3.77.68 \(\int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} (x+x^2)} (5+5 x+(5+3 x-7 x^2) \log (x)) \, dx\)

Optimal. Leaf size=24 \[ 5+x^{e^{\frac {1}{5} (5-2 x)-x} x (1+x)} \]

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Rubi [F]  time = 1.33, 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^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((5 - 7*x)/5)*x^(E^((5 - 7*x)/5)*(x + x^2))*(5 + 5*x + (5 + 3*x - 7*x^2)*Log[x]))/5,x]

[Out]

Defer[Int][E^(1 - (7*x)/5)*x^(E^(1 - (7*x)/5)*x*(1 + x)), x] + Log[x]*Defer[Int][E^(1 - (7*x)/5)*x^(E^(1 - (7*
x)/5)*x*(1 + x)), x] + Defer[Int][E^(1 - (7*x)/5)*x^(1 + E^(1 - (7*x)/5)*x*(1 + x)), x] + (3*Log[x]*Defer[Int]
[E^(1 - (7*x)/5)*x^(1 + E^(1 - (7*x)/5)*x*(1 + x)), x])/5 - (7*Log[x]*Defer[Int][E^(1 - (7*x)/5)*x^(2 + E^(1 -
(7*x)/5)*x*(1 + x)), x])/5 - Defer[Int][Defer[Int][E^(1 - (7*x)/5)*x^(E^(1 - (7*x)/5)*x*(1 + x)), x]/x, x] -
(3*Defer[Int][Defer[Int][E^(1 - (7*x)/5)*x^(1 + E^(1 - (7*x)/5)*x*(1 + x)), x]/x, x])/5 + (7*Defer[Int][Defer[
Int][E^(1 - (7*x)/5)*x^(2 + E^(1 - (7*x)/5)*x*(1 + x)), x]/x, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx\\ &=\frac {1}{5} \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx\\ &=\frac {1}{5} \int \left (5 e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)}+5 e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)}-e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \left (-5-3 x+7 x^2\right ) \log (x)\right ) \, dx\\ &=-\left (\frac {1}{5} \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \left (-5-3 x+7 x^2\right ) \log (x) \, dx\right )+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx\\ &=\frac {1}{5} \int \frac {-5 \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-3 \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+7 \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx+\frac {1}{5} (3 \log (x)) \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\log (x) \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\frac {1}{5} (7 \log (x)) \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx\\ &=\frac {1}{5} \int \left (\frac {-5 \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-3 \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x}+\frac {7 \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x}\right ) \, dx+\frac {1}{5} (3 \log (x)) \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\log (x) \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\frac {1}{5} (7 \log (x)) \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx\\ &=\frac {1}{5} \int \frac {-5 \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-3 \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx+\frac {7}{5} \int \frac {\int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx+\frac {1}{5} (3 \log (x)) \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\log (x) \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\frac {1}{5} (7 \log (x)) \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx\\ &=\frac {1}{5} \int \left (-\frac {5 \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x}-\frac {3 \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x}\right ) \, dx+\frac {7}{5} \int \frac {\int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx+\frac {1}{5} (3 \log (x)) \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\log (x) \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\frac {1}{5} (7 \log (x)) \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx\\ &=-\left (\frac {3}{5} \int \frac {\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx\right )+\frac {7}{5} \int \frac {\int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx+\frac {1}{5} (3 \log (x)) \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\log (x) \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\frac {1}{5} (7 \log (x)) \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\int \frac {\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.38, size = 16, normalized size = 0.67 \begin {gather*} x^{e^{1-\frac {7 x}{5}} x (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((5 - 7*x)/5)*x^(E^((5 - 7*x)/5)*(x + x^2))*(5 + 5*x + (5 + 3*x - 7*x^2)*Log[x]))/5,x]

[Out]

x^(E^(1 - (7*x)/5)*x*(1 + x))

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fricas [A]  time = 0.81, size = 14, normalized size = 0.58 \begin {gather*} x^{{\left (x^{2} + x\right )} e^{\left (-\frac {7}{5} \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((-7*x^2+3*x+5)*log(x)+5*x+5)*exp((x^2+x)*log(x)/exp(7/5*x-1))/exp(7/5*x-1),x, algorithm="fricas
")

[Out]

x^((x^2 + x)*e^(-7/5*x + 1))

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giac [A]  time = 0.33, size = 21, normalized size = 0.88 \begin {gather*} x^{x^{2} e^{\left (-\frac {7}{5} \, x + 1\right )} + x e^{\left (-\frac {7}{5} \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((-7*x^2+3*x+5)*log(x)+5*x+5)*exp((x^2+x)*log(x)/exp(7/5*x-1))/exp(7/5*x-1),x, algorithm="giac")

[Out]

x^(x^2*e^(-7/5*x + 1) + x*e^(-7/5*x + 1))

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maple [A]  time = 0.02, size = 14, normalized size = 0.58

method	result	size
risch	\(x^{\left (x +1\right ) x \,{\mathrm e}^{-\frac {7 x}{5}+1}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((-7*x^2+3*x+5)*ln(x)+5*x+5)*exp((x^2+x)*ln(x)/exp(7/5*x-1))/exp(7/5*x-1),x,method=_RETURNVERBOSE)

[Out]

x^((x+1)*x*exp(-7/5*x+1))

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maxima [A]  time = 0.49, size = 24, normalized size = 1.00 \begin {gather*} e^{\left (x^{2} e^{\left (-\frac {7}{5} \, x + 1\right )} \log \relax (x) + x e^{\left (-\frac {7}{5} \, x + 1\right )} \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((-7*x^2+3*x+5)*log(x)+5*x+5)*exp((x^2+x)*log(x)/exp(7/5*x-1))/exp(7/5*x-1),x, algorithm="maxima
")

[Out]

e^(x^2*e^(-7/5*x + 1)*log(x) + x*e^(-7/5*x + 1)*log(x))

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mupad [B]  time = 4.83, size = 14, normalized size = 0.58 \begin {gather*} x^{{\mathrm {e}}^{1-\frac {7\,x}{5}}\,\left (x^2+x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1 - (7*x)/5)*exp(exp(1 - (7*x)/5)*log(x)*(x + x^2))*(5*x + log(x)*(3*x - 7*x^2 + 5) + 5))/5,x)

[Out]

x^(exp(1 - (7*x)/5)*(x + x^2))

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sympy [A]  time = 0.56, size = 17, normalized size = 0.71 \begin {gather*} e^{\left (x^{2} + x\right ) e^{1 - \frac {7 x}{5}} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((-7*x**2+3*x+5)*ln(x)+5*x+5)*exp((x**2+x)*ln(x)/exp(7/5*x-1))/exp(7/5*x-1),x)

[Out]

exp((x**2 + x)*exp(1 - 7*x/5)*log(x))

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