Optimal. Leaf size=151 \[ \frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^2}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.171566, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {701, 634, 618, 206, 628} \[ \frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^2}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 701
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{a+b x+c x^2} \, dx &=\int \left (\frac{e^2 (3 c d-b e)}{c^2}+\frac{e^3 x}{c}+\frac{c^2 d^3-3 a c d e^2+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{e^2 (3 c d-b e) x}{c^2}+\frac{e^3 x^2}{2 c}+\frac{\int \frac{c^2 d^3-3 a c d e^2+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{a+b x+c x^2} \, dx}{c^2}\\ &=\frac{e^2 (3 c d-b e) x}{c^2}+\frac{e^3 x^2}{2 c}+\frac{\left (e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^3}+\frac{\left ((2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^3}\\ &=\frac{e^2 (3 c d-b e) x}{c^2}+\frac{e^3 x^2}{2 c}+\frac{e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{\left ((2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\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{e^2 (3 c d-b e) x}{c^2}+\frac{e^3 x^2}{2 c}-\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\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{e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.130507, size = 148, normalized size = 0.98 \[ \frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log (a+x (b+c x))+\frac{2 (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+2 c e^2 x (3 c d-b e)+c^2 e^3 x^2}{2 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.153, size = 366, normalized size = 2.4 \begin{align*}{\frac{{e}^{3}{x}^{2}}{2\,c}}-{\frac{{e}^{3}xb}{{c}^{2}}}+3\,{\frac{d{e}^{2}x}{c}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) a{e}^{3}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{e}^{3}}{2\,{c}^{3}}}-{\frac{3\,d\ln \left ( c{x}^{2}+bx+a \right ){e}^{2}b}{2\,{c}^{2}}}+{\frac{3\,\ln \left ( c{x}^{2}+bx+a \right ){d}^{2}e}{2\,c}}+3\,{\frac{ab{e}^{3}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-6\,{\frac{ad{e}^{2}}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{d}^{3}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}{e}^{3}}{{c}^{3}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+3\,{\frac{{b}^{2}d{e}^{2}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-3\,{\frac{b{d}^{2}e}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.91266, size = 1107, normalized size = 7.33 \begin{align*} \left [\frac{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{3} x^{2} +{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} -{\left (b^{3} - 3 \, a b c\right )} e^{3}\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 (3 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d e^{2} -{\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3}\right )} x +{\left (3 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 3 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} +{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{3}\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 )} e^{3} x^{2} - 2 \,{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} -{\left (b^{3} - 3 \, a b c\right )} e^{3}\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 (3 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d e^{2} -{\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3}\right )} x +{\left (3 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 3 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} +{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{3}\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 3.44919, size = 892, normalized size = 5.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13914, size = 217, normalized size = 1.44 \begin{align*} \frac{c x^{2} e^{3} + 6 \, c d x e^{2} - 2 \, b x e^{3}}{2 \, c^{2}} + \frac{{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3} - a c e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac{{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\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.
[In]
[Out]