3.34.24 \(\int e^{1-5 e^4} (3+e^{-1+5 e^4} (-18 x+18 x^2)) \, dx\)

Optimal. Leaf size=22 \[ 1+3 x \left (e^{1-5 e^4}+x (-3+2 x)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {12} \begin {gather*} 6 x^3-9 x^2+3 e^{1-5 e^4} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(1 - 5*E^4)*(3 + E^(-1 + 5*E^4)*(-18*x + 18*x^2)),x]

[Out]

3*E^(1 - 5*E^4)*x - 9*x^2 + 6*x^3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{1-5 e^4} \int \left (3+e^{-1+5 e^4} \left (-18 x+18 x^2\right )\right ) \, dx\\ &=3 e^{1-5 e^4} x+\int \left (-18 x+18 x^2\right ) \, dx\\ &=3 e^{1-5 e^4} x-9 x^2+6 x^3\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 24, normalized size = 1.09 \begin {gather*} 3 \left (e^{1-5 e^4} x-3 x^2+2 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(1 - 5*E^4)*(3 + E^(-1 + 5*E^4)*(-18*x + 18*x^2)),x]

[Out]

3*(E^(1 - 5*E^4)*x - 3*x^2 + 2*x^3)

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 30, normalized size = 1.36 \begin {gather*} 3 \, {\left ({\left (2 \, x^{3} - 3 \, x^{2}\right )} e^{\left (5 \, e^{4} - 1\right )} + x\right )} e^{\left (-5 \, e^{4} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^2-18*x)*exp(5*exp(4)-1)+3)/exp(5*exp(4)-1),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.18, size = 30, normalized size = 1.36 \begin {gather*} 3 \, {\left ({\left (2 \, x^{3} - 3 \, x^{2}\right )} e^{\left (5 \, e^{4} - 1\right )} + x\right )} e^{\left (-5 \, e^{4} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^2-18*x)*exp(5*exp(4)-1)+3)/exp(5*exp(4)-1),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 22, normalized size = 1.00

method	result	size
risch	\(6 x^{3}-9 x^{2}+3 x \,{\mathrm e}^{1-5 \,{\mathrm e}^{4}}\)	\(22\)
norman	\(-9 x^{2}+6 x^{3}+3 \,{\mathrm e}^{-5 \,{\mathrm e}^{4}} {\mathrm e} x\)	\(24\)
default	\({\mathrm e}^{1-5 \,{\mathrm e}^{4}} \left ({\mathrm e}^{5 \,{\mathrm e}^{4}-1} \left (6 x^{3}-9 x^{2}\right )+3 x \right )\)	\(34\)
gosper	\(3 x \left (2 \,{\mathrm e}^{5 \,{\mathrm e}^{4}-1} x^{2}-3 \,{\mathrm e}^{5 \,{\mathrm e}^{4}-1} x +1\right ) {\mathrm e}^{1-5 \,{\mathrm e}^{4}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((18*x^2-18*x)*exp(5*exp(4)-1)+3)/exp(5*exp(4)-1),x,method=_RETURNVERBOSE)

[Out]

6*x^3-9*x^2+3*x*exp(1-5*exp(4))

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 30, normalized size = 1.36 \begin {gather*} 3 \, {\left ({\left (2 \, x^{3} - 3 \, x^{2}\right )} e^{\left (5 \, e^{4} - 1\right )} + x\right )} e^{\left (-5 \, e^{4} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^2-18*x)*exp(5*exp(4)-1)+3)/exp(5*exp(4)-1),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 19, normalized size = 0.86 \begin {gather*} 3\,x\,\left (2\,x^2-3\,x+{\mathrm {e}}^{1-5\,{\mathrm {e}}^4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(1 - 5*exp(4))*(exp(5*exp(4) - 1)*(18*x - 18*x^2) - 3),x)

[Out]

3*x*(exp(1 - 5*exp(4)) - 3*x + 2*x^2)

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 22, normalized size = 1.00 \begin {gather*} 6 x^{3} - 9 x^{2} + \frac {3 e x}{e^{5 e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x**2-18*x)*exp(5*exp(4)-1)+3)/exp(5*exp(4)-1),x)

[Out]

6*x**3 - 9*x**2 + 3*E*x*exp(-5*exp(4))

________________________________________________________________________________________