3.37.77 \(\int \frac {1}{3} e^{2 x+e x} (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)) \, dx\)

Optimal. Leaf size=22 \[ 5 e^{(2+e) x} \left (e^{76 x^2}-\frac {2 x}{3}\right ) \]

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Rubi [A]  time = 0.22, antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps used = 9, number of rules used = 7, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {6, 12, 6741, 6742, 2194, 2176, 2236} \begin {gather*} 5 e^{76 x^2+(2+e) x}-\frac {10}{3} e^{(2+e) x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*x + E*x)*(-10 - 20*x - 10*E*x + E^(76*x^2)*(30 + 15*E + 2280*x)))/3,x]

[Out]

5*E^((2 + E)*x + 76*x^2) - (10*E^((2 + E)*x)*x)/3

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{3} e^{2 x+e x} \left (-10+(-20-10 e) x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx\\ &=\frac {1}{3} \int e^{2 x+e x} \left (-10+(-20-10 e) x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx\\ &=\frac {1}{3} \int e^{(2+e) x} \left (-10+(-20-10 e) x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx\\ &=\frac {1}{3} \int \left (-10 e^{(2+e) x}-10 e^{(2+e) x} (2+e) x+15 e^{(2+e) x+76 x^2} (2+e+152 x)\right ) \, dx\\ &=-\left (\frac {10}{3} \int e^{(2+e) x} \, dx\right )+5 \int e^{(2+e) x+76 x^2} (2+e+152 x) \, dx-\frac {1}{3} (10 (2+e)) \int e^{(2+e) x} x \, dx\\ &=5 e^{(2+e) x+76 x^2}-\frac {10 e^{(2+e) x}}{3 (2+e)}-\frac {10}{3} e^{(2+e) x} x+\frac {10}{3} \int e^{(2+e) x} \, dx\\ &=5 e^{(2+e) x+76 x^2}-\frac {10}{3} e^{(2+e) x} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 24, normalized size = 1.09 \begin {gather*} \frac {5}{3} e^{(2+e) x} \left (3 e^{76 x^2}-2 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x + E*x)*(-10 - 20*x - 10*E*x + E^(76*x^2)*(30 + 15*E + 2280*x)))/3,x]

[Out]

(5*E^((2 + E)*x)*(3*E^(76*x^2) - 2*x))/3

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fricas [A]  time = 0.66, size = 23, normalized size = 1.05 \begin {gather*} -\frac {5}{3} \, {\left (2 \, x - 3 \, e^{\left (76 \, x^{2}\right )}\right )} e^{\left (x e + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((15*exp(1)+2280*x+30)*exp(76*x^2)-10*x*exp(1)-20*x-10)*exp(x*exp(1)+2*x),x, algorithm="fricas")

[Out]

-5/3*(2*x - 3*e^(76*x^2))*e^(x*e + 2*x)

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giac [B]  time = 0.21, size = 80, normalized size = 3.64 \begin {gather*} -\frac {10 \, {\left (x e + 2 \, x - 1\right )} e^{\left (x e + 2 \, x + 1\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} - \frac {10 \, {\left (2 \, x e + 4 \, x + e\right )} e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} + 5 \, e^{\left (76 \, x^{2} + x e + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((15*exp(1)+2280*x+30)*exp(76*x^2)-10*x*exp(1)-20*x-10)*exp(x*exp(1)+2*x),x, algorithm="giac")

[Out]

-10/3*(x*e + 2*x - 1)*e^(x*e + 2*x + 1)/(e^2 + 4*e + 4) - 10/3*(2*x*e + 4*x + e)*e^(x*e + 2*x)/(e^2 + 4*e + 4)
+ 5*e^(76*x^2 + x*e + 2*x)

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maple [A]  time = 0.07, size = 22, normalized size = 1.00

method	result	size
risch	\(\frac {\left (-10 x +15 \,{\mathrm e}^{76 x^{2}}\right ) {\mathrm e}^{x \left ({\mathrm e}+2\right )}}{3}\)	\(22\)
norman	\(-\frac {10 x \,{\mathrm e}^{x \,{\mathrm e}+2 x}}{3}+5 \,{\mathrm e}^{76 x^{2}} {\mathrm e}^{x \,{\mathrm e}+2 x}\)	\(31\)
default	\(-\frac {10 \,{\mathrm e}^{x \,{\mathrm e}+2 x}}{3 \left ({\mathrm e}+2\right )}-\frac {20 \left ({\mathrm e}^{x \left ({\mathrm e}+2\right )} \left ({\mathrm e}+2\right ) x -{\mathrm e}^{x \left ({\mathrm e}+2\right )}\right )}{3 \left ({\mathrm e}+2\right )^{2}}-\frac {5 i \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \erf \left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{38}-\frac {10 \left ({\mathrm e}^{1+x \left ({\mathrm e}+2\right )} \left (1+x \left ({\mathrm e}+2\right )\right )-2 \,{\mathrm e}^{1+x \left ({\mathrm e}+2\right )}\right )}{3 \left ({\mathrm e}+2\right )^{2}}+5 \,{\mathrm e}^{76 x^{2}+x \left ({\mathrm e}+2\right )}+\frac {5 i \left ({\mathrm e}+2\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \erf \left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{76}-\frac {5 i \sqrt {\pi }\, {\mathrm e}^{1-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \erf \left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{76}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((15*exp(1)+2280*x+30)*exp(76*x^2)-10*x*exp(1)-20*x-10)*exp(x*exp(1)+2*x),x,method=_RETURNVERBOSE)

[Out]

1/3*(-10*x+15*exp(76*x^2))*exp(x*(exp(1)+2))

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maxima [C]  time = 0.54, size = 239, normalized size = 10.86 \begin {gather*} -\frac {5}{38} i \, \sqrt {19} \sqrt {\pi } \operatorname {erf}\left (2 i \, \sqrt {19} x + \frac {1}{76} i \, \sqrt {19} {\left (e + 2\right )}\right ) e^{\left (-\frac {1}{304} \, {\left (e + 2\right )}^{2}\right )} - \frac {5}{76} i \, \sqrt {19} \sqrt {\pi } \operatorname {erf}\left (2 i \, \sqrt {19} x + \frac {1}{76} i \, \sqrt {19} {\left (e + 2\right )}\right ) e^{\left (-\frac {1}{304} \, {\left (e + 2\right )}^{2} + 1\right )} - \frac {5}{76} \, \sqrt {19} {\left (\frac {\sqrt {19} \sqrt {\frac {1}{19}} \sqrt {\pi } {\left (152 \, x + e + 2\right )} {\left (\operatorname {erf}\left (\frac {1}{4} \, \sqrt {\frac {1}{19}} \sqrt {-{\left (152 \, x + e + 2\right )}^{2}}\right ) - 1\right )} {\left (e + 2\right )}}{\sqrt {-{\left (152 \, x + e + 2\right )}^{2}}} - 4 \, \sqrt {19} e^{\left (\frac {1}{304} \, {\left (152 \, x + e + 2\right )}^{2}\right )}\right )} e^{\left (-\frac {1}{304} \, {\left (e + 2\right )}^{2}\right )} - \frac {10 \, {\left (x {\left (e^{2} + 2 \, e\right )} - e\right )} e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} - \frac {20 \, {\left (x {\left (e + 2\right )} - 1\right )} e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} - \frac {10 \, e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((15*exp(1)+2280*x+30)*exp(76*x^2)-10*x*exp(1)-20*x-10)*exp(x*exp(1)+2*x),x, algorithm="maxima")

[Out]

-5/38*I*sqrt(19)*sqrt(pi)*erf(2*I*sqrt(19)*x + 1/76*I*sqrt(19)*(e + 2))*e^(-1/304*(e + 2)^2) - 5/76*I*sqrt(19)
*sqrt(pi)*erf(2*I*sqrt(19)*x + 1/76*I*sqrt(19)*(e + 2))*e^(-1/304*(e + 2)^2 + 1) - 5/76*sqrt(19)*(sqrt(19)*sqr
t(1/19)*sqrt(pi)*(152*x + e + 2)*(erf(1/4*sqrt(1/19)*sqrt(-(152*x + e + 2)^2)) - 1)*(e + 2)/sqrt(-(152*x + e +
2)^2) - 4*sqrt(19)*e^(1/304*(152*x + e + 2)^2))*e^(-1/304*(e + 2)^2) - 10/3*(x*(e^2 + 2*e) - e)*e^(x*e + 2*x)
/(e^2 + 4*e + 4) - 20/3*(x*(e + 2) - 1)*e^(x*e + 2*x)/(e^2 + 4*e + 4) - 10/3*e^(x*e + 2*x)/(e + 2)

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mupad [B]  time = 0.10, size = 23, normalized size = 1.05 \begin {gather*} -{\mathrm {e}}^{2\,x+x\,\mathrm {e}}\,\left (\frac {10\,x}{3}-5\,{\mathrm {e}}^{76\,x^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x + x*exp(1))*(20*x + 10*x*exp(1) - exp(76*x^2)*(2280*x + 15*exp(1) + 30) + 10))/3,x)

[Out]

-exp(2*x + x*exp(1))*((10*x)/3 - 5*exp(76*x^2))

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sympy [A]  time = 1.07, size = 22, normalized size = 1.00 \begin {gather*} \frac {\left (- 10 x + 15 e^{76 x^{2}}\right ) e^{2 x + e x}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((15*exp(1)+2280*x+30)*exp(76*x**2)-10*x*exp(1)-20*x-10)*exp(x*exp(1)+2*x),x)

[Out]

(-10*x + 15*exp(76*x**2))*exp(2*x + E*x)/3

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