3.49.55 \(\int \frac {5 x+e^{7+x} x (5+5 x)}{x} \, dx\)

Optimal. Leaf size=12 \[ 5 \left (2+x+e^{7+x} x\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.75, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {14, 2176, 2194} \begin {gather*} 5 x-5 e^{x+7}+5 e^{x+7} (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5*x + E^(7 + x)*x*(5 + 5*x))/x,x]

[Out]

-5*E^(7 + x) + 5*x + 5*E^(7 + x)*(1 + 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} &=\int \left (5+5 e^{7+x} (1+x)\right ) \, dx\\ &=5 x+5 \int e^{7+x} (1+x) \, dx\\ &=5 x+5 e^{7+x} (1+x)-5 \int e^{7+x} \, dx\\ &=-5 e^{7+x}+5 x+5 e^{7+x} (1+x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} 5 x+5 e^{7+x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5*x + E^(7 + x)*x*(5 + 5*x))/x,x]

[Out]

5*x + 5*E^(7 + x)*x

________________________________________________________________________________________

fricas [A]  time = 1.00, size = 12, normalized size = 1.00 \begin {gather*} 5 \, x + 5 \, e^{\left (x + \log \relax (x) + 7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*exp(log(x)+x+7)+5*x)/x,x, algorithm="fricas")

[Out]

5*x + 5*e^(x + log(x) + 7)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 11, normalized size = 0.92 \begin {gather*} 5 \, x e^{\left (x + 7\right )} + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*exp(log(x)+x+7)+5*x)/x,x, algorithm="giac")

[Out]

5*x*e^(x + 7) + 5*x

________________________________________________________________________________________

maple [A]  time = 0.02, size = 12, normalized size = 1.00

method	result	size
risch	\(5 x +5 x \,{\mathrm e}^{x +7}\)	\(12\)
norman	\(5 x +5 \,{\mathrm e}^{\ln \relax (x )+x +7}\)	\(13\)
default	\(5 x +5 \,{\mathrm e}^{x +7} \left (x +7\right )-35 \,{\mathrm e}^{x +7}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x+5)*exp(ln(x)+x+7)+5*x)/x,x,method=_RETURNVERBOSE)

[Out]

5*x+5*x*exp(x+7)

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 23, normalized size = 1.92 \begin {gather*} 5 \, {\left (x e^{7} - e^{7}\right )} e^{x} + 5 \, x + 5 \, e^{\left (x + 7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*exp(log(x)+x+7)+5*x)/x,x, algorithm="maxima")

[Out]

5*(x*e^7 - e^7)*e^x + 5*x + 5*e^(x + 7)

________________________________________________________________________________________

mupad [B]  time = 3.22, size = 9, normalized size = 0.75 \begin {gather*} 5\,x\,\left ({\mathrm {e}}^{x+7}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + exp(x + log(x) + 7)*(5*x + 5))/x,x)

[Out]

5*x*(exp(x + 7) + 1)

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 10, normalized size = 0.83 \begin {gather*} 5 x e^{x + 7} + 5 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*exp(ln(x)+x+7)+5*x)/x,x)

[Out]

5*x*exp(x + 7) + 5*x

________________________________________________________________________________________