3.34.25 \(\int (-24+e^x (-6-6 x)+e^{100+40 x+4 x^2} (6+240 x+48 x^2)) \, dx\)

Optimal. Leaf size=19 \[ 6 \left (-4-e^x+e^{4 (5+x)^2}\right ) x \]

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Rubi [B]  time = 0.04, antiderivative size = 43, normalized size of antiderivative = 2.26, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2176, 2194, 2288} \begin {gather*} \frac {6 e^{4 x^2+40 x+100} \left (x^2+5 x\right )}{x+5}-24 x+6 e^x-6 e^x (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-24 + E^x*(-6 - 6*x) + E^(100 + 40*x + 4*x^2)*(6 + 240*x + 48*x^2),x]

[Out]

6*E^x - 24*x - 6*E^x*(1 + x) + (6*E^(100 + 40*x + 4*x^2)*(5*x + x^2))/(5 + 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 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} &=-24 x+\int e^x (-6-6 x) \, dx+\int e^{100+40 x+4 x^2} \left (6+240 x+48 x^2\right ) \, dx\\ &=-24 x-6 e^x (1+x)+\frac {6 e^{100+40 x+4 x^2} \left (5 x+x^2\right )}{5+x}+6 \int e^x \, dx\\ &=6 e^x-24 x-6 e^x (1+x)+\frac {6 e^{100+40 x+4 x^2} \left (5 x+x^2\right )}{5+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 22, normalized size = 1.16 \begin {gather*} -24 x-6 e^x x+6 e^{4 (5+x)^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-24 + E^x*(-6 - 6*x) + E^(100 + 40*x + 4*x^2)*(6 + 240*x + 48*x^2),x]

[Out]

-24*x - 6*E^x*x + 6*E^(4*(5 + x)^2)*x

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fricas [A]  time = 0.66, size = 23, normalized size = 1.21 \begin {gather*} 6 \, x e^{\left (4 \, x^{2} + 40 \, x + 100\right )} - 6 \, x e^{x} - 24 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^2+240*x+6)*exp(4*x^2+40*x+100)+(-6*x-6)*exp(x)-24,x, algorithm="fricas")

[Out]

6*x*e^(4*x^2 + 40*x + 100) - 6*x*e^x - 24*x

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giac [A]  time = 0.23, size = 23, normalized size = 1.21 \begin {gather*} 6 \, x e^{\left (4 \, x^{2} + 40 \, x + 100\right )} - 6 \, x e^{x} - 24 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^2+240*x+6)*exp(4*x^2+40*x+100)+(-6*x-6)*exp(x)-24,x, algorithm="giac")

[Out]

6*x*e^(4*x^2 + 40*x + 100) - 6*x*e^x - 24*x

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maple [A]  time = 0.03, size = 21, normalized size = 1.11

method	result	size
risch	\(-24 x -6 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{4 \left (5+x \right )^{2}} x\)	\(21\)
default	\(-24 x -6 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{4 x^{2}+40 x +100} x\)	\(24\)
norman	\(-24 x -6 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{4 x^{2}+40 x +100} x\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((48*x^2+240*x+6)*exp(4*x^2+40*x+100)+(-6*x-6)*exp(x)-24,x,method=_RETURNVERBOSE)

[Out]

-24*x-6*exp(x)*x+6*exp(4*(5+x)^2)*x

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maxima [A]  time = 0.82, size = 29, normalized size = 1.53 \begin {gather*} 6 \, x e^{\left (4 \, x^{2} + 40 \, x + 100\right )} - 6 \, {\left (x - 1\right )} e^{x} - 24 \, x - 6 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^2+240*x+6)*exp(4*x^2+40*x+100)+(-6*x-6)*exp(x)-24,x, algorithm="maxima")

[Out]

6*x*e^(4*x^2 + 40*x + 100) - 6*(x - 1)*e^x - 24*x - 6*e^x

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mupad [B]  time = 0.08, size = 20, normalized size = 1.05 \begin {gather*} -6\,x\,\left ({\mathrm {e}}^x-{\mathrm {e}}^{4\,x^2+40\,x+100}+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(40*x + 4*x^2 + 100)*(240*x + 48*x^2 + 6) - exp(x)*(6*x + 6) - 24,x)

[Out]

-6*x*(exp(x) - exp(40*x + 4*x^2 + 100) + 4)

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sympy [A]  time = 0.15, size = 24, normalized size = 1.26 \begin {gather*} - 6 x e^{x} + 6 x e^{4 x^{2} + 40 x + 100} - 24 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x**2+240*x+6)*exp(4*x**2+40*x+100)+(-6*x-6)*exp(x)-24,x)

[Out]

-6*x*exp(x) + 6*x*exp(4*x**2 + 40*x + 100) - 24*x

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