3.87.51 \(\int \frac {3125000-906250 x+e^{x+4 x^2} (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4)+e^{2 x+8 x^2} (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4)}{-15625+1875 x-75 x^2+x^3} \, dx\)

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

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Rubi [B]  time = 0.51, antiderivative size = 59, normalized size of antiderivative = 2.36, number of steps used = 15, number of rules used = 8, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {6742, 2244, 2236, 37, 2234, 2204, 2242, 2240} \begin {gather*} -1450 e^{4 x^2+x}+25 e^{8 x^2+2 x}+\frac {31250 e^{4 x^2+x}}{25-x}+\frac {25 (100-29 x)^2}{(25-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3125000 - 906250*x + E^(x + 4*x^2)*(2343750 + 23875000*x - 9172500*x^2 + 618550*x^3 - 11600*x^4) + E^(2*x
+ 8*x^2)*(-781250 - 6156250*x + 746250*x^2 - 29950*x^3 + 400*x^4))/(-15625 + 1875*x - 75*x^2 + x^3),x]

[Out]

-1450*E^(x + 4*x^2) + 25*E^(2*x + 8*x^2) + (25*(100 - 29*x)^2)/(25 - x)^2 + (31250*E^(x + 4*x^2))/(25 - x)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2240

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 2242

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*F
^(a + b*x + c*x^2))/(e*(m + 1)), x] + (-Dist[(2*c*Log[F])/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*F^(a + b*x + c*
x^2), x], x] - Dist[((b*e - 2*c*d)*Log[F])/(e^2*(m + 1)), Int[(d + e*x)^(m + 1)*F^(a + b*x + c*x^2), x], x]) /
; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, 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 \left (50 e^{2 x (1+4 x)} (1+8 x)-\frac {31250 (-100+29 x)}{(-25+x)^3}-\frac {50 e^{x+4 x^2} \left (1875+19175 x-6571 x^2+232 x^3\right )}{(-25+x)^2}\right ) \, dx\\ &=50 \int e^{2 x (1+4 x)} (1+8 x) \, dx-50 \int \frac {e^{x+4 x^2} \left (1875+19175 x-6571 x^2+232 x^3\right )}{(-25+x)^2} \, dx-31250 \int \frac {-100+29 x}{(-25+x)^3} \, dx\\ &=\frac {25 (100-29 x)^2}{(25-x)^2}+50 \int e^{2 x+8 x^2} (1+8 x) \, dx-50 \int \left (5029 e^{x+4 x^2}-\frac {625 e^{x+4 x^2}}{(-25+x)^2}+\frac {125625 e^{x+4 x^2}}{-25+x}+232 e^{x+4 x^2} x\right ) \, dx\\ &=25 e^{2 x+8 x^2}+\frac {25 (100-29 x)^2}{(25-x)^2}-11600 \int e^{x+4 x^2} x \, dx+31250 \int \frac {e^{x+4 x^2}}{(-25+x)^2} \, dx-251450 \int e^{x+4 x^2} \, dx-6281250 \int \frac {e^{x+4 x^2}}{-25+x} \, dx\\ &=-1450 e^{x+4 x^2}+25 e^{2 x+8 x^2}+\frac {25 (100-29 x)^2}{(25-x)^2}+\frac {31250 e^{x+4 x^2}}{25-x}+1450 \int e^{x+4 x^2} \, dx+250000 \int e^{x+4 x^2} \, dx-\frac {251450 \int e^{\frac {1}{16} (1+8 x)^2} \, dx}{\sqrt [16]{e}}\\ &=-1450 e^{x+4 x^2}+25 e^{2 x+8 x^2}+\frac {25 (100-29 x)^2}{(25-x)^2}+\frac {31250 e^{x+4 x^2}}{25-x}-\frac {125725 \sqrt {\pi } \text {erfi}\left (\frac {1}{4} (1+8 x)\right )}{2 \sqrt [16]{e}}+\frac {1450 \int e^{\frac {1}{16} (1+8 x)^2} \, dx}{\sqrt [16]{e}}+\frac {250000 \int e^{\frac {1}{16} (1+8 x)^2} \, dx}{\sqrt [16]{e}}\\ &=-1450 e^{x+4 x^2}+25 e^{2 x+8 x^2}+\frac {25 (100-29 x)^2}{(25-x)^2}+\frac {31250 e^{x+4 x^2}}{25-x}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.64, size = 52, normalized size = 2.08 \begin {gather*} \frac {25 \left (e^{2 x (1+4 x)} (-25+x)^2+625 (-825+58 x)-2 e^{x+4 x^2} \left (2500-825 x+29 x^2\right )\right )}{(-25+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3125000 - 906250*x + E^(x + 4*x^2)*(2343750 + 23875000*x - 9172500*x^2 + 618550*x^3 - 11600*x^4) +
E^(2*x + 8*x^2)*(-781250 - 6156250*x + 746250*x^2 - 29950*x^3 + 400*x^4))/(-15625 + 1875*x - 75*x^2 + x^3),x]

[Out]

(25*(E^(2*x*(1 + 4*x))*(-25 + x)^2 + 625*(-825 + 58*x) - 2*E^(x + 4*x^2)*(2500 - 825*x + 29*x^2)))/(-25 + x)^2

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fricas [B]  time = 0.71, size = 56, normalized size = 2.24 \begin {gather*} \frac {25 \, {\left ({\left (x^{2} - 50 \, x + 625\right )} e^{\left (8 \, x^{2} + 2 \, x\right )} - 2 \, {\left (29 \, x^{2} - 825 \, x + 2500\right )} e^{\left (4 \, x^{2} + x\right )} + 36250 \, x - 515625\right )}}{x^{2} - 50 \, x + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x^4-29950*x^3+746250*x^2-6156250*x-781250)*exp(4*x^2+x)^2+(-11600*x^4+618550*x^3-9172500*x^2+2
3875000*x+2343750)*exp(4*x^2+x)-906250*x+3125000)/(x^3-75*x^2+1875*x-15625),x, algorithm="fricas")

[Out]

25*((x^2 - 50*x + 625)*e^(8*x^2 + 2*x) - 2*(29*x^2 - 825*x + 2500)*e^(4*x^2 + x) + 36250*x - 515625)/(x^2 - 50
*x + 625)

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giac [B]  time = 0.20, size = 90, normalized size = 3.60 \begin {gather*} \frac {25 \, {\left (x^{2} e^{\left (8 \, x^{2} + 2 \, x\right )} - 58 \, x^{2} e^{\left (4 \, x^{2} + x\right )} - 50 \, x e^{\left (8 \, x^{2} + 2 \, x\right )} + 1650 \, x e^{\left (4 \, x^{2} + x\right )} + 36250 \, x + 625 \, e^{\left (8 \, x^{2} + 2 \, x\right )} - 5000 \, e^{\left (4 \, x^{2} + x\right )} - 515625\right )}}{x^{2} - 50 \, x + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x^4-29950*x^3+746250*x^2-6156250*x-781250)*exp(4*x^2+x)^2+(-11600*x^4+618550*x^3-9172500*x^2+2
3875000*x+2343750)*exp(4*x^2+x)-906250*x+3125000)/(x^3-75*x^2+1875*x-15625),x, algorithm="giac")

[Out]

25*(x^2*e^(8*x^2 + 2*x) - 58*x^2*e^(4*x^2 + x) - 50*x*e^(8*x^2 + 2*x) + 1650*x*e^(4*x^2 + x) + 36250*x + 625*e
^(8*x^2 + 2*x) - 5000*e^(4*x^2 + x) - 515625)/(x^2 - 50*x + 625)

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maple [A]  time = 0.09, size = 49, normalized size = 1.96

method	result	size
risch	\(\frac {906250 x -12890625}{x^{2}-50 x +625}+25 \,{\mathrm e}^{2 x \left (4 x +1\right )}-\frac {50 \left (29 x -100\right ) {\mathrm e}^{x \left (4 x +1\right )}}{x -25}\)	\(49\)
norman	\(\frac {906250 x +15625 \,{\mathrm e}^{8 x^{2}+2 x}+41250 \,{\mathrm e}^{4 x^{2}+x} x -1450 \,{\mathrm e}^{4 x^{2}+x} x^{2}-1250 \,{\mathrm e}^{8 x^{2}+2 x} x +25 \,{\mathrm e}^{8 x^{2}+2 x} x^{2}-125000 \,{\mathrm e}^{4 x^{2}+x}-12890625}{\left (x -25\right )^{2}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((400*x^4-29950*x^3+746250*x^2-6156250*x-781250)*exp(4*x^2+x)^2+(-11600*x^4+618550*x^3-9172500*x^2+2387500
0*x+2343750)*exp(4*x^2+x)-906250*x+3125000)/(x^3-75*x^2+1875*x-15625),x,method=_RETURNVERBOSE)

[Out]

(906250*x-12890625)/(x^2-50*x+625)+25*exp(2*x*(4*x+1))-50*(29*x-100)/(x-25)*exp(x*(4*x+1))

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maxima [B]  time = 0.44, size = 67, normalized size = 2.68 \begin {gather*} \frac {453125 \, {\left (2 \, x - 25\right )}}{x^{2} - 50 \, x + 625} + \frac {25 \, {\left ({\left (x - 25\right )} e^{\left (8 \, x^{2} + 2 \, x\right )} - 2 \, {\left (29 \, x - 100\right )} e^{\left (4 \, x^{2} + x\right )}\right )}}{x - 25} - \frac {1562500}{x^{2} - 50 \, x + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x^4-29950*x^3+746250*x^2-6156250*x-781250)*exp(4*x^2+x)^2+(-11600*x^4+618550*x^3-9172500*x^2+2
3875000*x+2343750)*exp(4*x^2+x)-906250*x+3125000)/(x^3-75*x^2+1875*x-15625),x, algorithm="maxima")

[Out]

453125*(2*x - 25)/(x^2 - 50*x + 625) + 25*((x - 25)*e^(8*x^2 + 2*x) - 2*(29*x - 100)*e^(4*x^2 + x))/(x - 25) -
1562500/(x^2 - 50*x + 625)

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mupad [B]  time = 5.35, size = 56, normalized size = 2.24 \begin {gather*} 25\,{\mathrm {e}}^{8\,x^2+2\,x}-1450\,{\mathrm {e}}^{4\,x^2+x}-\frac {x\,\left (31250\,{\mathrm {e}}^{4\,x^2+x}-906250\right )-781250\,{\mathrm {e}}^{4\,x^2+x}[phone]}{{\left (x-25\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(906250*x + exp(2*x + 8*x^2)*(6156250*x - 746250*x^2 + 29950*x^3 - 400*x^4 + 781250) - exp(x + 4*x^2)*(23
875000*x - 9172500*x^2 + 618550*x^3 - 11600*x^4 + 2343750) - 3125000)/(1875*x - 75*x^2 + x^3 - 15625),x)

[Out]

25*exp(2*x + 8*x^2) - 1450*exp(x + 4*x^2) - (x*(31250*exp(x + 4*x^2) - 906250) - 781250*exp(x + 4*x^2) + 12890
625)/(x - 25)^2

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sympy [B]  time = 0.20, size = 44, normalized size = 1.76 \begin {gather*} - \frac {12890625 - 906250 x}{x^{2} - 50 x + 625} + \frac {\left (5000 - 1450 x\right ) e^{4 x^{2} + x} + \left (25 x - 625\right ) e^{8 x^{2} + 2 x}}{x - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x**4-29950*x**3+746250*x**2-6156250*x-781250)*exp(4*x**2+x)**2+(-11600*x**4+618550*x**3-917250
0*x**2+23875000*x+2343750)*exp(4*x**2+x)-906250*x+3125000)/(x**3-75*x**2+1875*x-15625),x)

[Out]

-(12890625 - 906250*x)/(x**2 - 50*x + 625) + ((5000 - 1450*x)*exp(4*x**2 + x) + (25*x - 625)*exp(8*x**2 + 2*x)
)/(x - 25)

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