Optimal. Leaf size=276 \[ -\frac {c x^2}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {2 c \text {Li}_2\left (-\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {2 c \text {Li}_2\left (-\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}} \]
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Rubi [A]
time = 0.28, 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 = {2295, 2215,
2221, 2317, 2438} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2295
Rule 2317
Rule 2438
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) \text {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) \text {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*}
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Mathematica [A]
time = 0.17, size = 205, normalized size = 0.74 \begin {gather*} \frac {x \left (\sqrt {b^2-4 a c} x-\left (b+\sqrt {b^2-4 a c}\right ) \log \left (1+\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )+\left (b-\sqrt {b^2-4 a c}\right ) \log \left (1+\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )\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 )+\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 )}{2 a \sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 376, normalized size = 1.36
method | result | size |
default | \(\frac {x^{2}}{2 a}+\frac {-\frac {x \left (\ln \left (\frac {-2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}+\ln \left (\frac {-2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b +\ln \left (\frac {2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}-\ln \left (\frac {2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b \right )}{2 \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {-2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}+\dilog \left (\frac {-2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b +\dilog \left (\frac {2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}-\dilog \left (\frac {2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 \sqrt {-4 c a +b^{2}}}}{a}\) | \(376\) |
risch | \(\frac {x^{2}}{2 a}-\frac {x \ln \left (\frac {2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right )}{2 a}-\frac {x \ln \left (\frac {-2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right )}{2 a}+\frac {x \ln \left (\frac {2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a \sqrt {-4 c a +b^{2}}}-\frac {x \ln \left (\frac {-2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {-2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right )}{2 a}-\frac {\dilog \left (\frac {2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right )}{2 a}-\frac {\dilog \left (\frac {-2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a \sqrt {-4 c a +b^{2}}}+\frac {\dilog \left (\frac {2 c \,{\mathrm e}^{x}+\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a \sqrt {-4 c a +b^{2}}}\) | \(378\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.41, size = 280, normalized size = 1.01 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{a + b e^{x} + c e^{2 x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x}{a+b\,{\mathrm {e}}^x+c\,{\mathrm {e}}^{2\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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