\(\int (-902+e^{5 x} (-1-5 x)+2 x-\log (x)) \, dx\) [7692]

Optimal result

Integrand size = 20, antiderivative size = 16 \[ \int \left (-902+e^{5 x} (-1-5 x)+2 x-\log (x)\right ) \, dx=x \left (-901-e^{5 x}+x-\log (x)\right ) \]

[Out]

x*(x-901-exp(5*x)-ln(x))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(35\) vs. \(2(16)=32\).

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.19, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2207, 2225, 2332} \[ \int \left (-902+e^{5 x} (-1-5 x)+2 x-\log (x)\right ) \, dx=x^2-901 x+\frac {e^{5 x}}{5}-\frac {1}{5} e^{5 x} (5 x+1)-x \log (x) \]

[In]

Int[-902 + E^(5*x)*(-1 - 5*x) + 2*x - Log[x],x]

[Out]

E^(5*x)/5 - 901*x + x^2 - (E^(5*x)*(1 + 5*x))/5 - x*Log[x]

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps \begin{align*} \text {integral}& = -902 x+x^2+\int e^{5 x} (-1-5 x) \, dx-\int \log (x) \, dx \\ & = -901 x+x^2-\frac {1}{5} e^{5 x} (1+5 x)-x \log (x)+\int e^{5 x} \, dx \\ & = \frac {e^{5 x}}{5}-901 x+x^2-\frac {1}{5} e^{5 x} (1+5 x)-x \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \left (-902+e^{5 x} (-1-5 x)+2 x-\log (x)\right ) \, dx=-901 x-e^{5 x} x+x^2-x \log (x) \]

[In]

Integrate[-902 + E^(5*x)*(-1 - 5*x) + 2*x - Log[x],x]

[Out]

-901*x - E^(5*x)*x + x^2 - x*Log[x]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25

method	result	size
default	\(-901 x -x \,{\mathrm e}^{5 x}+x^{2}-x \ln \left (x \right )\)	\(20\)
norman	\(-901 x -x \,{\mathrm e}^{5 x}+x^{2}-x \ln \left (x \right )\)	\(20\)
risch	\(-901 x -x \,{\mathrm e}^{5 x}+x^{2}-x \ln \left (x \right )\)	\(20\)
parallelrisch	\(-901 x -x \,{\mathrm e}^{5 x}+x^{2}-x \ln \left (x \right )\)	\(20\)
parts	\(-901 x -x \,{\mathrm e}^{5 x}+x^{2}-x \ln \left (x \right )\)	\(20\)

[In]

int(-ln(x)+(-5*x-1)*exp(5*x)+2*x-902,x,method=_RETURNVERBOSE)

[Out]

-901*x-x*exp(5*x)+x^2-x*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \left (-902+e^{5 x} (-1-5 x)+2 x-\log (x)\right ) \, dx=x^{2} - x e^{\left (5 \, x\right )} - x \log \left (x\right ) - 901 \, x \]

[In]

integrate(-log(x)+(-5*x-1)*exp(5*x)+2*x-902,x, algorithm="fricas")

[Out]

x^2 - x*e^(5*x) - x*log(x) - 901*x

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \left (-902+e^{5 x} (-1-5 x)+2 x-\log (x)\right ) \, dx=x^{2} - x e^{5 x} - x \log {\left (x \right )} - 901 x \]

[In]

integrate(-ln(x)+(-5*x-1)*exp(5*x)+2*x-902,x)

[Out]

x**2 - x*exp(5*x) - x*log(x) - 901*x

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \left (-902+e^{5 x} (-1-5 x)+2 x-\log (x)\right ) \, dx=x^{2} - x e^{\left (5 \, x\right )} - x \log \left (x\right ) - 901 \, x \]

[In]

integrate(-log(x)+(-5*x-1)*exp(5*x)+2*x-902,x, algorithm="maxima")

[Out]

x^2 - x*e^(5*x) - x*log(x) - 901*x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \left (-902+e^{5 x} (-1-5 x)+2 x-\log (x)\right ) \, dx=x^{2} - x e^{\left (5 \, x\right )} - x \log \left (x\right ) - 901 \, x \]

[In]

integrate(-log(x)+(-5*x-1)*exp(5*x)+2*x-902,x, algorithm="giac")

[Out]

x^2 - x*e^(5*x) - x*log(x) - 901*x

Mupad [B] (verification not implemented)

Time = 13.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (-902+e^{5 x} (-1-5 x)+2 x-\log (x)\right ) \, dx=-x\,\left ({\mathrm {e}}^{5\,x}-x+\ln \left (x\right )+901\right ) \]

[In]

int(2*x - log(x) - exp(5*x)*(5*x + 1) - 902,x)

[Out]

-x*(exp(5*x) - x + log(x) + 901)