3.103.96 \(\int \frac {50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} (2 x^2+x^3)+e^{10-x} (-45 x+25 x^2+x^3-14 x^4-2 x^5)+e^x (e^{10-x} (20 x-10 x^2-14 x^3-2 x^4)+e^{20-2 x} (-5 x^2+2 x^3+2 x^4))+(e^{20-x} (2 x^2+x^3)+e^{10-x} (-10+5 x-7 x^2-8 x^3-x^4)) \log (2+x)}{50-75 x-20 x^2+35 x^3+12 x^4+x^5+e^{20} (2 x^2+x^3)+e^{10} (20 x-10 x^2-14 x^3-2 x^4)} \, dx\)
Optimal. Leaf size=34 \[ x-\frac {2 x+\log (2+x)}{e^x+e^{-10+x} \left (-5+\frac {5}{x}-x\right )} \]
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Rubi [C] time = 14.38, antiderivative size = 4186, normalized size of antiderivative =
123.12, number of steps used = 62, number of rules used = 7, integrand size = 235, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030,
Rules used = {6688, 6742, 2178, 2177, 2270, 2194, 2554}
result too large to display
Antiderivative was successfully verified.
[In]
Int[(50 - 75*x - 20*x^2 + 35*x^3 + 12*x^4 + x^5 + E^20*(2*x^2 + x^3) + E^(10 - x)*(-45*x + 25*x^2 + x^3 - 14*x
^4 - 2*x^5) + E^x*(E^(10 - x)*(20*x - 10*x^2 - 14*x^3 - 2*x^4) + E^(20 - 2*x)*(-5*x^2 + 2*x^3 + 2*x^4)) + (E^(
20 - x)*(2*x^2 + x^3) + E^(10 - x)*(-10 + 5*x - 7*x^2 - 8*x^3 - x^4))*Log[2 + x])/(50 - 75*x - 20*x^2 + 35*x^3
+ 12*x^4 + x^5 + E^20*(2*x^2 + x^3) + E^10*(20*x - 10*x^2 - 14*x^3 - 2*x^4)),x]
[Out]
2*E^(10 - x) + x + (20*E^(20 - x)*(65 - 23*E^10 + 2*E^20))/((11 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 - Sq
rt[45 - 10*E^10 + E^20] + 2*x)) - (20*E^(10 - x)*(55 + 44*E^10 - 21*E^20 + 2*E^30))/((11 - 2*E^10)*(45 - 10*E^
10 + E^20)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20] + 2*x)) + (E^(20 - x)*(755 - 380*E^10 + 66*E^20 - 4*E^30)*(5
- E^10 - Sqrt[45 - 10*E^10 + E^20]))/((11 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20
] + 2*x)) - (E^(10 - x)*(770 + 395*E^10 - 318*E^20 + 62*E^30 - 4*E^40)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20]))
/((11 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20] + 2*x)) + (20*E^(20 - x)*(65 - 23*
E^10 + 2*E^20))/((11 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20] + 2*x)) - (20*E^(10
- x)*(55 + 44*E^10 - 21*E^20 + 2*E^30))/((11 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 + Sqrt[45 - 10*E^10 +
E^20] + 2*x)) + (E^(20 - x)*(755 - 380*E^10 + 66*E^20 - 4*E^30)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20]))/((11 -
2*E^10)*(45 - 10*E^10 + E^20)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20] + 2*x)) - (E^(10 - x)*(770 + 395*E^10 - 3
18*E^20 + 62*E^30 - 4*E^40)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20]))/((11 - 2*E^10)*(45 - 10*E^10 + E^20)*(5 -
E^10 + Sqrt[45 - 10*E^10 + E^20] + 2*x)) - (20*E^((45 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*(65 - 23*E^10 + 2
*E^20)*ExpIntegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/((11 - 2*E^10)*(45 - 10*E^10 + E^20)^(3
/2)) + (10*E^((45 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*(65 - 23*E^10 + 2*E^20)*ExpIntegralEi[(-5 + E^10 - Sq
rt[45 - 10*E^10 + E^20] - 2*x)/2])/((11 - 2*E^10)*(45 - 10*E^10 + E^20)) - (E^((45 - E^10 + Sqrt[45 - 10*E^10
+ E^20])/2)*(5 - E^10)*(755 - 380*E^10 + 66*E^20 - 4*E^30)*ExpIntegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20
] - 2*x)/2])/((11 - 2*E^10)*(45 - 10*E^10 + E^20)^(3/2)) + (20*E^((25 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*(
55 + 44*E^10 - 21*E^20 + 2*E^30)*ExpIntegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/((11 - 2*E^10
)*(45 - 10*E^10 + E^20)^(3/2)) - (10*E^((25 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*(55 + 44*E^10 - 21*E^20 + 2
*E^30)*ExpIntegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/((11 - 2*E^10)*(45 - 10*E^10 + E^20)) +
(E^((25 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*(5 - E^10)*(770 + 395*E^10 - 318*E^20 + 62*E^30 - 4*E^40)*ExpI
ntegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/((11 - 2*E^10)*(45 - 10*E^10 + E^20)^(3/2)) + (E^(
(25 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*(1 + (10 - E^10)/Sqrt[45 - 10*E^10 + E^20])*ExpIntegralEi[(-5 + E^1
0 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/(11 - 2*E^10) + (E^((45 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*(755 -
380*E^10 + 66*E^20 - 4*E^30)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20])*ExpIntegralEi[(-5 + E^10 - Sqrt[45 - 10*E
^10 + E^20] - 2*x)/2])/(2*(11 - 2*E^10)*(45 - 10*E^10 + E^20)) - (E^((25 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2
)*(770 + 395*E^10 - 318*E^20 + 62*E^30 - 4*E^40)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20])*ExpIntegralEi[(-5 + E^
10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/(2*(11 - 2*E^10)*(45 - 10*E^10 + E^20)) + (E^((45 - E^10 + Sqrt[45 -
10*E^10 + E^20])/2)*(123 - 44*E^10 + 4*E^20 + (2055 - 1109*E^10 + 200*E^20 - 12*E^30)/Sqrt[45 - 10*E^10 + E^2
0])*ExpIntegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/(11 - 2*E^10)^2 + (E^((25 - E^10 + Sqrt[45
- 10*E^10 + E^20])/2)*(594 - 462*E^10 + 108*E^20 - 8*E^30 + (2915 - 4092*E^10 + 1680*E^20 - 276*E^30 + 16*E^4
0)/Sqrt[45 - 10*E^10 + E^20])*ExpIntegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/(11 - 2*E^10)^2
+ (20*E^((45 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*(65 - 23*E^10 + 2*E^20)*ExpIntegralEi[(-5 + E^10 + Sqrt[45
- 10*E^10 + E^20] - 2*x)/2])/((11 - 2*E^10)*(45 - 10*E^10 + E^20)^(3/2)) + (10*E^((45 - E^10 - Sqrt[45 - 10*E
^10 + E^20])/2)*(65 - 23*E^10 + 2*E^20)*ExpIntegralEi[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/((11 -
2*E^10)*(45 - 10*E^10 + E^20)) + (E^((45 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*(5 - E^10)*(755 - 380*E^10 +
66*E^20 - 4*E^30)*ExpIntegralEi[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/((11 - 2*E^10)*(45 - 10*E^10
+ E^20)^(3/2)) - (20*E^((25 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*(55 + 44*E^10 - 21*E^20 + 2*E^30)*ExpInteg
ralEi[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/((11 - 2*E^10)*(45 - 10*E^10 + E^20)^(3/2)) - (10*E^((
25 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*(55 + 44*E^10 - 21*E^20 + 2*E^30)*ExpIntegralEi[(-5 + E^10 + Sqrt[45
- 10*E^10 + E^20] - 2*x)/2])/((11 - 2*E^10)*(45 - 10*E^10 + E^20)) - (E^((25 - E^10 - Sqrt[45 - 10*E^10 + E^2
0])/2)*(5 - E^10)*(770 + 395*E^10 - 318*E^20 + 62*E^30 - 4*E^40)*ExpIntegralEi[(-5 + E^10 + Sqrt[45 - 10*E^10
+ E^20] - 2*x)/2])/((11 - 2*E^10)*(45 - 10*E^10 + E^20)^(3/2)) + (E^((25 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2
)*(1 - (10 - E^10)/Sqrt[45 - 10*E^10 + E^20])*ExpIntegralEi[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/
(11 - 2*E^10) + (E^((45 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*(755 - 380*E^10 + 66*E^20 - 4*E^30)*(5 - E^10 -
Sqrt[45 - 10*E^10 + E^20])*ExpIntegralEi[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/(2*(11 - 2*E^10)*(
45 - 10*E^10 + E^20)) - (E^((25 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*(770 + 395*E^10 - 318*E^20 + 62*E^30 -
4*E^40)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20])*ExpIntegralEi[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])
/(2*(11 - 2*E^10)*(45 - 10*E^10 + E^20)) + (E^((45 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*(123 - 44*E^10 + 4*E
^20 - (2055 - 1109*E^10 + 200*E^20 - 12*E^30)/Sqrt[45 - 10*E^10 + E^20])*ExpIntegralEi[(-5 + E^10 + Sqrt[45 -
10*E^10 + E^20] - 2*x)/2])/(11 - 2*E^10)^2 + (E^((25 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*(594 - 462*E^10 +
108*E^20 - 8*E^30 - (2915 - 4092*E^10 + 1680*E^20 - 276*E^30 + 16*E^40)/Sqrt[45 - 10*E^10 + E^20])*ExpIntegral
Ei[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)/2])/(11 - 2*E^10)^2 + (22*E^12*ExpIntegralEi[-2 - x])/(11 - 2
*E^10)^2 - (4*E^22*ExpIntegralEi[-2 - x])/(11 - 2*E^10)^2 - (2*E^12*ExpIntegralEi[-2 - x])/(11 - 2*E^10) + (20
*E^(10 - x)*Log[2 + x])/((45 - 10*E^10 + E^20)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20] + 2*x)) + (E^(10 - x)*(5
- E^10)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20])*Log[2 + x])/((45 - 10*E^10 + E^20)*(5 - E^10 - Sqrt[45 - 10*E^1
0 + E^20] + 2*x)) + (20*E^(10 - x)*Log[2 + x])/((45 - 10*E^10 + E^20)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20] +
2*x)) + (E^(10 - x)*(5 - E^10)*(5 - E^10 + Sqrt[45 - 10*E^10 + E^20])*Log[2 + x])/((45 - 10*E^10 + E^20)*(5 -
E^10 + Sqrt[45 - 10*E^10 + E^20] + 2*x)) - (20*E^((25 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*ExpIntegralEi[(-5
+ E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2]*Log[2 + x])/(45 - 10*E^10 + E^20)^(3/2) - (E^((25 - E^10 + Sqrt[
45 - 10*E^10 + E^20])/2)*(5 - E^10)^2*ExpIntegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2]*Log[2 + x
])/(45 - 10*E^10 + E^20)^(3/2) + (10*E^((25 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*ExpIntegralEi[(-5 + E^10 -
Sqrt[45 - 10*E^10 + E^20] - 2*x)/2]*Log[2 + x])/(45 - 10*E^10 + E^20) - (E^((25 - E^10 + Sqrt[45 - 10*E^10 + E
^20])/2)*(1 + (3 - E^10)/Sqrt[45 - 10*E^10 + E^20])*ExpIntegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x
)/2]*Log[2 + x])/2 + (E^((25 - E^10 + Sqrt[45 - 10*E^10 + E^20])/2)*(5 - E^10)*(5 - E^10 + Sqrt[45 - 10*E^10 +
E^20])*ExpIntegralEi[(-5 + E^10 - Sqrt[45 - 10*E^10 + E^20] - 2*x)/2]*Log[2 + x])/(2*(45 - 10*E^10 + E^20)) +
(20*E^((25 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*ExpIntegralEi[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)
/2]*Log[2 + x])/(45 - 10*E^10 + E^20)^(3/2) + (E^((25 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*(5 - E^10)^2*ExpI
ntegralEi[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)/2]*Log[2 + x])/(45 - 10*E^10 + E^20)^(3/2) + (10*E^((2
5 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*ExpIntegralEi[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)/2]*Log[2
+ x])/(45 - 10*E^10 + E^20) - (E^((25 - E^10 - Sqrt[45 - 10*E^10 + E^20])/2)*(1 - (3 - E^10)/Sqrt[45 - 10*E^10
+ E^20])*ExpIntegralEi[(-5 + E^10 + Sqrt[45 - 10*E^10 + E^20] - 2*x)/2]*Log[2 + x])/2 + (E^((25 - E^10 - Sqrt
[45 - 10*E^10 + E^20])/2)*(5 - E^10)*(5 - E^10 - Sqrt[45 - 10*E^10 + E^20])*ExpIntegralEi[(-5 + E^10 + Sqrt[45
- 10*E^10 + E^20] - 2*x)/2]*Log[2 + x])/(2*(45 - 10*E^10 + E^20))
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 2270
Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_) + (c_)*(x_)^2), x_Symbol] :> Int[
ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Poly
nomialQ[u, x] && IntegerQ[m]
Rule 2554
Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, 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 {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (e^{20+x} x^2 (2+x)+e^x (2+x) \left (-5+5 x+x^2\right )^2+e^{20} x^2 \left (-5+2 x+2 x^2\right )-2 e^{10+x} x \left (-10+5 x+7 x^2+x^3\right )+e^{10} x \left (-45+25 x+x^2-14 x^3-2 x^4\right )+e^{10} \left (-10+5 x+\left (-7+2 e^{10}\right ) x^2+\left (-8+e^{10}\right ) x^3-x^4\right ) \log (2+x)\right )}{(2+x) \left (5-\left (5-e^{10}\right ) x-x^2\right )^2} \, dx\\ &=\int \left (1+\frac {e^{20-x} x^2 \left (-5+2 x+2 x^2\right )}{(2+x) \left (5-\left (5-e^{10}\right ) x-x^2\right )^2}+\frac {e^{10-x} x \left (-45+25 x+x^2-14 x^3-2 x^4\right )}{(2+x) \left (5-\left (5-e^{10}\right ) x-x^2\right )^2}+\frac {e^{10-x} \left (-5+5 x-\left (6-e^{10}\right ) x^2-x^3\right ) \log (2+x)}{\left (5-\left (5-e^{10}\right ) x-x^2\right )^2}\right ) \, dx\\ &=x+\int \frac {e^{20-x} x^2 \left (-5+2 x+2 x^2\right )}{(2+x) \left (5-\left (5-e^{10}\right ) x-x^2\right )^2} \, dx+\int \frac {e^{10-x} x \left (-45+25 x+x^2-14 x^3-2 x^4\right )}{(2+x) \left (5-\left (5-e^{10}\right ) x-x^2\right )^2} \, dx+\int \frac {e^{10-x} \left (-5+5 x-\left (6-e^{10}\right ) x^2-x^3\right ) \log (2+x)}{\left (5-\left (5-e^{10}\right ) x-x^2\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.20, size = 57, normalized size = 1.68 \begin {gather*} \frac {e^{-x} x \left (2 e^{10} x-e^{10+x} x+e^x \left (-5+5 x+x^2\right )+e^{10} \log (2+x)\right )}{-5-\left (-5+e^{10}\right ) x+x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(50 - 75*x - 20*x^2 + 35*x^3 + 12*x^4 + x^5 + E^20*(2*x^2 + x^3) + E^(10 - x)*(-45*x + 25*x^2 + x^3
- 14*x^4 - 2*x^5) + E^x*(E^(10 - x)*(20*x - 10*x^2 - 14*x^3 - 2*x^4) + E^(20 - 2*x)*(-5*x^2 + 2*x^3 + 2*x^4))
+ (E^(20 - x)*(2*x^2 + x^3) + E^(10 - x)*(-10 + 5*x - 7*x^2 - 8*x^3 - x^4))*Log[2 + x])/(50 - 75*x - 20*x^2 +
35*x^3 + 12*x^4 + x^5 + E^20*(2*x^2 + x^3) + E^10*(20*x - 10*x^2 - 14*x^3 - 2*x^4)),x]
[Out]
(x*(2*E^10*x - E^(10 + x)*x + E^x*(-5 + 5*x + x^2) + E^10*Log[2 + x]))/(E^x*(-5 - (-5 + E^10)*x + x^2))
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fricas [A] time = 0.68, size = 58, normalized size = 1.71 \begin {gather*} \frac {{\left (2 \, x^{2} e^{10} + x e^{10} \log \left (x + 2\right ) + {\left (x^{3} - x^{2} e^{10} + 5 \, x^{2} - 5 \, x\right )} e^{x}\right )} e^{\left (-x\right )}}{x^{2} - x e^{10} + 5 \, x - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((x^3+2*x^2)*exp(10-x)^2*exp(x)+(-x^4-8*x^3-7*x^2+5*x-10)*exp(10-x))*log(2+x)+(x^3+2*x^2)*exp(10-x)
^2*exp(x)^2+((2*x^4+2*x^3-5*x^2)*exp(10-x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x))*exp(x)+(-2*x^5-14*x^4+x^3+
25*x^2-45*x)*exp(10-x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50)/((x^3+2*x^2)*exp(10-x)^2*exp(x)^2+(-2*x^4-14*x^3-10*x
^2+20*x)*exp(10-x)*exp(x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50),x, algorithm="fricas")
[Out]
(2*x^2*e^10 + x*e^10*log(x + 2) + (x^3 - x^2*e^10 + 5*x^2 - 5*x)*e^x)*e^(-x)/(x^2 - x*e^10 + 5*x - 5)
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giac [B] time = 0.46, size = 221, normalized size = 6.50 \begin {gather*} \frac {x^{3} e^{\left (x + 20\right )} - 10 \, x^{3} e^{\left (x + 10\right )} + 45 \, x^{3} e^{x} + 2 \, x^{2} e^{30} - 20 \, x^{2} e^{20} + 90 \, x^{2} e^{10} - x^{2} e^{\left (x + 30\right )} + 15 \, x^{2} e^{\left (x + 20\right )} - 95 \, x^{2} e^{\left (x + 10\right )} + 225 \, x^{2} e^{x} + x e^{30} \log \left (x + 2\right ) - 10 \, x e^{20} \log \left (x + 2\right ) + 45 \, x e^{10} \log \left (x + 2\right ) - x e^{\left (x + 40\right )} + 10 \, x e^{\left (x + 30\right )} - 40 \, x e^{\left (x + 20\right )} + 50 \, x e^{\left (x + 10\right )} - 225 \, x e^{x} - 5 \, e^{\left (x + 30\right )} + 25 \, e^{\left (x + 20\right )}}{x^{2} e^{\left (x + 20\right )} - 10 \, x^{2} e^{\left (x + 10\right )} + 45 \, x^{2} e^{x} - x e^{\left (x + 30\right )} + 15 \, x e^{\left (x + 20\right )} - 95 \, x e^{\left (x + 10\right )} + 225 \, x e^{x} - 5 \, e^{\left (x + 20\right )} + 50 \, e^{\left (x + 10\right )} - 225 \, e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((x^3+2*x^2)*exp(10-x)^2*exp(x)+(-x^4-8*x^3-7*x^2+5*x-10)*exp(10-x))*log(2+x)+(x^3+2*x^2)*exp(10-x)
^2*exp(x)^2+((2*x^4+2*x^3-5*x^2)*exp(10-x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x))*exp(x)+(-2*x^5-14*x^4+x^3+
25*x^2-45*x)*exp(10-x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50)/((x^3+2*x^2)*exp(10-x)^2*exp(x)^2+(-2*x^4-14*x^3-10*x
^2+20*x)*exp(10-x)*exp(x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50),x, algorithm="giac")
[Out]
(x^3*e^(x + 20) - 10*x^3*e^(x + 10) + 45*x^3*e^x + 2*x^2*e^30 - 20*x^2*e^20 + 90*x^2*e^10 - x^2*e^(x + 30) + 1
5*x^2*e^(x + 20) - 95*x^2*e^(x + 10) + 225*x^2*e^x + x*e^30*log(x + 2) - 10*x*e^20*log(x + 2) + 45*x*e^10*log(
x + 2) - x*e^(x + 40) + 10*x*e^(x + 30) - 40*x*e^(x + 20) + 50*x*e^(x + 10) - 225*x*e^x - 5*e^(x + 30) + 25*e^
(x + 20))/(x^2*e^(x + 20) - 10*x^2*e^(x + 10) + 45*x^2*e^x - x*e^(x + 30) + 15*x*e^(x + 20) - 95*x*e^(x + 10)
+ 225*x*e^x - 5*e^(x + 20) + 50*e^(x + 10) - 225*e^x)
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (x^{3}+2 x^{2}\right ) {\mathrm e}^{-2 x +20} {\mathrm e}^{x}+\left (-x^{4}-8 x^{3}-7 x^{2}+5 x -10\right ) {\mathrm e}^{10-x}\right ) \ln \left (2+x \right )+\left (x^{3}+2 x^{2}\right ) {\mathrm e}^{-2 x +20} {\mathrm e}^{2 x}+\left (\left (2 x^{4}+2 x^{3}-5 x^{2}\right ) {\mathrm e}^{-2 x +20}+\left (-2 x^{4}-14 x^{3}-10 x^{2}+20 x \right ) {\mathrm e}^{10-x}\right ) {\mathrm e}^{x}+\left (-2 x^{5}-14 x^{4}+x^{3}+25 x^{2}-45 x \right ) {\mathrm e}^{10-x}+x^{5}+12 x^{4}+35 x^{3}-20 x^{2}-75 x +50}{\left (x^{3}+2 x^{2}\right ) {\mathrm e}^{-2 x +20} {\mathrm e}^{2 x}+\left (-2 x^{4}-14 x^{3}-10 x^{2}+20 x \right ) {\mathrm e}^{10-x} {\mathrm e}^{x}+x^{5}+12 x^{4}+35 x^{3}-20 x^{2}-75 x +50}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((x^3+2*x^2)*exp(10-x)^2*exp(x)+(-x^4-8*x^3-7*x^2+5*x-10)*exp(10-x))*ln(2+x)+(x^3+2*x^2)*exp(10-x)^2*exp(
x)^2+((2*x^4+2*x^3-5*x^2)*exp(10-x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x))*exp(x)+(-2*x^5-14*x^4+x^3+25*x^2-
45*x)*exp(10-x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50)/((x^3+2*x^2)*exp(10-x)^2*exp(x)^2+(-2*x^4-14*x^3-10*x^2+20*x
)*exp(10-x)*exp(x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50),x)
[Out]
int((((x^3+2*x^2)*exp(10-x)^2*exp(x)+(-x^4-8*x^3-7*x^2+5*x-10)*exp(10-x))*ln(2+x)+(x^3+2*x^2)*exp(10-x)^2*exp(
x)^2+((2*x^4+2*x^3-5*x^2)*exp(10-x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x))*exp(x)+(-2*x^5-14*x^4+x^3+25*x^2-
45*x)*exp(10-x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50)/((x^3+2*x^2)*exp(10-x)^2*exp(x)^2+(-2*x^4-14*x^3-10*x^2+20*x
)*exp(10-x)*exp(x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50),x)
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maxima [B] time = 0.52, size = 2515, normalized size = 73.97 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((x^3+2*x^2)*exp(10-x)^2*exp(x)+(-x^4-8*x^3-7*x^2+5*x-10)*exp(10-x))*log(2+x)+(x^3+2*x^2)*exp(10-x)
^2*exp(x)^2+((2*x^4+2*x^3-5*x^2)*exp(10-x)^2+(-2*x^4-14*x^3-10*x^2+20*x)*exp(10-x))*exp(x)+(-2*x^5-14*x^4+x^3+
25*x^2-45*x)*exp(10-x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50)/((x^3+2*x^2)*exp(10-x)^2*exp(x)^2+(-2*x^4-14*x^3-10*x
^2+20*x)*exp(10-x)*exp(x)+x^5+12*x^4+35*x^3-20*x^2-75*x+50),x, algorithm="maxima")
[Out]
(2*(2*e^30 - 30*e^20 + 135*e^10 - 350)*log((2*x - sqrt(e^20 - 10*e^10 + 45) - e^10 + 5)/(2*x + sqrt(e^20 - 10*
e^10 + 45) - e^10 + 5))/((4*e^40 - 84*e^30 + 741*e^20 - 3190*e^10 + 5445)*sqrt(e^20 - 10*e^10 + 45)) - (x*(2*e
^30 - 35*e^20 + 230*e^10 - 575) + 10*e^20 - 125*e^10 + 475)/(x^2*(2*e^30 - 31*e^20 + 200*e^10 - 495) - x*(2*e^
40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) - 10*e^30 + 155*e^20 - 1000*e^10 + 2475) + 4*log(x^2 - x*(e^10 - 5
) - 5)/(4*e^20 - 44*e^10 + 121) - 8*log(x + 2)/(4*e^20 - 44*e^10 + 121))*e^20 - 2*((2*e^30 - 50*e^20 + 355*e^1
0 - 955)*log((2*x - sqrt(e^20 - 10*e^10 + 45) - e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^
40 - 84*e^30 + 741*e^20 - 3190*e^10 + 5445)*sqrt(e^20 - 10*e^10 + 45)) + (x*(2*e^20 - 25*e^10 + 95) + 10*e^10
- 100)/(x^2*(2*e^30 - 31*e^20 + 200*e^10 - 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) - 10*e^30
+ 155*e^20 - 1000*e^10 + 2475) + 2*log(x^2 - x*(e^10 - 5) - 5)/(4*e^20 - 44*e^10 + 121) - 4*log(x + 2)/(4*e^2
0 - 44*e^10 + 121))*e^20 - ((4*e^20 - 44*e^10 + 105)*log(x^2 - x*(e^10 - 5) - 5)/(4*e^20 - 44*e^10 + 121) + (4
*e^50 - 104*e^40 + 1185*e^30 - 7295*e^20 + 23725*e^10 - 30475)*log((2*x - sqrt(e^20 - 10*e^10 + 45) - e^10 + 5
)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^40 - 84*e^30 + 741*e^20 - 3190*e^10 + 5445)*sqrt(e^20 -
10*e^10 + 45)) - 2*(x*(2*e^40 - 45*e^30 + 415*e^20 - 1850*e^10 + 3350) + 10*e^30 - 175*e^20 + 1150*e^10 - 2875
)/(x^2*(2*e^30 - 31*e^20 + 200*e^10 - 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) - 10*e^30 + 15
5*e^20 - 1000*e^10 + 2475) + 32*log(x + 2)/(4*e^20 - 44*e^10 + 121))*e^10 - 14*(2*(2*e^30 - 30*e^20 + 135*e^10
- 350)*log((2*x - sqrt(e^20 - 10*e^10 + 45) - e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^4
0 - 84*e^30 + 741*e^20 - 3190*e^10 + 5445)*sqrt(e^20 - 10*e^10 + 45)) - (x*(2*e^30 - 35*e^20 + 230*e^10 - 575)
+ 10*e^20 - 125*e^10 + 475)/(x^2*(2*e^30 - 31*e^20 + 200*e^10 - 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*
e^10 + 2475) - 10*e^30 + 155*e^20 - 1000*e^10 + 2475) + 4*log(x^2 - x*(e^10 - 5) - 5)/(4*e^20 - 44*e^10 + 121)
- 8*log(x + 2)/(4*e^20 - 44*e^10 + 121))*e^10 + 10*((2*e^30 - 50*e^20 + 355*e^10 - 955)*log((2*x - sqrt(e^20
- 10*e^10 + 45) - e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^40 - 84*e^30 + 741*e^20 - 3190
*e^10 + 5445)*sqrt(e^20 - 10*e^10 + 45)) + (x*(2*e^20 - 25*e^10 + 95) + 10*e^10 - 100)/(x^2*(2*e^30 - 31*e^20
+ 200*e^10 - 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) - 10*e^30 + 155*e^20 - 1000*e^10 + 2475
) + 2*log(x^2 - x*(e^10 - 5) - 5)/(4*e^20 - 44*e^10 + 121) - 4*log(x + 2)/(4*e^20 - 44*e^10 + 121))*e^10 - 20*
((e^30 - 7*e^20 - 7*e^10 + 175)*log((2*x - sqrt(e^20 - 10*e^10 + 45) - e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 +
45) - e^10 + 5))/((4*e^40 - 84*e^30 + 741*e^20 - 3190*e^10 + 5445)*sqrt(e^20 - 10*e^10 + 45)) + (2*x*(e^10 - 1
0) + 5*e^10 - 5)/(x^2*(2*e^30 - 31*e^20 + 200*e^10 - 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475)
- 10*e^30 + 155*e^20 - 1000*e^10 + 2475) - log(x^2 - x*(e^10 - 5) - 5)/(4*e^20 - 44*e^10 + 121) + 2*log(x + 2
)/(4*e^20 - 44*e^10 + 121))*e^10 + x + (4*e^30 - 68*e^20 + 385*e^10 - 710)*log(x^2 - x*(e^10 - 5) - 5)/(4*e^20
- 44*e^10 + 121) + 6*(4*e^20 - 44*e^10 + 105)*log(x^2 - x*(e^10 - 5) - 5)/(4*e^20 - 44*e^10 + 121) + (4*e^60
- 128*e^50 + 1825*e^40 - 14725*e^30 + 70375*e^20 - 187625*e^10 + 215000)*log((2*x - sqrt(e^20 - 10*e^10 + 45)
- e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^40 - 84*e^30 + 741*e^20 - 3190*e^10 + 5445)*sq
rt(e^20 - 10*e^10 + 45)) + 6*(4*e^50 - 104*e^40 + 1185*e^30 - 7295*e^20 + 23725*e^10 - 30475)*log((2*x - sqrt(
e^20 - 10*e^10 + 45) - e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^40 - 84*e^30 + 741*e^20 -
3190*e^10 + 5445)*sqrt(e^20 - 10*e^10 + 45)) + 70*(2*e^30 - 30*e^20 + 135*e^10 - 350)*log((2*x - sqrt(e^20 -
10*e^10 + 45) - e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^40 - 84*e^30 + 741*e^20 - 3190*e
^10 + 5445)*sqrt(e^20 - 10*e^10 + 45)) + 20*(2*e^30 - 50*e^20 + 355*e^10 - 955)*log((2*x - sqrt(e^20 - 10*e^10
+ 45) - e^10 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^40 - 84*e^30 + 741*e^20 - 3190*e^10 + 5
445)*sqrt(e^20 - 10*e^10 + 45)) + 75*(e^30 - 7*e^20 - 7*e^10 + 175)*log((2*x - sqrt(e^20 - 10*e^10 + 45) - e^1
0 + 5)/(2*x + sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^40 - 84*e^30 + 741*e^20 - 3190*e^10 + 5445)*sqrt(e^
20 - 10*e^10 + 45)) + 25*(e^30 - 15*e^20 + 81*e^10 - 67)*log((2*x - sqrt(e^20 - 10*e^10 + 45) - e^10 + 5)/(2*x
+ sqrt(e^20 - 10*e^10 + 45) - e^10 + 5))/((4*e^40 - 84*e^30 + 741*e^20 - 3190*e^10 + 5445)*sqrt(e^20 - 10*e^1
0 + 45)) + (2*x^2*e^(-x + 10) + x*e^(-x + 10)*log(x + 2))/(x^2 - x*(e^10 - 5) - 5) - (x*(2*e^50 - 55*e^40 + 65
0*e^30 - 4100*e^20 + 13750*e^10 - 19625) + 10*e^40 - 225*e^30 + 2075*e^20 - 9250*e^10 + 16750)/(x^2*(2*e^30 -
31*e^20 + 200*e^10 - 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) - 10*e^30 + 155*e^20 - 1000*e^1
0 + 2475) - 12*(x*(2*e^40 - 45*e^30 + 415*e^20 - 1850*e^10 + 3350) + 10*e^30 - 175*e^20 + 1150*e^10 - 2875)/(x
^2*(2*e^30 - 31*e^20 + 200*e^10 - 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) - 10*e^30 + 155*e^
20 - 1000*e^10 + 2475) - 35*(x*(2*e^30 - 35*e^20 + 230*e^10 - 575) + 10*e^20 - 125*e^10 + 475)/(x^2*(2*e^30 -
31*e^20 + 200*e^10 - 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) - 10*e^30 + 155*e^20 - 1000*e^1
0 + 2475) + 20*(x*(2*e^20 - 25*e^10 + 95) + 10*e^10 - 100)/(x^2*(2*e^30 - 31*e^20 + 200*e^10 - 495) - x*(2*e^4
0 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) - 10*e^30 + 155*e^20 - 1000*e^10 + 2475) - 50*(x*(e^10 - 1) - e^20
+ 8*e^10 - 25)/(x^2*(2*e^30 - 31*e^20 + 200*e^10 - 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) -
10*e^30 + 155*e^20 - 1000*e^10 + 2475) + 75*(2*x*(e^10 - 10) + 5*e^10 - 5)/(x^2*(2*e^30 - 31*e^20 + 200*e^10
- 495) - x*(2*e^40 - 41*e^30 + 355*e^20 - 1495*e^10 + 2475) - 10*e^30 + 155*e^20 - 1000*e^10 + 2475) + 80*log(
x^2 - x*(e^10 - 5) - 5)/(4*e^20 - 44*e^10 + 121)
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int -\frac {{\mathrm {e}}^{20}\,\left (x^3+2\,x^2\right )-{\mathrm {e}}^{10-x}\,\left (2\,x^5+14\,x^4-x^3-25\,x^2+45\,x\right )-75\,x-{\mathrm {e}}^x\,\left ({\mathrm {e}}^{10-x}\,\left (2\,x^4+14\,x^3+10\,x^2-20\,x\right )-{\mathrm {e}}^{20-2\,x}\,\left (2\,x^4+2\,x^3-5\,x^2\right )\right )-20\,x^2+35\,x^3+12\,x^4+x^5+\ln \left (x+2\right )\,\left ({\mathrm {e}}^{20-x}\,\left (x^3+2\,x^2\right )-{\mathrm {e}}^{10-x}\,\left (x^4+8\,x^3+7\,x^2-5\,x+10\right )\right )+50}{{\mathrm {e}}^{20}\,\left (x^3+2\,x^2\right )-75\,x-{\mathrm {e}}^{10}\,\left (2\,x^4+14\,x^3+10\,x^2-20\,x\right )-20\,x^2+35\,x^3+12\,x^4+x^5+50} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(75*x + exp(10 - x)*(45*x - 25*x^2 - x^3 + 14*x^4 + 2*x^5) + exp(x)*(exp(10 - x)*(10*x^2 - 20*x + 14*x^3
+ 2*x^4) - exp(20 - 2*x)*(2*x^3 - 5*x^2 + 2*x^4)) + log(x + 2)*(exp(10 - x)*(7*x^2 - 5*x + 8*x^3 + x^4 + 10) -
exp(20 - 2*x)*exp(x)*(2*x^2 + x^3)) + 20*x^2 - 35*x^3 - 12*x^4 - x^5 - exp(2*x)*exp(20 - 2*x)*(2*x^2 + x^3) -
50)/(35*x^3 - 20*x^2 - 75*x + 12*x^4 + x^5 + exp(2*x)*exp(20 - 2*x)*(2*x^2 + x^3) - exp(10 - x)*exp(x)*(10*x^
2 - 20*x + 14*x^3 + 2*x^4) + 50),x)
[Out]
-int(-(exp(20)*(2*x^2 + x^3) - exp(10 - x)*(45*x - 25*x^2 - x^3 + 14*x^4 + 2*x^5) - 75*x - exp(x)*(exp(10 - x)
*(10*x^2 - 20*x + 14*x^3 + 2*x^4) - exp(20 - 2*x)*(2*x^3 - 5*x^2 + 2*x^4)) - 20*x^2 + 35*x^3 + 12*x^4 + x^5 +
log(x + 2)*(exp(20 - x)*(2*x^2 + x^3) - exp(10 - x)*(7*x^2 - 5*x + 8*x^3 + x^4 + 10)) + 50)/(exp(20)*(2*x^2 +
x^3) - 75*x - exp(10)*(10*x^2 - 20*x + 14*x^3 + 2*x^4) - 20*x^2 + 35*x^3 + 12*x^4 + x^5 + 50), x)
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sympy [A] time = 0.68, size = 36, normalized size = 1.06 \begin {gather*} x + \frac {\left (2 x^{2} e^{10} + x e^{10} \log {\left (x + 2 \right )}\right ) e^{- x}}{x^{2} - x e^{10} + 5 x - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((x**3+2*x**2)*exp(10-x)**2*exp(x)+(-x**4-8*x**3-7*x**2+5*x-10)*exp(10-x))*ln(2+x)+(x**3+2*x**2)*ex
p(10-x)**2*exp(x)**2+((2*x**4+2*x**3-5*x**2)*exp(10-x)**2+(-2*x**4-14*x**3-10*x**2+20*x)*exp(10-x))*exp(x)+(-2
*x**5-14*x**4+x**3+25*x**2-45*x)*exp(10-x)+x**5+12*x**4+35*x**3-20*x**2-75*x+50)/((x**3+2*x**2)*exp(10-x)**2*e
xp(x)**2+(-2*x**4-14*x**3-10*x**2+20*x)*exp(10-x)*exp(x)+x**5+12*x**4+35*x**3-20*x**2-75*x+50),x)
[Out]
x + (2*x**2*exp(10) + x*exp(10)*log(x + 2))*exp(-x)/(x**2 - x*exp(10) + 5*x - 5)
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