3.93.50 \(\int (1+4 e^{4 x}-1000 x+600 x^3-120 x^5+8 x^7+e^{3 x} (60-8 x-12 x^2)+e^{2 x} (300-120 x-120 x^2+24 x^3+12 x^4)+e^x (500-600 x-300 x^2+240 x^3+60 x^4-24 x^5-4 x^6)) \, dx\)

Optimal. Leaf size=14 \[ x+\left (5+e^x-x^2\right )^4 \]

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Rubi [B]  time = 0.42, antiderivative size = 98, normalized size of antiderivative = 7.00, number of steps used = 57, number of rules used = 3, integrand size = 103, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2194, 2196, 2176} \begin {gather*} x^8-4 e^x x^6-20 x^6+60 e^x x^4+6 e^{2 x} x^4+150 x^4-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2-500 x^2+x+500 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + 4*E^(4*x) - 1000*x + 600*x^3 - 120*x^5 + 8*x^7 + E^(3*x)*(60 - 8*x - 12*x^2) + E^(2*x)*(300 - 120*x -
120*x^2 + 24*x^3 + 12*x^4) + E^x*(500 - 600*x - 300*x^2 + 240*x^3 + 60*x^4 - 24*x^5 - 4*x^6),x]

[Out]

500*E^x + 150*E^(2*x) + 20*E^(3*x) + E^(4*x) + x - 500*x^2 - 300*E^x*x^2 - 60*E^(2*x)*x^2 - 4*E^(3*x)*x^2 + 15
0*x^4 + 60*E^x*x^4 + 6*E^(2*x)*x^4 - 20*x^6 - 4*E^x*x^6 + x^8

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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x-500 x^2+150 x^4-20 x^6+x^8+4 \int e^{4 x} \, dx+\int e^{3 x} \left (60-8 x-12 x^2\right ) \, dx+\int e^{2 x} \left (300-120 x-120 x^2+24 x^3+12 x^4\right ) \, dx+\int e^x \left (500-600 x-300 x^2+240 x^3+60 x^4-24 x^5-4 x^6\right ) \, dx\\ &=e^{4 x}+x-500 x^2+150 x^4-20 x^6+x^8+\int \left (60 e^{3 x}-8 e^{3 x} x-12 e^{3 x} x^2\right ) \, dx+\int \left (300 e^{2 x}-120 e^{2 x} x-120 e^{2 x} x^2+24 e^{2 x} x^3+12 e^{2 x} x^4\right ) \, dx+\int \left (500 e^x-600 e^x x-300 e^x x^2+240 e^x x^3+60 e^x x^4-24 e^x x^5-4 e^x x^6\right ) \, dx\\ &=e^{4 x}+x-500 x^2+150 x^4-20 x^6+x^8-4 \int e^x x^6 \, dx-8 \int e^{3 x} x \, dx-12 \int e^{3 x} x^2 \, dx+12 \int e^{2 x} x^4 \, dx+24 \int e^{2 x} x^3 \, dx-24 \int e^x x^5 \, dx+60 \int e^{3 x} \, dx+60 \int e^x x^4 \, dx-120 \int e^{2 x} x \, dx-120 \int e^{2 x} x^2 \, dx+240 \int e^x x^3 \, dx+300 \int e^{2 x} \, dx-300 \int e^x x^2 \, dx+500 \int e^x \, dx-600 \int e^x x \, dx\\ &=500 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x}+x-600 e^x x-60 e^{2 x} x-\frac {8}{3} e^{3 x} x-500 x^2-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2+240 e^x x^3+12 e^{2 x} x^3+150 x^4+60 e^x x^4+6 e^{2 x} x^4-24 e^x x^5-20 x^6-4 e^x x^6+x^8+\frac {8}{3} \int e^{3 x} \, dx+8 \int e^{3 x} x \, dx-24 \int e^{2 x} x^3 \, dx+24 \int e^x x^5 \, dx-36 \int e^{2 x} x^2 \, dx+60 \int e^{2 x} \, dx+120 \int e^{2 x} x \, dx+120 \int e^x x^4 \, dx-240 \int e^x x^3 \, dx+600 \int e^x \, dx+600 \int e^x x \, dx-720 \int e^x x^2 \, dx\\ &=1100 e^x+180 e^{2 x}+\frac {188 e^{3 x}}{9}+e^{4 x}+x-500 x^2-1020 e^x x^2-78 e^{2 x} x^2-4 e^{3 x} x^2+150 x^4+180 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8-\frac {8}{3} \int e^{3 x} \, dx+36 \int e^{2 x} x \, dx+36 \int e^{2 x} x^2 \, dx-60 \int e^{2 x} \, dx-120 \int e^x x^4 \, dx-480 \int e^x x^3 \, dx-600 \int e^x \, dx+720 \int e^x x^2 \, dx+1440 \int e^x x \, dx\\ &=500 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x}+x+1440 e^x x+18 e^{2 x} x-500 x^2-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2-480 e^x x^3+150 x^4+60 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8-18 \int e^{2 x} \, dx-36 \int e^{2 x} x \, dx+480 \int e^x x^3 \, dx-1440 \int e^x \, dx-1440 \int e^x x \, dx+1440 \int e^x x^2 \, dx\\ &=-940 e^x+141 e^{2 x}+20 e^{3 x}+e^{4 x}+x-500 x^2+1140 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2+150 x^4+60 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8+18 \int e^{2 x} \, dx+1440 \int e^x \, dx-1440 \int e^x x^2 \, dx-2880 \int e^x x \, dx\\ &=500 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x}+x-2880 e^x x-500 x^2-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2+150 x^4+60 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8+2880 \int e^x \, dx+2880 \int e^x x \, dx\\ &=3380 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x}+x-500 x^2-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2+150 x^4+60 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8-2880 \int e^x \, dx\\ &=500 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x}+x-500 x^2-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2+150 x^4+60 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 70, normalized size = 5.00 \begin {gather*} e^{4 x}+x-500 x^2+150 x^4-20 x^6+x^8-4 e^x \left (-5+x^2\right )^3-\frac {4}{3} e^{3 x} \left (-15+3 x^2\right )+6 e^{2 x} \left (25-10 x^2+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + 4*E^(4*x) - 1000*x + 600*x^3 - 120*x^5 + 8*x^7 + E^(3*x)*(60 - 8*x - 12*x^2) + E^(2*x)*(300 - 12
0*x - 120*x^2 + 24*x^3 + 12*x^4) + E^x*(500 - 600*x - 300*x^2 + 240*x^3 + 60*x^4 - 24*x^5 - 4*x^6),x]

[Out]

E^(4*x) + x - 500*x^2 + 150*x^4 - 20*x^6 + x^8 - 4*E^x*(-5 + x^2)^3 - (4*E^(3*x)*(-15 + 3*x^2))/3 + 6*E^(2*x)*
(25 - 10*x^2 + x^4)

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fricas [B]  time = 0.80, size = 70, normalized size = 5.00 \begin {gather*} x^{8} - 20 \, x^{6} + 150 \, x^{4} - 500 \, x^{2} - 4 \, {\left (x^{2} - 5\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 10 \, x^{2} + 25\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{6} - 15 \, x^{4} + 75 \, x^{2} - 125\right )} e^{x} + x + e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)^4+(-12*x^2-8*x+60)*exp(x)^3+(12*x^4+24*x^3-120*x^2-120*x+300)*exp(x)^2+(-4*x^6-24*x^5+60*x^
4+240*x^3-300*x^2-600*x+500)*exp(x)+8*x^7-120*x^5+600*x^3-1000*x+1,x, algorithm="fricas")

[Out]

x^8 - 20*x^6 + 150*x^4 - 500*x^2 - 4*(x^2 - 5)*e^(3*x) + 6*(x^4 - 10*x^2 + 25)*e^(2*x) - 4*(x^6 - 15*x^4 + 75*
x^2 - 125)*e^x + x + e^(4*x)

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giac [B]  time = 0.14, size = 70, normalized size = 5.00 \begin {gather*} x^{8} - 20 \, x^{6} + 150 \, x^{4} - 500 \, x^{2} - 4 \, {\left (x^{2} - 5\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 10 \, x^{2} + 25\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{6} - 15 \, x^{4} + 75 \, x^{2} - 125\right )} e^{x} + x + e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)^4+(-12*x^2-8*x+60)*exp(x)^3+(12*x^4+24*x^3-120*x^2-120*x+300)*exp(x)^2+(-4*x^6-24*x^5+60*x^
4+240*x^3-300*x^2-600*x+500)*exp(x)+8*x^7-120*x^5+600*x^3-1000*x+1,x, algorithm="giac")

[Out]

x^8 - 20*x^6 + 150*x^4 - 500*x^2 - 4*(x^2 - 5)*e^(3*x) + 6*(x^4 - 10*x^2 + 25)*e^(2*x) - 4*(x^6 - 15*x^4 + 75*
x^2 - 125)*e^x + x + e^(4*x)

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maple [B]  time = 0.05, size = 74, normalized size = 5.29

method	result	size
risch	\({\mathrm e}^{4 x}+\left (-4 x^{2}+20\right ) {\mathrm e}^{3 x}+\left (6 x^{4}-60 x^{2}+150\right ) {\mathrm e}^{2 x}+\left (-4 x^{6}+60 x^{4}-300 x^{2}+500\right ) {\mathrm e}^{x}+x^{8}-20 x^{6}+150 x^{4}-500 x^{2}+x\)	\(74\)
default	\(x +20 \,{\mathrm e}^{3 x}-4 x^{2} {\mathrm e}^{3 x}+6 \,{\mathrm e}^{2 x} x^{4}-60 \,{\mathrm e}^{2 x} x^{2}+150 \,{\mathrm e}^{2 x}+60 \,{\mathrm e}^{x} x^{4}-300 \,{\mathrm e}^{x} x^{2}-4 x^{6} {\mathrm e}^{x}+500 \,{\mathrm e}^{x}-500 x^{2}+150 x^{4}-20 x^{6}+x^{8}+{\mathrm e}^{4 x}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*exp(x)^4+(-12*x^2-8*x+60)*exp(x)^3+(12*x^4+24*x^3-120*x^2-120*x+300)*exp(x)^2+(-4*x^6-24*x^5+60*x^4+240*
x^3-300*x^2-600*x+500)*exp(x)+8*x^7-120*x^5+600*x^3-1000*x+1,x,method=_RETURNVERBOSE)

[Out]

exp(4*x)+(-4*x^2+20)*exp(3*x)+(6*x^4-60*x^2+150)*exp(2*x)+(-4*x^6+60*x^4-300*x^2+500)*exp(x)+x^8-20*x^6+150*x^
4-500*x^2+x

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maxima [B]  time = 0.35, size = 70, normalized size = 5.00 \begin {gather*} x^{8} - 20 \, x^{6} + 150 \, x^{4} - 500 \, x^{2} - 4 \, {\left (x^{2} - 5\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 10 \, x^{2} + 25\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{6} - 15 \, x^{4} + 75 \, x^{2} - 125\right )} e^{x} + x + e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)^4+(-12*x^2-8*x+60)*exp(x)^3+(12*x^4+24*x^3-120*x^2-120*x+300)*exp(x)^2+(-4*x^6-24*x^5+60*x^
4+240*x^3-300*x^2-600*x+500)*exp(x)+8*x^7-120*x^5+600*x^3-1000*x+1,x, algorithm="maxima")

[Out]

x^8 - 20*x^6 + 150*x^4 - 500*x^2 - 4*(x^2 - 5)*e^(3*x) + 6*(x^4 - 10*x^2 + 25)*e^(2*x) - 4*(x^6 - 15*x^4 + 75*
x^2 - 125)*e^x + x + e^(4*x)

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mupad [B]  time = 5.87, size = 88, normalized size = 6.29 \begin {gather*} x+150\,{\mathrm {e}}^{2\,x}+20\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+500\,{\mathrm {e}}^x-300\,x^2\,{\mathrm {e}}^x+60\,x^4\,{\mathrm {e}}^x-4\,x^6\,{\mathrm {e}}^x-60\,x^2\,{\mathrm {e}}^{2\,x}-4\,x^2\,{\mathrm {e}}^{3\,x}+6\,x^4\,{\mathrm {e}}^{2\,x}-500\,x^2+150\,x^4-20\,x^6+x^8 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*exp(4*x) - 1000*x - exp(3*x)*(8*x + 12*x^2 - 60) + exp(2*x)*(24*x^3 - 120*x^2 - 120*x + 12*x^4 + 300) -
exp(x)*(600*x + 300*x^2 - 240*x^3 - 60*x^4 + 24*x^5 + 4*x^6 - 500) + 600*x^3 - 120*x^5 + 8*x^7 + 1,x)

[Out]

x + 150*exp(2*x) + 20*exp(3*x) + exp(4*x) + 500*exp(x) - 300*x^2*exp(x) + 60*x^4*exp(x) - 4*x^6*exp(x) - 60*x^
2*exp(2*x) - 4*x^2*exp(3*x) + 6*x^4*exp(2*x) - 500*x^2 + 150*x^4 - 20*x^6 + x^8

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sympy [B]  time = 0.18, size = 73, normalized size = 5.21 \begin {gather*} x^{8} - 20 x^{6} + 150 x^{4} - 500 x^{2} + x + \left (20 - 4 x^{2}\right ) e^{3 x} + \left (6 x^{4} - 60 x^{2} + 150\right ) e^{2 x} + \left (- 4 x^{6} + 60 x^{4} - 300 x^{2} + 500\right ) e^{x} + e^{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)**4+(-12*x**2-8*x+60)*exp(x)**3+(12*x**4+24*x**3-120*x**2-120*x+300)*exp(x)**2+(-4*x**6-24*x
**5+60*x**4+240*x**3-300*x**2-600*x+500)*exp(x)+8*x**7-120*x**5+600*x**3-1000*x+1,x)

[Out]

x**8 - 20*x**6 + 150*x**4 - 500*x**2 + x + (20 - 4*x**2)*exp(3*x) + (6*x**4 - 60*x**2 + 150)*exp(2*x) + (-4*x*
*6 + 60*x**4 - 300*x**2 + 500)*exp(x) + exp(4*x)

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