3.8.1 \(\int (150 x^5+e^{-12+10 x} (100 x^3+250 x^4)+e^{-6+5 x} (250 x^4+250 x^5)) \, dx\)

Optimal. Leaf size=26 \[ 2+25 x^4 \left (e^{-2 (3-2 x)+x}+x\right )^2-\log (4) \]

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Rubi [A]  time = 0.51, antiderivative size = 30, normalized size of antiderivative = 1.15, number of steps used = 27, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1593, 2196, 2176, 2194} \begin {gather*} 25 x^6+50 e^{5 x-6} x^5+25 e^{10 x-12} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[150*x^5 + E^(-12 + 10*x)*(100*x^3 + 250*x^4) + E^(-6 + 5*x)*(250*x^4 + 250*x^5),x]

[Out]

25*E^(-12 + 10*x)*x^4 + 50*E^(-6 + 5*x)*x^5 + 25*x^6

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} &=25 x^6+\int e^{-12+10 x} \left (100 x^3+250 x^4\right ) \, dx+\int e^{-6+5 x} \left (250 x^4+250 x^5\right ) \, dx\\ &=25 x^6+\int e^{-12+10 x} x^3 (100+250 x) \, dx+\int e^{-6+5 x} x^4 (250+250 x) \, dx\\ &=25 x^6+\int \left (100 e^{-12+10 x} x^3+250 e^{-12+10 x} x^4\right ) \, dx+\int \left (250 e^{-6+5 x} x^4+250 e^{-6+5 x} x^5\right ) \, dx\\ &=25 x^6+100 \int e^{-12+10 x} x^3 \, dx+250 \int e^{-6+5 x} x^4 \, dx+250 \int e^{-12+10 x} x^4 \, dx+250 \int e^{-6+5 x} x^5 \, dx\\ &=10 e^{-12+10 x} x^3+50 e^{-6+5 x} x^4+25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6-30 \int e^{-12+10 x} x^2 \, dx-100 \int e^{-12+10 x} x^3 \, dx-200 \int e^{-6+5 x} x^3 \, dx-250 \int e^{-6+5 x} x^4 \, dx\\ &=-3 e^{-12+10 x} x^2-40 e^{-6+5 x} x^3+25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6+6 \int e^{-12+10 x} x \, dx+30 \int e^{-12+10 x} x^2 \, dx+120 \int e^{-6+5 x} x^2 \, dx+200 \int e^{-6+5 x} x^3 \, dx\\ &=\frac {3}{5} e^{-12+10 x} x+24 e^{-6+5 x} x^2+25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6-\frac {3}{5} \int e^{-12+10 x} \, dx-6 \int e^{-12+10 x} x \, dx-48 \int e^{-6+5 x} x \, dx-120 \int e^{-6+5 x} x^2 \, dx\\ &=-\frac {3}{50} e^{-12+10 x}-\frac {48}{5} e^{-6+5 x} x+25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6+\frac {3}{5} \int e^{-12+10 x} \, dx+\frac {48}{5} \int e^{-6+5 x} \, dx+48 \int e^{-6+5 x} x \, dx\\ &=\frac {48}{25} e^{-6+5 x}+25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6-\frac {48}{5} \int e^{-6+5 x} \, dx\\ &=25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 21, normalized size = 0.81 \begin {gather*} \frac {25 x^4 \left (e^{5 x}+e^6 x\right )^2}{e^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[150*x^5 + E^(-12 + 10*x)*(100*x^3 + 250*x^4) + E^(-6 + 5*x)*(250*x^4 + 250*x^5),x]

[Out]

(25*x^4*(E^(5*x) + E^6*x)^2)/E^12

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fricas [A]  time = 0.84, size = 28, normalized size = 1.08 \begin {gather*} 25 \, x^{6} + 50 \, x^{5} e^{\left (5 \, x - 6\right )} + 25 \, x^{4} e^{\left (10 \, x - 12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((250*x^4+100*x^3)*exp(5*x-6)^2+(250*x^5+250*x^4)*exp(5*x-6)+150*x^5,x, algorithm="fricas")

[Out]

25*x^6 + 50*x^5*e^(5*x - 6) + 25*x^4*e^(10*x - 12)

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giac [A]  time = 0.26, size = 28, normalized size = 1.08 \begin {gather*} 25 \, x^{6} + 50 \, x^{5} e^{\left (5 \, x - 6\right )} + 25 \, x^{4} e^{\left (10 \, x - 12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((250*x^4+100*x^3)*exp(5*x-6)^2+(250*x^5+250*x^4)*exp(5*x-6)+150*x^5,x, algorithm="giac")

[Out]

25*x^6 + 50*x^5*e^(5*x - 6) + 25*x^4*e^(10*x - 12)

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maple [A]  time = 0.07, size = 29, normalized size = 1.12

method	result	size
risch	\(25 x^{6}+25 x^{4} {\mathrm e}^{10 x -12}+50 x^{5} {\mathrm e}^{5 x -6}\)	\(29\)
norman	\(25 x^{6}+25 x^{4} {\mathrm e}^{10 x -12}+50 x^{5} {\mathrm e}^{5 x -6}\)	\(31\)
derivativedivides	\(25 x^{6}+\frac {1296 \,{\mathrm e}^{10 x -12}}{25}+\frac {864 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )}{25}+\frac {216 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )^{2}}{25}+\frac {24 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )^{3}}{25}+\frac {{\mathrm e}^{10 x -12} \left (5 x -6\right )^{4}}{25}+\frac {2592 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )}{25}+\frac {15552 \,{\mathrm e}^{5 x -6}}{125}+\frac {864 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{2}}{25}+\frac {144 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{3}}{25}+\frac {12 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{4}}{25}+\frac {2 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{5}}{125}\)	\(164\)
default	\(25 x^{6}+\frac {1296 \,{\mathrm e}^{10 x -12}}{25}+\frac {864 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )}{25}+\frac {216 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )^{2}}{25}+\frac {24 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )^{3}}{25}+\frac {{\mathrm e}^{10 x -12} \left (5 x -6\right )^{4}}{25}+\frac {2592 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )}{25}+\frac {15552 \,{\mathrm e}^{5 x -6}}{125}+\frac {864 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{2}}{25}+\frac {144 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{3}}{25}+\frac {12 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{4}}{25}+\frac {2 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{5}}{125}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((250*x^4+100*x^3)*exp(5*x-6)^2+(250*x^5+250*x^4)*exp(5*x-6)+150*x^5,x,method=_RETURNVERBOSE)

[Out]

25*x^6+25*x^4*exp(10*x-12)+50*x^5*exp(5*x-6)

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maxima [B]  time = 0.61, size = 78, normalized size = 3.00 \begin {gather*} 25 \, x^{6} + 25 \, x^{4} e^{\left (10 \, x - 12\right )} + \frac {2}{25} \, {\left (625 \, x^{5} - 625 \, x^{4} + 500 \, x^{3} - 300 \, x^{2} + 120 \, x - 24\right )} e^{\left (5 \, x - 6\right )} + \frac {2}{25} \, {\left (625 \, x^{4} - 500 \, x^{3} + 300 \, x^{2} - 120 \, x + 24\right )} e^{\left (5 \, x - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((250*x^4+100*x^3)*exp(5*x-6)^2+(250*x^5+250*x^4)*exp(5*x-6)+150*x^5,x, algorithm="maxima")

[Out]

25*x^6 + 25*x^4*e^(10*x - 12) + 2/25*(625*x^5 - 625*x^4 + 500*x^3 - 300*x^2 + 120*x - 24)*e^(5*x - 6) + 2/25*(
625*x^4 - 500*x^3 + 300*x^2 - 120*x + 24)*e^(5*x - 6)

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mupad [B]  time = 0.53, size = 18, normalized size = 0.69 \begin {gather*} 25\,x^4\,{\mathrm {e}}^{-12}\,{\left ({\mathrm {e}}^{5\,x}+x\,{\mathrm {e}}^6\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(10*x - 12)*(100*x^3 + 250*x^4) + exp(5*x - 6)*(250*x^4 + 250*x^5) + 150*x^5,x)

[Out]

25*x^4*exp(-12)*(exp(5*x) + x*exp(6))^2

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sympy [A]  time = 0.11, size = 27, normalized size = 1.04 \begin {gather*} 25 x^{6} + 50 x^{5} e^{5 x - 6} + 25 x^{4} e^{10 x - 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((250*x**4+100*x**3)*exp(5*x-6)**2+(250*x**5+250*x**4)*exp(5*x-6)+150*x**5,x)

[Out]

25*x**6 + 50*x**5*exp(5*x - 6) + 25*x**4*exp(10*x - 12)

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