Optimal. Leaf size=186 \[ \frac {e^{d+e x}}{c e}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^2}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^2} \]
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Rubi [A]
time = 0.29, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2302, 2225,
2209} \begin {gather*} -\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^2}-\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^2}+\frac {e^{d+e x}}{c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2225
Rule 2302
Rubi steps
\begin {align*} \int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx &=\int \left (\frac {e^{d+e x}}{c}-\frac {e^{d+e x} (a+b x)}{c \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {\int e^{d+e x} \, dx}{c}-\frac {\int \frac {e^{d+e x} (a+b x)}{a+b x+c x^2} \, dx}{c}\\ &=\frac {e^{d+e x}}{c e}-\frac {\int \left (\frac {\left (b+\frac {-b^2+2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (b-\frac {-b^2+2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{c}\\ &=\frac {e^{d+e x}}{c e}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{c}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{c}\\ &=\frac {e^{d+e x}}{c e}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^2}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 217, normalized size = 1.17 \begin {gather*} -\frac {e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \left (-2 c \sqrt {b^2-4 a c} e^{\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}}+\left (-b^2+2 a c+b \sqrt {b^2-4 a c}\right ) e e^{\frac {\sqrt {b^2-4 a c} e}{c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )+\left (b^2-2 a c+b \sqrt {b^2-4 a c}\right ) e \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )\right )}{2 c^2 \sqrt {b^2-4 a c} e} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1729\) vs.
\(2(160)=320\).
time = 0.11, size = 1730, normalized size = 9.30
method | result | size |
risch | \(\frac {{\mathrm e}^{e x +d}}{c e}+\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) a}{c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2}}{2 c^{2} \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) a}{c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2}}{2 c^{2} \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b}{2 c^{2}}+\frac {{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b}{2 c^{2}}\) | \(561\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1730\) |
default | \(\text {Expression too large to display}\) | \(1730\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 266, normalized size = 1.43 \begin {gather*} \frac {{\left ({\left ({\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e - {\left (b^{3} - 4 \, a b c\right )} e\right )} {\rm Ei}\left (-\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e - {\left (2 \, c x + b\right )} e}{2 \, c}\right ) e^{\left (\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e + 2 \, c d - b e}{2 \, c}\right )} - {\left ({\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e + {\left (b^{3} - 4 \, a b c\right )} e\right )} {\rm Ei}\left (\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e + {\left (2 \, c x + b\right )} e}{2 \, c}\right ) e^{\left (-\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e - 2 \, c d + b e}{2 \, c}\right )} + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e^{\left (x e + d\right )}\right )} e^{\left (-1\right )}}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\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*} e^{d} \int \frac {x^{2} e^{e x}}{a + b x + c x^{2}}\, 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 = -0.01 \begin {gather*} \int \frac {x^2\,{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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