3.33.23 $\int \frac{1+x}{e^{10}{x}^3+2e^{10}{x}^2\log (x)+e^{10}x{\log }^2(x)}dx$

Optimal. Leaf size=16 $$4+\frac{1}{e^{10}(-x-\log (x))}$$

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 11, normalized size of antiderivative = 0.69, number of steps used = 3, number of rules used = 3, integrand size = 33, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.091, Rules used = {6688, 12, 6686} $$\begin{array}{c}-\frac{1}{e^{10}(x+\log (x))}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(1 + x)/(E^10*x^3 + 2*E^10*x^2*Log[x] + E^10*x*Log[x]^2),x]

[Out]

-(1/(E^10*(x + Log[x])))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{1+x}{e^{10}x(x+\log (x){)}^2}dx\\ & =\frac{\int \frac{1+x}{x(x+\log (x){)}^2}dx}{e^{10}}\\ & =-\frac{1}{e^{10}(x+\log (x))}\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 11, normalized size = 0.69 $$\begin{array}{c}-\frac{1}{e^{10}(x+\log (x))}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/(E^10*x^3 + 2*E^10*x^2*Log[x] + E^10*x*Log[x]^2),x]

[Out]

-(1/(E^10*(x + Log[x])))

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 14, normalized size = 0.88 $$\begin{array}{c}-\frac{1}{x{e}^{10}+e^{10}\log (x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)/(x*exp(10)*log(x)^2+2*x^2*exp(10)*log(x)+x^3*exp(10)),x, algorithm="fricas")

[Out]

-1/(x*e^10 + e^10*log(x))

________________________________________________________________________________________

giac [A]  time = 0.27, size = 14, normalized size = 0.88 $$\begin{array}{c}-\frac{1}{x{e}^{10}+e^{10}\log (x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)/(x*exp(10)*log(x)^2+2*x^2*exp(10)*log(x)+x^3*exp(10)),x, algorithm="giac")

[Out]

-1/(x*e^10 + e^10*log(x))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 11, normalized size = 0.69

method	result	size
risch	$-\frac{{\mathrm{e}}^{-10}}{x+\ln (x)}$	$11$
norman	$-\frac{{\mathrm{e}}^{-10}}{x+\ln (x)}$	$13$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)/(x*exp(10)*ln(x)^2+2*x^2*exp(10)*ln(x)+x^3*exp(10)),x,method=_RETURNVERBOSE)

[Out]

-exp(-10)/(x+ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 14, normalized size = 0.88 $$\begin{array}{c}-\frac{1}{x{e}^{10}+e^{10}\log (x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)/(x*exp(10)*log(x)^2+2*x^2*exp(10)*log(x)+x^3*exp(10)),x, algorithm="maxima")

[Out]

-1/(x*e^10 + e^10*log(x))

________________________________________________________________________________________

mupad [B]  time = 2.10, size = 10, normalized size = 0.62 $$\begin{array}{c}-\frac{{\mathrm{e}}^{-10}}{x+\ln (x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(x^3*exp(10) + x*exp(10)*log(x)^2 + 2*x^2*exp(10)*log(x)),x)

[Out]

-exp(-10)/(x + log(x))

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 14, normalized size = 0.88 $$\begin{array}{c}-\frac{1}{x{e}^{10}+e^{10}\log (x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)/(x*exp(10)*ln(x)**2+2*x**2*exp(10)*ln(x)+x**3*exp(10)),x)

[Out]

-1/(x*exp(10) + exp(10)*log(x))

________________________________________________________________________________________