3.58.1 \(\int (8 x+24 x^2+e^x (4-12 x^2-4 x^3)+e^{4+3 x+x^2} (8 x+12 x^2+8 x^3)) \, dx\)

Optimal. Leaf size=30 \[ 4 \left (e^x+x^2 \left (1+e^{4+x (3+x)}+2 x-e^x x\right )\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 53, normalized size of antiderivative = 1.77, number of steps used = 13, number of rules used = 5, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2196, 2194, 2176, 1594, 2288} \begin {gather*} -4 e^x x^3+8 x^3+4 x^2+\frac {4 e^{x^2+3 x+4} \left (2 x^2+3 x\right ) x}{2 x+3}+4 e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[8*x + 24*x^2 + E^x*(4 - 12*x^2 - 4*x^3) + E^(4 + 3*x + x^2)*(8*x + 12*x^2 + 8*x^3),x]

[Out]

4*E^x + 4*x^2 + 8*x^3 - 4*E^x*x^3 + (4*E^(4 + 3*x + x^2)*x*(3*x + 2*x^2))/(3 + 2*x)

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 x^2+8 x^3+\int e^x \left (4-12 x^2-4 x^3\right ) \, dx+\int e^{4+3 x+x^2} \left (8 x+12 x^2+8 x^3\right ) \, dx\\ &=4 x^2+8 x^3+\int e^{4+3 x+x^2} x \left (8+12 x+8 x^2\right ) \, dx+\int \left (4 e^x-12 e^x x^2-4 e^x x^3\right ) \, dx\\ &=4 x^2+8 x^3+\frac {4 e^{4+3 x+x^2} x \left (3 x+2 x^2\right )}{3+2 x}+4 \int e^x \, dx-4 \int e^x x^3 \, dx-12 \int e^x x^2 \, dx\\ &=4 e^x+4 x^2-12 e^x x^2+8 x^3-4 e^x x^3+\frac {4 e^{4+3 x+x^2} x \left (3 x+2 x^2\right )}{3+2 x}+12 \int e^x x^2 \, dx+24 \int e^x x \, dx\\ &=4 e^x+24 e^x x+4 x^2+8 x^3-4 e^x x^3+\frac {4 e^{4+3 x+x^2} x \left (3 x+2 x^2\right )}{3+2 x}-24 \int e^x \, dx-24 \int e^x x \, dx\\ &=-20 e^x+4 x^2+8 x^3-4 e^x x^3+\frac {4 e^{4+3 x+x^2} x \left (3 x+2 x^2\right )}{3+2 x}+24 \int e^x \, dx\\ &=4 e^x+4 x^2+8 x^3-4 e^x x^3+\frac {4 e^{4+3 x+x^2} x \left (3 x+2 x^2\right )}{3+2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 50, normalized size = 1.67 \begin {gather*} 4 x^2+8 x^3+\frac {4 e^{4+3 x+x^2} x \left (3 x+2 x^2\right )}{3+2 x}-4 e^x \left (-1+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[8*x + 24*x^2 + E^x*(4 - 12*x^2 - 4*x^3) + E^(4 + 3*x + x^2)*(8*x + 12*x^2 + 8*x^3),x]

[Out]

4*x^2 + 8*x^3 + (4*E^(4 + 3*x + x^2)*x*(3*x + 2*x^2))/(3 + 2*x) - 4*E^x*(-1 + x^3)

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fricas [A]  time = 0.51, size = 34, normalized size = 1.13 \begin {gather*} 8 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 3 \, x + 4\right )} + 4 \, x^{2} - 4 \, {\left (x^{3} - 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+12*x^2+8*x)*exp(x^2+3*x+4)+(-4*x^3-12*x^2+4)*exp(x)+24*x^2+8*x,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 42, normalized size = 1.40 \begin {gather*} 8 \, x^{3} + 4 \, x^{2} + {\left ({\left (2 \, x + 3\right )}^{2} - 12 \, x - 9\right )} e^{\left (x^{2} + 3 \, x + 4\right )} - 4 \, {\left (x^{3} - 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+12*x^2+8*x)*exp(x^2+3*x+4)+(-4*x^3-12*x^2+4)*exp(x)+24*x^2+8*x,x, algorithm="giac")

[Out]

8*x^3 + 4*x^2 + ((2*x + 3)^2 - 12*x - 9)*e^(x^2 + 3*x + 4) - 4*(x^3 - 1)*e^x

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maple [A]  time = 0.04, size = 37, normalized size = 1.23

method	result	size
default	\(-4 \,{\mathrm e}^{x} x^{3}+4 \,{\mathrm e}^{x}+4 \,{\mathrm e}^{x^{2}+3 x +4} x^{2}+4 x^{2}+8 x^{3}\)	\(37\)
norman	\(-4 \,{\mathrm e}^{x} x^{3}+4 \,{\mathrm e}^{x}+4 \,{\mathrm e}^{x^{2}+3 x +4} x^{2}+4 x^{2}+8 x^{3}\)	\(37\)
risch	\(-4 \,{\mathrm e}^{x} x^{3}+4 \,{\mathrm e}^{x}+4 \,{\mathrm e}^{x^{2}+3 x +4} x^{2}+4 x^{2}+8 x^{3}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^3+12*x^2+8*x)*exp(x^2+3*x+4)+(-4*x^3-12*x^2+4)*exp(x)+24*x^2+8*x,x,method=_RETURNVERBOSE)

[Out]

-4*exp(x)*x^3+4*exp(x)+4*exp(x^2+3*x+4)*x^2+4*x^2+8*x^3

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maxima [A]  time = 0.37, size = 34, normalized size = 1.13 \begin {gather*} 8 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 3 \, x + 4\right )} + 4 \, x^{2} - 4 \, {\left (x^{3} - 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+12*x^2+8*x)*exp(x^2+3*x+4)+(-4*x^3-12*x^2+4)*exp(x)+24*x^2+8*x,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.11, size = 36, normalized size = 1.20 \begin {gather*} 4\,{\mathrm {e}}^x-4\,x^3\,{\mathrm {e}}^x+4\,x^2\,{\mathrm {e}}^{x^2+3\,x+4}+4\,x^2+8\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8*x - exp(x)*(12*x^2 + 4*x^3 - 4) + exp(3*x + x^2 + 4)*(8*x + 12*x^2 + 8*x^3) + 24*x^2,x)

[Out]

4*exp(x) - 4*x^3*exp(x) + 4*x^2*exp(3*x + x^2 + 4) + 4*x^2 + 8*x^3

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sympy [A]  time = 0.16, size = 34, normalized size = 1.13 \begin {gather*} 8 x^{3} + 4 x^{2} e^{x^{2} + 3 x + 4} + 4 x^{2} + \left (4 - 4 x^{3}\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**3+12*x**2+8*x)*exp(x**2+3*x+4)+(-4*x**3-12*x**2+4)*exp(x)+24*x**2+8*x,x)

[Out]

8*x**3 + 4*x**2*exp(x**2 + 3*x + 4) + 4*x**2 + (4 - 4*x**3)*exp(x)

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