3.87.7 \(\int \frac {1248+3250 x+2900 x^2+1060 x^3+170 x^4+10 x^5+(-625-1000 x-450 x^2-80 x^3-5 x^4) \log (5)+e^{x^2} (4 x+4 x^2-2 x \log (5))}{-2-2 x+\log (5)} \, dx\)

Optimal. Leaf size=26 \[ -e^{x^2}-x (5+x)^4+\log (2+2 x-\log (5)) \]

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Rubi [B]  time = 0.57, antiderivative size = 367, normalized size of antiderivative = 14.12, number of steps used = 15, number of rules used = 4, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6742, 2209, 43, 77} \begin {gather*} -x^5-\frac {85 x^4}{4}+\frac {5}{8} x^4 (2-\log (5))-\frac {530 x^3}{3}-\frac {5}{12} x^3 (2-\log (5))^2+\frac {85}{6} x^3 (2-\log (5))-725 x^2-e^{x^2}+\frac {5}{16} x^2 (2-\log (5))^3-\frac {85}{8} x^2 (2-\log (5))^2+\frac {265}{2} x^2 (2-\log (5))-1625 x+\frac {5}{16} x \log ^2(5) (8+\log (5))^2+\frac {5}{32} \log ^2(5) (8+\log (5))^3 \log (2 x+2-\log (5))+\frac {5}{16} (x+5)^2 \log ^2(5) (8+\log (5))+\frac {5}{12} (x+5)^3 \log ^2(5)-\frac {5}{16} x (2-\log (5))^4+\frac {85}{8} x (2-\log (5))^3-\frac {265}{2} x (2-\log (5))^2+725 x (2-\log (5))+\frac {5}{32} (2-\log (5))^5 \log (2 x+2-\log (5))-\frac {85}{16} (2-\log (5))^4 \log (2 x+2-\log (5))+\frac {265}{4} (2-\log (5))^3 \log (2 x+2-\log (5))-\frac {725}{2} (2-\log (5))^2 \log (2 x+2-\log (5))+\frac {1625}{2} (2-\log (5)) \log (2 x+2-\log (5))-624 \log (2 x+2-\log (5))+\frac {5}{8} (x+5)^4 \log (5) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1248 + 3250*x + 2900*x^2 + 1060*x^3 + 170*x^4 + 10*x^5 + (-625 - 1000*x - 450*x^2 - 80*x^3 - 5*x^4)*Log[5
] + E^x^2*(4*x + 4*x^2 - 2*x*Log[5]))/(-2 - 2*x + Log[5]),x]

[Out]

-E^x^2 - 1625*x - 725*x^2 - (530*x^3)/3 - (85*x^4)/4 - x^5 + 725*x*(2 - Log[5]) + (265*x^2*(2 - Log[5]))/2 + (
85*x^3*(2 - Log[5]))/6 + (5*x^4*(2 - Log[5]))/8 - (265*x*(2 - Log[5])^2)/2 - (85*x^2*(2 - Log[5])^2)/8 - (5*x^
3*(2 - Log[5])^2)/12 + (85*x*(2 - Log[5])^3)/8 + (5*x^2*(2 - Log[5])^3)/16 - (5*x*(2 - Log[5])^4)/16 + (5*(5 +
x)^4*Log[5])/8 + (5*(5 + x)^3*Log[5]^2)/12 + (5*(5 + x)^2*Log[5]^2*(8 + Log[5]))/16 + (5*x*Log[5]^2*(8 + Log[
5])^2)/16 - 624*Log[2 + 2*x - Log[5]] + (1625*(2 - Log[5])*Log[2 + 2*x - Log[5]])/2 - (725*(2 - Log[5])^2*Log[
2 + 2*x - Log[5]])/2 + (265*(2 - Log[5])^3*Log[2 + 2*x - Log[5]])/4 - (85*(2 - Log[5])^4*Log[2 + 2*x - Log[5]]
)/16 + (5*(2 - Log[5])^5*Log[2 + 2*x - Log[5]])/32 + (5*Log[5]^2*(8 + Log[5])^3*Log[2 + 2*x - Log[5]])/32

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2209

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

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 (-2 e^{x^2} x-\frac {1248}{2+2 x-\log (5)}-\frac {3250 x}{2+2 x-\log (5)}-\frac {2900 x^2}{2+2 x-\log (5)}-\frac {1060 x^3}{2+2 x-\log (5)}-\frac {170 x^4}{2+2 x-\log (5)}-\frac {10 x^5}{2+2 x-\log (5)}+\frac {5 (1+x) (5+x)^3 \log (5)}{2+2 x-\log (5)}\right ) \, dx\\ &=-624 \log (2+2 x-\log (5))-2 \int e^{x^2} x \, dx-10 \int \frac {x^5}{2+2 x-\log (5)} \, dx-170 \int \frac {x^4}{2+2 x-\log (5)} \, dx-1060 \int \frac {x^3}{2+2 x-\log (5)} \, dx-2900 \int \frac {x^2}{2+2 x-\log (5)} \, dx-3250 \int \frac {x}{2+2 x-\log (5)} \, dx+(5 \log (5)) \int \frac {(1+x) (5+x)^3}{2+2 x-\log (5)} \, dx\\ &=-e^{x^2}-624 \log (2+2 x-\log (5))-10 \int \left (\frac {x^4}{2}+\frac {1}{4} x^3 (-2+\log (5))+\frac {1}{8} x^2 (-2+\log (5))^2+\frac {1}{16} x (-2+\log (5))^3+\frac {1}{32} (-2+\log (5))^4+\frac {(-2+\log (5))^5}{32 (2+2 x-\log (5))}\right ) \, dx-170 \int \left (\frac {x^3}{2}+\frac {1}{4} x^2 (-2+\log (5))+\frac {1}{8} x (-2+\log (5))^2+\frac {1}{16} (-2+\log (5))^3+\frac {(-2+\log (5))^4}{16 (2+2 x-\log (5))}\right ) \, dx-1060 \int \left (\frac {x^2}{2}+\frac {1}{4} x (-2+\log (5))+\frac {1}{8} (-2+\log (5))^2+\frac {(-2+\log (5))^3}{8 (2+2 x-\log (5))}\right ) \, dx-2900 \int \left (\frac {x}{2}+\frac {1}{4} (-2+\log (5))+\frac {(-2+\log (5))^2}{4 (2+2 x-\log (5))}\right ) \, dx-3250 \int \left (\frac {1}{2}+\frac {-2+\log (5)}{2 (2+2 x-\log (5))}\right ) \, dx+(5 \log (5)) \int \left (\frac {1}{2} (5+x)^3+\frac {1}{4} (5+x)^2 \log (5)+\frac {1}{8} (5+x) \log (5) (8+\log (5))+\frac {1}{16} \log (5) (8+\log (5))^2+\frac {\log (5) (8+\log (5))^3}{16 (2+2 x-\log (5))}\right ) \, dx\\ &=-e^{x^2}-1625 x-725 x^2-\frac {530 x^3}{3}-\frac {85 x^4}{4}-x^5+725 x (2-\log (5))+\frac {265}{2} x^2 (2-\log (5))+\frac {85}{6} x^3 (2-\log (5))+\frac {5}{8} x^4 (2-\log (5))-\frac {265}{2} x (2-\log (5))^2-\frac {85}{8} x^2 (2-\log (5))^2-\frac {5}{12} x^3 (2-\log (5))^2+\frac {85}{8} x (2-\log (5))^3+\frac {5}{16} x^2 (2-\log (5))^3-\frac {5}{16} x (2-\log (5))^4+\frac {5}{8} (5+x)^4 \log (5)+\frac {5}{12} (5+x)^3 \log ^2(5)+\frac {5}{16} (5+x)^2 \log ^2(5) (8+\log (5))+\frac {5}{16} x \log ^2(5) (8+\log (5))^2-624 \log (2+2 x-\log (5))+\frac {1625}{2} (2-\log (5)) \log (2+2 x-\log (5))-\frac {725}{2} (2-\log (5))^2 \log (2+2 x-\log (5))+\frac {265}{4} (2-\log (5))^3 \log (2+2 x-\log (5))-\frac {85}{16} (2-\log (5))^4 \log (2+2 x-\log (5))+\frac {5}{32} (2-\log (5))^5 \log (2+2 x-\log (5))+\frac {5}{32} \log ^2(5) (8+\log (5))^3 \log (2+2 x-\log (5))\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.13, size = 128, normalized size = 4.92 \begin {gather*} -e^{x^2}-\frac {1}{2} x \left (300 x^2+40 x^3+2 x^4+5 x \left (200+4 \log ^2(5)-\log (5) \log (625)\right )+5 \left (250+4 \log ^3(5)+\log ^2(625)-\log ^2(5) (16+\log (625))\right )\right )-\frac {1}{4} \left (-4+20 \log ^4(5)+540 \log (625)-5 \log ^3(5) (24+\log (625))+10 \log ^2(5) (124+3 \log (625))-10 \log (5) (216+31 \log (625))\right ) \log (2+2 x-\log (5)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1248 + 3250*x + 2900*x^2 + 1060*x^3 + 170*x^4 + 10*x^5 + (-625 - 1000*x - 450*x^2 - 80*x^3 - 5*x^4)
*Log[5] + E^x^2*(4*x + 4*x^2 - 2*x*Log[5]))/(-2 - 2*x + Log[5]),x]

[Out]

-E^x^2 - (x*(300*x^2 + 40*x^3 + 2*x^4 + 5*x*(200 + 4*Log[5]^2 - Log[5]*Log[625]) + 5*(250 + 4*Log[5]^3 + Log[6
25]^2 - Log[5]^2*(16 + Log[625]))))/2 - ((-4 + 20*Log[5]^4 + 540*Log[625] - 5*Log[5]^3*(24 + Log[625]) + 10*Lo
g[5]^2*(124 + 3*Log[625]) - 10*Log[5]*(216 + 31*Log[625]))*Log[2 + 2*x - Log[5]])/4

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fricas [A]  time = 1.67, size = 40, normalized size = 1.54 \begin {gather*} -x^{5} - 20 \, x^{4} - 150 \, x^{3} - 500 \, x^{2} - 625 \, x - e^{\left (x^{2}\right )} + \log \left (2 \, x - \log \relax (5) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(5)+4*x^2+4*x)*exp(x^2)+(-5*x^4-80*x^3-450*x^2-1000*x-625)*log(5)+10*x^5+170*x^4+1060*x^3+
2900*x^2+3250*x+1248)/(log(5)-2*x-2),x, algorithm="fricas")

[Out]

-x^5 - 20*x^4 - 150*x^3 - 500*x^2 - 625*x - e^(x^2) + log(2*x - log(5) + 2)

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giac [A]  time = 0.16, size = 40, normalized size = 1.54 \begin {gather*} -x^{5} - 20 \, x^{4} - 150 \, x^{3} - 500 \, x^{2} - 625 \, x - e^{\left (x^{2}\right )} + \log \left (2 \, x - \log \relax (5) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(5)+4*x^2+4*x)*exp(x^2)+(-5*x^4-80*x^3-450*x^2-1000*x-625)*log(5)+10*x^5+170*x^4+1060*x^3+
2900*x^2+3250*x+1248)/(log(5)-2*x-2),x, algorithm="giac")

[Out]

-x^5 - 20*x^4 - 150*x^3 - 500*x^2 - 625*x - e^(x^2) + log(2*x - log(5) + 2)

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maple [A]  time = 0.41, size = 39, normalized size = 1.50

method	result	size
norman	\(-625 x -500 x^{2}-150 x^{3}-20 x^{4}-x^{5}-{\mathrm e}^{x^{2}}+\ln \left (\ln \relax (5)-2 x -2\right )\)	\(39\)
risch	\(-x^{5}-20 x^{4}-150 x^{3}-500 x^{2}-625 x +\ln \left (2-\ln \relax (5)+2 x \right )-{\mathrm e}^{x^{2}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x*ln(5)+4*x^2+4*x)*exp(x^2)+(-5*x^4-80*x^3-450*x^2-1000*x-625)*ln(5)+10*x^5+170*x^4+1060*x^3+2900*x^2
+3250*x+1248)/(ln(5)-2*x-2),x,method=_RETURNVERBOSE)

[Out]

-625*x-500*x^2-150*x^3-20*x^4-x^5-exp(x^2)+ln(ln(5)-2*x-2)

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maxima [B]  time = 0.50, size = 546, normalized size = 21.00 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(5)+4*x^2+4*x)*exp(x^2)+(-5*x^4-80*x^3-450*x^2-1000*x-625)*log(5)+10*x^5+170*x^4+1060*x^3+
2900*x^2+3250*x+1248)/(log(5)-2*x-2),x, algorithm="maxima")

[Out]

-x^5 - 5/8*x^4*(log(5) - 2) - 5/12*(log(5)^2 - 4*log(5) + 4)*x^3 - 85/4*x^4 - 85/6*x^3*(log(5) - 2) - 5/16*(lo
g(5)^3 - 6*log(5)^2 + 12*log(5) - 8)*x^2 - 85/8*(log(5)^2 - 4*log(5) + 4)*x^2 - 530/3*x^3 - 265/2*x^2*(log(5)
- 2) - 5/16*(log(5)^4 - 8*log(5)^3 + 24*log(5)^2 - 32*log(5) + 16)*x - 85/8*(log(5)^3 - 6*log(5)^2 + 12*log(5)
- 8)*x - 265/2*(log(5)^2 - 4*log(5) + 4)*x - 725*x^2 - 725*x*(log(5) - 2) + 5/96*(12*x^4 + 8*x^3*(log(5) - 2)
+ 6*(log(5)^2 - 4*log(5) + 4)*x^2 + 6*(log(5)^3 - 6*log(5)^2 + 12*log(5) - 8)*x + 3*(log(5)^4 - 8*log(5)^3 +
24*log(5)^2 - 32*log(5) + 16)*log(2*x - log(5) + 2))*log(5) + 5/3*(8*x^3 + 6*x^2*(log(5) - 2) + 6*(log(5)^2 -
4*log(5) + 4)*x + 3*(log(5)^3 - 6*log(5)^2 + 12*log(5) - 8)*log(2*x - log(5) + 2))*log(5) + 225/4*(2*x^2 + 2*x
*(log(5) - 2) + (log(5)^2 - 4*log(5) + 4)*log(2*x - log(5) + 2))*log(5) + 250*((log(5) - 2)*log(2*x - log(5) +
2) + 2*x)*log(5) - 5/32*(log(5)^5 - 10*log(5)^4 + 40*log(5)^3 - 80*log(5)^2 + 80*log(5) - 32)*log(2*x - log(5
) + 2) - 85/16*(log(5)^4 - 8*log(5)^3 + 24*log(5)^2 - 32*log(5) + 16)*log(2*x - log(5) + 2) - 265/4*(log(5)^3
- 6*log(5)^2 + 12*log(5) - 8)*log(2*x - log(5) + 2) - 725/2*(log(5)^2 - 4*log(5) + 4)*log(2*x - log(5) + 2) -
1625/2*(log(5) - 2)*log(2*x - log(5) + 2) + 625/2*log(5)*log(2*x - log(5) + 2) - 1625*x - e^(x^2) - 624*log(2*
x - log(5) + 2)

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mupad [B]  time = 0.12, size = 40, normalized size = 1.54 \begin {gather*} \ln \left (2\,x-\ln \relax (5)+2\right )-{\mathrm {e}}^{x^2}-625\,x-500\,x^2-150\,x^3-20\,x^4-x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3250*x + exp(x^2)*(4*x - 2*x*log(5) + 4*x^2) - log(5)*(1000*x + 450*x^2 + 80*x^3 + 5*x^4 + 625) + 2900*x
^2 + 1060*x^3 + 170*x^4 + 10*x^5 + 1248)/(2*x - log(5) + 2),x)

[Out]

log(2*x - log(5) + 2) - exp(x^2) - 625*x - 500*x^2 - 150*x^3 - 20*x^4 - x^5

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sympy [A]  time = 0.26, size = 36, normalized size = 1.38 \begin {gather*} - x^{5} - 20 x^{4} - 150 x^{3} - 500 x^{2} - 625 x - e^{x^{2}} + \log {\left (2 x - \log {\relax (5 )} + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*ln(5)+4*x**2+4*x)*exp(x**2)+(-5*x**4-80*x**3-450*x**2-1000*x-625)*ln(5)+10*x**5+170*x**4+1060
*x**3+2900*x**2+3250*x+1248)/(ln(5)-2*x-2),x)

[Out]

-x**5 - 20*x**4 - 150*x**3 - 500*x**2 - 625*x - exp(x**2) + log(2*x - log(5) + 2)

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