3.41.54 $\int \frac{e^{2x}(8-20x+24x^2-14x^3+2x^4)+e^{3+x}(-5184x^3+3888x^4-972x^5+81x^6)}{-5184x^3+3888x^4-972x^5+81x^6}dx$

Optimal. Leaf size=28 $$e^{3+x}+\frac{e^{2x}(-1+x{)}^2}{81(-4+x{)}^2{x}^2}$$

________________________________________________________________________________________

Rubi [B]  time = 0.49, antiderivative size = 62, normalized size of antiderivative = 2.21, number of steps used = 17, number of rules used = 5, integrand size = 78, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.064, Rules used = {6688, 2194, 6742, 2177, 2178} $$\begin{array}{c}\frac{e^{2x}}{1296x^2}+e^{x+3}-\frac{e^{2x}}{864x}-\frac{e^{2x}}{864(4-x)}+\frac{e^{2x}}{144(4-x{)}^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^(2*x)*(8 - 20*x + 24*x^2 - 14*x^3 + 2*x^4) + E^(3 + x)*(-5184*x^3 + 3888*x^4 - 972*x^5 + 81*x^6))/(-518
4*x^3 + 3888*x^4 - 972*x^5 + 81*x^6),x]

[Out]

E^(3 + x) + E^(2*x)/(144*(4 - x)^2) - E^(2*x)/(864*(4 - x)) + E^(2*x)/(1296*x^2) - E^(2*x)/(864*x)

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 6688

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

Rule 6742

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (e^{3+x}+\frac{2e^{2x}(4-10x+12x^2-7x^3+x^4)}{81(-4+x{)}^3{x}^3})dx\\ & =\frac{2}{81}\int \frac{e^{2x}(4-10x+12x^2-7x^3+x^4)}{(-4+x{)}^3{x}^3}dx+\int e^{3+x}dx\\ & =e^{3+x}+\frac{2}{81}\int (-\frac{9e^{2x}}{16(-4+x{)}^3}+\frac{33e^{2x}}{64(-4+x{)}^2}+\frac{3e^{2x}}{32(-4+x)}-\frac{e^{2x}}{16x^3}+\frac{7e^{2x}}{64x^2}-\frac{3e^{2x}}{32x})dx\\ & =e^{3+x}-\frac{1}{648}\int \frac{e^{2x}}{x^3}dx+\frac{1}{432}\int \frac{e^{2x}}{-4+x}dx-\frac{1}{432}\int \frac{e^{2x}}{x}dx+\frac{7\int \frac{e^{2x}}{x^2}dx}{2592}+\frac{11}{864}\int \frac{e^{2x}}{(-4+x{)}^2}dx-\frac{1}{72}\int \frac{e^{2x}}{(-4+x{)}^3}dx\\ & =e^{3+x}+\frac{e^{2x}}{144(4-x{)}^2}+\frac{11e^{2x}}{864(4-x)}+\frac{e^{2x}}{1296x^2}-\frac{7e^{2x}}{2592x}+\frac{1}{432}{e}^8\text{Ei}(-2(4-x))-\frac{\text{Ei}(2x)}{432}-\frac{1}{648}\int \frac{e^{2x}}{x^2}dx+\frac{7\int \frac{e^{2x}}{x}dx}{1296}-\frac{1}{72}\int \frac{e^{2x}}{(-4+x{)}^2}dx+\frac{11}{432}\int \frac{e^{2x}}{-4+x}dx\\ & =e^{3+x}+\frac{e^{2x}}{144(4-x{)}^2}-\frac{e^{2x}}{864(4-x)}+\frac{e^{2x}}{1296x^2}-\frac{e^{2x}}{864x}+\frac{1}{36}{e}^8\text{Ei}(-2(4-x))+\frac{\text{Ei}(2x)}{324}-\frac{1}{324}\int \frac{e^{2x}}{x}dx-\frac{1}{36}\int \frac{e^{2x}}{-4+x}dx\\ & =e^{3+x}+\frac{e^{2x}}{144(4-x{)}^2}-\frac{e^{2x}}{864(4-x)}+\frac{e^{2x}}{1296x^2}-\frac{e^{2x}}{864x}\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 38, normalized size = 1.36 $$\begin{array}{c}\frac{e^x(e^x(-1+x{)}^2+81e^3(-4+x{)}^2{x}^2)}{81(-4+x{)}^2{x}^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(8 - 20*x + 24*x^2 - 14*x^3 + 2*x^4) + E^(3 + x)*(-5184*x^3 + 3888*x^4 - 972*x^5 + 81*x^6))
/(-5184*x^3 + 3888*x^4 - 972*x^5 + 81*x^6),x]

[Out]

(E^x*(E^x*(-1 + x)^2 + 81*E^3*(-4 + x)^2*x^2))/(81*(-4 + x)^2*x^2)

________________________________________________________________________________________

fricas [B]  time = 0.58, size = 56, normalized size = 2.00 $$\begin{array}{c}\frac{((x^2-2x+1)e^{(2x+6)}+81(x^4-8x^3+16x^2)e^{(x+9)})e^{(-6)}}{81(x^4-8x^3+16x^2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+
3888*x^4-5184*x^3),x, algorithm="fricas")

[Out]

1/81*((x^2 - 2*x + 1)*e^(2*x + 6) + 81*(x^4 - 8*x^3 + 16*x^2)*e^(x + 9))*e^(-6)/(x^4 - 8*x^3 + 16*x^2)

________________________________________________________________________________________

giac [B]  time = 0.16, size = 65, normalized size = 2.32 $$\begin{array}{c}\frac{81x^4{e}^{(x+3)}-648x^3{e}^{(x+3)}+x^2{e}^{\left(2x\right)}+1296x^2{e}^{(x+3)}-2x{e}^{\left(2x\right)}+e^{\left(2x\right)}}{81(x^4-8x^3+16x^2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+
3888*x^4-5184*x^3),x, algorithm="giac")

[Out]

1/81*(81*x^4*e^(x + 3) - 648*x^3*e^(x + 3) + x^2*e^(2*x) + 1296*x^2*e^(x + 3) - 2*x*e^(2*x) + e^(2*x))/(x^4 -
8*x^3 + 16*x^2)

________________________________________________________________________________________

maple [A]  time = 0.09, size = 28, normalized size = 1.00

method	result	size
risch	$\frac{(x^2-2x+1){\mathrm{e}}^{2x}}{81{(x-4)}^2{x}^2}+{\mathrm{e}}^{3+x}$	$28$
norman	$\frac{{\mathrm{e}}^x{\mathrm{e}}^3{x}^4+\frac{{\mathrm{e}}^{2x}}{81}-\frac{2x{\mathrm{e}}^{2x}}{81}+\frac{{\mathrm{e}}^{2x}{x}^2}{81}+16x^2{\mathrm{e}}^3{\mathrm{e}}^x-8{\mathrm{e}}^x{\mathrm{e}}^3{x}^3}{{(x-4)}^2{x}^2}$	$59$
default	$\frac{{\mathrm{e}}^{2x}}{144{(x-4)}^2}+\frac{{\mathrm{e}}^{2x}}{864x-3456}-\frac{{\mathrm{e}}^{2x}}{864x}+\frac{{\mathrm{e}}^{2x}}{1296x^2}-64{\mathrm{e}}^3(-\frac{{\mathrm{e}}^x}{2{(x-4)}^2}-\frac{{\mathrm{e}}^x}{2(x-4)}-\frac{{\mathrm{e}}^4\text{expIntegralEi}(1,-x+4)}{2})+48{\mathrm{e}}^3(-\frac{2{\mathrm{e}}^x}{{(x-4)}^2}-\frac{3{\mathrm{e}}^x}{x-4}-3{\mathrm{e}}^4\text{expIntegralEi}(1,-x+4))-12{\mathrm{e}}^3(-\frac{8{\mathrm{e}}^x}{{(x-4)}^2}-\frac{16{\mathrm{e}}^x}{x-4}-17{\mathrm{e}}^4\text{expIntegralEi}(1,-x+4))+{\mathrm{e}}^3({\mathrm{e}}^x-\frac{32{\mathrm{e}}^x}{{(x-4)}^2}-\frac{80{\mathrm{e}}^x}{x-4}-92{\mathrm{e}}^4\text{expIntegralEi}(1,-x+4))$	$179$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+3888*x
^4-5184*x^3),x,method=_RETURNVERBOSE)

[Out]

1/81*(x^2-2*x+1)/(x-4)^2/x^2*exp(2*x)+exp(3+x)

________________________________________________________________________________________

maxima [B]  time = 0.40, size = 57, normalized size = 2.04 $$\begin{array}{c}\frac{(x^2-2x+1)e^{\left(2x\right)}+81(x^4{e}^3-8x^3{e}^3+16x^2{e}^3)e^x}{81(x^4-8x^3+16x^2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+
3888*x^4-5184*x^3),x, algorithm="maxima")

[Out]

1/81*((x^2 - 2*x + 1)*e^(2*x) + 81*(x^4*e^3 - 8*x^3*e^3 + 16*x^2*e^3)*e^x)/(x^4 - 8*x^3 + 16*x^2)

________________________________________________________________________________________

mupad [B]  time = 3.21, size = 55, normalized size = 1.96 $$\begin{array}{c}\frac{{\mathrm{e}}^{x-3}({\mathrm{e}}^{x+3}-2x{\mathrm{e}}^{x+3}+x^2{\mathrm{e}}^{x+3}+1296x^2{\mathrm{e}}^6-648x^3{\mathrm{e}}^6+81x^4{\mathrm{e}}^6)}{81x^2{(x-4)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(24*x^2 - 20*x - 14*x^3 + 2*x^4 + 8) - exp(x + 3)*(5184*x^3 - 3888*x^4 + 972*x^5 - 81*x^6))/(51
84*x^3 - 3888*x^4 + 972*x^5 - 81*x^6),x)

[Out]

(exp(x - 3)*(exp(x + 3) - 2*x*exp(x + 3) + x^2*exp(x + 3) + 1296*x^2*exp(6) - 648*x^3*exp(6) + 81*x^4*exp(6)))
/(81*x^2*(x - 4)^2)

________________________________________________________________________________________

sympy [B]  time = 0.19, size = 61, normalized size = 2.18 $$\begin{array}{c}\frac{(x^2-2x+1)e^{2x}+(81x^4{e}^3-648x^3{e}^3+1296x^2{e}^3)\sqrt{e^{2x}}}{81x^4-648x^3+1296x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x**6-972*x**5+3888*x**4-5184*x**3)*exp(3+x)+(2*x**4-14*x**3+24*x**2-20*x+8)*exp(x)**2)/(81*x**6
-972*x**5+3888*x**4-5184*x**3),x)

[Out]

((x**2 - 2*x + 1)*exp(2*x) + (81*x**4*exp(3) - 648*x**3*exp(3) + 1296*x**2*exp(3))*sqrt(exp(2*x)))/(81*x**4 -
648*x**3 + 1296*x**2)

________________________________________________________________________________________