3.31.28 \(\int \frac {-e^{3+x} x+e^3 (-1+x+2 x^2)}{x} \, dx\)

Optimal. Leaf size=21 \[ e^3 \left (-1-e^x+x+x^2+\log \left (\frac {2}{x}\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {14, 2194, 76} \begin {gather*} e^3 x^2+e^3 x-e^{x+3}-e^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-(E^(3 + x)*x) + E^3*(-1 + x + 2*x^2))/x,x]

[Out]

-E^(3 + x) + E^3*x + E^3*x^2 - E^3*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.86 \begin {gather*} e^3 \left (-e^x+x+x^2-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(E^(3 + x)*x) + E^3*(-1 + x + 2*x^2))/x,x]

[Out]

E^3*(-E^x + x + x^2 - Log[x])

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fricas [A]  time = 0.90, size = 21, normalized size = 1.00 \begin {gather*} {\left (x^{2} + x\right )} e^{3} - e^{3} \log \relax (x) - e^{\left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 + x)*e^3 - e^3*log(x) - e^(x + 3)

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giac [A]  time = 0.20, size = 23, normalized size = 1.10 \begin {gather*} x^{2} e^{3} + x e^{3} - e^{3} \log \relax (x) - e^{\left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^3 + x*e^3 - e^3*log(x) - e^(x + 3)

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maple [A]  time = 0.04, size = 24, normalized size = 1.14

method	result	size
default	\(x^{2} {\mathrm e}^{3}-\ln \relax (x ) {\mathrm e}^{3}-{\mathrm e}^{x} {\mathrm e}^{3}+x \,{\mathrm e}^{3}\)	\(24\)
norman	\(x^{2} {\mathrm e}^{3}-\ln \relax (x ) {\mathrm e}^{3}-{\mathrm e}^{x} {\mathrm e}^{3}+x \,{\mathrm e}^{3}\)	\(24\)
risch	\(x^{2} {\mathrm e}^{3}-\ln \relax (x ) {\mathrm e}^{3}-{\mathrm e}^{3+x}+x \,{\mathrm e}^{3}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*exp(3)-ln(x)*exp(3)-exp(x)*exp(3)+x*exp(3)

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maxima [A]  time = 0.89, size = 23, normalized size = 1.10 \begin {gather*} x^{2} e^{3} + x e^{3} - e^{3} \log \relax (x) - e^{\left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^3 + x*e^3 - e^3*log(x) - e^(x + 3)

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mupad [B]  time = 0.07, size = 16, normalized size = 0.76 \begin {gather*} {\mathrm {e}}^3\,\left (x-{\mathrm {e}}^x-\ln \relax (x)+x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3)*(x + 2*x^2 - 1) - x*exp(3)*exp(x))/x,x)

[Out]

exp(3)*(x - exp(x) - log(x) + x^2)

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sympy [A]  time = 0.13, size = 24, normalized size = 1.14 \begin {gather*} x^{2} e^{3} + x e^{3} - e^{3} e^{x} - e^{3} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*exp(3) + x*exp(3) - exp(3)*exp(x) - exp(3)*log(x)

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