3.43.1 \(\int e^{-10 \sqrt {\frac {2}{3}}} (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} (4 x^3+2 x^4)+e^x (10 x^4+2 x^5+e^3 (8 x^3+2 x^4))) \, dx\)

Optimal. Leaf size=25 \[ e^{-10 \sqrt {\frac {2}{3}}} x^4 \left (e^3+e^x+x\right )^2 \]

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Rubi [B]  time = 0.40, antiderivative size = 110, normalized size of antiderivative = 4.40, number of steps used = 38, number of rules used = 5, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {12, 1593, 2196, 2176, 2194} \begin {gather*} e^{-10 \sqrt {\frac {2}{3}}} x^6+2 e^{x-10 \sqrt {\frac {2}{3}}} x^5+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{x+\frac {1}{3} \left (9-10 \sqrt {6}\right )} x^4+e^{2 x-10 \sqrt {\frac {2}{3}}} x^4+e^{6-10 \sqrt {\frac {2}{3}}} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^6*x^3 + 10*E^3*x^4 + 6*x^5 + E^(2*x)*(4*x^3 + 2*x^4) + E^x*(10*x^4 + 2*x^5 + E^3*(8*x^3 + 2*x^4)))/E^
(10*Sqrt[2/3]),x]

[Out]

E^(6 - 10*Sqrt[2/3])*x^4 + 2*E^((9 - 10*Sqrt[6])/3 + x)*x^4 + E^(-10*Sqrt[2/3] + 2*x)*x^4 + 2*E^(3 - 10*Sqrt[2
/3])*x^5 + 2*E^(-10*Sqrt[2/3] + x)*x^5 + x^6/E^(10*Sqrt[2/3])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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} &=e^{-10 \sqrt {\frac {2}{3}}} \int \left (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right )\right ) \, dx\\ &=e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+e^{-10 \sqrt {\frac {2}{3}}} \int e^{2 x} \left (4 x^3+2 x^4\right ) \, dx+e^{-10 \sqrt {\frac {2}{3}}} \int e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right ) \, dx\\ &=e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+e^{-10 \sqrt {\frac {2}{3}}} \int e^{2 x} x^3 (4+2 x) \, dx+e^{-10 \sqrt {\frac {2}{3}}} \int \left (10 e^x x^4+2 e^x x^5+2 e^{3+x} x^3 (4+x)\right ) \, dx\\ &=e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+e^{-10 \sqrt {\frac {2}{3}}} \int \left (4 e^{2 x} x^3+2 e^{2 x} x^4\right ) \, dx+\left (2 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^5 \, dx+\left (2 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^3 (4+x) \, dx+\left (10 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^4 \, dx\\ &=e^{6-10 \sqrt {\frac {2}{3}}} x^4+10 e^{-10 \sqrt {\frac {2}{3}}+x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+\left (2 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x^4 \, dx+\left (2 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int \left (4 e^{3+x} x^3+e^{3+x} x^4\right ) \, dx+\left (4 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x^3 \, dx-\left (10 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^4 \, dx-\left (40 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^3 \, dx\\ &=-40 e^{-10 \sqrt {\frac {2}{3}}+x} x^3+2 e^{-10 \sqrt {\frac {2}{3}}+2 x} x^3+e^{6-10 \sqrt {\frac {2}{3}}} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+\left (2 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^4 \, dx-\left (4 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x^3 \, dx-\left (6 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x^2 \, dx+\left (8 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^3 \, dx+\left (40 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^3 \, dx+\left (120 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^2 \, dx\\ &=120 e^{-10 \sqrt {\frac {2}{3}}+x} x^2-3 e^{-10 \sqrt {\frac {2}{3}}+2 x} x^2+8 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^3+e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+\left (6 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x \, dx+\left (6 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x^2 \, dx-\left (8 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^3 \, dx-\left (24 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^2 \, dx-\left (120 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^2 \, dx-\left (240 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x \, dx\\ &=-240 e^{-10 \sqrt {\frac {2}{3}}+x} x+3 e^{-10 \sqrt {\frac {2}{3}}+2 x} x-24 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^2+e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6-\left (3 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} \, dx-\left (6 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x \, dx+\left (24 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^2 \, dx+\left (48 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x \, dx+\left (240 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x \, dx+\left (240 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x \, dx\\ &=240 e^{-10 \sqrt {\frac {2}{3}}+x}-\frac {3}{2} e^{-10 \sqrt {\frac {2}{3}}+2 x}+48 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x+e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+\left (3 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} \, dx-\left (48 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} \, dx-\left (48 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x \, dx-\left (240 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x \, dx\\ &=-48 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x}+e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+\left (48 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} \, dx\\ &=e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 25, normalized size = 1.00 \begin {gather*} e^{-10 \sqrt {\frac {2}{3}}} x^4 \left (e^3+e^x+x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^6*x^3 + 10*E^3*x^4 + 6*x^5 + E^(2*x)*(4*x^3 + 2*x^4) + E^x*(10*x^4 + 2*x^5 + E^3*(8*x^3 + 2*x^4
)))/E^(10*Sqrt[2/3]),x]

[Out]

(x^4*(E^3 + E^x + x)^2)/E^(10*Sqrt[2/3])

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fricas [B]  time = 1.28, size = 49, normalized size = 1.96 \begin {gather*} {\left (x^{6} + 2 \, x^{5} e^{3} + x^{4} e^{6} + x^{4} e^{\left (2 \, x\right )} + 2 \, {\left (x^{5} + x^{4} e^{3}\right )} e^{x}\right )} e^{\left (-\frac {10}{3} \, \sqrt {3} \sqrt {2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+4*x^3)*exp(x)^2+((2*x^4+8*x^3)*exp(3)+2*x^5+10*x^4)*exp(x)+4*x^3*exp(3)^2+10*x^4*exp(3)+6*x^
5)/exp(5/3*3^(1/2)*2^(1/2))^2,x, algorithm="fricas")

[Out]

(x^6 + 2*x^5*e^3 + x^4*e^6 + x^4*e^(2*x) + 2*(x^5 + x^4*e^3)*e^x)*e^(-10/3*sqrt(3)*sqrt(2))

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giac [B]  time = 0.25, size = 51, normalized size = 2.04 \begin {gather*} {\left (x^{6} + 2 \, x^{5} e^{3} + 2 \, x^{5} e^{x} + x^{4} e^{6} + x^{4} e^{\left (2 \, x\right )} + 2 \, x^{4} e^{\left (x + 3\right )}\right )} e^{\left (-\frac {10}{3} \, \sqrt {3} \sqrt {2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+4*x^3)*exp(x)^2+((2*x^4+8*x^3)*exp(3)+2*x^5+10*x^4)*exp(x)+4*x^3*exp(3)^2+10*x^4*exp(3)+6*x^
5)/exp(5/3*3^(1/2)*2^(1/2))^2,x, algorithm="giac")

[Out]

(x^6 + 2*x^5*e^3 + 2*x^5*e^x + x^4*e^6 + x^4*e^(2*x) + 2*x^4*e^(x + 3))*e^(-10/3*sqrt(3)*sqrt(2))

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maple [B]  time = 0.12, size = 88, normalized size = 3.52

method	result	size
risch	\(x^{4} {\mathrm e}^{-\frac {10 \sqrt {3}\, \sqrt {2}}{3}+6}+2 x^{5} {\mathrm e}^{-\frac {10 \sqrt {3}\, \sqrt {2}}{3}+3}+{\mathrm e}^{-\frac {10 \sqrt {3}\, \sqrt {2}}{3}} x^{6}+x^{4} {\mathrm e}^{-\frac {10 \sqrt {3}\, \sqrt {2}}{3}+2 x}+\left (2 x^{4} {\mathrm e}^{3}+2 x^{5}\right ) {\mathrm e}^{-\frac {10 \sqrt {3}\, \sqrt {2}}{3}+x}\)	\(88\)
default	\({\mathrm e}^{-\frac {10 \sqrt {6}}{3}} \left (2 x^{5} {\mathrm e}^{x}+8 \,{\mathrm e}^{3} \left ({\mathrm e}^{x} x^{3}-3 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x}\right )+2 \,{\mathrm e}^{3} \left ({\mathrm e}^{x} x^{4}-4 \,{\mathrm e}^{x} x^{3}+12 \,{\mathrm e}^{x} x^{2}-24 \,{\mathrm e}^{x} x +24 \,{\mathrm e}^{x}\right )+{\mathrm e}^{2 x} x^{4}+x^{6}+x^{4} {\mathrm e}^{6}+2 x^{5} {\mathrm e}^{3}\right )\)	\(108\)
norman	\(\left ({\mathrm e}^{-\frac {5 \sqrt {6}}{3}} x^{6}+{\mathrm e}^{-\frac {5 \sqrt {6}}{3}} x^{4} {\mathrm e}^{2 x}+{\mathrm e}^{-\frac {5 \sqrt {6}}{3}} {\mathrm e}^{6} x^{4}+2 \,{\mathrm e}^{-\frac {5 \sqrt {6}}{3}} x^{5} {\mathrm e}^{x}+2 \,{\mathrm e}^{-\frac {5 \sqrt {6}}{3}} {\mathrm e}^{3} x^{5}+2 \,{\mathrm e}^{-\frac {5 \sqrt {6}}{3}} {\mathrm e}^{3} x^{4} {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {5 \sqrt {6}}{3}}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4+4*x^3)*exp(x)^2+((2*x^4+8*x^3)*exp(3)+2*x^5+10*x^4)*exp(x)+4*x^3*exp(3)^2+10*x^4*exp(3)+6*x^5)/exp
(5/3*3^(1/2)*2^(1/2))^2,x,method=_RETURNVERBOSE)

[Out]

x^4*exp(-10/3*3^(1/2)*2^(1/2)+6)+2*x^5*exp(-10/3*3^(1/2)*2^(1/2)+3)+exp(-10/3*3^(1/2)*2^(1/2))*x^6+x^4*exp(-10
/3*3^(1/2)*2^(1/2)+2*x)+(2*x^4*exp(3)+2*x^5)*exp(-10/3*3^(1/2)*2^(1/2)+x)

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maxima [B]  time = 0.43, size = 49, normalized size = 1.96 \begin {gather*} {\left (x^{6} + 2 \, x^{5} e^{3} + x^{4} e^{6} + x^{4} e^{\left (2 \, x\right )} + 2 \, {\left (x^{5} + x^{4} e^{3}\right )} e^{x}\right )} e^{\left (-\frac {10}{3} \, \sqrt {3} \sqrt {2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+4*x^3)*exp(x)^2+((2*x^4+8*x^3)*exp(3)+2*x^5+10*x^4)*exp(x)+4*x^3*exp(3)^2+10*x^4*exp(3)+6*x^
5)/exp(5/3*3^(1/2)*2^(1/2))^2,x, algorithm="maxima")

[Out]

(x^6 + 2*x^5*e^3 + x^4*e^6 + x^4*e^(2*x) + 2*(x^5 + x^4*e^3)*e^x)*e^(-10/3*sqrt(3)*sqrt(2))

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mupad [B]  time = 0.18, size = 18, normalized size = 0.72 \begin {gather*} x^4\,{\mathrm {e}}^{-\frac {10\,\sqrt {6}}{3}}\,{\left (x+{\mathrm {e}}^3+{\mathrm {e}}^x\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-(10*2^(1/2)*3^(1/2))/3)*(exp(2*x)*(4*x^3 + 2*x^4) + exp(x)*(exp(3)*(8*x^3 + 2*x^4) + 10*x^4 + 2*x^5)
+ 10*x^4*exp(3) + 4*x^3*exp(6) + 6*x^5),x)

[Out]

x^4*exp(-(10*6^(1/2))/3)*(x + exp(3) + exp(x))^2

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sympy [B]  time = 0.20, size = 114, normalized size = 4.56 \begin {gather*} \frac {x^{6}}{e^{\frac {10 \sqrt {6}}{3}}} + \frac {2 x^{5} e^{3}}{e^{\frac {10 \sqrt {6}}{3}}} + \frac {x^{4} e^{6}}{e^{\frac {10 \sqrt {6}}{3}}} + \frac {x^{4} e^{\frac {10 \sqrt {6}}{3}} e^{2 x} + \left (2 x^{5} e^{\frac {10 \sqrt {6}}{3}} + 2 x^{4} e^{3} e^{\frac {10 \sqrt {6}}{3}}\right ) e^{x}}{e^{\frac {20 \sqrt {6}}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4+4*x**3)*exp(x)**2+((2*x**4+8*x**3)*exp(3)+2*x**5+10*x**4)*exp(x)+4*x**3*exp(3)**2+10*x**4*e
xp(3)+6*x**5)/exp(5/3*3**(1/2)*2**(1/2))**2,x)

[Out]

x**6*exp(-10*sqrt(6)/3) + 2*x**5*exp(3)*exp(-10*sqrt(6)/3) + x**4*exp(6)*exp(-10*sqrt(6)/3) + (x**4*exp(10*sqr
t(6)/3)*exp(2*x) + (2*x**5*exp(10*sqrt(6)/3) + 2*x**4*exp(3)*exp(10*sqrt(6)/3))*exp(x))*exp(-20*sqrt(6)/3)

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