3.26.44 \(\int (101-2 x+e^{1+e^x (5+x)} (25+e^x (150 x+25 x^2))) \, dx\)

Optimal. Leaf size=25 \[ x-x \left (-25 \left (4+e^{1+e^x (5+x)}+\frac {1}{x}\right )+x\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.72, number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2288} \begin {gather*} -x^2+\frac {25 e^{x+e^x (x+5)+1} \left (x^2+6 x\right )}{e^x (x+5)+e^x}+101 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[101 - 2*x + E^(1 + E^x*(5 + x))*(25 + E^x*(150*x + 25*x^2)),x]

[Out]

101*x - x^2 + (25*E^(1 + x + E^x*(5 + x))*(6*x + x^2))/(E^x + E^x*(5 + 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} &=101 x-x^2+\int e^{1+e^x (5+x)} \left (25+e^x \left (150 x+25 x^2\right )\right ) \, dx\\ &=101 x-x^2+\frac {25 e^{1+x+e^x (5+x)} \left (6 x+x^2\right )}{e^x+e^x (5+x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 19, normalized size = 0.76 \begin {gather*} -x \left (-101-25 e^{1+e^x (5+x)}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[101 - 2*x + E^(1 + E^x*(5 + x))*(25 + E^x*(150*x + 25*x^2)),x]

[Out]

-(x*(-101 - 25*E^(1 + E^x*(5 + x)) + x))

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 21, normalized size = 0.84 \begin {gather*} -x^{2} + 25 \, x e^{\left ({\left (x + 5\right )} e^{x} + 1\right )} + 101 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^2+150*x)*exp(x)+25)*exp((5+x)*exp(x)+1)-2*x+101,x, algorithm="fricas")

[Out]

-x^2 + 25*x*e^((x + 5)*e^x + 1) + 101*x

________________________________________________________________________________________

giac [A]  time = 0.28, size = 23, normalized size = 0.92 \begin {gather*} -x^{2} + 25 \, x e^{\left (x e^{x} + 5 \, e^{x} + 1\right )} + 101 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^2+150*x)*exp(x)+25)*exp((5+x)*exp(x)+1)-2*x+101,x, algorithm="giac")

[Out]

-x^2 + 25*x*e^(x*e^x + 5*e^x + 1) + 101*x

________________________________________________________________________________________

maple [A]  time = 0.04, size = 22, normalized size = 0.88

method	result	size
default	\(101 x +25 x \,{\mathrm e}^{\left (5+x \right ) {\mathrm e}^{x}+1}-x^{2}\)	\(22\)
norman	\(101 x +25 x \,{\mathrm e}^{\left (5+x \right ) {\mathrm e}^{x}+1}-x^{2}\)	\(22\)
risch	\(101 x +25 x \,{\mathrm e}^{{\mathrm e}^{x} x +5 \,{\mathrm e}^{x}+1}-x^{2}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((25*x^2+150*x)*exp(x)+25)*exp((5+x)*exp(x)+1)-2*x+101,x,method=_RETURNVERBOSE)

[Out]

101*x+25*x*exp((5+x)*exp(x)+1)-x^2

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 23, normalized size = 0.92 \begin {gather*} -x^{2} + 25 \, x e^{\left (x e^{x} + 5 \, e^{x} + 1\right )} + 101 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^2+150*x)*exp(x)+25)*exp((5+x)*exp(x)+1)-2*x+101,x, algorithm="maxima")

[Out]

-x^2 + 25*x*e^(x*e^x + 5*e^x + 1) + 101*x

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 21, normalized size = 0.84 \begin {gather*} x\,\left (25\,{\mathrm {e}}^{x\,{\mathrm {e}}^x}\,\mathrm {e}\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}-x+101\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(x)*(x + 5) + 1)*(exp(x)*(150*x + 25*x^2) + 25) - 2*x + 101,x)

[Out]

x*(25*exp(x*exp(x))*exp(1)*exp(5*exp(x)) - x + 101)

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 19, normalized size = 0.76 \begin {gather*} - x^{2} + 25 x e^{\left (x + 5\right ) e^{x} + 1} + 101 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x**2+150*x)*exp(x)+25)*exp((5+x)*exp(x)+1)-2*x+101,x)

[Out]

-x**2 + 25*x*exp((x + 5)*exp(x) + 1) + 101*x

________________________________________________________________________________________