3.33.47 $\int \frac{1-36x-3x^2+e^x(-18-18x-3x^2)}{36+12x+x^2+e(36+12x+x^2)}dx$

Optimal. Leaf size=26 $$4-\frac{x(\frac{1}{x}+3(e^x+x))}{(1+e)(6+x)}$$

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Rubi [A]  time = 0.29, antiderivative size = 46, normalized size of antiderivative = 1.77, number of steps used = 13, number of rules used = 9, integrand size = 45, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.200, Rules used = {6741, 27, 12, 6742, 683, 2199, 2194, 2177, 2178} $$\begin{array}{c}-\frac{3x}{1+e}+\frac{18e^x}{(1+e)(x+6)}-\frac{109}{(1+e)(x+6)}-\frac{3e^x}{1+e}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(1 - 36*x - 3*x^2 + E^x*(-18 - 18*x - 3*x^2))/(36 + 12*x + x^2 + E*(36 + 12*x + x^2)),x]

[Out]

(-3*E^x)/(1 + E) - (3*x)/(1 + E) - 109/((1 + E)*(6 + x)) + (18*E^x)/((1 + E)*(6 + 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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), 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] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$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]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 6741

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

Rule 6742

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{1-36x-3x^2+e^x(-18-18x-3x^2)}{36(1+e)+12(1+e)x+(1+e)x^2}dx\\ & =\int \frac{1-36x-3x^2+e^x(-18-18x-3x^2)}{(1+e)(6+x{)}^2}dx\\ & =\frac{\int \frac{1-36x-3x^2+e^x(-18-18x-3x^2)}{(6+x{)}^2}dx}{1+e}\\ & =\frac{\int (\frac{1-36x-3x^2}{(6+x{)}^2}-\frac{3e^x(6+6x+x^2)}{(6+x{)}^2})dx}{1+e}\\ & =\frac{\int \frac{1-36x-3x^2}{(6+x{)}^2}dx}{1+e}-\frac{3\int \frac{e^x(6+6x+x^2)}{(6+x{)}^2}dx}{1+e}\\ & =\frac{\int (-3+\frac{109}{(6+x{)}^2})dx}{1+e}-\frac{3\int (e^x+\frac{6e^x}{(6+x{)}^2}-\frac{6e^x}{6+x})dx}{1+e}\\ & =-\frac{3x}{1+e}-\frac{109}{(1+e)(6+x)}-\frac{3\int e^{x}dx}{1+e}-\frac{18\int \frac{e^x}{(6+x{)}^2}dx}{1+e}+\frac{18\int \frac{e^x}{6+x}dx}{1+e}\\ & =-\frac{3e^x}{1+e}-\frac{3x}{1+e}-\frac{109}{(1+e)(6+x)}+\frac{18e^x}{(1+e)(6+x)}+\frac{18\text{Ei}(6+x)}{e^6(1+e)}-\frac{18\int \frac{e^x}{6+x}dx}{1+e}\\ & =-\frac{3e^x}{1+e}-\frac{3x}{1+e}-\frac{109}{(1+e)(6+x)}+\frac{18e^x}{(1+e)(6+x)}\end{array}\end{array}$$

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Mathematica [A]  time = 0.06, size = 27, normalized size = 1.04 $$\begin{array}{c}-\frac{109+3(6+e^x)x+3x^2}{(1+e)(6+x)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(1 - 36*x - 3*x^2 + E^x*(-18 - 18*x - 3*x^2))/(36 + 12*x + x^2 + E*(36 + 12*x + x^2)),x]

[Out]

-((109 + 3*(6 + E^x)*x + 3*x^2)/((1 + E)*(6 + x)))

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fricas [A]  time = 0.61, size = 28, normalized size = 1.08 $$\begin{array}{c}-\frac{3x^2+3x{e}^x+18x+109}{(x+6)e+x+6}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2-18*x-18)*exp(x)-3*x^2-36*x+1)/((x^2+12*x+36)*exp(1)+x^2+12*x+36),x, algorithm="fricas")

[Out]

-(3*x^2 + 3*x*e^x + 18*x + 109)/((x + 6)*e + x + 6)

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giac [A]  time = 0.31, size = 30, normalized size = 1.15 $$\begin{array}{c}-\frac{3x^2+3x{e}^x+18x+109}{xe+x+6e+6}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2-18*x-18)*exp(x)-3*x^2-36*x+1)/((x^2+12*x+36)*exp(1)+x^2+12*x+36),x, algorithm="giac")

[Out]

-(3*x^2 + 3*x*e^x + 18*x + 109)/(x*e + x + 6*e + 6)

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maple [A]  time = 0.59, size = 38, normalized size = 1.46

method	result	size
norman	$\frac{-\frac{3x^2}{1+\mathrm{e}}-\frac{3x{\mathrm{e}}^x}{1+\mathrm{e}}-\frac{1}{1+\mathrm{e}}}{x+6}$	$38$
risch	$-\frac{3x}{1+\mathrm{e}}-\frac{109\mathrm{e}}{(1+\mathrm{e})(x\mathrm{e}+6\mathrm{e}+x+6)}-\frac{109}{(1+\mathrm{e})(x\mathrm{e}+6\mathrm{e}+x+6)}-\frac{3x{\mathrm{e}}^x}{(1+\mathrm{e})(x+6)}$	$71$
default	$-\frac{109}{(1+\mathrm{e})(x+6)}-\frac{3x}{1+\mathrm{e}}-\frac{18(-\frac{{\mathrm{e}}^x}{x+6}-{\mathrm{e}}^{-6}\text{expIntegralEi}(1,-x-6))}{1+\mathrm{e}}-\frac{18{\mathrm{e}}^{-6}\text{expIntegralEi}(1,-x-6)}{1+\mathrm{e}}-\frac{3{\mathrm{e}}^x}{1+\mathrm{e}}$	$80$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^2-18*x-18)*exp(x)-3*x^2-36*x+1)/((x^2+12*x+36)*exp(1)+x^2+12*x+36),x,method=_RETURNVERBOSE)

[Out]

(-3/(1+exp(1))*x^2-3*x/(1+exp(1))*exp(x)-1/(1+exp(1)))/(x+6)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}-\frac{3x{e}^x}{x(e+1)+6e+6}-\frac{3x}{e+1}+\frac{18e^{(-6)}{E}_2(-x-6)}{(x+6)(e+1)}-\frac{109}{x(e+1)+6e+6}+18\int \frac{e^x}{x^2(e+1)+12x(e+1)+36e+36}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2-18*x-18)*exp(x)-3*x^2-36*x+1)/((x^2+12*x+36)*exp(1)+x^2+12*x+36),x, algorithm="maxima")

[Out]

-3*x*e^x/(x*(e + 1) + 6*e + 6) - 3*x/(e + 1) + 18*e^(-6)*exp_integral_e(2, -x - 6)/((x + 6)*(e + 1)) - 109/(x*
(e + 1) + 6*e + 6) + 18*integrate(e^x/(x^2*(e + 1) + 12*x*(e + 1) + 36*e + 36), x)

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mupad [B]  time = 0.27, size = 23, normalized size = 0.88 $$\begin{array}{c}-\frac{x(18x+18{\mathrm{e}}^x-1)}{6(\mathrm{e}+1)(x+6)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(36*x + exp(x)*(18*x + 3*x^2 + 18) + 3*x^2 - 1)/(12*x + exp(1)*(12*x + x^2 + 36) + x^2 + 36),x)

[Out]

-(x*(18*x + 18*exp(x) - 1))/(6*(exp(1) + 1)*(x + 6))

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sympy [A]  time = 0.29, size = 44, normalized size = 1.69 $$\begin{array}{c}-\frac{3x}{1+e}-\frac{3x{e}^x}{x+ex+6+6e}-\frac{109}{x(1+e)+6+6e}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**2-18*x-18)*exp(x)-3*x**2-36*x+1)/((x**2+12*x+36)*exp(1)+x**2+12*x+36),x)

[Out]

-3*x/(1 + E) - 3*x*exp(x)/(x + E*x + 6 + 6*E) - 109/(x*(1 + E) + 6 + 6*E)

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