Optimal. Leaf size=276 \[ \frac {2 c \text {Li}_2\left (-\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{-b \sqrt {b^2-4 a c}-4 a c+b^2}+\frac {2 c \text {Li}_2\left (-\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {c x^2}{-b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {c x^2}{b \sqrt {b^2-4 a c}-4 a c+b^2}+\frac {2 c x \log \left (\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}+1\right )}{-b \sqrt {b^2-4 a c}-4 a c+b^2}+\frac {2 c x \log \left (\frac {2 c e^x}{\sqrt {b^2-4 a c}+b}+1\right )}{b \sqrt {b^2-4 a c}-4 a c+b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.43, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2263, 2184, 2190, 2279, 2391} \[ \frac {2 c \text {PolyLog}\left (2,-\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{-b \sqrt {b^2-4 a c}-4 a c+b^2}+\frac {2 c \text {PolyLog}\left (2,-\frac {2 c e^x}{\sqrt {b^2-4 a c}+b}\right )}{b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {c x^2}{-b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {c x^2}{b \sqrt {b^2-4 a c}-4 a c+b^2}+\frac {2 c x \log \left (\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}+1\right )}{-b \sqrt {b^2-4 a c}-4 a c+b^2}+\frac {2 c x \log \left (\frac {2 c e^x}{\sqrt {b^2-4 a c}+b}+1\right )}{b \sqrt {b^2-4 a c}-4 a c+b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2184
Rule 2190
Rule 2263
Rule 2279
Rule 2391
Rubi steps
\begin {align*} \int \frac {x}{a+b e^x+c e^{2 x}} \, dx &=\frac {(2 c) \int \frac {x}{b-\sqrt {b^2-4 a c}+2 c e^x} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {x}{b+\sqrt {b^2-4 a c}+2 c e^x} \, dx}{\sqrt {b^2-4 a c}}\\ &=-\frac {c x^2}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {\left (4 c^2\right ) \int \frac {e^x x}{b-\sqrt {b^2-4 a c}+2 c e^x} \, dx}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {\left (4 c^2\right ) \int \frac {e^x x}{b+\sqrt {b^2-4 a c}+2 c e^x} \, dx}{b^2-4 a c+b \sqrt {b^2-4 a c}}\\ &=-\frac {c x^2}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}-\frac {(2 c) \int \log \left (1+\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {(2 c) \int \log \left (1+\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{b^2-4 a c+b \sqrt {b^2-4 a c}}\\ &=-\frac {c x^2}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,e^x\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,e^x\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}\\ &=-\frac {c x^2}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {2 c \text {Li}_2\left (-\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {2 c \text {Li}_2\left (-\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 205, normalized size = 0.74 \[ \frac {-\left (\sqrt {b^2-4 a c}+b\right ) \text {Li}_2\left (\frac {2 c e^x}{\sqrt {b^2-4 a c}-b}\right )+\left (b-\sqrt {b^2-4 a c}\right ) \text {Li}_2\left (-\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )+x \left (x \sqrt {b^2-4 a c}-\left (\sqrt {b^2-4 a c}+b\right ) \log \left (\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}+1\right )+\left (b-\sqrt {b^2-4 a c}\right ) \log \left (\frac {2 c e^x}{\sqrt {b^2-4 a c}+b}+1\right )\right )}{2 a \sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 280, normalized size = 1.01 \[ \frac {{\left (b^{2} - 4 \, a c\right )} x^{2} - {\left (a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 4 \, a c\right )} {\rm Li}_2\left (-\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{x} + b e^{x} + 2 \, a}{2 \, a} + 1\right ) + {\left (a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b^{2} + 4 \, a c\right )} {\rm Li}_2\left (\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{x} - b e^{x} - 2 \, a}{2 \, a} + 1\right ) - {\left (a b x \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + {\left (b^{2} - 4 \, a c\right )} x\right )} \log \left (\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{x} + b e^{x} + 2 \, a}{2 \, a}\right ) + {\left (a b x \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - {\left (b^{2} - 4 \, a c\right )} x\right )} \log \left (-\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{x} - b e^{x} - 2 \, a}{2 \, a}\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{c e^{\left (2 \, x\right )} + b e^{x} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 378, normalized size = 1.37 \[ -\frac {b x \ln \left (\frac {-2 c \,{\mathrm e}^{x}-b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a}+\frac {b x \ln \left (\frac {2 c \,{\mathrm e}^{x}+b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a}-\frac {b \dilog \left (\frac {-2 c \,{\mathrm e}^{x}-b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a}+\frac {b \dilog \left (\frac {2 c \,{\mathrm e}^{x}+b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a}+\frac {x^{2}}{2 a}-\frac {x \ln \left (\frac {-2 c \,{\mathrm e}^{x}-b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 a}-\frac {x \ln \left (\frac {2 c \,{\mathrm e}^{x}+b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 a}-\frac {\dilog \left (\frac {-2 c \,{\mathrm e}^{x}-b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 a}-\frac {\dilog \left (\frac {2 c \,{\mathrm e}^{x}+b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{a+b\,{\mathrm {e}}^x+c\,{\mathrm {e}}^{2\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{a + b e^{x} + c e^{2 x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________