int \left (-\frac{b c d-b^2 e+a c e}{c^3}+\frac{(c d-b e) x}{c^2}+\frac{e x^2}{c}+\frac{a \left (b c d-b^2 e+a c e\right )+\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac{\left (b c d-b^2 e+a c e\right ) x}{c^3}+\frac{(c d-b e) x^2}{2 c^2}+\frac{e x^3}{3 c}+\frac{\int \frac{a \left (b c d-b^2 e+a c e\right )+\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{a+b x+c x^2} \, dx}{c^3}\\ &=-\frac{\left (b c d-b^2 e+a c e\right ) x}{c^3}+\frac{(c d-b e) x^2}{2 c^2}+\frac{e x^3}{3 c}+\frac{\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}-\frac{\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^4}\\ &=-\frac{\left (b c d-b^2 e+a c e\right ) x}{c^3}+\frac{(c d-b e) x^2}{2 c^2}+\frac{e x^3}{3 c}+\frac{\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\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 c d-b^2 e+a c e\right ) x}{c^3}+\frac{(c d-b e) x^2}{2 c^2}+\frac{e x^3}{3 c}+\frac{\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\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{\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end{align*}
Mathematica [A] time = 0.101373, size = 165, normalized size = 0.98 \[ \frac{\frac{6 \left (2 a^2 c^2 e-4 a b^2 c e+3 a b c^2 d-b^3 c d+b^4 e\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-3 \left (-2 a b c e+a c^2 d-b^2 c d+b^3 e\right ) \log (a+x (b+c x))-6 c x \left (a c e+b^2 (-e)+b c d\right )+3 c^2 x^2 (c d-b e)+2 c^3 e x^3}{6 c^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 335, normalized size = 2. \begin{align*}{\frac{e{x}^{3}}{3\,c}}-{\frac{b{x}^{2}e}{2\,{c}^{2}}}+{\frac{d{x}^{2}}{2\,c}}-{\frac{aex}{{c}^{2}}}+{\frac{{b}^{2}ex}{{c}^{3}}}-{\frac{bdx}{{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) abe}{{c}^{3}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) ad}{2\,{c}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}e}{2\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}d}{2\,{c}^{3}}}+2\,{\frac{{a}^{2}e}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{a{b}^{2}e}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+3\,{\frac{abd}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}e}{{c}^{4}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}d}{{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 = 1.66374, size = 1180, normalized size = 6.98 \begin{align*} \left [\frac{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e x^{3} + 3 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d -{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e\right )} x^{2} - 3 \, \sqrt{b^{2} - 4 \, a c}{\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} d -{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e\right )} \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 ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d -{\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e\right )} x + 3 \,{\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d -{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e\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 )} e x^{3} + 3 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d -{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e\right )} x^{2} + 6 \, \sqrt{-b^{2} + 4 \, a c}{\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} d -{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 6 \,{\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d -{\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e\right )} x + 3 \,{\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d -{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e\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 = 3.00058, size = 835, normalized size = 4.94 \begin{align*} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) \log{\left (x + \frac{- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + 4 a c^{4} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) - b^{2} c^{3} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right )}{2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) \log{\left (x + \frac{- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + 4 a c^{4} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) - b^{2} c^{3} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right )}{2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d} \right )} + \frac{e x^{3}}{3 c} - \frac{x^{2} \left (b e - c d\right )}{2 c^{2}} - \frac{x \left (a c e - b^{2} e + b c d\right )}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31173, size = 240, normalized size = 1.42 \begin{align*} \frac{2 \, c^{2} x^{3} e + 3 \, c^{2} d x^{2} - 3 \, b c x^{2} e - 6 \, b c d x + 6 \, b^{2} x e - 6 \, a c x e}{6 \, c^{3}} + \frac{{\left (b^{2} c d - a c^{2} d - b^{3} e + 2 \, a b c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} - \frac{{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\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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