3.63.31 \(\int (5000 x-12000 x^3+8400 x^5-2400 x^7+250 x^9+e^{2 x} (50 x+50 x^2)+e^x (1000 x+500 x^2-1200 x^3-300 x^4+300 x^5+50 x^6)) \, dx\)

Optimal. Leaf size=19 \[ 25 \left (x+x \left (e^x+\left (-3+x^2\right )^2\right )\right )^2 \]

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Rubi [B]  time = 0.30, antiderivative size = 60, normalized size of antiderivative = 3.16, number of steps used = 38, number of rules used = 4, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1593, 2196, 2176, 2194} \begin {gather*} 25 x^{10}-300 x^8+50 e^x x^6+1400 x^6-300 e^x x^4-3000 x^4+500 e^x x^2+25 e^{2 x} x^2+2500 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[5000*x - 12000*x^3 + 8400*x^5 - 2400*x^7 + 250*x^9 + E^(2*x)*(50*x + 50*x^2) + E^x*(1000*x + 500*x^2 - 120
0*x^3 - 300*x^4 + 300*x^5 + 50*x^6),x]

[Out]

2500*x^2 + 500*E^x*x^2 + 25*E^(2*x)*x^2 - 3000*x^4 - 300*E^x*x^4 + 1400*x^6 + 50*E^x*x^6 - 300*x^8 + 25*x^10

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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} &=2500 x^2-3000 x^4+1400 x^6-300 x^8+25 x^{10}+\int e^{2 x} \left (50 x+50 x^2\right ) \, dx+\int e^x \left (1000 x+500 x^2-1200 x^3-300 x^4+300 x^5+50 x^6\right ) \, dx\\ &=2500 x^2-3000 x^4+1400 x^6-300 x^8+25 x^{10}+\int e^{2 x} x (50+50 x) \, dx+\int \left (1000 e^x x+500 e^x x^2-1200 e^x x^3-300 e^x x^4+300 e^x x^5+50 e^x x^6\right ) \, dx\\ &=2500 x^2-3000 x^4+1400 x^6-300 x^8+25 x^{10}+50 \int e^x x^6 \, dx-300 \int e^x x^4 \, dx+300 \int e^x x^5 \, dx+500 \int e^x x^2 \, dx+1000 \int e^x x \, dx-1200 \int e^x x^3 \, dx+\int \left (50 e^{2 x} x+50 e^{2 x} x^2\right ) \, dx\\ &=1000 e^x x+2500 x^2+500 e^x x^2-1200 e^x x^3-3000 x^4-300 e^x x^4+300 e^x x^5+1400 x^6+50 e^x x^6-300 x^8+25 x^{10}+50 \int e^{2 x} x \, dx+50 \int e^{2 x} x^2 \, dx-300 \int e^x x^5 \, dx-1000 \int e^x \, dx-1000 \int e^x x \, dx+1200 \int e^x x^3 \, dx-1500 \int e^x x^4 \, dx+3600 \int e^x x^2 \, dx\\ &=-1000 e^x+25 e^{2 x} x+2500 x^2+4100 e^x x^2+25 e^{2 x} x^2-3000 x^4-1800 e^x x^4+1400 x^6+50 e^x x^6-300 x^8+25 x^{10}-25 \int e^{2 x} \, dx-50 \int e^{2 x} x \, dx+1000 \int e^x \, dx+1500 \int e^x x^4 \, dx-3600 \int e^x x^2 \, dx+6000 \int e^x x^3 \, dx-7200 \int e^x x \, dx\\ &=-\frac {25 e^{2 x}}{2}-7200 e^x x+2500 x^2+500 e^x x^2+25 e^{2 x} x^2+6000 e^x x^3-3000 x^4-300 e^x x^4+1400 x^6+50 e^x x^6-300 x^8+25 x^{10}+25 \int e^{2 x} \, dx-6000 \int e^x x^3 \, dx+7200 \int e^x \, dx+7200 \int e^x x \, dx-18000 \int e^x x^2 \, dx\\ &=7200 e^x+2500 x^2-17500 e^x x^2+25 e^{2 x} x^2-3000 x^4-300 e^x x^4+1400 x^6+50 e^x x^6-300 x^8+25 x^{10}-7200 \int e^x \, dx+18000 \int e^x x^2 \, dx+36000 \int e^x x \, dx\\ &=36000 e^x x+2500 x^2+500 e^x x^2+25 e^{2 x} x^2-3000 x^4-300 e^x x^4+1400 x^6+50 e^x x^6-300 x^8+25 x^{10}-36000 \int e^x \, dx-36000 \int e^x x \, dx\\ &=-36000 e^x+2500 x^2+500 e^x x^2+25 e^{2 x} x^2-3000 x^4-300 e^x x^4+1400 x^6+50 e^x x^6-300 x^8+25 x^{10}+36000 \int e^x \, dx\\ &=2500 x^2+500 e^x x^2+25 e^{2 x} x^2-3000 x^4-300 e^x x^4+1400 x^6+50 e^x x^6-300 x^8+25 x^{10}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 20, normalized size = 1.05 \begin {gather*} 25 x^2 \left (10+e^x-6 x^2+x^4\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[5000*x - 12000*x^3 + 8400*x^5 - 2400*x^7 + 250*x^9 + E^(2*x)*(50*x + 50*x^2) + E^x*(1000*x + 500*x^2
- 1200*x^3 - 300*x^4 + 300*x^5 + 50*x^6),x]

[Out]

25*x^2*(10 + E^x - 6*x^2 + x^4)^2

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fricas [B]  time = 0.59, size = 53, normalized size = 2.79 \begin {gather*} 25 \, x^{10} - 300 \, x^{8} + 1400 \, x^{6} - 3000 \, x^{4} + 25 \, x^{2} e^{\left (2 \, x\right )} + 2500 \, x^{2} + 50 \, {\left (x^{6} - 6 \, x^{4} + 10 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^2+50*x)*exp(x)^2+(50*x^6+300*x^5-300*x^4-1200*x^3+500*x^2+1000*x)*exp(x)+250*x^9-2400*x^7+8400
*x^5-12000*x^3+5000*x,x, algorithm="fricas")

[Out]

25*x^10 - 300*x^8 + 1400*x^6 - 3000*x^4 + 25*x^2*e^(2*x) + 2500*x^2 + 50*(x^6 - 6*x^4 + 10*x^2)*e^x

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giac [B]  time = 0.21, size = 53, normalized size = 2.79 \begin {gather*} 25 \, x^{10} - 300 \, x^{8} + 1400 \, x^{6} - 3000 \, x^{4} + 25 \, x^{2} e^{\left (2 \, x\right )} + 2500 \, x^{2} + 50 \, {\left (x^{6} - 6 \, x^{4} + 10 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^2+50*x)*exp(x)^2+(50*x^6+300*x^5-300*x^4-1200*x^3+500*x^2+1000*x)*exp(x)+250*x^9-2400*x^7+8400
*x^5-12000*x^3+5000*x,x, algorithm="giac")

[Out]

25*x^10 - 300*x^8 + 1400*x^6 - 3000*x^4 + 25*x^2*e^(2*x) + 2500*x^2 + 50*(x^6 - 6*x^4 + 10*x^2)*e^x

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maple [A]  time = 0.04, size = 55, normalized size = 2.89

method	result	size
risch	\(25 \,{\mathrm e}^{2 x} x^{2}+\left (50 x^{6}-300 x^{4}+500 x^{2}\right ) {\mathrm e}^{x}+25 x^{10}-300 x^{8}+1400 x^{6}-3000 x^{4}+2500 x^{2}\)	\(55\)
default	\(25 \,{\mathrm e}^{2 x} x^{2}+50 x^{6} {\mathrm e}^{x}+500 \,{\mathrm e}^{x} x^{2}-300 \,{\mathrm e}^{x} x^{4}+2500 x^{2}-3000 x^{4}+1400 x^{6}-300 x^{8}+25 x^{10}\)	\(57\)
norman	\(25 \,{\mathrm e}^{2 x} x^{2}+50 x^{6} {\mathrm e}^{x}+500 \,{\mathrm e}^{x} x^{2}-300 \,{\mathrm e}^{x} x^{4}+2500 x^{2}-3000 x^{4}+1400 x^{6}-300 x^{8}+25 x^{10}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x^2+50*x)*exp(x)^2+(50*x^6+300*x^5-300*x^4-1200*x^3+500*x^2+1000*x)*exp(x)+250*x^9-2400*x^7+8400*x^5-1
2000*x^3+5000*x,x,method=_RETURNVERBOSE)

[Out]

25*exp(2*x)*x^2+(50*x^6-300*x^4+500*x^2)*exp(x)+25*x^10-300*x^8+1400*x^6-3000*x^4+2500*x^2

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maxima [B]  time = 0.35, size = 53, normalized size = 2.79 \begin {gather*} 25 \, x^{10} - 300 \, x^{8} + 1400 \, x^{6} - 3000 \, x^{4} + 25 \, x^{2} e^{\left (2 \, x\right )} + 2500 \, x^{2} + 50 \, {\left (x^{6} - 6 \, x^{4} + 10 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^2+50*x)*exp(x)^2+(50*x^6+300*x^5-300*x^4-1200*x^3+500*x^2+1000*x)*exp(x)+250*x^9-2400*x^7+8400
*x^5-12000*x^3+5000*x,x, algorithm="maxima")

[Out]

25*x^10 - 300*x^8 + 1400*x^6 - 3000*x^4 + 25*x^2*e^(2*x) + 2500*x^2 + 50*(x^6 - 6*x^4 + 10*x^2)*e^x

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mupad [B]  time = 0.12, size = 19, normalized size = 1.00 \begin {gather*} 25\,x^2\,{\left ({\mathrm {e}}^x-6\,x^2+x^4+10\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5000*x + exp(2*x)*(50*x + 50*x^2) + exp(x)*(1000*x + 500*x^2 - 1200*x^3 - 300*x^4 + 300*x^5 + 50*x^6) - 12
000*x^3 + 8400*x^5 - 2400*x^7 + 250*x^9,x)

[Out]

25*x^2*(exp(x) - 6*x^2 + x^4 + 10)^2

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sympy [A]  time = 0.13, size = 53, normalized size = 2.79 \begin {gather*} 25 x^{10} - 300 x^{8} + 1400 x^{6} - 3000 x^{4} + 25 x^{2} e^{2 x} + 2500 x^{2} + \left (50 x^{6} - 300 x^{4} + 500 x^{2}\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x**2+50*x)*exp(x)**2+(50*x**6+300*x**5-300*x**4-1200*x**3+500*x**2+1000*x)*exp(x)+250*x**9-2400*
x**7+8400*x**5-12000*x**3+5000*x,x)

[Out]

25*x**10 - 300*x**8 + 1400*x**6 - 3000*x**4 + 25*x**2*exp(2*x) + 2500*x**2 + (50*x**6 - 300*x**4 + 500*x**2)*e
xp(x)

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