3.46.69 \(\int \frac {-250+e^{4 x} (9000+49500 x+60750 x^2+28125 x^3+4500 x^4+(1200+6600 x+8100 x^2+3750 x^3+600 x^4) \log (4)+(40+220 x+270 x^2+125 x^3+20 x^4) \log ^2(4))}{1800+2700 x+1350 x^2+225 x^3+(240+360 x+180 x^2+30 x^3) \log (4)+(8+12 x+6 x^2+x^3) \log ^2(4)} \, dx\)

Optimal. Leaf size=28 \[ 5 \left (-5 e^2+e^{4 x} x+\frac {25}{(2+x)^2 (15+\log (4))^2}\right ) \]

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Rubi [A]  time = 0.19, antiderivative size = 37, normalized size of antiderivative = 1.32, number of steps used = 4, number of rules used = 3, integrand size = 130, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6688, 2176, 2194} \begin {gather*} \frac {5}{4} e^{4 x} (4 x+1)-\frac {5 e^{4 x}}{4}+\frac {125}{(x+2)^2 (15+\log (4))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-250 + E^(4*x)*(9000 + 49500*x + 60750*x^2 + 28125*x^3 + 4500*x^4 + (1200 + 6600*x + 8100*x^2 + 3750*x^3
+ 600*x^4)*Log[4] + (40 + 220*x + 270*x^2 + 125*x^3 + 20*x^4)*Log[4]^2))/(1800 + 2700*x + 1350*x^2 + 225*x^3 +
(240 + 360*x + 180*x^2 + 30*x^3)*Log[4] + (8 + 12*x + 6*x^2 + x^3)*Log[4]^2),x]

[Out]

(-5*E^(4*x))/4 + (5*E^(4*x)*(1 + 4*x))/4 + 125/((2 + x)^2*(15 + Log[4])^2)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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 (5 e^{4 x} (1+4 x)-\frac {250}{(2+x)^3 (15+\log (4))^2}\right ) \, dx\\ &=\frac {125}{(2+x)^2 (15+\log (4))^2}+5 \int e^{4 x} (1+4 x) \, dx\\ &=\frac {5}{4} e^{4 x} (1+4 x)+\frac {125}{(2+x)^2 (15+\log (4))^2}-5 \int e^{4 x} \, dx\\ &=-\frac {5 e^{4 x}}{4}+\frac {5}{4} e^{4 x} (1+4 x)+\frac {125}{(2+x)^2 (15+\log (4))^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 22, normalized size = 0.79 \begin {gather*} 5 e^{4 x} x+\frac {125}{(2+x)^2 (15+\log (4))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-250 + E^(4*x)*(9000 + 49500*x + 60750*x^2 + 28125*x^3 + 4500*x^4 + (1200 + 6600*x + 8100*x^2 + 375
0*x^3 + 600*x^4)*Log[4] + (40 + 220*x + 270*x^2 + 125*x^3 + 20*x^4)*Log[4]^2))/(1800 + 2700*x + 1350*x^2 + 225
*x^3 + (240 + 360*x + 180*x^2 + 30*x^3)*Log[4] + (8 + 12*x + 6*x^2 + x^3)*Log[4]^2),x]

[Out]

5*E^(4*x)*x + 125/((2 + x)^2*(15 + Log[4])^2)

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fricas [B]  time = 0.70, size = 95, normalized size = 3.39 \begin {gather*} \frac {5 \, {\left ({\left (225 \, x^{3} + 4 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \relax (2)^{2} + 900 \, x^{2} + 60 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \relax (2) + 900 \, x\right )} e^{\left (4 \, x\right )} + 25\right )}}{4 \, {\left (x^{2} + 4 \, x + 4\right )} \log \relax (2)^{2} + 225 \, x^{2} + 60 \, {\left (x^{2} + 4 \, x + 4\right )} \log \relax (2) + 900 \, x + 900} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(20*x^4+125*x^3+270*x^2+220*x+40)*log(2)^2+2*(600*x^4+3750*x^3+8100*x^2+6600*x+1200)*log(2)+4500
*x^4+28125*x^3+60750*x^2+49500*x+9000)*exp(4*x)-250)/(4*(x^3+6*x^2+12*x+8)*log(2)^2+2*(30*x^3+180*x^2+360*x+24
0)*log(2)+225*x^3+1350*x^2+2700*x+1800),x, algorithm="fricas")

[Out]

5*((225*x^3 + 4*(x^3 + 4*x^2 + 4*x)*log(2)^2 + 900*x^2 + 60*(x^3 + 4*x^2 + 4*x)*log(2) + 900*x)*e^(4*x) + 25)/
(4*(x^2 + 4*x + 4)*log(2)^2 + 225*x^2 + 60*(x^2 + 4*x + 4)*log(2) + 900*x + 900)

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giac [B]  time = 0.14, size = 147, normalized size = 5.25 \begin {gather*} \frac {5 \, {\left (4 \, x^{3} e^{\left (4 \, x\right )} \log \relax (2)^{2} + 60 \, x^{3} e^{\left (4 \, x\right )} \log \relax (2) + 16 \, x^{2} e^{\left (4 \, x\right )} \log \relax (2)^{2} + 225 \, x^{3} e^{\left (4 \, x\right )} + 240 \, x^{2} e^{\left (4 \, x\right )} \log \relax (2) + 16 \, x e^{\left (4 \, x\right )} \log \relax (2)^{2} + 900 \, x^{2} e^{\left (4 \, x\right )} + 240 \, x e^{\left (4 \, x\right )} \log \relax (2) + 900 \, x e^{\left (4 \, x\right )} + 25\right )}}{4 \, x^{2} \log \relax (2)^{2} + 60 \, x^{2} \log \relax (2) + 16 \, x \log \relax (2)^{2} + 225 \, x^{2} + 240 \, x \log \relax (2) + 16 \, \log \relax (2)^{2} + 900 \, x + 240 \, \log \relax (2) + 900} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(20*x^4+125*x^3+270*x^2+220*x+40)*log(2)^2+2*(600*x^4+3750*x^3+8100*x^2+6600*x+1200)*log(2)+4500
*x^4+28125*x^3+60750*x^2+49500*x+9000)*exp(4*x)-250)/(4*(x^3+6*x^2+12*x+8)*log(2)^2+2*(30*x^3+180*x^2+360*x+24
0)*log(2)+225*x^3+1350*x^2+2700*x+1800),x, algorithm="giac")

[Out]

5*(4*x^3*e^(4*x)*log(2)^2 + 60*x^3*e^(4*x)*log(2) + 16*x^2*e^(4*x)*log(2)^2 + 225*x^3*e^(4*x) + 240*x^2*e^(4*x
)*log(2) + 16*x*e^(4*x)*log(2)^2 + 900*x^2*e^(4*x) + 240*x*e^(4*x)*log(2) + 900*x*e^(4*x) + 25)/(4*x^2*log(2)^
2 + 60*x^2*log(2) + 16*x*log(2)^2 + 225*x^2 + 240*x*log(2) + 16*log(2)^2 + 900*x + 240*log(2) + 900)

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maple [B]  time = 0.15, size = 60, normalized size = 2.14

method	result	size
risch	\(\frac {125}{4 \left (x^{2} \ln \relax (2)^{2}+4 x \ln \relax (2)^{2}+15 x^{2} \ln \relax (2)+4 \ln \relax (2)^{2}+60 x \ln \relax (2)+\frac {225 x^{2}}{4}+60 \ln \relax (2)+225 x +225\right )}+5 x \,{\mathrm e}^{4 x}\)	\(60\)
norman	\(\frac {\left (10 \ln \relax (2)+75\right ) x^{3} {\mathrm e}^{4 x}+\left (40 \ln \relax (2)+300\right ) x \,{\mathrm e}^{4 x}+\left (40 \ln \relax (2)+300\right ) x^{2} {\mathrm e}^{4 x}+\frac {125}{2 \ln \relax (2)+15}}{\left (2+x \right )^{2} \left (2 \ln \relax (2)+15\right )}\)	\(66\)
derivativedivides	\(\frac {2000}{\left (4 \ln \relax (2)^{2}+60 \ln \relax (2)+225\right ) \left (4 x +8\right )^{2}}+\frac {1125 \,{\mathrm e}^{4 x} x}{4 \ln \relax (2)^{2}+60 \ln \relax (2)+225}+\frac {300 \ln \relax (2) {\mathrm e}^{4 x} x}{4 \ln \relax (2)^{2}+60 \ln \relax (2)+225}+\frac {20 \ln \relax (2)^{2} {\mathrm e}^{4 x} x}{4 \ln \relax (2)^{2}+60 \ln \relax (2)+225}\)	\(94\)
default	\(\frac {2000}{\left (4 \ln \relax (2)^{2}+60 \ln \relax (2)+225\right ) \left (4 x +8\right )^{2}}+\frac {1125 \,{\mathrm e}^{4 x} x}{4 \ln \relax (2)^{2}+60 \ln \relax (2)+225}+\frac {300 \ln \relax (2) {\mathrm e}^{4 x} x}{4 \ln \relax (2)^{2}+60 \ln \relax (2)+225}+\frac {20 \ln \relax (2)^{2} {\mathrm e}^{4 x} x}{4 \ln \relax (2)^{2}+60 \ln \relax (2)+225}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*(20*x^4+125*x^3+270*x^2+220*x+40)*ln(2)^2+2*(600*x^4+3750*x^3+8100*x^2+6600*x+1200)*ln(2)+4500*x^4+281
25*x^3+60750*x^2+49500*x+9000)*exp(4*x)-250)/(4*(x^3+6*x^2+12*x+8)*ln(2)^2+2*(30*x^3+180*x^2+360*x+240)*ln(2)+
225*x^3+1350*x^2+2700*x+1800),x,method=_RETURNVERBOSE)

[Out]

125/4/(x^2*ln(2)^2+4*x*ln(2)^2+15*x^2*ln(2)+4*ln(2)^2+60*x*ln(2)+225/4*x^2+60*ln(2)+225*x+225)+5*x*exp(4*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 5 \, x e^{\left (4 \, x\right )} - \frac {160 \, e^{\left (-8\right )} E_{3}\left (-4 \, x - 8\right ) \log \relax (2)^{2}}{{\left (x + 2\right )}^{2} {\left (2 \, \log \relax (2) + 15\right )}^{2}} + \frac {125}{{\left (4 \, \log \relax (2)^{2} + 60 \, \log \relax (2) + 225\right )} x^{2} + 4 \, {\left (4 \, \log \relax (2)^{2} + 60 \, \log \relax (2) + 225\right )} x + 16 \, \log \relax (2)^{2} + 240 \, \log \relax (2) + 900} - \frac {2400 \, e^{\left (-8\right )} E_{3}\left (-4 \, x - 8\right ) \log \relax (2)}{{\left (x + 2\right )}^{2} {\left (2 \, \log \relax (2) + 15\right )}^{2}} - \frac {9000 \, e^{\left (-8\right )} E_{3}\left (-4 \, x - 8\right )}{{\left (x + 2\right )}^{2} {\left (2 \, \log \relax (2) + 15\right )}^{2}} - 40 \, \int \frac {e^{\left (4 \, x\right )}}{x^{3} + 6 \, x^{2} + 12 \, x + 8}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(20*x^4+125*x^3+270*x^2+220*x+40)*log(2)^2+2*(600*x^4+3750*x^3+8100*x^2+6600*x+1200)*log(2)+4500
*x^4+28125*x^3+60750*x^2+49500*x+9000)*exp(4*x)-250)/(4*(x^3+6*x^2+12*x+8)*log(2)^2+2*(30*x^3+180*x^2+360*x+24
0)*log(2)+225*x^3+1350*x^2+2700*x+1800),x, algorithm="maxima")

[Out]

5*x*e^(4*x) - 160*e^(-8)*exp_integral_e(3, -4*x - 8)*log(2)^2/((x + 2)^2*(2*log(2) + 15)^2) + 125/((4*log(2)^2
+ 60*log(2) + 225)*x^2 + 4*(4*log(2)^2 + 60*log(2) + 225)*x + 16*log(2)^2 + 240*log(2) + 900) - 2400*e^(-8)*e
xp_integral_e(3, -4*x - 8)*log(2)/((x + 2)^2*(2*log(2) + 15)^2) - 9000*e^(-8)*exp_integral_e(3, -4*x - 8)/((x
+ 2)^2*(2*log(2) + 15)^2) - 40*integrate(e^(4*x)/(x^3 + 6*x^2 + 12*x + 8), x)

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mupad [B]  time = 3.48, size = 52, normalized size = 1.86 \begin {gather*} \frac {125}{\left (30\,\ln \relax (4)+{\ln \relax (4)}^2+225\right )\,x^2+\left (120\,\ln \relax (4)+4\,{\ln \relax (4)}^2+900\right )\,x+120\,\ln \relax (4)+4\,{\ln \relax (4)}^2+900}+5\,x\,{\mathrm {e}}^{4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x)*(49500*x + 4*log(2)^2*(220*x + 270*x^2 + 125*x^3 + 20*x^4 + 40) + 2*log(2)*(6600*x + 8100*x^2 +
3750*x^3 + 600*x^4 + 1200) + 60750*x^2 + 28125*x^3 + 4500*x^4 + 9000) - 250)/(2700*x + 2*log(2)*(360*x + 180*x
^2 + 30*x^3 + 240) + 4*log(2)^2*(12*x + 6*x^2 + x^3 + 8) + 1350*x^2 + 225*x^3 + 1800),x)

[Out]

125/(120*log(4) + x*(120*log(4) + 4*log(4)^2 + 900) + x^2*(30*log(4) + log(4)^2 + 225) + 4*log(4)^2 + 900) + 5
*x*exp(4*x)

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sympy [A]  time = 0.43, size = 54, normalized size = 1.93 \begin {gather*} 5 x e^{4 x} + \frac {250}{x^{2} \left (8 \log {\relax (2 )}^{2} + 120 \log {\relax (2 )} + 450\right ) + x \left (32 \log {\relax (2 )}^{2} + 480 \log {\relax (2 )} + 1800\right ) + 32 \log {\relax (2 )}^{2} + 480 \log {\relax (2 )} + 1800} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(20*x**4+125*x**3+270*x**2+220*x+40)*ln(2)**2+2*(600*x**4+3750*x**3+8100*x**2+6600*x+1200)*ln(2)
+4500*x**4+28125*x**3+60750*x**2+49500*x+9000)*exp(4*x)-250)/(4*(x**3+6*x**2+12*x+8)*ln(2)**2+2*(30*x**3+180*x
**2+360*x+240)*ln(2)+225*x**3+1350*x**2+2700*x+1800),x)

[Out]

5*x*exp(4*x) + 250/(x**2*(8*log(2)**2 + 120*log(2) + 450) + x*(32*log(2)**2 + 480*log(2) + 1800) + 32*log(2)**
2 + 480*log(2) + 1800)

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