Optimal. Leaf size=64 \[ \frac {(b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {B \log \left (a+b x+c x^2\right )}{2 c} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {634, 618, 206, 628} \begin {gather*} \frac {(b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {B \log \left (a+b x+c x^2\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {A+B x}{a+b x+c x^2} \, dx &=\frac {B \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c}+\frac {(-b B+2 A c) \int \frac {1}{a+b x+c x^2} \, dx}{2 c}\\ &=\frac {B \log \left (a+b x+c x^2\right )}{2 c}-\frac {(-b B+2 A c) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c}\\ &=\frac {(b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {B \log \left (a+b x+c x^2\right )}{2 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 66, normalized size = 1.03 \begin {gather*} \frac {B \log (a+x (b+c x))-\frac {2 (b B-2 A c) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}}{2 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{a+b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 207, normalized size = 3.23 \begin {gather*} \left [-\frac {{\left (B b - 2 \, A 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 ) - {\left (B b^{2} - 4 \, B a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac {2 \, {\left (B b - 2 \, A 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 ) + {\left (B b^{2} - 4 \, B a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 63, normalized size = 0.98 \begin {gather*} \frac {B \log \left (c x^{2} + b x + a\right )}{2 \, c} - \frac {{\left (B b - 2 \, A c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 93, normalized size = 1.45 \begin {gather*} \frac {2 A \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {B b \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {B \ln \left (c \,x^{2}+b x +a \right )}{2 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, size = 162, normalized size = 2.53 \begin {gather*} \frac {2\,A\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}}-\frac {B\,b^2\,\ln \left (c\,x^2+b\,x+a\right )}{2\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,B\,a\,c\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^2-b^2\,c}-\frac {B\,b\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.69, size = 280, normalized size = 4.38 \begin {gather*} \left (\frac {B}{2 c} - \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- A b + 2 B a - 4 a c \left (\frac {B}{2 c} - \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right ) + b^{2} \left (\frac {B}{2 c} - \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right )}{- 2 A c + B b} \right )} + \left (\frac {B}{2 c} + \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- A b + 2 B a - 4 a c \left (\frac {B}{2 c} + \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right ) + b^{2} \left (\frac {B}{2 c} + \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right )}{- 2 A c + B b} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________