3.23.62 \(\int (50 e x+75 x^2+e^{-1+e^{2 x}} (e+2 x+e^{2 x} (2 e x+2 x^2))) \, dx\)

Optimal. Leaf size=22 \[ \left (25+\frac {e^{-1+e^{2 x}}}{x}\right ) x^2 (e+x) \]

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Rubi [A]  time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.32, number of steps used = 2, number of rules used = 1, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2288} \begin {gather*} 25 x^3+25 e x^2+e^{e^{2 x}-1} \left (x^2+e x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[50*E*x + 75*x^2 + E^(-1 + E^(2*x))*(E + 2*x + E^(2*x)*(2*E*x + 2*x^2)),x]

[Out]

25*E*x^2 + 25*x^3 + E^(-1 + E^(2*x))*(E*x + x^2)

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} &=25 e x^2+25 x^3+\int e^{-1+e^{2 x}} \left (e+2 x+e^{2 x} \left (2 e x+2 x^2\right )\right ) \, dx\\ &=25 e x^2+25 x^3+e^{-1+e^{2 x}} \left (e x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 26, normalized size = 1.18 \begin {gather*} 25 e x^2+25 x^3+e^{-1+e^{2 x}} x (e+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[50*E*x + 75*x^2 + E^(-1 + E^(2*x))*(E + 2*x + E^(2*x)*(2*E*x + 2*x^2)),x]

[Out]

25*E*x^2 + 25*x^3 + E^(-1 + E^(2*x))*x*(E + x)

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fricas [A]  time = 1.74, size = 29, normalized size = 1.32 \begin {gather*} 25 \, x^{3} + 25 \, x^{2} e + {\left (x^{2} + x e\right )} e^{\left (e^{\left (2 \, x\right )} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(1)+2*x^2)*exp(x)^2+exp(1)+2*x)*exp(exp(x)^2-1)+50*x*exp(1)+75*x^2,x, algorithm="fricas")

[Out]

25*x^3 + 25*x^2*e + (x^2 + x*e)*e^(e^(2*x) - 1)

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giac [B]  time = 0.34, size = 46, normalized size = 2.09 \begin {gather*} 25 \, x^{3} + 25 \, x^{2} e + {\left (x^{2} e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )} + x e^{\left (2 \, x + e^{\left (2 \, x\right )} + 1\right )}\right )} e^{\left (-2 \, x - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(1)+2*x^2)*exp(x)^2+exp(1)+2*x)*exp(exp(x)^2-1)+50*x*exp(1)+75*x^2,x, algorithm="giac")

[Out]

25*x^3 + 25*x^2*e + (x^2*e^(2*x + e^(2*x)) + x*e^(2*x + e^(2*x) + 1))*e^(-2*x - 1)

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maple [A]  time = 0.09, size = 30, normalized size = 1.36

method	result	size
risch	\(\left (x \,{\mathrm e}+x^{2}\right ) {\mathrm e}^{{\mathrm e}^{2 x}-1}+25 x^{2} {\mathrm e}+25 x^{3}\)	\(30\)
default	\(x^{2} {\mathrm e}^{{\mathrm e}^{2 x}-1}+{\mathrm e} x \,{\mathrm e}^{{\mathrm e}^{2 x}-1}+25 x^{3}+25 x^{2} {\mathrm e}\)	\(36\)
norman	\(x^{2} {\mathrm e}^{{\mathrm e}^{2 x}-1}+{\mathrm e} x \,{\mathrm e}^{{\mathrm e}^{2 x}-1}+25 x^{3}+25 x^{2} {\mathrm e}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*exp(1)+2*x^2)*exp(x)^2+exp(1)+2*x)*exp(exp(x)^2-1)+50*x*exp(1)+75*x^2,x,method=_RETURNVERBOSE)

[Out]

(x*exp(1)+x^2)*exp(exp(2*x)-1)+25*x^2*exp(1)+25*x^3

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maxima [A]  time = 0.35, size = 29, normalized size = 1.32 \begin {gather*} 25 \, x^{3} + 25 \, x^{2} e + {\left (x^{2} + x e\right )} e^{\left (e^{\left (2 \, x\right )} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(1)+2*x^2)*exp(x)^2+exp(1)+2*x)*exp(exp(x)^2-1)+50*x*exp(1)+75*x^2,x, algorithm="maxima")

[Out]

25*x^3 + 25*x^2*e + (x^2 + x*e)*e^(e^(2*x) - 1)

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mupad [B]  time = 1.29, size = 19, normalized size = 0.86 \begin {gather*} x\,{\mathrm {e}}^{-1}\,\left (x+\mathrm {e}\right )\,\left ({\mathrm {e}}^{{\mathrm {e}}^{2\,x}}+25\,x\,\mathrm {e}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(50*x*exp(1) + exp(exp(2*x) - 1)*(2*x + exp(1) + exp(2*x)*(2*x*exp(1) + 2*x^2)) + 75*x^2,x)

[Out]

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

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sympy [A]  time = 0.17, size = 29, normalized size = 1.32 \begin {gather*} 25 x^{3} + 25 e x^{2} + \left (x^{2} + e x\right ) e^{e^{2 x} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(1)+2*x**2)*exp(x)**2+exp(1)+2*x)*exp(exp(x)**2-1)+50*x*exp(1)+75*x**2,x)

[Out]

25*x**3 + 25*E*x**2 + (x**2 + E*x)*exp(exp(2*x) - 1)

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