3.12.94 \(\int \frac {6400 x+4800 x^2+2480 x^3+660 x^4+108 x^5+8 x^6+e^x (-1600-1200 x-300 x^2-25 x^3)}{1600+1200 x+300 x^2+25 x^3} \, dx\)

Optimal. Leaf size=27 \[ 1-e^x+\frac {2}{25} x^2 \left (5+\frac {x^2}{4+x}\right )^2 \]

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Rubi [A]  time = 0.16, antiderivative size = 50, normalized size of antiderivative = 1.85, number of steps used = 5, number of rules used = 3, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6688, 2194, 1620} \begin {gather*} \frac {2 x^4}{25}+\frac {4 x^3}{25}+\frac {66 x^2}{25}-\frac {192 x}{25}-e^x-\frac {7168}{25 (x+4)}+\frac {8192}{25 (x+4)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6400*x + 4800*x^2 + 2480*x^3 + 660*x^4 + 108*x^5 + 8*x^6 + E^x*(-1600 - 1200*x - 300*x^2 - 25*x^3))/(1600
+ 1200*x + 300*x^2 + 25*x^3),x]

[Out]

-E^x - (192*x)/25 + (66*x^2)/25 + (4*x^3)/25 + (2*x^4)/25 + 8192/(25*(4 + x)^2) - 7168/(25*(4 + x))

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^x+\frac {4 x \left (1600+1200 x+620 x^2+165 x^3+27 x^4+2 x^5\right )}{25 (4+x)^3}\right ) \, dx\\ &=\frac {4}{25} \int \frac {x \left (1600+1200 x+620 x^2+165 x^3+27 x^4+2 x^5\right )}{(4+x)^3} \, dx-\int e^x \, dx\\ &=-e^x+\frac {4}{25} \int \left (-48+33 x+3 x^2+2 x^3-\frac {4096}{(4+x)^3}+\frac {1792}{(4+x)^2}\right ) \, dx\\ &=-e^x-\frac {192 x}{25}+\frac {66 x^2}{25}+\frac {4 x^3}{25}+\frac {2 x^4}{25}+\frac {8192}{25 (4+x)^2}-\frac {7168}{25 (4+x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 58, normalized size = 2.15 \begin {gather*} -e^x+\frac {8192}{25 (4+x)^2}-\frac {7168}{25 (4+x)}-\frac {208 (4+x)}{5}+\frac {42}{5} (4+x)^2-\frac {28}{25} (4+x)^3+\frac {2}{25} (4+x)^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6400*x + 4800*x^2 + 2480*x^3 + 660*x^4 + 108*x^5 + 8*x^6 + E^x*(-1600 - 1200*x - 300*x^2 - 25*x^3))
/(1600 + 1200*x + 300*x^2 + 25*x^3),x]

[Out]

-E^x + 8192/(25*(4 + x)^2) - 7168/(25*(4 + x)) - (208*(4 + x))/5 + (42*(4 + x)^2)/5 - (28*(4 + x)^3)/25 + (2*(
4 + x)^4)/25

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fricas [B]  time = 0.61, size = 54, normalized size = 2.00 \begin {gather*} \frac {2 \, x^{6} + 20 \, x^{5} + 130 \, x^{4} + 400 \, x^{3} - 480 \, x^{2} - 25 \, {\left (x^{2} + 8 \, x + 16\right )} e^{x} - 10240 \, x - 20480}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^3-300*x^2-1200*x-1600)*exp(x)+8*x^6+108*x^5+660*x^4+2480*x^3+4800*x^2+6400*x)/(25*x^3+300*x^
2+1200*x+1600),x, algorithm="fricas")

[Out]

1/25*(2*x^6 + 20*x^5 + 130*x^4 + 400*x^3 - 480*x^2 - 25*(x^2 + 8*x + 16)*e^x - 10240*x - 20480)/(x^2 + 8*x + 1
6)

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giac [B]  time = 0.33, size = 58, normalized size = 2.15 \begin {gather*} \frac {2 \, x^{6} + 20 \, x^{5} + 130 \, x^{4} + 400 \, x^{3} - 25 \, x^{2} e^{x} - 480 \, x^{2} - 200 \, x e^{x} - 10240 \, x - 400 \, e^{x} - 20480}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^3-300*x^2-1200*x-1600)*exp(x)+8*x^6+108*x^5+660*x^4+2480*x^3+4800*x^2+6400*x)/(25*x^3+300*x^
2+1200*x+1600),x, algorithm="giac")

[Out]

1/25*(2*x^6 + 20*x^5 + 130*x^4 + 400*x^3 - 25*x^2*e^x - 480*x^2 - 200*x*e^x - 10240*x - 400*e^x - 20480)/(x^2
+ 8*x + 16)

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maple [A]  time = 0.06, size = 38, normalized size = 1.41

method	result	size
default	\(-\frac {7168}{25 \left (4+x \right )}+\frac {8192}{25 \left (4+x \right )^{2}}-\frac {192 x}{25}+\frac {66 x^{2}}{25}+\frac {4 x^{3}}{25}+\frac {2 x^{4}}{25}-{\mathrm e}^{x}\)	\(38\)
risch	\(\frac {2 x^{4}}{25}+\frac {4 x^{3}}{25}+\frac {66 x^{2}}{25}-\frac {192 x}{25}+\frac {-\frac {7168 x}{25}-\frac {4096}{5}}{x^{2}+8 x +16}-{\mathrm e}^{x}\)	\(40\)
norman	\(\frac {-256 x +16 x^{3}+\frac {26 x^{4}}{5}+\frac {4 x^{5}}{5}+\frac {2 x^{6}}{25}-8 \,{\mathrm e}^{x} x -{\mathrm e}^{x} x^{2}-16 \,{\mathrm e}^{x}-512}{\left (4+x \right )^{2}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-25*x^3-300*x^2-1200*x-1600)*exp(x)+8*x^6+108*x^5+660*x^4+2480*x^3+4800*x^2+6400*x)/(25*x^3+300*x^2+1200
*x+1600),x,method=_RETURNVERBOSE)

[Out]

-7168/25/(4+x)+8192/25/(4+x)^2-192/25*x+66/25*x^2+4/25*x^3+2/25*x^4-exp(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2}{25} \, x^{4} + \frac {4}{25} \, x^{3} + \frac {66}{25} \, x^{2} - \frac {192}{25} \, x - \frac {{\left (x^{3} + 12 \, x^{2} + 48 \, x\right )} e^{x}}{x^{3} + 12 \, x^{2} + 48 \, x + 64} - \frac {27648 \, {\left (5 \, x + 18\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} + \frac {16384 \, {\left (3 \, x + 11\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} - \frac {7936 \, {\left (3 \, x + 10\right )}}{5 \, {\left (x^{2} + 8 \, x + 16\right )}} + \frac {16896 \, {\left (2 \, x + 7\right )}}{5 \, {\left (x^{2} + 8 \, x + 16\right )}} + \frac {1536 \, {\left (x + 3\right )}}{x^{2} + 8 \, x + 16} - \frac {256 \, {\left (x + 2\right )}}{x^{2} + 8 \, x + 16} + \frac {64 \, e^{\left (-4\right )} E_{3}\left (-x - 4\right )}{{\left (x + 4\right )}^{2}} + 192 \, \int \frac {e^{x}}{x^{4} + 16 \, x^{3} + 96 \, x^{2} + 256 \, x + 256}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^3-300*x^2-1200*x-1600)*exp(x)+8*x^6+108*x^5+660*x^4+2480*x^3+4800*x^2+6400*x)/(25*x^3+300*x^
2+1200*x+1600),x, algorithm="maxima")

[Out]

2/25*x^4 + 4/25*x^3 + 66/25*x^2 - 192/25*x - (x^3 + 12*x^2 + 48*x)*e^x/(x^3 + 12*x^2 + 48*x + 64) - 27648/25*(
5*x + 18)/(x^2 + 8*x + 16) + 16384/25*(3*x + 11)/(x^2 + 8*x + 16) - 7936/5*(3*x + 10)/(x^2 + 8*x + 16) + 16896
/5*(2*x + 7)/(x^2 + 8*x + 16) + 1536*(x + 3)/(x^2 + 8*x + 16) - 256*(x + 2)/(x^2 + 8*x + 16) + 64*e^(-4)*exp_i
ntegral_e(3, -x - 4)/(x + 4)^2 + 192*integrate(e^x/(x^4 + 16*x^3 + 96*x^2 + 256*x + 256), x)

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mupad [B]  time = 0.89, size = 35, normalized size = 1.30 \begin {gather*} \frac {66\,x^2}{25}-{\mathrm {e}}^x-\frac {\frac {7168\,x}{25}+\frac {4096}{5}}{{\left (x+4\right )}^2}-\frac {192\,x}{25}+\frac {4\,x^3}{25}+\frac {2\,x^4}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6400*x + 4800*x^2 + 2480*x^3 + 660*x^4 + 108*x^5 + 8*x^6 - exp(x)*(1200*x + 300*x^2 + 25*x^3 + 1600))/(12
00*x + 300*x^2 + 25*x^3 + 1600),x)

[Out]

(66*x^2)/25 - exp(x) - ((7168*x)/25 + 4096/5)/(x + 4)^2 - (192*x)/25 + (4*x^3)/25 + (2*x^4)/25

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sympy [B]  time = 0.16, size = 44, normalized size = 1.63 \begin {gather*} \frac {2 x^{4}}{25} + \frac {4 x^{3}}{25} + \frac {66 x^{2}}{25} - \frac {192 x}{25} + \frac {- 7168 x - 20480}{25 x^{2} + 200 x + 400} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x**3-300*x**2-1200*x-1600)*exp(x)+8*x**6+108*x**5+660*x**4+2480*x**3+4800*x**2+6400*x)/(25*x**
3+300*x**2+1200*x+1600),x)

[Out]

2*x**4/25 + 4*x**3/25 + 66*x**2/25 - 192*x/25 + (-7168*x - 20480)/(25*x**2 + 200*x + 400) - exp(x)

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