3.43.84 \(\int e^{-20 x} (-7812500-2812500 e^{5 x}+e^{20 x} (-21+2 x)+e^{15 x} (-15300+1500 x)+e^{10 x} (-351250+12500 x)) \, dx\)

Optimal. Leaf size=23 \[ -2-x+\left (-1-\left (3+25 e^{-5 x}\right )^2+x\right )^2 \]

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Rubi [B]  time = 0.26, antiderivative size = 59, normalized size of antiderivative = 2.57, number of steps used = 8, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6742, 2194, 2176} \begin {gather*} x^2-21 x+390625 e^{-20 x}+187500 e^{-15 x}-125 e^{-10 x}-60 e^{-5 x}+125 e^{-10 x} (281-10 x)+60 e^{-5 x} (51-5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-7812500 - 2812500*E^(5*x) + E^(20*x)*(-21 + 2*x) + E^(15*x)*(-15300 + 1500*x) + E^(10*x)*(-351250 + 1250
0*x))/E^(20*x),x]

[Out]

390625/E^(20*x) + 187500/E^(15*x) - 125/E^(10*x) - 60/E^(5*x) + (125*(281 - 10*x))/E^(10*x) + (60*(51 - 5*x))/
E^(5*x) - 21*x + x^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 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 (-21-7812500 e^{-20 x}-2812500 e^{-15 x}+2 x+300 e^{-5 x} (-51+5 x)+1250 e^{-10 x} (-281+10 x)\right ) \, dx\\ &=-21 x+x^2+300 \int e^{-5 x} (-51+5 x) \, dx+1250 \int e^{-10 x} (-281+10 x) \, dx-2812500 \int e^{-15 x} \, dx-7812500 \int e^{-20 x} \, dx\\ &=390625 e^{-20 x}+187500 e^{-15 x}+125 e^{-10 x} (281-10 x)+60 e^{-5 x} (51-5 x)-21 x+x^2+300 \int e^{-5 x} \, dx+1250 \int e^{-10 x} \, dx\\ &=390625 e^{-20 x}+187500 e^{-15 x}-125 e^{-10 x}-60 e^{-5 x}+125 e^{-10 x} (281-10 x)+60 e^{-5 x} (51-5 x)-21 x+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 45, normalized size = 1.96 \begin {gather*} 390625 e^{-20 x}+187500 e^{-15 x}+300 e^{-5 x} (10-x)+1250 e^{-10 x} (28-x)-21 x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7812500 - 2812500*E^(5*x) + E^(20*x)*(-21 + 2*x) + E^(15*x)*(-15300 + 1500*x) + E^(10*x)*(-351250
+ 12500*x))/E^(20*x),x]

[Out]

390625/E^(20*x) + 187500/E^(15*x) + (300*(10 - x))/E^(5*x) + (1250*(28 - x))/E^(10*x) - 21*x + x^2

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fricas [A]  time = 0.77, size = 43, normalized size = 1.87 \begin {gather*} {\left ({\left (x^{2} - 21 \, x\right )} e^{\left (20 \, x\right )} - 300 \, {\left (x - 10\right )} e^{\left (15 \, x\right )} - 1250 \, {\left (x - 28\right )} e^{\left (10 \, x\right )} + 187500 \, e^{\left (5 \, x\right )} + 390625\right )} e^{\left (-20 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-21)*exp(5*x)^4+(1500*x-15300)*exp(5*x)^3+(12500*x-351250)*exp(5*x)^2-2812500*exp(5*x)-7812500)
/exp(5*x)^4,x, algorithm="fricas")

[Out]

((x^2 - 21*x)*e^(20*x) - 300*(x - 10)*e^(15*x) - 1250*(x - 28)*e^(10*x) + 187500*e^(5*x) + 390625)*e^(-20*x)

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giac [B]  time = 0.13, size = 45, normalized size = 1.96 \begin {gather*} x^{2} - 300 \, x e^{\left (-5 \, x\right )} - 1250 \, x e^{\left (-10 \, x\right )} - 21 \, x + 3000 \, e^{\left (-5 \, x\right )} + 35000 \, e^{\left (-10 \, x\right )} + 187500 \, e^{\left (-15 \, x\right )} + 390625 \, e^{\left (-20 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-21)*exp(5*x)^4+(1500*x-15300)*exp(5*x)^3+(12500*x-351250)*exp(5*x)^2-2812500*exp(5*x)-7812500)
/exp(5*x)^4,x, algorithm="giac")

[Out]

x^2 - 300*x*e^(-5*x) - 1250*x*e^(-10*x) - 21*x + 3000*e^(-5*x) + 35000*e^(-10*x) + 187500*e^(-15*x) + 390625*e
^(-20*x)

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maple [A]  time = 0.04, size = 40, normalized size = 1.74

method	result	size
risch	\(x^{2}-21 x +\left (3000-300 x \right ) {\mathrm e}^{-5 x}+\left (35000-1250 x \right ) {\mathrm e}^{-10 x}+187500 \,{\mathrm e}^{-15 x}+390625 \,{\mathrm e}^{-20 x}\)	\(40\)
derivativedivides	\(-21 x +x^{2}+390625 \,{\mathrm e}^{-20 x}+187500 \,{\mathrm e}^{-15 x}+35000 \,{\mathrm e}^{-10 x}+3000 \,{\mathrm e}^{-5 x}-1250 \,{\mathrm e}^{-10 x} x -300 x \,{\mathrm e}^{-5 x}\)	\(58\)
default	\(-21 x +x^{2}+390625 \,{\mathrm e}^{-20 x}+187500 \,{\mathrm e}^{-15 x}+35000 \,{\mathrm e}^{-10 x}+3000 \,{\mathrm e}^{-5 x}-1250 \,{\mathrm e}^{-10 x} x -300 x \,{\mathrm e}^{-5 x}\)	\(58\)
norman	\(\left (390625+x^{2} {\mathrm e}^{20 x}+35000 \,{\mathrm e}^{10 x}+3000 \,{\mathrm e}^{15 x}-1250 \,{\mathrm e}^{10 x} x -300 \,{\mathrm e}^{15 x} x -21 x \,{\mathrm e}^{20 x}+187500 \,{\mathrm e}^{5 x}\right ) {\mathrm e}^{-20 x}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x-21)*exp(5*x)^4+(1500*x-15300)*exp(5*x)^3+(12500*x-351250)*exp(5*x)^2-2812500*exp(5*x)-7812500)/exp(5
*x)^4,x,method=_RETURNVERBOSE)

[Out]

x^2-21*x+(3000-300*x)*exp(-5*x)+(35000-1250*x)*exp(-10*x)+187500*exp(-15*x)+390625*exp(-20*x)

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maxima [B]  time = 0.35, size = 53, normalized size = 2.30 \begin {gather*} x^{2} - 60 \, {\left (5 \, x + 1\right )} e^{\left (-5 \, x\right )} - 125 \, {\left (10 \, x + 1\right )} e^{\left (-10 \, x\right )} - 21 \, x + 3060 \, e^{\left (-5 \, x\right )} + 35125 \, e^{\left (-10 \, x\right )} + 187500 \, e^{\left (-15 \, x\right )} + 390625 \, e^{\left (-20 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-21)*exp(5*x)^4+(1500*x-15300)*exp(5*x)^3+(12500*x-351250)*exp(5*x)^2-2812500*exp(5*x)-7812500)
/exp(5*x)^4,x, algorithm="maxima")

[Out]

x^2 - 60*(5*x + 1)*e^(-5*x) - 125*(10*x + 1)*e^(-10*x) - 21*x + 3060*e^(-5*x) + 35125*e^(-10*x) + 187500*e^(-1
5*x) + 390625*e^(-20*x)

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mupad [B]  time = 3.18, size = 41, normalized size = 1.78 \begin {gather*} 187500\,{\mathrm {e}}^{-15\,x}-21\,x+390625\,{\mathrm {e}}^{-20\,x}-{\mathrm {e}}^{-5\,x}\,\left (300\,x-3000\right )-{\mathrm {e}}^{-10\,x}\,\left (1250\,x-35000\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-20*x)*(exp(20*x)*(2*x - 21) - 2812500*exp(5*x) + exp(15*x)*(1500*x - 15300) + exp(10*x)*(12500*x - 35
1250) - 7812500),x)

[Out]

187500*exp(-15*x) - 21*x + 390625*exp(-20*x) - exp(-5*x)*(300*x - 3000) - exp(-10*x)*(1250*x - 35000) + x^2

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sympy [B]  time = 0.15, size = 39, normalized size = 1.70 \begin {gather*} x^{2} - 21 x + \left (3000 - 300 x\right ) e^{- 5 x} + \left (35000 - 1250 x\right ) e^{- 10 x} + 187500 e^{- 15 x} + 390625 e^{- 20 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-21)*exp(5*x)**4+(1500*x-15300)*exp(5*x)**3+(12500*x-351250)*exp(5*x)**2-2812500*exp(5*x)-78125
00)/exp(5*x)**4,x)

[Out]

x**2 - 21*x + (3000 - 300*x)*exp(-5*x) + (35000 - 1250*x)*exp(-10*x) + 187500*exp(-15*x) + 390625*exp(-20*x)

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