\(\int \frac {1}{9} (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} (4 x^3+2 x^4)+e^x (-162 x^2-94 x^3-20 x^4-2 x^5)) \, dx\) [3444]

Optimal result

Integrand size = 70, antiderivative size = 26 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {1}{9} x^2 \left (-11+3 x+e^x x-(4+x)^2\right )^2 \]

[Out]

1/9*x^2*(exp(x)*x-(4+x)^2-11+3*x)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(26)=52\).

Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77, number of steps used = 34, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {12, 1607, 2227, 2207, 2225} \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {x^6}{9}-\frac {2 e^x x^5}{9}+\frac {10 x^5}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {79 x^4}{9}-6 e^x x^3+30 x^3+81 x^2 \]

[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 1607

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 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 2227

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] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \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{align*}

Mathematica [A] (verified)

Time = 2.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {1}{9} x^2 \left (27-\left (-5+e^x\right ) x+x^2\right )^2 \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(23)=46\).

Time = 0.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 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	\(\frac {{\mathrm e}^{2 x} x^{4}}{9}-\frac {2 x^{5} {\mathrm e}^{x}}{9}-\frac {10 \,{\mathrm e}^{x} x^{4}}{9}-6 \,{\mathrm e}^{x} x^{3}+\frac {x^{6}}{9}+\frac {10 x^{5}}{9}+\frac {79 x^{4}}{9}+30 x^{3}+81 x^{2}\)	\(57\)
norman	\(\frac {{\mathrm e}^{2 x} x^{4}}{9}-\frac {2 x^{5} {\mathrm e}^{x}}{9}-\frac {10 \,{\mathrm e}^{x} x^{4}}{9}-6 \,{\mathrm e}^{x} x^{3}+\frac {x^{6}}{9}+\frac {10 x^{5}}{9}+\frac {79 x^{4}}{9}+30 x^{3}+81 x^{2}\)	\(57\)
parallelrisch	\(\frac {{\mathrm e}^{2 x} x^{4}}{9}-\frac {2 x^{5} {\mathrm e}^{x}}{9}-\frac {10 \,{\mathrm e}^{x} x^{4}}{9}-6 \,{\mathrm e}^{x} x^{3}+\frac {x^{6}}{9}+\frac {10 x^{5}}{9}+\frac {79 x^{4}}{9}+30 x^{3}+81 x^{2}\)	\(57\)
parts	\(\frac {{\mathrm e}^{2 x} x^{4}}{9}-\frac {2 x^{5} {\mathrm e}^{x}}{9}-\frac {10 \,{\mathrm e}^{x} x^{4}}{9}-6 \,{\mathrm e}^{x} x^{3}+\frac {x^{6}}{9}+\frac {10 x^{5}}{9}+\frac {79 x^{4}}{9}+30 x^{3}+81 x^{2}\)	\(57\)

[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(x)^2*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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {1}{9} \, x^{6} + \frac {10}{9} \, x^{5} + \frac {1}{9} \, x^{4} e^{\left (2 \, x\right )} + \frac {79}{9} \, x^{4} + 30 \, x^{3} + 81 \, x^{2} - \frac {2}{9} \, {\left (x^{5} + 5 \, x^{4} + 27 \, x^{3}\right )} e^{x} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).

Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {x^{6}}{9} + \frac {10 x^{5}}{9} + \frac {x^{4} e^{2 x}}{9} + \frac {79 x^{4}}{9} + 30 x^{3} + 81 x^{2} + \frac {\left (- 18 x^{5} - 90 x^{4} - 486 x^{3}\right ) e^{x}}{81} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {1}{9} \, x^{6} + \frac {10}{9} \, x^{5} + \frac {1}{9} \, x^{4} e^{\left (2 \, x\right )} + \frac {79}{9} \, x^{4} + 30 \, x^{3} + 81 \, x^{2} - \frac {2}{9} \, {\left (x^{5} + 5 \, x^{4} + 27 \, x^{3}\right )} e^{x} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {1}{9} \, x^{6} + \frac {10}{9} \, x^{5} + \frac {1}{9} \, x^{4} e^{\left (2 \, x\right )} + \frac {79}{9} \, x^{4} + 30 \, x^{3} + 81 \, x^{2} - \frac {2}{9} \, {\left (x^{5} + 5 \, x^{4} + 27 \, x^{3}\right )} e^{x} \]

[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

Mupad [B] (verification not implemented)

Time = 9.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {x^2\,{\left (5\,x-x\,{\mathrm {e}}^x+x^2+27\right )}^2}{9} \]

[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