3.5.63 \(\int \frac {e^{4 x} (-2+x)+1944 x^{14}+e^{2 x} (18 x^7-36 x^8)}{324 x^9} \, dx\)

Optimal. Leaf size=18 \[ \left (-\frac {e^{2 x}}{36 x^4}+x^3\right )^2 \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 28, normalized size of antiderivative = 1.56, number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 14, 2197} \begin {gather*} \frac {e^{4 x}}{1296 x^8}+x^6-\frac {e^{2 x}}{18 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4*x)*(-2 + x) + 1944*x^14 + E^(2*x)*(18*x^7 - 36*x^8))/(324*x^9),x]

[Out]

E^(4*x)/(1296*x^8) - E^(2*x)/(18*x) + x^6

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{324} \int \frac {e^{4 x} (-2+x)+1944 x^{14}+e^{2 x} \left (18 x^7-36 x^8\right )}{x^9} \, dx\\ &=\frac {1}{324} \int \left (\frac {e^{4 x} (-2+x)}{x^9}+1944 x^5-\frac {18 e^{2 x} (-1+2 x)}{x^2}\right ) \, dx\\ &=x^6+\frac {1}{324} \int \frac {e^{4 x} (-2+x)}{x^9} \, dx-\frac {1}{18} \int \frac {e^{2 x} (-1+2 x)}{x^2} \, dx\\ &=\frac {e^{4 x}}{1296 x^8}-\frac {e^{2 x}}{18 x}+x^6\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 28, normalized size = 1.56 \begin {gather*} \frac {e^{4 x}}{1296 x^8}-\frac {e^{2 x}}{18 x}+x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*x)*(-2 + x) + 1944*x^14 + E^(2*x)*(18*x^7 - 36*x^8))/(324*x^9),x]

[Out]

E^(4*x)/(1296*x^8) - E^(2*x)/(18*x) + x^6

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 24, normalized size = 1.33 \begin {gather*} \frac {1296 \, x^{14} - 72 \, x^{7} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}}{1296 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/324*((x-2)*exp(x)^4+(-36*x^8+18*x^7)*exp(x)^2+1944*x^14)/x^9,x, algorithm="fricas")

[Out]

1/1296*(1296*x^14 - 72*x^7*e^(2*x) + e^(4*x))/x^8

________________________________________________________________________________________

giac [A]  time = 0.32, size = 24, normalized size = 1.33 \begin {gather*} \frac {1296 \, x^{14} - 72 \, x^{7} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}}{1296 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/324*((x-2)*exp(x)^4+(-36*x^8+18*x^7)*exp(x)^2+1944*x^14)/x^9,x, algorithm="giac")

[Out]

1/1296*(1296*x^14 - 72*x^7*e^(2*x) + e^(4*x))/x^8

________________________________________________________________________________________

maple [A]  time = 0.11, size = 23, normalized size = 1.28

method	result	size
default	\(x^{6}+\frac {{\mathrm e}^{4 x}}{1296 x^{8}}-\frac {{\mathrm e}^{2 x}}{18 x}\)	\(23\)
risch	\(x^{6}+\frac {{\mathrm e}^{4 x}}{1296 x^{8}}-\frac {{\mathrm e}^{2 x}}{18 x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/324*((x-2)*exp(x)^4+(-36*x^8+18*x^7)*exp(x)^2+1944*x^14)/x^9,x,method=_RETURNVERBOSE)

[Out]

x^6+1/1296*exp(x)^4/x^8-1/18*exp(x)^2/x

________________________________________________________________________________________

maxima [C]  time = 0.51, size = 31, normalized size = 1.72 \begin {gather*} x^{6} - \frac {1}{9} \, {\rm Ei}\left (2 \, x\right ) + \frac {1}{9} \, \Gamma \left (-1, -2 \, x\right ) + \frac {4096}{81} \, \Gamma \left (-7, -4 \, x\right ) + \frac {32768}{81} \, \Gamma \left (-8, -4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/324*((x-2)*exp(x)^4+(-36*x^8+18*x^7)*exp(x)^2+1944*x^14)/x^9,x, algorithm="maxima")

[Out]

x^6 - 1/9*Ei(2*x) + 1/9*gamma(-1, -2*x) + 4096/81*gamma(-7, -4*x) + 32768/81*gamma(-8, -4*x)

________________________________________________________________________________________

mupad [B]  time = 0.52, size = 17, normalized size = 0.94 \begin {gather*} \frac {{\left ({\mathrm {e}}^{2\,x}-36\,x^7\right )}^2}{1296\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(2*x)*(18*x^7 - 36*x^8))/324 + (exp(4*x)*(x - 2))/324 + 6*x^14)/x^9,x)

[Out]

(exp(2*x) - 36*x^7)^2/(1296*x^8)

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 26, normalized size = 1.44 \begin {gather*} x^{6} + \frac {- 1296 x^{8} e^{2 x} + 18 x e^{4 x}}{23328 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/324*((x-2)*exp(x)**4+(-36*x**8+18*x**7)*exp(x)**2+1944*x**14)/x**9,x)

[Out]

x**6 + (-1296*x**8*exp(2*x) + 18*x*exp(4*x))/(23328*x**9)

________________________________________________________________________________________