3.2.32 \(\int \frac {e^{-2 x} (-14 x+16 x^2-2 x^3+8 x \log (7-x)+(-28+60 x-8 x^2) \log ^2(7-x))}{81 (-7+x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{81} e^{-2 x} x \left (x+4 \log ^2(7-x)\right ) \]

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Rubi [F]  time = 0.71, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-2 x} \left (-14 x+16 x^2-2 x^3+8 x \log (7-x)+\left (-28+60 x-8 x^2\right ) \log ^2(7-x)\right )}{81 (-7+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-14*x + 16*x^2 - 2*x^3 + 8*x*Log[7 - x] + (-28 + 60*x - 8*x^2)*Log[7 - x]^2)/(81*E^(2*x)*(-7 + x)),x]

[Out]

x^2/(81*E^(2*x)) + (4*ExpIntegralEi[2*(7 - x)])/(81*E^14) - (4*Log[7 - x])/(81*E^(2*x)) + (56*ExpIntegralEi[2*
(7 - x)]*Log[7 - x])/(81*E^14) - (56*Defer[Int][ExpIntegralEi[14 - 2*x]/(-7 + x), x])/(81*E^14) - (52*Defer[In
t][Log[7 - x]^2/E^(2*x), x])/81 + (8*Defer[Int][((7 - x)*Log[7 - x]^2)/E^(2*x), x])/81

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{81} \int \frac {e^{-2 x} \left (-14 x+16 x^2-2 x^3+8 x \log (7-x)+\left (-28+60 x-8 x^2\right ) \log ^2(7-x)\right )}{-7+x} \, dx\\ &=\frac {1}{81} \int \left (-2 e^{-2 x} (-1+x) x+\frac {8 e^{-2 x} x \log (7-x)}{-7+x}-4 e^{-2 x} (-1+2 x) \log ^2(7-x)\right ) \, dx\\ &=-\left (\frac {2}{81} \int e^{-2 x} (-1+x) x \, dx\right )-\frac {4}{81} \int e^{-2 x} (-1+2 x) \log ^2(7-x) \, dx+\frac {8}{81} \int \frac {e^{-2 x} x \log (7-x)}{-7+x} \, dx\\ &=-\frac {4}{81} e^{-2 x} \log (7-x)+\frac {56 \text {Ei}(2 (7-x)) \log (7-x)}{81 e^{14}}-\frac {2}{81} \int \left (-e^{-2 x} x+e^{-2 x} x^2\right ) \, dx-\frac {4}{81} \int \left (13 e^{-2 x} \log ^2(7-x)-2 e^{-2 x} (7-x) \log ^2(7-x)\right ) \, dx-\frac {8}{81} \int \frac {e^{-2 x}-\frac {14 \text {Ei}(14-2 x)}{e^{14}}}{14-2 x} \, dx\\ &=-\frac {4}{81} e^{-2 x} \log (7-x)+\frac {56 \text {Ei}(2 (7-x)) \log (7-x)}{81 e^{14}}+\frac {2}{81} \int e^{-2 x} x \, dx-\frac {2}{81} \int e^{-2 x} x^2 \, dx-\frac {8}{81} \int \left (-\frac {e^{-2 x}}{2 (-7+x)}+\frac {7 \text {Ei}(14-2 x)}{e^{14} (-7+x)}\right ) \, dx+\frac {8}{81} \int e^{-2 x} (7-x) \log ^2(7-x) \, dx-\frac {52}{81} \int e^{-2 x} \log ^2(7-x) \, dx\\ &=-\frac {1}{81} e^{-2 x} x+\frac {1}{81} e^{-2 x} x^2-\frac {4}{81} e^{-2 x} \log (7-x)+\frac {56 \text {Ei}(2 (7-x)) \log (7-x)}{81 e^{14}}+\frac {1}{81} \int e^{-2 x} \, dx-\frac {2}{81} \int e^{-2 x} x \, dx+\frac {4}{81} \int \frac {e^{-2 x}}{-7+x} \, dx+\frac {8}{81} \int e^{-2 x} (7-x) \log ^2(7-x) \, dx-\frac {52}{81} \int e^{-2 x} \log ^2(7-x) \, dx-\frac {56 \int \frac {\text {Ei}(14-2 x)}{-7+x} \, dx}{81 e^{14}}\\ &=-\frac {1}{162} e^{-2 x}+\frac {1}{81} e^{-2 x} x^2+\frac {4 \text {Ei}(2 (7-x))}{81 e^{14}}-\frac {4}{81} e^{-2 x} \log (7-x)+\frac {56 \text {Ei}(2 (7-x)) \log (7-x)}{81 e^{14}}-\frac {1}{81} \int e^{-2 x} \, dx+\frac {8}{81} \int e^{-2 x} (7-x) \log ^2(7-x) \, dx-\frac {52}{81} \int e^{-2 x} \log ^2(7-x) \, dx-\frac {56 \int \frac {\text {Ei}(14-2 x)}{-7+x} \, dx}{81 e^{14}}\\ &=\frac {1}{81} e^{-2 x} x^2+\frac {4 \text {Ei}(2 (7-x))}{81 e^{14}}-\frac {4}{81} e^{-2 x} \log (7-x)+\frac {56 \text {Ei}(2 (7-x)) \log (7-x)}{81 e^{14}}+\frac {8}{81} \int e^{-2 x} (7-x) \log ^2(7-x) \, dx-\frac {52}{81} \int e^{-2 x} \log ^2(7-x) \, dx-\frac {56 \int \frac {\text {Ei}(14-2 x)}{-7+x} \, dx}{81 e^{14}}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.36, size = 59, normalized size = 2.68 \begin {gather*} \frac {1}{81} \int \frac {e^{-2 x} \left (-14 x+16 x^2-2 x^3+8 x \log (7-x)+\left (-28+60 x-8 x^2\right ) \log ^2(7-x)\right )}{-7+x} \, dx \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-14*x + 16*x^2 - 2*x^3 + 8*x*Log[7 - x] + (-28 + 60*x - 8*x^2)*Log[7 - x]^2)/(81*E^(2*x)*(-7 + x)),
x]

[Out]

Integrate[(-14*x + 16*x^2 - 2*x^3 + 8*x*Log[7 - x] + (-28 + 60*x - 8*x^2)*Log[7 - x]^2)/(E^(2*x)*(-7 + x)), x]
/81

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fricas [A]  time = 0.64, size = 34, normalized size = 1.55 \begin {gather*} 4 \, x e^{\left (-2 \, x - 4 \, \log \relax (3)\right )} \log \left (-x + 7\right )^{2} + x^{2} e^{\left (-2 \, x - 4 \, \log \relax (3)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+60*x-28)*log(-x+7)^2+8*x*log(-x+7)-2*x^3+16*x^2-14*x)/(x-7)/exp(2*log(3)+x)^2,x, algorithm=
"fricas")

[Out]

4*x*e^(-2*x - 4*log(3))*log(-x + 7)^2 + x^2*e^(-2*x - 4*log(3))

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giac [A]  time = 0.38, size = 25, normalized size = 1.14 \begin {gather*} \frac {4}{81} \, x e^{\left (-2 \, x\right )} \log \left (-x + 7\right )^{2} + \frac {1}{81} \, x^{2} e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+60*x-28)*log(-x+7)^2+8*x*log(-x+7)-2*x^3+16*x^2-14*x)/(x-7)/exp(2*log(3)+x)^2,x, algorithm=
"giac")

[Out]

4/81*x*e^(-2*x)*log(-x + 7)^2 + 1/81*x^2*e^(-2*x)

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maple [A]  time = 0.06, size = 26, normalized size = 1.18

method	result	size
risch	\(\frac {4 x \,{\mathrm e}^{-2 x} \ln \left (-x +7\right )^{2}}{81}+\frac {x^{2} {\mathrm e}^{-2 x}}{81}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^2+60*x-28)*ln(-x+7)^2+8*x*ln(-x+7)-2*x^3+16*x^2-14*x)/(x-7)/exp(2*ln(3)+x)^2,x,method=_RETURNVERBOS
E)

[Out]

4/81*x*exp(-2*x)*ln(-x+7)^2+1/81*x^2*exp(-2*x)

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maxima [A]  time = 0.54, size = 25, normalized size = 1.14 \begin {gather*} \frac {4}{81} \, x e^{\left (-2 \, x\right )} \log \left (-x + 7\right )^{2} + \frac {1}{81} \, x^{2} e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+60*x-28)*log(-x+7)^2+8*x*log(-x+7)-2*x^3+16*x^2-14*x)/(x-7)/exp(2*log(3)+x)^2,x, algorithm=
"maxima")

[Out]

4/81*x*e^(-2*x)*log(-x + 7)^2 + 1/81*x^2*e^(-2*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {{\mathrm {e}}^{-2\,x-4\,\ln \relax (3)}\,\left (14\,x-8\,x\,\ln \left (7-x\right )+{\ln \left (7-x\right )}^2\,\left (8\,x^2-60\,x+28\right )-16\,x^2+2\,x^3\right )}{x-7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(- 2*x - 4*log(3))*(14*x - 8*x*log(7 - x) + log(7 - x)^2*(8*x^2 - 60*x + 28) - 16*x^2 + 2*x^3))/(x -
7),x)

[Out]

int(-(exp(- 2*x - 4*log(3))*(14*x - 8*x*log(7 - x) + log(7 - x)^2*(8*x^2 - 60*x + 28) - 16*x^2 + 2*x^3))/(x -
7), x)

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sympy [A]  time = 0.41, size = 19, normalized size = 0.86 \begin {gather*} \frac {\left (x^{2} + 4 x \log {\left (7 - x \right )}^{2}\right ) e^{- 2 x}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**2+60*x-28)*ln(-x+7)**2+8*x*ln(-x+7)-2*x**3+16*x**2-14*x)/(x-7)/exp(2*ln(3)+x)**2,x)

[Out]

(x**2 + 4*x*log(7 - x)**2)*exp(-2*x)/81

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