3.35.44 \(\int \frac {1}{9} (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} (4 x^3+2 x^4)+e^x (-162 x^2-94 x^3-20 x^4-2 x^5)) \, dx\)

Optimal. Leaf size=26 \[ \frac {1}{9} x^2 \left (-11+3 x+e^x x-(4+x)^2\right )^2 \]

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Rubi [B]  time = 0.30, antiderivative size = 72, normalized size of antiderivative = 2.77, number of steps used = 34, number of rules used = 5, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {12, 1593, 2196, 2176, 2194} \begin {gather*} \frac {x^6}{9}-\frac {2 e^x x^5}{9}+\frac {10 x^5}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {79 x^4}{9}-6 e^x x^3+30 x^3+81 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1458*x + 810*x^2 + 316*x^3 + 50*x^4 + 6*x^5 + E^(2*x)*(4*x^3 + 2*x^4) + E^x*(-162*x^2 - 94*x^3 - 20*x^4 -
2*x^5))/9,x]

[Out]

81*x^2 + 30*x^3 - 6*E^x*x^3 + (79*x^4)/9 - (10*E^x*x^4)/9 + (E^(2*x)*x^4)/9 + (10*x^5)/9 - (2*E^x*x^5)/9 + x^6
/9

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} &=\frac {1}{9} \int \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx\\ &=81 x^2+30 x^3+\frac {79 x^4}{9}+\frac {10 x^5}{9}+\frac {x^6}{9}+\frac {1}{9} \int e^{2 x} \left (4 x^3+2 x^4\right ) \, dx+\frac {1}{9} \int e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right ) \, dx\\ &=81 x^2+30 x^3+\frac {79 x^4}{9}+\frac {10 x^5}{9}+\frac {x^6}{9}+\frac {1}{9} \int e^{2 x} x^3 (4+2 x) \, dx+\frac {1}{9} \int \left (-162 e^x x^2-94 e^x x^3-20 e^x x^4-2 e^x x^5\right ) \, dx\\ &=81 x^2+30 x^3+\frac {79 x^4}{9}+\frac {10 x^5}{9}+\frac {x^6}{9}+\frac {1}{9} \int \left (4 e^{2 x} x^3+2 e^{2 x} x^4\right ) \, dx-\frac {2}{9} \int e^x x^5 \, dx-\frac {20}{9} \int e^x x^4 \, dx-\frac {94}{9} \int e^x x^3 \, dx-18 \int e^x x^2 \, dx\\ &=81 x^2-18 e^x x^2+30 x^3-\frac {94 e^x x^3}{9}+\frac {79 x^4}{9}-\frac {20 e^x x^4}{9}+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}+\frac {2}{9} \int e^{2 x} x^4 \, dx+\frac {4}{9} \int e^{2 x} x^3 \, dx+\frac {10}{9} \int e^x x^4 \, dx+\frac {80}{9} \int e^x x^3 \, dx+\frac {94}{3} \int e^x x^2 \, dx+36 \int e^x x \, dx\\ &=36 e^x x+81 x^2+\frac {40 e^x x^2}{3}+30 x^3-\frac {14 e^x x^3}{9}+\frac {2}{9} e^{2 x} x^3+\frac {79 x^4}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}-\frac {4}{9} \int e^{2 x} x^3 \, dx-\frac {2}{3} \int e^{2 x} x^2 \, dx-\frac {40}{9} \int e^x x^3 \, dx-\frac {80}{3} \int e^x x^2 \, dx-36 \int e^x \, dx-\frac {188}{3} \int e^x x \, dx\\ &=-36 e^x-\frac {80 e^x x}{3}+81 x^2-\frac {40 e^x x^2}{3}-\frac {1}{3} e^{2 x} x^2+30 x^3-6 e^x x^3+\frac {79 x^4}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}+\frac {2}{3} \int e^{2 x} x \, dx+\frac {2}{3} \int e^{2 x} x^2 \, dx+\frac {40}{3} \int e^x x^2 \, dx+\frac {160}{3} \int e^x x \, dx+\frac {188 \int e^x \, dx}{3}\\ &=\frac {80 e^x}{3}+\frac {80 e^x x}{3}+\frac {1}{3} e^{2 x} x+81 x^2+30 x^3-6 e^x x^3+\frac {79 x^4}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}-\frac {1}{3} \int e^{2 x} \, dx-\frac {2}{3} \int e^{2 x} x \, dx-\frac {80}{3} \int e^x x \, dx-\frac {160 \int e^x \, dx}{3}\\ &=-\frac {80 e^x}{3}-\frac {e^{2 x}}{6}+81 x^2+30 x^3-6 e^x x^3+\frac {79 x^4}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}+\frac {1}{3} \int e^{2 x} \, dx+\frac {80 \int e^x \, dx}{3}\\ &=81 x^2+30 x^3-6 e^x x^3+\frac {79 x^4}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{9} x^2 \left (27-\left (-5+e^x\right ) x+x^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1458*x + 810*x^2 + 316*x^3 + 50*x^4 + 6*x^5 + E^(2*x)*(4*x^3 + 2*x^4) + E^x*(-162*x^2 - 94*x^3 - 20
*x^4 - 2*x^5))/9,x]

[Out]

(x^2*(27 - (-5 + E^x)*x + x^2)^2)/9

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fricas [B]  time = 0.57, size = 53, normalized size = 2.04 \begin {gather*} \frac {1}{9} \, x^{6} + \frac {10}{9} \, x^{5} + \frac {1}{9} \, x^{4} e^{\left (2 \, x\right )} + \frac {79}{9} \, x^{4} + 30 \, x^{3} + 81 \, x^{2} - \frac {2}{9} \, {\left (x^{5} + 5 \, x^{4} + 27 \, x^{3}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(2*x^4+4*x^3)*exp(x)^2+1/9*(-2*x^5-20*x^4-94*x^3-162*x^2)*exp(x)+2/3*x^5+50/9*x^4+316/9*x^3+90*x
^2+162*x,x, algorithm="fricas")

[Out]

1/9*x^6 + 10/9*x^5 + 1/9*x^4*e^(2*x) + 79/9*x^4 + 30*x^3 + 81*x^2 - 2/9*(x^5 + 5*x^4 + 27*x^3)*e^x

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giac [B]  time = 0.23, size = 53, normalized size = 2.04 \begin {gather*} \frac {1}{9} \, x^{6} + \frac {10}{9} \, x^{5} + \frac {1}{9} \, x^{4} e^{\left (2 \, x\right )} + \frac {79}{9} \, x^{4} + 30 \, x^{3} + 81 \, x^{2} - \frac {2}{9} \, {\left (x^{5} + 5 \, x^{4} + 27 \, x^{3}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(2*x^4+4*x^3)*exp(x)^2+1/9*(-2*x^5-20*x^4-94*x^3-162*x^2)*exp(x)+2/3*x^5+50/9*x^4+316/9*x^3+90*x
^2+162*x,x, algorithm="giac")

[Out]

1/9*x^6 + 10/9*x^5 + 1/9*x^4*e^(2*x) + 79/9*x^4 + 30*x^3 + 81*x^2 - 2/9*(x^5 + 5*x^4 + 27*x^3)*e^x

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maple [B]  time = 0.08, size = 56, normalized size = 2.15

method	result	size
risch	\(\frac {{\mathrm e}^{2 x} x^{4}}{9}+\frac {\left (-2 x^{5}-10 x^{4}-54 x^{3}\right ) {\mathrm e}^{x}}{9}+\frac {x^{6}}{9}+\frac {10 x^{5}}{9}+\frac {79 x^{4}}{9}+30 x^{3}+81 x^{2}\)	\(56\)
default	\(30 x^{3}+81 x^{2}+\frac {79 x^{4}}{9}+\frac {10 x^{5}}{9}+\frac {x^{6}}{9}+\frac {{\mathrm e}^{2 x} x^{4}}{9}-6 \,{\mathrm e}^{x} x^{3}-\frac {10 \,{\mathrm e}^{x} x^{4}}{9}-\frac {2 x^{5} {\mathrm e}^{x}}{9}\)	\(57\)
norman	\(30 x^{3}+81 x^{2}+\frac {79 x^{4}}{9}+\frac {10 x^{5}}{9}+\frac {x^{6}}{9}+\frac {{\mathrm e}^{2 x} x^{4}}{9}-6 \,{\mathrm e}^{x} x^{3}-\frac {10 \,{\mathrm e}^{x} x^{4}}{9}-\frac {2 x^{5} {\mathrm e}^{x}}{9}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(2*x^4+4*x^3)*exp(x)^2+1/9*(-2*x^5-20*x^4-94*x^3-162*x^2)*exp(x)+2/3*x^5+50/9*x^4+316/9*x^3+90*x^2+162
*x,x,method=_RETURNVERBOSE)

[Out]

1/9*exp(2*x)*x^4+1/9*(-2*x^5-10*x^4-54*x^3)*exp(x)+1/9*x^6+10/9*x^5+79/9*x^4+30*x^3+81*x^2

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maxima [B]  time = 0.70, size = 53, normalized size = 2.04 \begin {gather*} \frac {1}{9} \, x^{6} + \frac {10}{9} \, x^{5} + \frac {1}{9} \, x^{4} e^{\left (2 \, x\right )} + \frac {79}{9} \, x^{4} + 30 \, x^{3} + 81 \, x^{2} - \frac {2}{9} \, {\left (x^{5} + 5 \, x^{4} + 27 \, x^{3}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(2*x^4+4*x^3)*exp(x)^2+1/9*(-2*x^5-20*x^4-94*x^3-162*x^2)*exp(x)+2/3*x^5+50/9*x^4+316/9*x^3+90*x
^2+162*x,x, algorithm="maxima")

[Out]

1/9*x^6 + 10/9*x^5 + 1/9*x^4*e^(2*x) + 79/9*x^4 + 30*x^3 + 81*x^2 - 2/9*(x^5 + 5*x^4 + 27*x^3)*e^x

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mupad [B]  time = 2.07, size = 20, normalized size = 0.77 \begin {gather*} \frac {x^2\,{\left (5\,x-x\,{\mathrm {e}}^x+x^2+27\right )}^2}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(162*x + (exp(2*x)*(4*x^3 + 2*x^4))/9 - (exp(x)*(162*x^2 + 94*x^3 + 20*x^4 + 2*x^5))/9 + 90*x^2 + (316*x^3)
/9 + (50*x^4)/9 + (2*x^5)/3,x)

[Out]

(x^2*(5*x - x*exp(x) + x^2 + 27)^2)/9

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sympy [B]  time = 0.15, size = 60, normalized size = 2.31 \begin {gather*} \frac {x^{6}}{9} + \frac {10 x^{5}}{9} + \frac {x^{4} e^{2 x}}{9} + \frac {79 x^{4}}{9} + 30 x^{3} + 81 x^{2} + \frac {\left (- 18 x^{5} - 90 x^{4} - 486 x^{3}\right ) e^{x}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(2*x**4+4*x**3)*exp(x)**2+1/9*(-2*x**5-20*x**4-94*x**3-162*x**2)*exp(x)+2/3*x**5+50/9*x**4+316/9
*x**3+90*x**2+162*x,x)

[Out]

x**6/9 + 10*x**5/9 + x**4*exp(2*x)/9 + 79*x**4/9 + 30*x**3 + 81*x**2 + (-18*x**5 - 90*x**4 - 486*x**3)*exp(x)/
81

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