3.71.89 \(\int \frac {-7 x+e^x x (4+4 x)}{3 x} \, dx\)

Optimal. Leaf size=26 \[ 3-x+\frac {1}{3} \left (\frac {80}{e^3}+4 \left (-x+e^x x\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {12, 14, 2176, 2194} \begin {gather*} -\frac {7 x}{3}-\frac {4 e^x}{3}+\frac {4}{3} e^x (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-7*x + E^x*x*(4 + 4*x))/(3*x),x]

[Out]

(-4*E^x)/3 - (7*x)/3 + (4*E^x*(1 + 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]]

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 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-7 x+e^x x (4+4 x)}{x} \, dx\\ &=\frac {1}{3} \int \left (-7+4 e^x (1+x)\right ) \, dx\\ &=-\frac {7 x}{3}+\frac {4}{3} \int e^x (1+x) \, dx\\ &=-\frac {7 x}{3}+\frac {4}{3} e^x (1+x)-\frac {4 \int e^x \, dx}{3}\\ &=-\frac {4 e^x}{3}-\frac {7 x}{3}+\frac {4}{3} e^x (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.54 \begin {gather*} \frac {1}{3} \left (-7 x+4 e^x x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7*x + E^x*x*(4 + 4*x))/(3*x),x]

[Out]

(-7*x + 4*E^x*x)/3

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fricas [A]  time = 1.13, size = 11, normalized size = 0.42 \begin {gather*} -\frac {7}{3} \, x + \frac {4}{3} \, e^{\left (x + \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((4*x+4)*exp(x+log(x))-7*x)/x,x, algorithm="fricas")

[Out]

-7/3*x + 4/3*e^(x + log(x))

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giac [A]  time = 0.13, size = 9, normalized size = 0.35 \begin {gather*} \frac {4}{3} \, x e^{x} - \frac {7}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((4*x+4)*exp(x+log(x))-7*x)/x,x, algorithm="giac")

[Out]

4/3*x*e^x - 7/3*x

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maple [A]  time = 0.02, size = 10, normalized size = 0.38

method	result	size
default	\(-\frac {7 x}{3}+\frac {4 \,{\mathrm e}^{x} x}{3}\)	\(10\)
risch	\(-\frac {7 x}{3}+\frac {4 \,{\mathrm e}^{x} x}{3}\)	\(10\)
norman	\(-\frac {7 x}{3}+\frac {4 \,{\mathrm e}^{x +\ln \relax (x )}}{3}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((4*x+4)*exp(x+ln(x))-7*x)/x,x,method=_RETURNVERBOSE)

[Out]

-7/3*x+4/3*exp(x)*x

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maxima [A]  time = 0.37, size = 15, normalized size = 0.58 \begin {gather*} \frac {4}{3} \, {\left (x - 1\right )} e^{x} - \frac {7}{3} \, x + \frac {4}{3} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((4*x+4)*exp(x+log(x))-7*x)/x,x, algorithm="maxima")

[Out]

4/3*(x - 1)*e^x - 7/3*x + 4/3*e^x

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mupad [B]  time = 4.17, size = 9, normalized size = 0.35 \begin {gather*} \frac {x\,\left (4\,{\mathrm {e}}^x-7\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((7*x)/3 - (exp(x + log(x))*(4*x + 4))/3)/x,x)

[Out]

(x*(4*exp(x) - 7))/3

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sympy [A]  time = 0.10, size = 12, normalized size = 0.46 \begin {gather*} \frac {4 x e^{x}}{3} - \frac {7 x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((4*x+4)*exp(x+ln(x))-7*x)/x,x)

[Out]

4*x*exp(x)/3 - 7*x/3

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