3.27.20 \(\int (-1+e^{1+4 e^{10}+8 x+16 x^2+e^5 (4+16 x)} (8+16 e^5+32 x)) \, dx\)

Optimal. Leaf size=18 \[ e^{\left (1+2 e^5+4 x\right )^2}-x \]

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Rubi [A]  time = 0.09, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2227, 2209} \begin {gather*} e^{\left (4 x+2 e^5+1\right )^2}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + E^(1 + 4*E^10 + 8*x + 16*x^2 + E^5*(4 + 16*x))*(8 + 16*E^5 + 32*x),x]

[Out]

E^(1 + 2*E^5 + 4*x)^2 - x

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2227

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-x+\int e^{1+4 e^{10}+8 x+16 x^2+e^5 (4+16 x)} \left (8+16 e^5+32 x\right ) \, dx\\ &=-x+\int e^{\left (1+2 e^5+4 x\right )^2} \left (8+16 e^5+32 x\right ) \, dx\\ &=e^{\left (1+2 e^5+4 x\right )^2}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} e^{\left (1+2 e^5+4 x\right )^2}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + E^(1 + 4*E^10 + 8*x + 16*x^2 + E^5*(4 + 16*x))*(8 + 16*E^5 + 32*x),x]

[Out]

E^(1 + 2*E^5 + 4*x)^2 - x

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fricas [A]  time = 0.46, size = 28, normalized size = 1.56 \begin {gather*} -x + e^{\left (16 \, x^{2} + 4 \, {\left (4 \, x + 1\right )} e^{5} + 8 \, x + 4 \, e^{10} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp(5)+32*x+8)*exp(4*exp(5)^2+(16*x+4)*exp(5)+16*x^2+8*x+1)-1,x, algorithm="fricas")

[Out]

-x + e^(16*x^2 + 4*(4*x + 1)*e^5 + 8*x + 4*e^10 + 1)

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giac [A]  time = 1.12, size = 28, normalized size = 1.56 \begin {gather*} -x + e^{\left (16 \, x^{2} + 16 \, x e^{5} + 8 \, x + 4 \, e^{10} + 4 \, e^{5} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp(5)+32*x+8)*exp(4*exp(5)^2+(16*x+4)*exp(5)+16*x^2+8*x+1)-1,x, algorithm="giac")

[Out]

-x + e^(16*x^2 + 16*x*e^5 + 8*x + 4*e^10 + 4*e^5 + 1)

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maple [A]  time = 0.04, size = 29, normalized size = 1.61

method	result	size
risch	\(-x +{\mathrm e}^{16 x \,{\mathrm e}^{5}+16 x^{2}+4 \,{\mathrm e}^{5}+4 \,{\mathrm e}^{10}+8 x +1}\)	\(29\)
norman	\(-x +{\mathrm e}^{4 \,{\mathrm e}^{10}+\left (16 x +4\right ) {\mathrm e}^{5}+16 x^{2}+8 x +1}\)	\(30\)
default	\(-x -i \sqrt {\pi }\, {\mathrm e}^{4 \,{\mathrm e}^{5}+4 \,{\mathrm e}^{10}+1-\frac {\left (16 \,{\mathrm e}^{5}+8\right )^{2}}{64}} \erf \left (4 i x +\frac {i \left (16 \,{\mathrm e}^{5}+8\right )}{8}\right )+{\mathrm e}^{16 x^{2}+\left (16 \,{\mathrm e}^{5}+8\right ) x +4 \,{\mathrm e}^{5}+4 \,{\mathrm e}^{10}+1}+\frac {i \left (16 \,{\mathrm e}^{5}+8\right ) \sqrt {\pi }\, {\mathrm e}^{4 \,{\mathrm e}^{5}+4 \,{\mathrm e}^{10}+1-\frac {\left (16 \,{\mathrm e}^{5}+8\right )^{2}}{64}} \erf \left (4 i x +\frac {i \left (16 \,{\mathrm e}^{5}+8\right )}{8}\right )}{8}-2 i \sqrt {\pi }\, {\mathrm e}^{4 \,{\mathrm e}^{5}+4 \,{\mathrm e}^{10}+6-\frac {\left (16 \,{\mathrm e}^{5}+8\right )^{2}}{64}} \erf \left (4 i x +\frac {i \left (16 \,{\mathrm e}^{5}+8\right )}{8}\right )\)	\(161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*exp(5)+32*x+8)*exp(4*exp(5)^2+(16*x+4)*exp(5)+16*x^2+8*x+1)-1,x,method=_RETURNVERBOSE)

[Out]

-x+exp(16*x*exp(5)+16*x^2+4*exp(5)+4*exp(10)+8*x+1)

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maxima [A]  time = 0.46, size = 28, normalized size = 1.56 \begin {gather*} -x + e^{\left (16 \, x^{2} + 4 \, {\left (4 \, x + 1\right )} e^{5} + 8 \, x + 4 \, e^{10} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp(5)+32*x+8)*exp(4*exp(5)^2+(16*x+4)*exp(5)+16*x^2+8*x+1)-1,x, algorithm="maxima")

[Out]

-x + e^(16*x^2 + 4*(4*x + 1)*e^5 + 8*x + 4*e^10 + 1)

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mupad [B]  time = 0.14, size = 33, normalized size = 1.83 \begin {gather*} {\mathrm {e}}^{4\,{\mathrm {e}}^5}\,{\mathrm {e}}^{4\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{8\,x}\,\mathrm {e}\,{\mathrm {e}}^{16\,x^2}\,{\mathrm {e}}^{16\,x\,{\mathrm {e}}^5}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(8*x + 4*exp(10) + 16*x^2 + exp(5)*(16*x + 4) + 1)*(32*x + 16*exp(5) + 8) - 1,x)

[Out]

exp(4*exp(5))*exp(4*exp(10))*exp(8*x)*exp(1)*exp(16*x^2)*exp(16*x*exp(5)) - x

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sympy [A]  time = 0.12, size = 26, normalized size = 1.44 \begin {gather*} - x + e^{16 x^{2} + 8 x + \left (16 x + 4\right ) e^{5} + 1 + 4 e^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp(5)+32*x+8)*exp(4*exp(5)**2+(16*x+4)*exp(5)+16*x**2+8*x+1)-1,x)

[Out]

-x + exp(16*x**2 + 8*x + (16*x + 4)*exp(5) + 1 + 4*exp(10))

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