3.26.19 \(\int \frac {159+640 x+272 x^2+32 x^3+e^2 (10+40 x+17 x^2+2 x^3)+e (80+320 x+136 x^2+16 x^3)}{256+128 x+16 x^2+e^2 (16+8 x+x^2)+e (128+64 x+8 x^2)} \, dx\)

Optimal. Leaf size=23 \[ 4+x+x^2+\frac {6+\frac {5+x}{(4+e)^2}}{4+x} \]

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Rubi [A]  time = 0.10, antiderivative size = 26, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 4, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1986, 27, 12, 1850} \begin {gather*} x^2+x+\frac {97+48 e+6 e^2}{(4+e)^2 (x+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(159 + 640*x + 272*x^2 + 32*x^3 + E^2*(10 + 40*x + 17*x^2 + 2*x^3) + E*(80 + 320*x + 136*x^2 + 16*x^3))/(2
56 + 128*x + 16*x^2 + E^2*(16 + 8*x + x^2) + E*(128 + 64*x + 8*x^2)),x]

[Out]

x + x^2 + (97 + 48*E + 6*E^2)/((4 + E)^2*(4 + 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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 1986

Int[(Pq_)*(u_)^(p_.), x_Symbol] :> Int[Pq*ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && PolyQ[Pq, x] && QuadraticQ
[u, x] &&  !QuadraticMatchQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {159+640 x+272 x^2+32 x^3+e^2 \left (10+40 x+17 x^2+2 x^3\right )+e \left (80+320 x+136 x^2+16 x^3\right )}{16 (4+e)^2+8 (4+e)^2 x+(4+e)^2 x^2} \, dx\\ &=\int \frac {159+640 x+272 x^2+32 x^3+e^2 \left (10+40 x+17 x^2+2 x^3\right )+e \left (80+320 x+136 x^2+16 x^3\right )}{(4+e)^2 (4+x)^2} \, dx\\ &=\frac {\int \frac {159+640 x+272 x^2+32 x^3+e^2 \left (10+40 x+17 x^2+2 x^3\right )+e \left (80+320 x+136 x^2+16 x^3\right )}{(4+x)^2} \, dx}{(4+e)^2}\\ &=\frac {\int \left ((4+e)^2+2 (4+e)^2 x+\frac {-97-48 e-6 e^2}{(4+x)^2}\right ) \, dx}{(4+e)^2}\\ &=x+x^2+\frac {97+48 e+6 e^2}{(4+e)^2 (4+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 44, normalized size = 1.91 \begin {gather*} \frac {\frac {97+48 e+6 e^2}{4+x}-7 (4+e)^2 (4+x)+(4+e)^2 (4+x)^2}{(4+e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(159 + 640*x + 272*x^2 + 32*x^3 + E^2*(10 + 40*x + 17*x^2 + 2*x^3) + E*(80 + 320*x + 136*x^2 + 16*x^
3))/(256 + 128*x + 16*x^2 + E^2*(16 + 8*x + x^2) + E*(128 + 64*x + 8*x^2)),x]

[Out]

((97 + 48*E + 6*E^2)/(4 + x) - 7*(4 + E)^2*(4 + x) + (4 + E)^2*(4 + x)^2)/(4 + E)^2

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fricas [B]  time = 0.70, size = 69, normalized size = 3.00 \begin {gather*} \frac {16 \, x^{3} + 80 \, x^{2} + {\left (x^{3} + 5 \, x^{2} + 4 \, x + 6\right )} e^{2} + 8 \, {\left (x^{3} + 5 \, x^{2} + 4 \, x + 6\right )} e + 64 \, x + 97}{{\left (x + 4\right )} e^{2} + 8 \, {\left (x + 4\right )} e + 16 \, x + 64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+17*x^2+40*x+10)*exp(1)^2+(16*x^3+136*x^2+320*x+80)*exp(1)+32*x^3+272*x^2+640*x+159)/((x^2+8*
x+16)*exp(1)^2+(8*x^2+64*x+128)*exp(1)+16*x^2+128*x+256),x, algorithm="fricas")

[Out]

(16*x^3 + 80*x^2 + (x^3 + 5*x^2 + 4*x + 6)*e^2 + 8*(x^3 + 5*x^2 + 4*x + 6)*e + 64*x + 97)/((x + 4)*e^2 + 8*(x
+ 4)*e + 16*x + 64)

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giac [B]  time = 0.27, size = 101, normalized size = 4.39 \begin {gather*} \frac {x^{2} e^{4} + 16 \, x^{2} e^{3} + 96 \, x^{2} e^{2} + 256 \, x^{2} e + 256 \, x^{2} + x e^{4} + 16 \, x e^{3} + 96 \, x e^{2} + 256 \, x e + 256 \, x}{e^{4} + 16 \, e^{3} + 96 \, e^{2} + 256 \, e + 256} + \frac {6 \, e^{2} + 48 \, e + 97}{{\left (x + 4\right )} {\left (e^{2} + 8 \, e + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+17*x^2+40*x+10)*exp(1)^2+(16*x^3+136*x^2+320*x+80)*exp(1)+32*x^3+272*x^2+640*x+159)/((x^2+8*
x+16)*exp(1)^2+(8*x^2+64*x+128)*exp(1)+16*x^2+128*x+256),x, algorithm="giac")

[Out]

(x^2*e^4 + 16*x^2*e^3 + 96*x^2*e^2 + 256*x^2*e + 256*x^2 + x*e^4 + 16*x*e^3 + 96*x*e^2 + 256*x*e + 256*x)/(e^4
+ 16*e^3 + 96*e^2 + 256*e + 256) + (6*e^2 + 48*e + 97)/((x + 4)*(e^2 + 8*e + 16))

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maple [B]  time = 0.42, size = 52, normalized size = 2.26

method	result	size
norman	\(\frac {\left ({\mathrm e}+4\right ) x^{3}+\left (5 \,{\mathrm e}+20\right ) x^{2}-\frac {10 \,{\mathrm e}^{2}+80 \,{\mathrm e}+159}{{\mathrm e}+4}}{\left (4+x \right ) \left ({\mathrm e}+4\right )}\)	\(52\)
default	\(\frac {x^{2} {\mathrm e}^{2}+8 x^{2} {\mathrm e}+{\mathrm e}^{2} x +8 x \,{\mathrm e}+16 x^{2}+16 x -\frac {-6 \,{\mathrm e}^{2}-48 \,{\mathrm e}-97}{4+x}}{{\mathrm e}^{2}+8 \,{\mathrm e}+16}\)	\(62\)
gosper	\(\frac {x^{3} {\mathrm e}^{2}+5 x^{2} {\mathrm e}^{2}+8 x^{3} {\mathrm e}+40 x^{2} {\mathrm e}+16 x^{3}-10 \,{\mathrm e}^{2}+80 x^{2}-80 \,{\mathrm e}-159}{{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}+8 x \,{\mathrm e}+32 \,{\mathrm e}+16 x +64}\)	\(83\)
risch	\(x^{2}+x +\frac {6 \,{\mathrm e}^{2}}{{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}+8 x \,{\mathrm e}+32 \,{\mathrm e}+16 x +64}+\frac {48 \,{\mathrm e}}{{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}+8 x \,{\mathrm e}+32 \,{\mathrm e}+16 x +64}+\frac {97}{{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}+8 x \,{\mathrm e}+32 \,{\mathrm e}+16 x +64}\)	\(88\)
meijerg	\(\frac {159 x}{16 \left ({\mathrm e}^{2}+8 \,{\mathrm e}+16\right ) \left (1+\frac {x}{4}\right )}+\frac {16 \left (2 \,{\mathrm e}^{2}+16 \,{\mathrm e}+32\right ) \left (-\frac {x \left (-\frac {1}{8} x^{2}+\frac {3}{2} x +12\right )}{16 \left (1+\frac {x}{4}\right )}+3 \ln \left (1+\frac {x}{4}\right )\right )}{{\mathrm e}^{2}+8 \,{\mathrm e}+16}+\frac {4 \left (17 \,{\mathrm e}^{2}+136 \,{\mathrm e}+272\right ) \left (\frac {x \left (\frac {3 x}{4}+6\right )}{3 x +12}-2 \ln \left (1+\frac {x}{4}\right )\right )}{{\mathrm e}^{2}+8 \,{\mathrm e}+16}+\frac {\left (40 \,{\mathrm e}^{2}+320 \,{\mathrm e}+640\right ) \left (-\frac {x}{4 \left (1+\frac {x}{4}\right )}+\ln \left (1+\frac {x}{4}\right )\right )}{{\mathrm e}^{2}+8 \,{\mathrm e}+16}+\frac {5 \,{\mathrm e}^{2} x}{8 \left ({\mathrm e}^{2}+8 \,{\mathrm e}+16\right ) \left (1+\frac {x}{4}\right )}+\frac {5 \,{\mathrm e} x}{\left ({\mathrm e}^{2}+8 \,{\mathrm e}+16\right ) \left (1+\frac {x}{4}\right )}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+17*x^2+40*x+10)*exp(1)^2+(16*x^3+136*x^2+320*x+80)*exp(1)+32*x^3+272*x^2+640*x+159)/((x^2+8*x+16)*
exp(1)^2+(8*x^2+64*x+128)*exp(1)+16*x^2+128*x+256),x,method=_RETURNVERBOSE)

[Out]

((exp(1)+4)*x^3+(5*exp(1)+20)*x^2-(10*exp(1)^2+80*exp(1)+159)/(exp(1)+4))/(4+x)/(exp(1)+4)

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maxima [A]  time = 0.42, size = 38, normalized size = 1.65 \begin {gather*} x^{2} + x + \frac {6 \, e^{2} + 48 \, e + 97}{x {\left (e^{2} + 8 \, e + 16\right )} + 4 \, e^{2} + 32 \, e + 64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+17*x^2+40*x+10)*exp(1)^2+(16*x^3+136*x^2+320*x+80)*exp(1)+32*x^3+272*x^2+640*x+159)/((x^2+8*
x+16)*exp(1)^2+(8*x^2+64*x+128)*exp(1)+16*x^2+128*x+256),x, algorithm="maxima")

[Out]

x^2 + x + (6*e^2 + 48*e + 97)/(x*(e^2 + 8*e + 16) + 4*e^2 + 32*e + 64)

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mupad [B]  time = 1.46, size = 38, normalized size = 1.65 \begin {gather*} x+x^2+\frac {48\,\mathrm {e}+6\,{\mathrm {e}}^2+97}{32\,\mathrm {e}+4\,{\mathrm {e}}^2+x\,\left (8\,\mathrm {e}+{\mathrm {e}}^2+16\right )+64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((640*x + exp(2)*(40*x + 17*x^2 + 2*x^3 + 10) + exp(1)*(320*x + 136*x^2 + 16*x^3 + 80) + 272*x^2 + 32*x^3 +
159)/(128*x + exp(1)*(64*x + 8*x^2 + 128) + exp(2)*(8*x + x^2 + 16) + 16*x^2 + 256),x)

[Out]

x + x^2 + (48*exp(1) + 6*exp(2) + 97)/(32*exp(1) + 4*exp(2) + x*(8*exp(1) + exp(2) + 16) + 64)

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sympy [A]  time = 0.43, size = 39, normalized size = 1.70 \begin {gather*} x^{2} + x + \frac {6 e^{2} + 97 + 48 e}{x \left (e^{2} + 16 + 8 e\right ) + 4 e^{2} + 64 + 32 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+17*x**2+40*x+10)*exp(1)**2+(16*x**3+136*x**2+320*x+80)*exp(1)+32*x**3+272*x**2+640*x+159)/(
(x**2+8*x+16)*exp(1)**2+(8*x**2+64*x+128)*exp(1)+16*x**2+128*x+256),x)

[Out]

x**2 + x + (6*exp(2) + 97 + 48*E)/(x*(exp(2) + 16 + 8*E) + 4*exp(2) + 64 + 32*E)

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