3.89.94 \(\int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} (1944 x-5508 x^2+864 x^3)+e^{\frac {2 (3+x)}{-3+2 x}} (1944 x^3-3564 x^4+864 x^5)+e^{\frac {3+x}{-3+2 x}} (648 x^5-972 x^6+288 x^7)}{729-972 x+324 x^2} \, dx\)

Optimal. Leaf size=23 \[ \left (e^{\frac {3+x}{-3+2 x}}+\frac {x^2}{3}\right )^4 \]

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Rubi [A]  time = 0.40, antiderivative size = 26, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 4, integrand size = 135, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {27, 12, 6688, 6686} \begin {gather*} \frac {1}{81} \left (x^2+3 e^{-\frac {x+3}{3-2 x}}\right )^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2916*E^((4*(3 + x))/(-3 + 2*x)) + 72*x^7 - 96*x^8 + 32*x^9 + E^((3*(3 + x))/(-3 + 2*x))*(1944*x - 5508*x
^2 + 864*x^3) + E^((2*(3 + x))/(-3 + 2*x))*(1944*x^3 - 3564*x^4 + 864*x^5) + E^((3 + x)/(-3 + 2*x))*(648*x^5 -
972*x^6 + 288*x^7))/(729 - 972*x + 324*x^2),x]

[Out]

(3/E^((3 + x)/(3 - 2*x)) + x^2)^4/81

Rule 12

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} \left (1944 x-5508 x^2+864 x^3\right )+e^{\frac {2 (3+x)}{-3+2 x}} \left (1944 x^3-3564 x^4+864 x^5\right )+e^{\frac {3+x}{-3+2 x}} \left (648 x^5-972 x^6+288 x^7\right )}{81 (-3+2 x)^2} \, dx\\ &=\frac {1}{81} \int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} \left (1944 x-5508 x^2+864 x^3\right )+e^{\frac {2 (3+x)}{-3+2 x}} \left (1944 x^3-3564 x^4+864 x^5\right )+e^{\frac {3+x}{-3+2 x}} \left (648 x^5-972 x^6+288 x^7\right )}{(-3+2 x)^2} \, dx\\ &=\frac {1}{81} \int \frac {4 \left (-27 e^{\frac {3+x}{-3+2 x}}+2 (3-2 x)^2 x\right ) \left (3 e^{\frac {3+x}{-3+2 x}}+x^2\right )^3}{(3-2 x)^2} \, dx\\ &=\frac {4}{81} \int \frac {\left (-27 e^{\frac {3+x}{-3+2 x}}+2 (3-2 x)^2 x\right ) \left (3 e^{\frac {3+x}{-3+2 x}}+x^2\right )^3}{(3-2 x)^2} \, dx\\ &=\frac {1}{81} \left (3 e^{-\frac {3+x}{3-2 x}}+x^2\right )^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 25, normalized size = 1.09 \begin {gather*} \frac {1}{81} \left (3 e^{\frac {3+x}{-3+2 x}}+x^2\right )^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2916*E^((4*(3 + x))/(-3 + 2*x)) + 72*x^7 - 96*x^8 + 32*x^9 + E^((3*(3 + x))/(-3 + 2*x))*(1944*x -
5508*x^2 + 864*x^3) + E^((2*(3 + x))/(-3 + 2*x))*(1944*x^3 - 3564*x^4 + 864*x^5) + E^((3 + x)/(-3 + 2*x))*(648
*x^5 - 972*x^6 + 288*x^7))/(729 - 972*x + 324*x^2),x]

[Out]

(3*E^((3 + x)/(-3 + 2*x)) + x^2)^4/81

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fricas [B]  time = 0.49, size = 72, normalized size = 3.13 \begin {gather*} \frac {1}{81} \, x^{8} + \frac {4}{27} \, x^{6} e^{\left (\frac {x + 3}{2 \, x - 3}\right )} + \frac {2}{3} \, x^{4} e^{\left (\frac {2 \, {\left (x + 3\right )}}{2 \, x - 3}\right )} + \frac {4}{3} \, x^{2} e^{\left (\frac {3 \, {\left (x + 3\right )}}{2 \, x - 3}\right )} + e^{\left (\frac {4 \, {\left (x + 3\right )}}{2 \, x - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2916*exp((3+x)/(2*x-3))^4+(864*x^3-5508*x^2+1944*x)*exp((3+x)/(2*x-3))^3+(864*x^5-3564*x^4+1944*x^
3)*exp((3+x)/(2*x-3))^2+(288*x^7-972*x^6+648*x^5)*exp((3+x)/(2*x-3))+32*x^9-96*x^8+72*x^7)/(324*x^2-972*x+729)
,x, algorithm="fricas")

[Out]

1/81*x^8 + 4/27*x^6*e^((x + 3)/(2*x - 3)) + 2/3*x^4*e^(2*(x + 3)/(2*x - 3)) + 4/3*x^2*e^(3*(x + 3)/(2*x - 3))
+ e^(4*(x + 3)/(2*x - 3))

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giac [B]  time = 0.36, size = 87, normalized size = 3.78 \begin {gather*} \frac {1}{81} \, {\left (x^{8} e^{6} + 12 \, x^{6} e^{\left (\frac {3 \, x}{2 \, x - 3} + 5\right )} + 54 \, x^{4} e^{\left (\frac {6 \, x}{2 \, x - 3} + 4\right )} + 108 \, x^{2} e^{\left (\frac {9 \, x}{2 \, x - 3} + 3\right )}\right )} e^{\left (-6\right )} + e^{\left (\frac {4 \, x}{2 \, x - 3} + \frac {12}{2 \, x - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2916*exp((3+x)/(2*x-3))^4+(864*x^3-5508*x^2+1944*x)*exp((3+x)/(2*x-3))^3+(864*x^5-3564*x^4+1944*x^
3)*exp((3+x)/(2*x-3))^2+(288*x^7-972*x^6+648*x^5)*exp((3+x)/(2*x-3))+32*x^9-96*x^8+72*x^7)/(324*x^2-972*x+729)
,x, algorithm="giac")

[Out]

1/81*(x^8*e^6 + 12*x^6*e^(3*x/(2*x - 3) + 5) + 54*x^4*e^(6*x/(2*x - 3) + 4) + 108*x^2*e^(9*x/(2*x - 3) + 3))*e
^(-6) + e^(4*x/(2*x - 3) + 12/(2*x - 3))

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maple [B]  time = 0.57, size = 73, normalized size = 3.17

method	result	size
risch	\(\frac {x^{8}}{81}+{\mathrm e}^{\frac {4 x +12}{2 x -3}}+\frac {4 \,{\mathrm e}^{\frac {3 x +9}{2 x -3}} x^{2}}{3}+\frac {2 \,{\mathrm e}^{\frac {2 x +6}{2 x -3}} x^{4}}{3}+\frac {4 \,{\mathrm e}^{\frac {3+x}{2 x -3}} x^{6}}{27}\)	\(73\)
derivativedivides	\(\frac {27 x}{16}+3 \,{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (2 x -3\right )}}+\frac {63 \left (2 x -3\right )^{2}}{64}+2 \,{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (2 x -3\right )}} \left (2 x -3\right )+\frac {9 \,{\mathrm e}^{1+\frac {9}{2 x -3}} \left (2 x -3\right )^{2}}{4}+\frac {{\mathrm e}^{1+\frac {9}{2 x -3}} \left (2 x -3\right )^{3}}{2}+\frac {{\mathrm e}^{1+\frac {9}{2 x -3}} \left (2 x -3\right )^{4}}{24}+\frac {9 \,{\mathrm e}^{1+\frac {9}{2 x -3}} \left (2 x -3\right )}{2}+\frac {{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{2}}{3}+\frac {27 \,{\mathrm e}^{1+\frac {9}{2 x -3}}}{8}+\frac {27 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )}{8}+\frac {45 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{2}}{16}+\frac {5 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{3}}{4}+\frac {5 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{4}}{16}+\frac {{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{5}}{24}+\frac {{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{6}}{432}+{\mathrm e}^{2+\frac {18}{2 x -3}}+\frac {21 \left (2 x -3\right )^{3}}{32}+\frac {35 \left (2 x -3\right )^{4}}{128}+\frac {\left (2 x -3\right )^{7}}{864}+\frac {\left (2 x -3\right )^{8}}{20736}+\frac {7 \left (2 x -3\right )^{5}}{96}+\frac {7 \left (2 x -3\right )^{6}}{576}-\frac {81}{32}+\frac {27 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}}}{16}\)	\(387\)
default	\(\frac {27 x}{16}+3 \,{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (2 x -3\right )}}+\frac {63 \left (2 x -3\right )^{2}}{64}+2 \,{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (2 x -3\right )}} \left (2 x -3\right )+\frac {9 \,{\mathrm e}^{1+\frac {9}{2 x -3}} \left (2 x -3\right )^{2}}{4}+\frac {{\mathrm e}^{1+\frac {9}{2 x -3}} \left (2 x -3\right )^{3}}{2}+\frac {{\mathrm e}^{1+\frac {9}{2 x -3}} \left (2 x -3\right )^{4}}{24}+\frac {9 \,{\mathrm e}^{1+\frac {9}{2 x -3}} \left (2 x -3\right )}{2}+\frac {{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{2}}{3}+\frac {27 \,{\mathrm e}^{1+\frac {9}{2 x -3}}}{8}+\frac {27 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )}{8}+\frac {45 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{2}}{16}+\frac {5 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{3}}{4}+\frac {5 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{4}}{16}+\frac {{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{5}}{24}+\frac {{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}} \left (2 x -3\right )^{6}}{432}+{\mathrm e}^{2+\frac {18}{2 x -3}}+\frac {21 \left (2 x -3\right )^{3}}{32}+\frac {35 \left (2 x -3\right )^{4}}{128}+\frac {\left (2 x -3\right )^{7}}{864}+\frac {\left (2 x -3\right )^{8}}{20736}+\frac {7 \left (2 x -3\right )^{5}}{96}+\frac {7 \left (2 x -3\right )^{6}}{576}-\frac {81}{32}+\frac {27 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (2 x -3\right )}}}{16}\)	\(387\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2916*exp((3+x)/(2*x-3))^4+(864*x^3-5508*x^2+1944*x)*exp((3+x)/(2*x-3))^3+(864*x^5-3564*x^4+1944*x^3)*exp
((3+x)/(2*x-3))^2+(288*x^7-972*x^6+648*x^5)*exp((3+x)/(2*x-3))+32*x^9-96*x^8+72*x^7)/(324*x^2-972*x+729),x,met
hod=_RETURNVERBOSE)

[Out]

1/81*x^8+exp(4*(3+x)/(2*x-3))+4/3*exp(3*(3+x)/(2*x-3))*x^2+2/3*exp(2*(3+x)/(2*x-3))*x^4+4/27*exp((3+x)/(2*x-3)
)*x^6

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maxima [B]  time = 0.45, size = 69, normalized size = 3.00 \begin {gather*} \frac {1}{81} \, x^{8} + \frac {4}{27} \, x^{6} e^{\left (\frac {9}{2 \, {\left (2 \, x - 3\right )}} + \frac {1}{2}\right )} + \frac {2}{3} \, x^{4} e^{\left (\frac {9}{2 \, x - 3} + 1\right )} + \frac {4}{3} \, x^{2} e^{\left (\frac {27}{2 \, {\left (2 \, x - 3\right )}} + \frac {3}{2}\right )} + e^{\left (\frac {18}{2 \, x - 3} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2916*exp((3+x)/(2*x-3))^4+(864*x^3-5508*x^2+1944*x)*exp((3+x)/(2*x-3))^3+(864*x^5-3564*x^4+1944*x^
3)*exp((3+x)/(2*x-3))^2+(288*x^7-972*x^6+648*x^5)*exp((3+x)/(2*x-3))+32*x^9-96*x^8+72*x^7)/(324*x^2-972*x+729)
,x, algorithm="maxima")

[Out]

1/81*x^8 + 4/27*x^6*e^(9/2/(2*x - 3) + 1/2) + 2/3*x^4*e^(9/(2*x - 3) + 1) + 4/3*x^2*e^(27/2/(2*x - 3) + 3/2) +
e^(18/(2*x - 3) + 2)

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mupad [B]  time = 5.66, size = 104, normalized size = 4.52 \begin {gather*} {\mathrm {e}}^{\frac {4\,x}{2\,x-3}+\frac {12}{2\,x-3}}+\frac {4\,x^6\,{\mathrm {e}}^{\frac {x}{2\,x-3}+\frac {3}{2\,x-3}}}{27}+\frac {2\,x^4\,{\mathrm {e}}^{\frac {2\,x}{2\,x-3}+\frac {6}{2\,x-3}}}{3}+\frac {4\,x^2\,{\mathrm {e}}^{\frac {3\,x}{2\,x-3}+\frac {9}{2\,x-3}}}{3}+\frac {x^8}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((3*(x + 3))/(2*x - 3))*(1944*x - 5508*x^2 + 864*x^3) - 2916*exp((4*(x + 3))/(2*x - 3)) + exp((x + 3)/
(2*x - 3))*(648*x^5 - 972*x^6 + 288*x^7) + exp((2*(x + 3))/(2*x - 3))*(1944*x^3 - 3564*x^4 + 864*x^5) + 72*x^7
- 96*x^8 + 32*x^9)/(324*x^2 - 972*x + 729),x)

[Out]

exp((4*x)/(2*x - 3) + 12/(2*x - 3)) + (4*x^6*exp(x/(2*x - 3) + 3/(2*x - 3)))/27 + (2*x^4*exp((2*x)/(2*x - 3) +
6/(2*x - 3)))/3 + (4*x^2*exp((3*x)/(2*x - 3) + 9/(2*x - 3)))/3 + x^8/81

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sympy [B]  time = 0.30, size = 70, normalized size = 3.04 \begin {gather*} \frac {x^{8}}{81} + \frac {4 x^{6} e^{\frac {x + 3}{2 x - 3}}}{27} + \frac {2 x^{4} e^{\frac {2 \left (x + 3\right )}{2 x - 3}}}{3} + \frac {4 x^{2} e^{\frac {3 \left (x + 3\right )}{2 x - 3}}}{3} + e^{\frac {4 \left (x + 3\right )}{2 x - 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2916*exp((3+x)/(2*x-3))**4+(864*x**3-5508*x**2+1944*x)*exp((3+x)/(2*x-3))**3+(864*x**5-3564*x**4+1
944*x**3)*exp((3+x)/(2*x-3))**2+(288*x**7-972*x**6+648*x**5)*exp((3+x)/(2*x-3))+32*x**9-96*x**8+72*x**7)/(324*
x**2-972*x+729),x)

[Out]

x**8/81 + 4*x**6*exp((x + 3)/(2*x - 3))/27 + 2*x**4*exp(2*(x + 3)/(2*x - 3))/3 + 4*x**2*exp(3*(x + 3)/(2*x - 3
))/3 + exp(4*(x + 3)/(2*x - 3))

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