3.88.11 \(\int \frac {e^{5 x/2} (66+5 x+e^2 (20-32 x+20 x^2))}{5488+1176 x+84 x^2+2 x^3+e^2 (4704+672 x+4728 x^2+672 x^3+24 x^4)+e^4 (1344+96 x+2688 x^2+192 x^3+1344 x^4+96 x^5)+e^6 (128+384 x^2+384 x^4+128 x^6)} \, dx\)

Optimal. Leaf size=26 \[ \frac {e^{5 x/2}}{\left (-2+x+4 \left (4+e^2 \left (1+x^2\right )\right )\right )^2} \]

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Rubi [A]  time = 0.27, antiderivative size = 29, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, integrand size = 118, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6688, 12, 2289} \begin {gather*} \frac {e^{5 x/2}}{\left (4 e^2 x^2+x+2 \left (7+2 e^2\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((5*x)/2)*(66 + 5*x + E^2*(20 - 32*x + 20*x^2)))/(5488 + 1176*x + 84*x^2 + 2*x^3 + E^2*(4704 + 672*x +
4728*x^2 + 672*x^3 + 24*x^4) + E^4*(1344 + 96*x + 2688*x^2 + 192*x^3 + 1344*x^4 + 96*x^5) + E^6*(128 + 384*x^2
+ 384*x^4 + 128*x^6)),x]

[Out]

E^((5*x)/2)/(2*(7 + 2*E^2) + x + 4*E^2*x^2)^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 2289

Int[(F_)^(u_)*(v_)^(n_.)*(w_), x_Symbol] :> With[{z = Log[F]*v*D[u, x] + (n + 1)*D[v, x]}, Simp[(Coefficient[w
, x, Exponent[w, x]]*F^u*v^(n + 1))/Coefficient[z, x, Exponent[z, x]], x] /; EqQ[Exponent[w, x], Exponent[z, x
]] && EqQ[w*Coefficient[z, x, Exponent[z, x]], z*Coefficient[w, x, Exponent[w, x]]]] /; FreeQ[{F, n}, x] && Po
lynomialQ[u, x] && PolynomialQ[v, x] && PolynomialQ[w, x]

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 {e^{5 x/2} \left (2 \left (33+10 e^2\right )+\left (5-32 e^2\right ) x+20 e^2 x^2\right )}{2 \left (2 \left (7+2 e^2\right )+x+4 e^2 x^2\right )^3} \, dx\\ &=\frac {1}{2} \int \frac {e^{5 x/2} \left (2 \left (33+10 e^2\right )+\left (5-32 e^2\right ) x+20 e^2 x^2\right )}{\left (2 \left (7+2 e^2\right )+x+4 e^2 x^2\right )^3} \, dx\\ &=\frac {e^{5 x/2}}{\left (2 \left (7+2 e^2\right )+x+4 e^2 x^2\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.93, size = 23, normalized size = 0.88 \begin {gather*} \frac {e^{5 x/2}}{\left (14+x+4 e^2 \left (1+x^2\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((5*x)/2)*(66 + 5*x + E^2*(20 - 32*x + 20*x^2)))/(5488 + 1176*x + 84*x^2 + 2*x^3 + E^2*(4704 + 67
2*x + 4728*x^2 + 672*x^3 + 24*x^4) + E^4*(1344 + 96*x + 2688*x^2 + 192*x^3 + 1344*x^4 + 96*x^5) + E^6*(128 + 3
84*x^2 + 384*x^4 + 128*x^6)),x]

[Out]

E^((5*x)/2)/(14 + x + 4*E^2*(1 + x^2))^2

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fricas [B]  time = 0.63, size = 44, normalized size = 1.69 \begin {gather*} \frac {e^{\left (\frac {5}{2} \, x\right )}}{x^{2} + 16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} e^{4} + 8 \, {\left (x^{3} + 14 \, x^{2} + x + 14\right )} e^{2} + 28 \, x + 196} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^2-32*x+20)*exp(2)+5*x+66)*exp(5/4*x)^2/((128*x^6+384*x^4+384*x^2+128)*exp(2)^3+(96*x^5+1344*x
^4+192*x^3+2688*x^2+96*x+1344)*exp(2)^2+(24*x^4+672*x^3+4728*x^2+672*x+4704)*exp(2)+2*x^3+84*x^2+1176*x+5488),
x, algorithm="fricas")

[Out]

e^(5/2*x)/(x^2 + 16*(x^4 + 2*x^2 + 1)*e^4 + 8*(x^3 + 14*x^2 + x + 14)*e^2 + 28*x + 196)

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giac [B]  time = 0.86, size = 57, normalized size = 2.19 \begin {gather*} \frac {2 \, e^{\left (\frac {5}{2} \, x\right )}}{16 \, x^{4} e^{4} + 8 \, x^{3} e^{2} + 32 \, x^{2} e^{4} + 112 \, x^{2} e^{2} + x^{2} + 8 \, x e^{2} + 28 \, x + 16 \, e^{4} + 112 \, e^{2} + 196} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^2-32*x+20)*exp(2)+5*x+66)*exp(5/4*x)^2/((128*x^6+384*x^4+384*x^2+128)*exp(2)^3+(96*x^5+1344*x
^4+192*x^3+2688*x^2+96*x+1344)*exp(2)^2+(24*x^4+672*x^3+4728*x^2+672*x+4704)*exp(2)+2*x^3+84*x^2+1176*x+5488),
x, algorithm="giac")

[Out]

2*e^(5/2*x)/(16*x^4*e^4 + 8*x^3*e^2 + 32*x^2*e^4 + 112*x^2*e^2 + x^2 + 8*x*e^2 + 28*x + 16*e^4 + 112*e^2 + 196
)

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maple [A]  time = 0.47, size = 22, normalized size = 0.85

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {5 x}{2}}}{\left (4 x^{2} {\mathrm e}^{2}+4 \,{\mathrm e}^{2}+x +14\right )^{2}}\)	\(22\)
norman	\(\frac {{\mathrm e}^{\frac {5 x}{2}}}{\left (4 x^{2} {\mathrm e}^{2}+4 \,{\mathrm e}^{2}+x +14\right )^{2}}\)	\(24\)
gosper	\(\frac {{\mathrm e}^{\frac {5 x}{2}}}{16 x^{4} {\mathrm e}^{4}+32 x^{2} {\mathrm e}^{4}+8 x^{3} {\mathrm e}^{2}+112 x^{2} {\mathrm e}^{2}+16 \,{\mathrm e}^{4}+8 \,{\mathrm e}^{2} x +x^{2}+112 \,{\mathrm e}^{2}+28 x +196}\)	\(65\)
derivativedivides	\(\text {Expression too large to display}\)	\(6102\)
default	\(\text {Expression too large to display}\)	\(6102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^2-32*x+20)*exp(2)+5*x+66)*exp(5/4*x)^2/((128*x^6+384*x^4+384*x^2+128)*exp(2)^3+(96*x^5+1344*x^4+192
*x^3+2688*x^2+96*x+1344)*exp(2)^2+(24*x^4+672*x^3+4728*x^2+672*x+4704)*exp(2)+2*x^3+84*x^2+1176*x+5488),x,meth
od=_RETURNVERBOSE)

[Out]

exp(5/2*x)/(4*x^2*exp(2)+4*exp(2)+x+14)^2

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maxima [B]  time = 0.41, size = 54, normalized size = 2.08 \begin {gather*} \frac {e^{\left (\frac {5}{2} \, x\right )}}{16 \, x^{4} e^{4} + 8 \, x^{3} e^{2} + x^{2} {\left (32 \, e^{4} + 112 \, e^{2} + 1\right )} + 4 \, x {\left (2 \, e^{2} + 7\right )} + 16 \, e^{4} + 112 \, e^{2} + 196} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^2-32*x+20)*exp(2)+5*x+66)*exp(5/4*x)^2/((128*x^6+384*x^4+384*x^2+128)*exp(2)^3+(96*x^5+1344*x
^4+192*x^3+2688*x^2+96*x+1344)*exp(2)^2+(24*x^4+672*x^3+4728*x^2+672*x+4704)*exp(2)+2*x^3+84*x^2+1176*x+5488),
x, algorithm="maxima")

[Out]

e^(5/2*x)/(16*x^4*e^4 + 8*x^3*e^2 + x^2*(32*e^4 + 112*e^2 + 1) + 4*x*(2*e^2 + 7) + 16*e^4 + 112*e^2 + 196)

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mupad [B]  time = 5.97, size = 60, normalized size = 2.31 \begin {gather*} \frac {{\mathrm {e}}^{\frac {5\,x}{2}-4}}{16\,\left (x^4+\frac {{\mathrm {e}}^{-2}\,x^3}{2}+\frac {{\mathrm {e}}^{-4}\,\left (112\,{\mathrm {e}}^2+32\,{\mathrm {e}}^4+1\right )\,x^2}{16}+\frac {{\mathrm {e}}^{-4}\,\left (8\,{\mathrm {e}}^2+28\right )\,x}{16}+\frac {{\mathrm {e}}^{-4}\,{\left (2\,{\mathrm {e}}^2+7\right )}^2}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((5*x)/2)*(5*x + exp(2)*(20*x^2 - 32*x + 20) + 66))/(1176*x + exp(2)*(672*x + 4728*x^2 + 672*x^3 + 24*
x^4 + 4704) + exp(6)*(384*x^2 + 384*x^4 + 128*x^6 + 128) + exp(4)*(96*x + 2688*x^2 + 192*x^3 + 1344*x^4 + 96*x
^5 + 1344) + 84*x^2 + 2*x^3 + 5488),x)

[Out]

exp((5*x)/2 - 4)/(16*((exp(-4)*(2*exp(2) + 7)^2)/4 + (x^3*exp(-2))/2 + x^4 + (x*exp(-4)*(8*exp(2) + 28))/16 +
(x^2*exp(-4)*(112*exp(2) + 32*exp(4) + 1))/16))

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sympy [B]  time = 0.28, size = 65, normalized size = 2.50 \begin {gather*} \frac {e^{\frac {5 x}{2}}}{16 x^{4} e^{4} + 8 x^{3} e^{2} + x^{2} + 112 x^{2} e^{2} + 32 x^{2} e^{4} + 28 x + 8 x e^{2} + 196 + 112 e^{2} + 16 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**2-32*x+20)*exp(2)+5*x+66)*exp(5/4*x)**2/((128*x**6+384*x**4+384*x**2+128)*exp(2)**3+(96*x**5
+1344*x**4+192*x**3+2688*x**2+96*x+1344)*exp(2)**2+(24*x**4+672*x**3+4728*x**2+672*x+4704)*exp(2)+2*x**3+84*x*
*2+1176*x+5488),x)

[Out]

exp(5*x/2)/(16*x**4*exp(4) + 8*x**3*exp(2) + x**2 + 112*x**2*exp(2) + 32*x**2*exp(4) + 28*x + 8*x*exp(2) + 196
+ 112*exp(2) + 16*exp(4))

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