3.2.2 \(\int (3 x^2+e^{2 x} (3 x^2+2 x^3+e^2 (2 x+2 x^2))) \, dx\)

Optimal. Leaf size=17 \[ x^2 \left (x+e^{2 x} \left (e^2+x\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 24, normalized size of antiderivative = 1.41, number of steps used = 17, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2196, 2176, 2194} \begin {gather*} e^{2 x} x^3+x^3+e^{2 x+2} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[3*x^2 + E^(2*x)*(3*x^2 + 2*x^3 + E^2*(2*x + 2*x^2)),x]

[Out]

E^(2 + 2*x)*x^2 + x^3 + E^(2*x)*x^3

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} &=x^3+\int e^{2 x} \left (3 x^2+2 x^3+e^2 \left (2 x+2 x^2\right )\right ) \, dx\\ &=x^3+\int \left (3 e^{2 x} x^2+2 e^{2 x} x^3+2 e^{2+2 x} x (1+x)\right ) \, dx\\ &=x^3+2 \int e^{2 x} x^3 \, dx+2 \int e^{2+2 x} x (1+x) \, dx+3 \int e^{2 x} x^2 \, dx\\ &=\frac {3}{2} e^{2 x} x^2+x^3+e^{2 x} x^3+2 \int \left (e^{2+2 x} x+e^{2+2 x} x^2\right ) \, dx-3 \int e^{2 x} x \, dx-3 \int e^{2 x} x^2 \, dx\\ &=-\frac {3}{2} e^{2 x} x+x^3+e^{2 x} x^3+\frac {3}{2} \int e^{2 x} \, dx+2 \int e^{2+2 x} x \, dx+2 \int e^{2+2 x} x^2 \, dx+3 \int e^{2 x} x \, dx\\ &=\frac {3 e^{2 x}}{4}+e^{2+2 x} x+e^{2+2 x} x^2+x^3+e^{2 x} x^3-\frac {3}{2} \int e^{2 x} \, dx-2 \int e^{2+2 x} x \, dx-\int e^{2+2 x} \, dx\\ &=-\frac {1}{2} e^{2+2 x}+e^{2+2 x} x^2+x^3+e^{2 x} x^3+\int e^{2+2 x} \, dx\\ &=e^{2+2 x} x^2+x^3+e^{2 x} x^3\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 21, normalized size = 1.24 \begin {gather*} x^3+e^{2 x} \left (e^2 x^2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[3*x^2 + E^(2*x)*(3*x^2 + 2*x^3 + E^2*(2*x + 2*x^2)),x]

[Out]

x^3 + E^(2*x)*(E^2*x^2 + x^3)

________________________________________________________________________________________

fricas [A]  time = 1.12, size = 19, normalized size = 1.12 \begin {gather*} x^{3} + {\left (x^{3} + x^{2} e^{2}\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(2)+2*x^3+3*x^2)*exp(x)^2+3*x^2,x, algorithm="fricas")

[Out]

x^3 + (x^3 + x^2*e^2)*e^(2*x)

________________________________________________________________________________________

giac [A]  time = 0.33, size = 22, normalized size = 1.29 \begin {gather*} x^{3} e^{\left (2 \, x\right )} + x^{3} + x^{2} e^{\left (2 \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(2)+2*x^3+3*x^2)*exp(x)^2+3*x^2,x, algorithm="giac")

[Out]

x^3*e^(2*x) + x^3 + x^2*e^(2*x + 2)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 20, normalized size = 1.18

method	result	size
risch	\(\left (x^{2} {\mathrm e}^{2}+x^{3}\right ) {\mathrm e}^{2 x}+x^{3}\)	\(20\)
norman	\(x^{3}+{\mathrm e}^{2 x} x^{3}+x^{2} {\mathrm e}^{2} {\mathrm e}^{2 x}\)	\(23\)
default	\({\mathrm e}^{2 x} x^{3}+2 \,{\mathrm e}^{2} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )+2 \,{\mathrm e}^{2} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )+x^{3}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+2*x)*exp(2)+2*x^3+3*x^2)*exp(x)^2+3*x^2,x,method=_RETURNVERBOSE)

[Out]

(x^2*exp(2)+x^3)*exp(2*x)+x^3

________________________________________________________________________________________

maxima [B]  time = 0.42, size = 78, normalized size = 4.59 \begin {gather*} x^{3} + \frac {1}{4} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (2 \, x^{2} e^{2} - 2 \, x e^{2} + e^{2}\right )} e^{\left (2 \, x\right )} + \frac {3}{4} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (2 \, x e^{2} - e^{2}\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(2)+2*x^3+3*x^2)*exp(x)^2+3*x^2,x, algorithm="maxima")

[Out]

x^3 + 1/4*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) + 1/2*(2*x^2*e^2 - 2*x*e^2 + e^2)*e^(2*x) + 3/4*(2*x^2 - 2*x + 1)*
e^(2*x) + 1/2*(2*x*e^2 - e^2)*e^(2*x)

________________________________________________________________________________________

mupad [B]  time = 0.24, size = 19, normalized size = 1.12 \begin {gather*} x^2\,\left (x+{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^2+x\,{\mathrm {e}}^{2\,x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)*(exp(2)*(2*x + 2*x^2) + 3*x^2 + 2*x^3) + 3*x^2,x)

[Out]

x^2*(x + exp(2*x)*exp(2) + x*exp(2*x))

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 17, normalized size = 1.00 \begin {gather*} x^{3} + \left (x^{3} + x^{2} e^{2}\right ) e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+2*x)*exp(2)+2*x**3+3*x**2)*exp(x)**2+3*x**2,x)

[Out]

x**3 + (x**3 + x**2*exp(2))*exp(2*x)

________________________________________________________________________________________