3.81.73 \(\int \frac {1}{3} (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} (e^x+6 x)} (51-24 x-18 x^2+e^x (9-x-3 x^2))) \, dx\)

Optimal. Leaf size=34 \[ 3+\left (3-e^{\frac {e^x}{3}+2 x}+x (3+x)\right ) \left (x+3 \left (-3+x^2\right )\right ) \]

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Rubi [F]  time = 0.53, 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}{3} \left (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-72 + 18*x + 90*x^2 + 36*x^3 + E^((E^x + 6*x)/3)*(51 - 24*x - 18*x^2 + E^x*(9 - x - 3*x^2)))/3,x]

[Out]

-24*x + 3*x^2 + 10*x^3 + 3*x^4 + 17*Defer[Int][E^((E^x + 6*x)/3), x] + 3*Defer[Int][E^((E^x + 9*x)/3), x] - 8*
Defer[Int][E^((E^x + 6*x)/3)*x, x] - Defer[Int][E^((E^x + 9*x)/3)*x, x]/3 - 6*Defer[Int][E^((E^x + 6*x)/3)*x^2
, x] - Defer[Int][E^((E^x + 9*x)/3)*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right )\right ) \, dx\\ &=-24 x+3 x^2+10 x^3+3 x^4+\frac {1}{3} \int e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right ) \, dx\\ &=-24 x+3 x^2+10 x^3+3 x^4+\frac {1}{3} \int \left (51 e^{\frac {1}{3} \left (e^x+6 x\right )}-24 e^{\frac {1}{3} \left (e^x+6 x\right )} x-18 e^{\frac {1}{3} \left (e^x+6 x\right )} x^2-e^{x+\frac {1}{3} \left (e^x+6 x\right )} \left (-9+x+3 x^2\right )\right ) \, dx\\ &=-24 x+3 x^2+10 x^3+3 x^4-\frac {1}{3} \int e^{x+\frac {1}{3} \left (e^x+6 x\right )} \left (-9+x+3 x^2\right ) \, dx-6 \int e^{\frac {1}{3} \left (e^x+6 x\right )} x^2 \, dx-8 \int e^{\frac {1}{3} \left (e^x+6 x\right )} x \, dx+17 \int e^{\frac {1}{3} \left (e^x+6 x\right )} \, dx\\ &=-24 x+3 x^2+10 x^3+3 x^4-\frac {1}{3} \int e^{\frac {1}{3} \left (e^x+9 x\right )} \left (-9+x+3 x^2\right ) \, dx-6 \int e^{\frac {1}{3} \left (e^x+6 x\right )} x^2 \, dx-8 \int e^{\frac {1}{3} \left (e^x+6 x\right )} x \, dx+17 \int e^{\frac {1}{3} \left (e^x+6 x\right )} \, dx\\ &=-24 x+3 x^2+10 x^3+3 x^4-\frac {1}{3} \int \left (-9 e^{\frac {1}{3} \left (e^x+9 x\right )}+e^{\frac {1}{3} \left (e^x+9 x\right )} x+3 e^{\frac {1}{3} \left (e^x+9 x\right )} x^2\right ) \, dx-6 \int e^{\frac {1}{3} \left (e^x+6 x\right )} x^2 \, dx-8 \int e^{\frac {1}{3} \left (e^x+6 x\right )} x \, dx+17 \int e^{\frac {1}{3} \left (e^x+6 x\right )} \, dx\\ &=-24 x+3 x^2+10 x^3+3 x^4-\frac {1}{3} \int e^{\frac {1}{3} \left (e^x+9 x\right )} x \, dx+3 \int e^{\frac {1}{3} \left (e^x+9 x\right )} \, dx-6 \int e^{\frac {1}{3} \left (e^x+6 x\right )} x^2 \, dx-8 \int e^{\frac {1}{3} \left (e^x+6 x\right )} x \, dx+17 \int e^{\frac {1}{3} \left (e^x+6 x\right )} \, dx-\int e^{\frac {1}{3} \left (e^x+9 x\right )} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 42, normalized size = 1.24 \begin {gather*} -24 x+3 x^2+10 x^3+3 x^4-e^{\frac {e^x}{3}+2 x} \left (-9+x+3 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-72 + 18*x + 90*x^2 + 36*x^3 + E^((E^x + 6*x)/3)*(51 - 24*x - 18*x^2 + E^x*(9 - x - 3*x^2)))/3,x]

[Out]

-24*x + 3*x^2 + 10*x^3 + 3*x^4 - E^(E^x/3 + 2*x)*(-9 + x + 3*x^2)

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fricas [A]  time = 0.87, size = 38, normalized size = 1.12 \begin {gather*} 3 \, x^{4} + 10 \, x^{3} + 3 \, x^{2} - {\left (3 \, x^{2} + x - 9\right )} e^{\left (2 \, x + \frac {1}{3} \, e^{x}\right )} - 24 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-3*x^2-x+9)*exp(x)-18*x^2-24*x+51)*exp(1/3*exp(x)+2*x)+12*x^3+30*x^2+6*x-24,x, algorithm="fric
as")

[Out]

3*x^4 + 10*x^3 + 3*x^2 - (3*x^2 + x - 9)*e^(2*x + 1/3*e^x) - 24*x

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giac [A]  time = 1.11, size = 56, normalized size = 1.65 \begin {gather*} 3 \, x^{4} + 10 \, x^{3} - 3 \, x^{2} e^{\left (2 \, x + \frac {1}{3} \, e^{x}\right )} + 3 \, x^{2} - x e^{\left (2 \, x + \frac {1}{3} \, e^{x}\right )} - 24 \, x + 9 \, e^{\left (2 \, x + \frac {1}{3} \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-3*x^2-x+9)*exp(x)-18*x^2-24*x+51)*exp(1/3*exp(x)+2*x)+12*x^3+30*x^2+6*x-24,x, algorithm="giac
")

[Out]

3*x^4 + 10*x^3 - 3*x^2*e^(2*x + 1/3*e^x) + 3*x^2 - x*e^(2*x + 1/3*e^x) - 24*x + 9*e^(2*x + 1/3*e^x)

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maple [A]  time = 0.06, size = 41, normalized size = 1.21

method	result	size
risch	\(\frac {\left (-9 x^{2}-3 x +27\right ) {\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x}}{3}+3 x^{4}+10 x^{3}+3 x^{2}-24 x\)	\(41\)
default	\(3 x^{4}+10 x^{3}+3 x^{2}-24 x -{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x} x -3 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x} x^{2}+9 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x}\)	\(57\)
norman	\(3 x^{4}+10 x^{3}+3 x^{2}-24 x -{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x} x -3 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x} x^{2}+9 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((-3*x^2-x+9)*exp(x)-18*x^2-24*x+51)*exp(1/3*exp(x)+2*x)+12*x^3+30*x^2+6*x-24,x,method=_RETURNVERBOSE)

[Out]

1/3*(-9*x^2-3*x+27)*exp(1/3*exp(x)+2*x)+3*x^4+10*x^3+3*x^2-24*x

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maxima [A]  time = 0.36, size = 38, normalized size = 1.12 \begin {gather*} 3 \, x^{4} + 10 \, x^{3} + 3 \, x^{2} - {\left (3 \, x^{2} + x - 9\right )} e^{\left (2 \, x + \frac {1}{3} \, e^{x}\right )} - 24 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-3*x^2-x+9)*exp(x)-18*x^2-24*x+51)*exp(1/3*exp(x)+2*x)+12*x^3+30*x^2+6*x-24,x, algorithm="maxi
ma")

[Out]

3*x^4 + 10*x^3 + 3*x^2 - (3*x^2 + x - 9)*e^(2*x + 1/3*e^x) - 24*x

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mupad [B]  time = 5.34, size = 56, normalized size = 1.65 \begin {gather*} 9\,{\mathrm {e}}^{2\,x+\frac {{\mathrm {e}}^x}{3}}-24\,x-3\,x^2\,{\mathrm {e}}^{2\,x+\frac {{\mathrm {e}}^x}{3}}-x\,{\mathrm {e}}^{2\,x+\frac {{\mathrm {e}}^x}{3}}+3\,x^2+10\,x^3+3\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*x + 30*x^2 + 12*x^3 - (exp(2*x + exp(x)/3)*(24*x + exp(x)*(x + 3*x^2 - 9) + 18*x^2 - 51))/3 - 24,x)

[Out]

9*exp(2*x + exp(x)/3) - 24*x - 3*x^2*exp(2*x + exp(x)/3) - x*exp(2*x + exp(x)/3) + 3*x^2 + 10*x^3 + 3*x^4

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sympy [A]  time = 0.21, size = 36, normalized size = 1.06 \begin {gather*} 3 x^{4} + 10 x^{3} + 3 x^{2} - 24 x + \left (- 3 x^{2} - x + 9\right ) e^{2 x + \frac {e^{x}}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-3*x**2-x+9)*exp(x)-18*x**2-24*x+51)*exp(1/3*exp(x)+2*x)+12*x**3+30*x**2+6*x-24,x)

[Out]

3*x**4 + 10*x**3 + 3*x**2 - 24*x + (-3*x**2 - x + 9)*exp(2*x + exp(x)/3)

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