3.30.19 \(\int \frac {1}{3} (18 e^{2 x}+18 x+12 x^2+4 x^3+e^x (117+56 x+10 x^2)) \, dx\)

Optimal. Leaf size=23 \[ 3 \left (e^x+\frac {x^2}{9}\right ) \left (5+e^x+(2+x)^2\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.05, antiderivative size = 48, normalized size of antiderivative = 2.09, number of steps used = 11, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 2194, 2196, 2176} \begin {gather*} \frac {x^4}{3}+\frac {4 x^3}{3}+\frac {10 e^x x^2}{3}+3 x^2+12 e^x x+27 e^x+3 e^{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(18*E^(2*x) + 18*x + 12*x^2 + 4*x^3 + E^x*(117 + 56*x + 10*x^2))/3,x]

[Out]

27*E^x + 3*E^(2*x) + 12*E^x*x + 3*x^2 + (10*E^x*x^2)/3 + (4*x^3)/3 + x^4/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (18 e^{2 x}+18 x+12 x^2+4 x^3+e^x \left (117+56 x+10 x^2\right )\right ) \, dx\\ &=3 x^2+\frac {4 x^3}{3}+\frac {x^4}{3}+\frac {1}{3} \int e^x \left (117+56 x+10 x^2\right ) \, dx+6 \int e^{2 x} \, dx\\ &=3 e^{2 x}+3 x^2+\frac {4 x^3}{3}+\frac {x^4}{3}+\frac {1}{3} \int \left (117 e^x+56 e^x x+10 e^x x^2\right ) \, dx\\ &=3 e^{2 x}+3 x^2+\frac {4 x^3}{3}+\frac {x^4}{3}+\frac {10}{3} \int e^x x^2 \, dx+\frac {56}{3} \int e^x x \, dx+39 \int e^x \, dx\\ &=39 e^x+3 e^{2 x}+\frac {56 e^x x}{3}+3 x^2+\frac {10 e^x x^2}{3}+\frac {4 x^3}{3}+\frac {x^4}{3}-\frac {20}{3} \int e^x x \, dx-\frac {56 \int e^x \, dx}{3}\\ &=\frac {61 e^x}{3}+3 e^{2 x}+12 e^x x+3 x^2+\frac {10 e^x x^2}{3}+\frac {4 x^3}{3}+\frac {x^4}{3}+\frac {20 \int e^x \, dx}{3}\\ &=27 e^x+3 e^{2 x}+12 e^x x+3 x^2+\frac {10 e^x x^2}{3}+\frac {4 x^3}{3}+\frac {x^4}{3}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 39, normalized size = 1.70 \begin {gather*} \frac {1}{3} \left (9 e^{2 x}+9 x^2+4 x^3+x^4+e^x \left (81+36 x+10 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18*E^(2*x) + 18*x + 12*x^2 + 4*x^3 + E^x*(117 + 56*x + 10*x^2))/3,x]

[Out]

(9*E^(2*x) + 9*x^2 + 4*x^3 + x^4 + E^x*(81 + 36*x + 10*x^2))/3

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 36, normalized size = 1.57 \begin {gather*} \frac {1}{3} \, x^{4} + \frac {4}{3} \, x^{3} + 3 \, x^{2} + \frac {1}{3} \, {\left (10 \, x^{2} + 36 \, x + 81\right )} e^{x} + 3 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(x)^2+1/3*(10*x^2+56*x+117)*exp(x)+4/3*x^3+4*x^2+6*x,x, algorithm="fricas")

[Out]

1/3*x^4 + 4/3*x^3 + 3*x^2 + 1/3*(10*x^2 + 36*x + 81)*e^x + 3*e^(2*x)

________________________________________________________________________________________

giac [A]  time = 0.29, size = 36, normalized size = 1.57 \begin {gather*} \frac {1}{3} \, x^{4} + \frac {4}{3} \, x^{3} + 3 \, x^{2} + \frac {1}{3} \, {\left (10 \, x^{2} + 36 \, x + 81\right )} e^{x} + 3 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(x)^2+1/3*(10*x^2+56*x+117)*exp(x)+4/3*x^3+4*x^2+6*x,x, algorithm="giac")

[Out]

1/3*x^4 + 4/3*x^3 + 3*x^2 + 1/3*(10*x^2 + 36*x + 81)*e^x + 3*e^(2*x)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 37, normalized size = 1.61

method	result	size
risch	\(3 \,{\mathrm e}^{2 x}+\frac {\left (10 x^{2}+36 x +81\right ) {\mathrm e}^{x}}{3}+\frac {x^{4}}{3}+\frac {4 x^{3}}{3}+3 x^{2}\)	\(37\)
default	\(3 x^{2}+\frac {4 x^{3}}{3}+\frac {x^{4}}{3}+3 \,{\mathrm e}^{2 x}+12 \,{\mathrm e}^{x} x +27 \,{\mathrm e}^{x}+\frac {10 \,{\mathrm e}^{x} x^{2}}{3}\)	\(39\)
norman	\(3 x^{2}+\frac {4 x^{3}}{3}+\frac {x^{4}}{3}+3 \,{\mathrm e}^{2 x}+12 \,{\mathrm e}^{x} x +27 \,{\mathrm e}^{x}+\frac {10 \,{\mathrm e}^{x} x^{2}}{3}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*exp(x)^2+1/3*(10*x^2+56*x+117)*exp(x)+4/3*x^3+4*x^2+6*x,x,method=_RETURNVERBOSE)

[Out]

3*exp(2*x)+1/3*(10*x^2+36*x+81)*exp(x)+1/3*x^4+4/3*x^3+3*x^2

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 36, normalized size = 1.57 \begin {gather*} \frac {1}{3} \, x^{4} + \frac {4}{3} \, x^{3} + 3 \, x^{2} + \frac {1}{3} \, {\left (10 \, x^{2} + 36 \, x + 81\right )} e^{x} + 3 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(x)^2+1/3*(10*x^2+56*x+117)*exp(x)+4/3*x^3+4*x^2+6*x,x, algorithm="maxima")

[Out]

1/3*x^4 + 4/3*x^3 + 3*x^2 + 1/3*(10*x^2 + 36*x + 81)*e^x + 3*e^(2*x)

________________________________________________________________________________________

mupad [B]  time = 1.87, size = 38, normalized size = 1.65 \begin {gather*} 3\,{\mathrm {e}}^{2\,x}+27\,{\mathrm {e}}^x+\frac {10\,x^2\,{\mathrm {e}}^x}{3}+12\,x\,{\mathrm {e}}^x+3\,x^2+\frac {4\,x^3}{3}+\frac {x^4}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*x + 6*exp(2*x) + (exp(x)*(56*x + 10*x^2 + 117))/3 + 4*x^2 + (4*x^3)/3,x)

[Out]

3*exp(2*x) + 27*exp(x) + (10*x^2*exp(x))/3 + 12*x*exp(x) + 3*x^2 + (4*x^3)/3 + x^4/3

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 37, normalized size = 1.61 \begin {gather*} \frac {x^{4}}{3} + \frac {4 x^{3}}{3} + 3 x^{2} + \frac {\left (10 x^{2} + 36 x + 81\right ) e^{x}}{3} + 3 e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(x)**2+1/3*(10*x**2+56*x+117)*exp(x)+4/3*x**3+4*x**2+6*x,x)

[Out]

x**4/3 + 4*x**3/3 + 3*x**2 + (10*x**2 + 36*x + 81)*exp(x)/3 + 3*exp(2*x)

________________________________________________________________________________________