Optimal. Leaf size=146 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(B d-A e) \log (d+e x)}{a e^2-b d e+c d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.224456, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {800, 634, 618, 206, 628} \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(B d-A e) \log (d+e x)}{a e^2-b d e+c d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x) \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac{e (-B d+A e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{A c d-A b e+a B e+c (B d-A e) x}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac{(B d-A e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac{\int \frac{A c d-A b e+a B e+c (B d-A e) x}{a+b x+c x^2} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac{(B d-A e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac{(B d-A e) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )}-\frac{(b B d-2 A c d+A b e-2 a B e) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(B d-A e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac{(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\frac{(b B d-2 A c d+A b e-2 a B e) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c d^2-b d e+a e^2}\\ &=\frac{(b B d-2 A c d+A b e-2 a B e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )}-\frac{(B d-A e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac{(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.149245, size = 125, normalized size = 0.86 \[ \frac{\sqrt{4 a c-b^2} (B d-A e) (2 \log (d+e x)-\log (a+x (b+c x)))+2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{2 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.006, size = 343, normalized size = 2.4 \begin{align*}{\frac{\ln \left ( ex+d \right ) Ae}{a{e}^{2}-bde+c{d}^{2}}}-{\frac{\ln \left ( ex+d \right ) Bd}{a{e}^{2}-bde+c{d}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Ae}{2\,a{e}^{2}-2\,bde+2\,c{d}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Bd}{2\,a{e}^{2}-2\,bde+2\,c{d}^{2}}}-{\frac{Abe}{a{e}^{2}-bde+c{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{Acd}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{aBe}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{Bbd}{a{e}^{2}-bde+c{d}^{2}}\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.
[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 = 23.9728, size = 905, normalized size = 6.2 \begin{align*} \left [\frac{\sqrt{b^{2} - 4 \, a c}{\left ({\left (B b - 2 \, A c\right )} d -{\left (2 \, B a - A b\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 ) +{\left ({\left (B b^{2} - 4 \, B a c\right )} d -{\left (A b^{2} - 4 \, A a c\right )} e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left ({\left (B b^{2} - 4 \, B a c\right )} d -{\left (A b^{2} - 4 \, A a c\right )} e\right )} \log \left (e x + d\right )}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} -{\left (b^{3} - 4 \, a b c\right )} d e +{\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c}{\left ({\left (B b - 2 \, A c\right )} d -{\left (2 \, B a - A b\right )} e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (B b^{2} - 4 \, B a c\right )} d -{\left (A b^{2} - 4 \, A a c\right )} e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left ({\left (B b^{2} - 4 \, B a c\right )} d -{\left (A b^{2} - 4 \, A a c\right )} e\right )} \log \left (e x + d\right )}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} -{\left (b^{3} - 4 \, a b c\right )} d e +{\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12122, size = 209, normalized size = 1.43 \begin{align*} \frac{{\left (B d - A e\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}} - \frac{{\left (B d e - A e^{2}\right )} \log \left ({\left | x e + d \right |}\right )}{c d^{2} e - b d e^{2} + a e^{3}} - \frac{{\left (B b d - 2 \, A c d - 2 \, B a e + A b e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]