3.8.11 \(\int \frac {-75-30 e^{10+x}-10 e^{10+2 x}-90 x-15 x^2}{81 e^{20}+12 e^{20+3 x}+e^{20+4 x}+225 x^2+270 x^3+111 x^4+18 x^5+x^6+e^{10} (270 x+162 x^2+18 x^3)+e^{2 x} (54 e^{20}+e^{10} (30 x+18 x^2+2 x^3))+e^x (108 e^{20}+e^{10} (180 x+108 x^2+12 x^3))} \, dx\)

Optimal. Leaf size=28 \[ \frac {5}{e^{10} \left (3+e^x\right )^2-x+x \left (x+(4+x)^2\right )} \]

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Rubi [A]  time = 0.46, antiderivative size = 34, normalized size of antiderivative = 1.21, number of steps used = 3, number of rules used = 3, integrand size = 150, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6688, 12, 6686} \begin {gather*} \frac {5}{x \left (x^2+9 x+15\right )+e^{2 (x+5)}+6 e^{x+10}+9 e^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-75 - 30*E^(10 + x) - 10*E^(10 + 2*x) - 90*x - 15*x^2)/(81*E^20 + 12*E^(20 + 3*x) + E^(20 + 4*x) + 225*x^
2 + 270*x^3 + 111*x^4 + 18*x^5 + x^6 + E^10*(270*x + 162*x^2 + 18*x^3) + E^(2*x)*(54*E^20 + E^10*(30*x + 18*x^
2 + 2*x^3)) + E^x*(108*E^20 + E^10*(180*x + 108*x^2 + 12*x^3))),x]

[Out]

5/(9*E^10 + E^(2*(5 + x)) + 6*E^(10 + x) + x*(15 + 9*x + x^2))

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 {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (-2 e^{2 (5+x)}-6 e^{10+x}-3 \left (5+6 x+x^2\right )\right )}{\left (9 e^{10}+e^{2 (5+x)}+6 e^{10+x}+x \left (15+9 x+x^2\right )\right )^2} \, dx\\ &=5 \int \frac {-2 e^{2 (5+x)}-6 e^{10+x}-3 \left (5+6 x+x^2\right )}{\left (9 e^{10}+e^{2 (5+x)}+6 e^{10+x}+x \left (15+9 x+x^2\right )\right )^2} \, dx\\ &=\frac {5}{9 e^{10}+e^{2 (5+x)}+6 e^{10+x}+x \left (15+9 x+x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 34, normalized size = 1.21 \begin {gather*} \frac {5}{9 e^{10}+e^{2 (5+x)}+6 e^{10+x}+x \left (15+9 x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-75 - 30*E^(10 + x) - 10*E^(10 + 2*x) - 90*x - 15*x^2)/(81*E^20 + 12*E^(20 + 3*x) + E^(20 + 4*x) +
225*x^2 + 270*x^3 + 111*x^4 + 18*x^5 + x^6 + E^10*(270*x + 162*x^2 + 18*x^3) + E^(2*x)*(54*E^20 + E^10*(30*x +
18*x^2 + 2*x^3)) + E^x*(108*E^20 + E^10*(180*x + 108*x^2 + 12*x^3))),x]

[Out]

5/(9*E^10 + E^(2*(5 + x)) + 6*E^(10 + x) + x*(15 + 9*x + x^2))

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fricas [A]  time = 1.38, size = 38, normalized size = 1.36 \begin {gather*} \frac {5 \, e^{10}}{{\left (x^{3} + 9 \, x^{2} + 15 \, x\right )} e^{10} + 9 \, e^{20} + e^{\left (2 \, x + 20\right )} + 6 \, e^{\left (x + 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(5)^2*exp(x)^2-30*exp(5)^2*exp(x)-15*x^2-90*x-75)/(exp(5)^4*exp(x)^4+12*exp(5)^4*exp(x)^3+(5
4*exp(5)^4+(2*x^3+18*x^2+30*x)*exp(5)^2)*exp(x)^2+(108*exp(5)^4+(12*x^3+108*x^2+180*x)*exp(5)^2)*exp(x)+81*exp
(5)^4+(18*x^3+162*x^2+270*x)*exp(5)^2+x^6+18*x^5+111*x^4+270*x^3+225*x^2),x, algorithm="fricas")

[Out]

5*e^10/((x^3 + 9*x^2 + 15*x)*e^10 + 9*e^20 + e^(2*x + 20) + 6*e^(x + 20))

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giac [A]  time = 7.57, size = 32, normalized size = 1.14 \begin {gather*} \frac {10}{x^{3} + 9 \, x^{2} + 15 \, x + 9 \, e^{10} + e^{\left (2 \, x + 10\right )} + 6 \, e^{\left (x + 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(5)^2*exp(x)^2-30*exp(5)^2*exp(x)-15*x^2-90*x-75)/(exp(5)^4*exp(x)^4+12*exp(5)^4*exp(x)^3+(5
4*exp(5)^4+(2*x^3+18*x^2+30*x)*exp(5)^2)*exp(x)^2+(108*exp(5)^4+(12*x^3+108*x^2+180*x)*exp(5)^2)*exp(x)+81*exp
(5)^4+(18*x^3+162*x^2+270*x)*exp(5)^2+x^6+18*x^5+111*x^4+270*x^3+225*x^2),x, algorithm="giac")

[Out]

10/(x^3 + 9*x^2 + 15*x + 9*e^10 + e^(2*x + 10) + 6*e^(x + 10))

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maple [A]  time = 0.14, size = 33, normalized size = 1.18

method	result	size
risch	\(\frac {5}{{\mathrm e}^{2 x +10}+6 \,{\mathrm e}^{x +10}+x^{3}+9 \,{\mathrm e}^{10}+9 x^{2}+15 x}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-10*exp(5)^2*exp(x)^2-30*exp(5)^2*exp(x)-15*x^2-90*x-75)/(exp(5)^4*exp(x)^4+12*exp(5)^4*exp(x)^3+(54*exp(
5)^4+(2*x^3+18*x^2+30*x)*exp(5)^2)*exp(x)^2+(108*exp(5)^4+(12*x^3+108*x^2+180*x)*exp(5)^2)*exp(x)+81*exp(5)^4+
(18*x^3+162*x^2+270*x)*exp(5)^2+x^6+18*x^5+111*x^4+270*x^3+225*x^2),x,method=_RETURNVERBOSE)

[Out]

5/(exp(2*x+10)+6*exp(x+10)+x^3+9*exp(10)+9*x^2+15*x)

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maxima [A]  time = 0.68, size = 32, normalized size = 1.14 \begin {gather*} \frac {5}{x^{3} + 9 \, x^{2} + 15 \, x + 9 \, e^{10} + e^{\left (2 \, x + 10\right )} + 6 \, e^{\left (x + 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(5)^2*exp(x)^2-30*exp(5)^2*exp(x)-15*x^2-90*x-75)/(exp(5)^4*exp(x)^4+12*exp(5)^4*exp(x)^3+(5
4*exp(5)^4+(2*x^3+18*x^2+30*x)*exp(5)^2)*exp(x)^2+(108*exp(5)^4+(12*x^3+108*x^2+180*x)*exp(5)^2)*exp(x)+81*exp
(5)^4+(18*x^3+162*x^2+270*x)*exp(5)^2+x^6+18*x^5+111*x^4+270*x^3+225*x^2),x, algorithm="maxima")

[Out]

5/(x^3 + 9*x^2 + 15*x + 9*e^10 + e^(2*x + 10) + 6*e^(x + 10))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {90\,x+10\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{10}+30\,{\mathrm {e}}^{10}\,{\mathrm {e}}^x+15\,x^2+75}{81\,{\mathrm {e}}^{20}+12\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{20}+{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{20}+{\mathrm {e}}^{2\,x}\,\left (54\,{\mathrm {e}}^{20}+{\mathrm {e}}^{10}\,\left (2\,x^3+18\,x^2+30\,x\right )\right )+{\mathrm {e}}^{10}\,\left (18\,x^3+162\,x^2+270\,x\right )+{\mathrm {e}}^x\,\left (108\,{\mathrm {e}}^{20}+{\mathrm {e}}^{10}\,\left (12\,x^3+108\,x^2+180\,x\right )\right )+225\,x^2+270\,x^3+111\,x^4+18\,x^5+x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(90*x + 10*exp(2*x)*exp(10) + 30*exp(10)*exp(x) + 15*x^2 + 75)/(81*exp(20) + 12*exp(3*x)*exp(20) + exp(4*
x)*exp(20) + exp(2*x)*(54*exp(20) + exp(10)*(30*x + 18*x^2 + 2*x^3)) + exp(10)*(270*x + 162*x^2 + 18*x^3) + ex
p(x)*(108*exp(20) + exp(10)*(180*x + 108*x^2 + 12*x^3)) + 225*x^2 + 270*x^3 + 111*x^4 + 18*x^5 + x^6),x)

[Out]

int(-(90*x + 10*exp(2*x)*exp(10) + 30*exp(10)*exp(x) + 15*x^2 + 75)/(81*exp(20) + 12*exp(3*x)*exp(20) + exp(4*
x)*exp(20) + exp(2*x)*(54*exp(20) + exp(10)*(30*x + 18*x^2 + 2*x^3)) + exp(10)*(270*x + 162*x^2 + 18*x^3) + ex
p(x)*(108*exp(20) + exp(10)*(180*x + 108*x^2 + 12*x^3)) + 225*x^2 + 270*x^3 + 111*x^4 + 18*x^5 + x^6), x)

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sympy [A]  time = 0.23, size = 34, normalized size = 1.21 \begin {gather*} \frac {5}{x^{3} + 9 x^{2} + 15 x + e^{10} e^{2 x} + 6 e^{10} e^{x} + 9 e^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(5)**2*exp(x)**2-30*exp(5)**2*exp(x)-15*x**2-90*x-75)/(exp(5)**4*exp(x)**4+12*exp(5)**4*exp(
x)**3+(54*exp(5)**4+(2*x**3+18*x**2+30*x)*exp(5)**2)*exp(x)**2+(108*exp(5)**4+(12*x**3+108*x**2+180*x)*exp(5)*
*2)*exp(x)+81*exp(5)**4+(18*x**3+162*x**2+270*x)*exp(5)**2+x**6+18*x**5+111*x**4+270*x**3+225*x**2),x)

[Out]

5/(x**3 + 9*x**2 + 15*x + exp(10)*exp(2*x) + 6*exp(10)*exp(x) + 9*exp(10))

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