3.92.19 \(\int (5+e^{e (-35-5 x)+e^x (-35-5 x)+7 x+x^2} (14+6 x+7 x^2+2 x^3+e (-10-5 x^2)+e^x (-80-10 x-40 x^2-5 x^3))) \, dx\)

Optimal. Leaf size=25 \[ 5 x+e^{(7+x) \left (-5 \left (e+e^x\right )+x\right )} \left (2+x^2\right ) \]

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Rubi [F]  time = 3.44, 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 \left (5+\exp \left (e (-35-5 x)+e^x (-35-5 x)+7 x+x^2\right ) \left (14+6 x+7 x^2+2 x^3+e \left (-10-5 x^2\right )+e^x \left (-80-10 x-40 x^2-5 x^3\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[5 + E^(E*(-35 - 5*x) + E^x*(-35 - 5*x) + 7*x + x^2)*(14 + 6*x + 7*x^2 + 2*x^3 + E*(-10 - 5*x^2) + E^x*(-80
- 10*x - 40*x^2 - 5*x^3)),x]

[Out]

5*x + 14*Defer[Int][E^(-((5*E + 5*E^x - x)*(7 + x))), x] - 10*Defer[Int][E^(1 - (5*E + 5*E^x - x)*(7 + x)), x]
- 80*Defer[Int][E^(x - (5*E + 5*E^x - x)*(7 + x)), x] + 6*Defer[Int][x/E^((5*E + 5*E^x - x)*(7 + x)), x] - 10
*Defer[Int][E^(x - (5*E + 5*E^x - x)*(7 + x))*x, x] + 7*Defer[Int][x^2/E^((5*E + 5*E^x - x)*(7 + x)), x] - 5*D
efer[Int][E^(1 - (5*E + 5*E^x - x)*(7 + x))*x^2, x] - 40*Defer[Int][E^(x - (5*E + 5*E^x - x)*(7 + x))*x^2, x]
+ 2*Defer[Int][x^3/E^((5*E + 5*E^x - x)*(7 + x)), x] - 5*Defer[Int][E^(x - (5*E + 5*E^x - x)*(7 + x))*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=5 x+\int \exp \left (e (-35-5 x)+e^x (-35-5 x)+7 x+x^2\right ) \left (14+6 x+7 x^2+2 x^3+e \left (-10-5 x^2\right )+e^x \left (-80-10 x-40 x^2-5 x^3\right )\right ) \, dx\\ &=5 x+\int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} \left (14+6 x+7 x^2+2 x^3+e \left (-10-5 x^2\right )+e^x \left (-80-10 x-40 x^2-5 x^3\right )\right ) \, dx\\ &=5 x+\int \left (14 e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )}+6 e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x+7 e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x^2+2 e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x^3-5 e^{1-\left (5 e+5 e^x-x\right ) (7+x)} \left (2+x^2\right )-5 e^{x-\left (5 e+5 e^x-x\right ) (7+x)} \left (16+2 x+8 x^2+x^3\right )\right ) \, dx\\ &=5 x+2 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x^3 \, dx-5 \int e^{1-\left (5 e+5 e^x-x\right ) (7+x)} \left (2+x^2\right ) \, dx-5 \int e^{x-\left (5 e+5 e^x-x\right ) (7+x)} \left (16+2 x+8 x^2+x^3\right ) \, dx+6 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x \, dx+7 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x^2 \, dx+14 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} \, dx\\ &=5 x+2 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x^3 \, dx-5 \int \left (2 e^{1-\left (5 e+5 e^x-x\right ) (7+x)}+e^{1-\left (5 e+5 e^x-x\right ) (7+x)} x^2\right ) \, dx-5 \int \left (16 e^{x-\left (5 e+5 e^x-x\right ) (7+x)}+2 e^{x-\left (5 e+5 e^x-x\right ) (7+x)} x+8 e^{x-\left (5 e+5 e^x-x\right ) (7+x)} x^2+e^{x-\left (5 e+5 e^x-x\right ) (7+x)} x^3\right ) \, dx+6 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x \, dx+7 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x^2 \, dx+14 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} \, dx\\ &=5 x+2 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x^3 \, dx-5 \int e^{1-\left (5 e+5 e^x-x\right ) (7+x)} x^2 \, dx-5 \int e^{x-\left (5 e+5 e^x-x\right ) (7+x)} x^3 \, dx+6 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x \, dx+7 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} x^2 \, dx-10 \int e^{1-\left (5 e+5 e^x-x\right ) (7+x)} \, dx-10 \int e^{x-\left (5 e+5 e^x-x\right ) (7+x)} x \, dx+14 \int e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} \, dx-40 \int e^{x-\left (5 e+5 e^x-x\right ) (7+x)} x^2 \, dx-80 \int e^{x-\left (5 e+5 e^x-x\right ) (7+x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.33, size = 29, normalized size = 1.16 \begin {gather*} 5 x+e^{-\left (\left (5 e+5 e^x-x\right ) (7+x)\right )} \left (2+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[5 + E^(E*(-35 - 5*x) + E^x*(-35 - 5*x) + 7*x + x^2)*(14 + 6*x + 7*x^2 + 2*x^3 + E*(-10 - 5*x^2) + E^
x*(-80 - 10*x - 40*x^2 - 5*x^3)),x]

[Out]

5*x + (2 + x^2)/E^((5*E + 5*E^x - x)*(7 + x))

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fricas [A]  time = 0.71, size = 32, normalized size = 1.28 \begin {gather*} {\left (x^{2} + 2\right )} e^{\left (x^{2} - 5 \, {\left (x + 7\right )} e - 5 \, {\left (x + 7\right )} e^{x} + 7 \, x\right )} + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^3-40*x^2-10*x-80)*exp(x)+(-5*x^2-10)*exp(1)+2*x^3+7*x^2+6*x+14)*exp((-5*x-35)*exp(x)+(-5*x-35
)*exp(1)+x^2+7*x)+5,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.23, size = 68, normalized size = 2.72 \begin {gather*} {\left (x^{2} e^{\left (x^{2} - 5 \, x e - 5 \, x e^{x} + 7 \, x - 35 \, e - 35 \, e^{x} + 1\right )} + 2 \, e^{\left (x^{2} - 5 \, x e - 5 \, x e^{x} + 7 \, x - 35 \, e - 35 \, e^{x} + 1\right )}\right )} e^{\left (-1\right )} + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^3-40*x^2-10*x-80)*exp(x)+(-5*x^2-10)*exp(1)+2*x^3+7*x^2+6*x+14)*exp((-5*x-35)*exp(x)+(-5*x-35
)*exp(1)+x^2+7*x)+5,x, algorithm="giac")

[Out]

(x^2*e^(x^2 - 5*x*e - 5*x*e^x + 7*x - 35*e - 35*e^x + 1) + 2*e^(x^2 - 5*x*e - 5*x*e^x + 7*x - 35*e - 35*e^x +
1))*e^(-1) + 5*x

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maple [A]  time = 0.14, size = 29, normalized size = 1.16

method	result	size
risch	\(\left (x^{2}+2\right ) {\mathrm e}^{-\left (x +7\right ) \left (-x +5 \,{\mathrm e}+5 \,{\mathrm e}^{x}\right )}+5 x\)	\(29\)
default	\(5 x +x^{2} {\mathrm e}^{\left (-5 x -35\right ) {\mathrm e}^{x}+\left (-5 x -35\right ) {\mathrm e}+x^{2}+7 x}+2 \,{\mathrm e}^{\left (-5 x -35\right ) {\mathrm e}^{x}+\left (-5 x -35\right ) {\mathrm e}+x^{2}+7 x}\)	\(59\)
norman	\(5 x +x^{2} {\mathrm e}^{\left (-5 x -35\right ) {\mathrm e}^{x}+\left (-5 x -35\right ) {\mathrm e}+x^{2}+7 x}+2 \,{\mathrm e}^{\left (-5 x -35\right ) {\mathrm e}^{x}+\left (-5 x -35\right ) {\mathrm e}+x^{2}+7 x}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x^3-40*x^2-10*x-80)*exp(x)+(-5*x^2-10)*exp(1)+2*x^3+7*x^2+6*x+14)*exp((-5*x-35)*exp(x)+(-5*x-35)*exp(
1)+x^2+7*x)+5,x,method=_RETURNVERBOSE)

[Out]

(x^2+2)*exp(-(x+7)*(-x+5*exp(1)+5*exp(x)))+5*x

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maxima [A]  time = 0.51, size = 36, normalized size = 1.44 \begin {gather*} {\left (x^{2} + 2\right )} e^{\left (x^{2} - 5 \, x e - 5 \, x e^{x} + 7 \, x - 35 \, e - 35 \, e^{x}\right )} + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^3-40*x^2-10*x-80)*exp(x)+(-5*x^2-10)*exp(1)+2*x^3+7*x^2+6*x+14)*exp((-5*x-35)*exp(x)+(-5*x-35
)*exp(1)+x^2+7*x)+5,x, algorithm="maxima")

[Out]

(x^2 + 2)*e^(x^2 - 5*x*e - 5*x*e^x + 7*x - 35*e - 35*e^x) + 5*x

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mupad [B]  time = 5.33, size = 70, normalized size = 2.80 \begin {gather*} 5\,x+2\,{\mathrm {e}}^{-5\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-35\,\mathrm {e}}\,{\mathrm {e}}^{7\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-5\,x\,\mathrm {e}}\,{\mathrm {e}}^{-35\,{\mathrm {e}}^x}+x^2\,{\mathrm {e}}^{-5\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-35\,\mathrm {e}}\,{\mathrm {e}}^{7\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-5\,x\,\mathrm {e}}\,{\mathrm {e}}^{-35\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(7*x - exp(x)*(5*x + 35) + x^2 - exp(1)*(5*x + 35))*(6*x - exp(1)*(5*x^2 + 10) + 7*x^2 + 2*x^3 - exp(x)
*(10*x + 40*x^2 + 5*x^3 + 80) + 14) + 5,x)

[Out]

5*x + 2*exp(-5*x*exp(x))*exp(-35*exp(1))*exp(7*x)*exp(x^2)*exp(-5*x*exp(1))*exp(-35*exp(x)) + x^2*exp(-5*x*exp
(x))*exp(-35*exp(1))*exp(7*x)*exp(x^2)*exp(-5*x*exp(1))*exp(-35*exp(x))

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sympy [A]  time = 0.26, size = 36, normalized size = 1.44 \begin {gather*} 5 x + \left (x^{2} + 2\right ) e^{x^{2} + 7 x + \left (- 5 x - 35\right ) e^{x} + e \left (- 5 x - 35\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x**3-40*x**2-10*x-80)*exp(x)+(-5*x**2-10)*exp(1)+2*x**3+7*x**2+6*x+14)*exp((-5*x-35)*exp(x)+(-5
*x-35)*exp(1)+x**2+7*x)+5,x)

[Out]

5*x + (x**2 + 2)*exp(x**2 + 7*x + (-5*x - 35)*exp(x) + E*(-5*x - 35))

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