3.31.41 \(\int \frac {1}{5} e^{-x} x^{-1+\frac {1}{5} e^{-x} (6+24 x+75 e^x x^2)} (6+24 x+75 e^x x^2+(18 x-24 x^2+150 e^x x^2) \log (x)) \, dx\)

Optimal. Leaf size=23 \[ x^{3 x \left (\frac {2}{5} e^{-x} \left (4+\frac {1}{x}\right )+5 x\right )} \]

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Rubi [F]  time = 2.14, 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 {1}{5} e^{-x} x^{-1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \left (6+24 x+75 e^x x^2+\left (18 x-24 x^2+150 e^x x^2\right ) \log (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^(-1 + (6 + 24*x + 75*E^x*x^2)/(5*E^x))*(6 + 24*x + 75*E^x*x^2 + (18*x - 24*x^2 + 150*E^x*x^2)*Log[x]))/
(5*E^x),x]

[Out]

(24*Defer[Int][x^((3*(2 + 8*x + 25*E^x*x^2))/(5*E^x))/E^x, x])/5 + (18*Log[x]*Defer[Int][x^((3*(2 + 8*x + 25*E
^x*x^2))/(5*E^x))/E^x, x])/5 + 30*Log[x]*Defer[Int][x^(1 + (3*(2 + 8*x + 25*E^x*x^2))/(5*E^x)), x] - (24*Log[x
]*Defer[Int][x^(1 + (3*(2 + 8*x + 25*E^x*x^2))/(5*E^x))/E^x, x])/5 + (6*Defer[Int][x^(-1 + (6 + 24*x + 75*E^x*
x^2)/(5*E^x))/E^x, x])/5 + 15*Defer[Int][x^(1 + (6 + 24*x + 75*E^x*x^2)/(5*E^x)), x] - (18*Defer[Int][Defer[In
t][x^((3*(2 + 8*x + 25*E^x*x^2))/(5*E^x))/E^x, x]/x, x])/5 - 30*Defer[Int][Defer[Int][x^(1 + 15*x^2 + (6*(1 +
4*x))/(5*E^x)), x]/x, x] + (24*Defer[Int][Defer[Int][x^(1 + 15*x^2 + (6*(1 + 4*x))/(5*E^x))/E^x, x]/x, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int e^{-x} x^{-1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \left (6+24 x+75 e^x x^2+\left (18 x-24 x^2+150 e^x x^2\right ) \log (x)\right ) \, dx\\ &=\frac {1}{5} \int \left (24 e^{-x} x^{\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )}+6 e^{-x} x^{-1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )}+75 x^{1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )}+6 e^{-x} x^{\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \left (3-4 x+25 e^x x\right ) \log (x)\right ) \, dx\\ &=\frac {6}{5} \int e^{-x} x^{-1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx+\frac {6}{5} \int e^{-x} x^{\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \left (3-4 x+25 e^x x\right ) \log (x) \, dx+\frac {24}{5} \int e^{-x} x^{\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx+15 \int x^{1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx\\ &=\frac {6}{5} \int e^{-x} x^{-1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx-\frac {6}{5} \int \frac {3 \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+25 \int x^{1+15 x^2+\frac {6}{5} e^{-x} (1+4 x)} \, dx-4 \int e^{-x} x^{1+15 x^2+\frac {6}{5} e^{-x} (1+4 x)} \, dx}{x} \, dx+\frac {24}{5} \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+15 \int x^{1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx+\frac {1}{5} (18 \log (x)) \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx-\frac {1}{5} (24 \log (x)) \int e^{-x} x^{1+\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+(30 \log (x)) \int x^{1+\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx\\ &=\frac {6}{5} \int e^{-x} x^{-1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx-\frac {6}{5} \int \left (\frac {3 \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+25 \int x^{1+15 x^2+\frac {6}{5} e^{-x} (1+4 x)} \, dx}{x}-\frac {4 \int e^{-x} x^{1+15 x^2+\frac {6}{5} e^{-x} (1+4 x)} \, dx}{x}\right ) \, dx+\frac {24}{5} \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+15 \int x^{1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx+\frac {1}{5} (18 \log (x)) \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx-\frac {1}{5} (24 \log (x)) \int e^{-x} x^{1+\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+(30 \log (x)) \int x^{1+\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx\\ &=\frac {6}{5} \int e^{-x} x^{-1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx-\frac {6}{5} \int \frac {3 \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+25 \int x^{1+15 x^2+\frac {6}{5} e^{-x} (1+4 x)} \, dx}{x} \, dx+\frac {24}{5} \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+\frac {24}{5} \int \frac {\int e^{-x} x^{1+15 x^2+\frac {6}{5} e^{-x} (1+4 x)} \, dx}{x} \, dx+15 \int x^{1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx+\frac {1}{5} (18 \log (x)) \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx-\frac {1}{5} (24 \log (x)) \int e^{-x} x^{1+\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+(30 \log (x)) \int x^{1+\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx\\ &=\frac {6}{5} \int e^{-x} x^{-1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx-\frac {6}{5} \int \left (\frac {3 \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx}{x}+\frac {25 \int x^{1+15 x^2+\frac {6}{5} e^{-x} (1+4 x)} \, dx}{x}\right ) \, dx+\frac {24}{5} \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+\frac {24}{5} \int \frac {\int e^{-x} x^{1+15 x^2+\frac {6}{5} e^{-x} (1+4 x)} \, dx}{x} \, dx+15 \int x^{1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx+\frac {1}{5} (18 \log (x)) \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx-\frac {1}{5} (24 \log (x)) \int e^{-x} x^{1+\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+(30 \log (x)) \int x^{1+\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx\\ &=\frac {6}{5} \int e^{-x} x^{-1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx-\frac {18}{5} \int \frac {\int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx}{x} \, dx+\frac {24}{5} \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+\frac {24}{5} \int \frac {\int e^{-x} x^{1+15 x^2+\frac {6}{5} e^{-x} (1+4 x)} \, dx}{x} \, dx+15 \int x^{1+\frac {1}{5} e^{-x} \left (6+24 x+75 e^x x^2\right )} \, dx-30 \int \frac {\int x^{1+15 x^2+\frac {6}{5} e^{-x} (1+4 x)} \, dx}{x} \, dx+\frac {1}{5} (18 \log (x)) \int e^{-x} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx-\frac {1}{5} (24 \log (x)) \int e^{-x} x^{1+\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx+(30 \log (x)) \int x^{1+\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.45, size = 24, normalized size = 1.04 \begin {gather*} x^{\frac {3}{5} e^{-x} \left (2+8 x+25 e^x x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(-1 + (6 + 24*x + 75*E^x*x^2)/(5*E^x))*(6 + 24*x + 75*E^x*x^2 + (18*x - 24*x^2 + 150*E^x*x^2)*Log
[x]))/(5*E^x),x]

[Out]

x^((3*(2 + 8*x + 25*E^x*x^2))/(5*E^x))

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fricas [A]  time = 0.58, size = 20, normalized size = 0.87 \begin {gather*} x^{\frac {3}{5} \, {\left (25 \, x^{2} e^{x} + 8 \, x + 2\right )} e^{\left (-x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((150*exp(x)*x^2-24*x^2+18*x)*log(x)+75*exp(x)*x^2+24*x+6)*exp(1/5*(75*exp(x)*x^2+24*x+6)*log(x)
/exp(x))/exp(x)/x,x, algorithm="fricas")

[Out]

x^(3/5*(25*x^2*e^x + 8*x + 2)*e^(-x))

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giac [A]  time = 1.07, size = 26, normalized size = 1.13 \begin {gather*} e^{\left (15 \, x^{2} \log \relax (x) + \frac {24}{5} \, x e^{\left (-x\right )} \log \relax (x) + \frac {6}{5} \, e^{\left (-x\right )} \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((150*exp(x)*x^2-24*x^2+18*x)*log(x)+75*exp(x)*x^2+24*x+6)*exp(1/5*(75*exp(x)*x^2+24*x+6)*log(x)
/exp(x))/exp(x)/x,x, algorithm="giac")

[Out]

e^(15*x^2*log(x) + 24/5*x*e^(-x)*log(x) + 6/5*e^(-x)*log(x))

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maple [A]  time = 0.09, size = 22, normalized size = 0.96

method	result	size
risch	\(x^{15 x^{2}+\frac {24 x \,{\mathrm e}^{-x}}{5}+\frac {6 \,{\mathrm e}^{-x}}{5}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((150*exp(x)*x^2-24*x^2+18*x)*ln(x)+75*exp(x)*x^2+24*x+6)*exp(1/5*(75*exp(x)*x^2+24*x+6)*ln(x)/exp(x))
/exp(x)/x,x,method=_RETURNVERBOSE)

[Out]

x^(15*x^2+24/5*x*exp(-x)+6/5*exp(-x))

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maxima [A]  time = 0.57, size = 26, normalized size = 1.13 \begin {gather*} e^{\left (15 \, x^{2} \log \relax (x) + \frac {24}{5} \, x e^{\left (-x\right )} \log \relax (x) + \frac {6}{5} \, e^{\left (-x\right )} \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((150*exp(x)*x^2-24*x^2+18*x)*log(x)+75*exp(x)*x^2+24*x+6)*exp(1/5*(75*exp(x)*x^2+24*x+6)*log(x)
/exp(x))/exp(x)/x,x, algorithm="maxima")

[Out]

e^(15*x^2*log(x) + 24/5*x*e^(-x)*log(x) + 6/5*e^(-x)*log(x))

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mupad [B]  time = 1.92, size = 20, normalized size = 0.87 \begin {gather*} x^{\frac {3\,{\mathrm {e}}^{-x}\,\left (8\,x+25\,x^2\,{\mathrm {e}}^x+2\right )}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((exp(-x)*log(x)*(24*x + 75*x^2*exp(x) + 6))/5)*exp(-x)*(24*x + 75*x^2*exp(x) + log(x)*(18*x + 150*x^2
*exp(x) - 24*x^2) + 6))/(5*x),x)

[Out]

x^((3*exp(-x)*(8*x + 25*x^2*exp(x) + 2))/5)

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sympy [A]  time = 0.61, size = 24, normalized size = 1.04 \begin {gather*} e^{\left (15 x^{2} e^{x} + \frac {24 x}{5} + \frac {6}{5}\right ) e^{- x} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((150*exp(x)*x**2-24*x**2+18*x)*ln(x)+75*exp(x)*x**2+24*x+6)*exp(1/5*(75*exp(x)*x**2+24*x+6)*ln(
x)/exp(x))/exp(x)/x,x)

[Out]

exp((15*x**2*exp(x) + 24*x/5 + 6/5)*exp(-x)*log(x))

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