3.73.93 \(\int \frac {-27 x^5-27 x^6+45 x^7-17 x^8+2 x^9+e^2 (48+48 x-16 x^2-16 x^3)+e (36 x^3-36 x^4-4 x^5+4 x^6)}{-27 x^5+27 x^6-9 x^7+x^8} \, dx\)

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

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Rubi [B]  time = 0.13, antiderivative size = 79, normalized size of antiderivative = 3.04, number of steps used = 2, number of rules used = 1, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {2074} \begin {gather*} \frac {4 e^2}{9 x^4}+\frac {32 e^2}{27 x^3}+x^2+\frac {32 e^2}{27 x^2}+x+\frac {16 e (81+10 e)}{243 (3-x)}+\frac {64 e^2}{81 (3-x)^2}+\frac {4 e (81+40 e)}{243 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-27*x^5 - 27*x^6 + 45*x^7 - 17*x^8 + 2*x^9 + E^2*(48 + 48*x - 16*x^2 - 16*x^3) + E*(36*x^3 - 36*x^4 - 4*x
^5 + 4*x^6))/(-27*x^5 + 27*x^6 - 9*x^7 + x^8),x]

[Out]

(64*E^2)/(81*(3 - x)^2) + (16*E*(81 + 10*E))/(243*(3 - x)) + (4*E^2)/(9*x^4) + (32*E^2)/(27*x^3) + (32*E^2)/(2
7*x^2) + (4*E*(81 + 40*E))/(243*x) + x + x^2

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {128 e^2}{81 (-3+x)^3}+\frac {16 e (81+10 e)}{243 (-3+x)^2}-\frac {16 e^2}{9 x^5}-\frac {32 e^2}{9 x^4}-\frac {64 e^2}{27 x^3}-\frac {4 e (81+40 e)}{243 x^2}+2 x\right ) \, dx\\ &=\frac {64 e^2}{81 (3-x)^2}+\frac {16 e (81+10 e)}{243 (3-x)}+\frac {4 e^2}{9 x^4}+\frac {32 e^2}{27 x^3}+\frac {32 e^2}{27 x^2}+\frac {4 e (81+40 e)}{243 x}+x+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 39, normalized size = 1.50 \begin {gather*} \frac {(1+x) \left (-4 e (-3+x) x^3+(-3+x)^2 x^5+4 e^2 (1+x)\right )}{(-3+x)^2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-27*x^5 - 27*x^6 + 45*x^7 - 17*x^8 + 2*x^9 + E^2*(48 + 48*x - 16*x^2 - 16*x^3) + E*(36*x^3 - 36*x^4
- 4*x^5 + 4*x^6))/(-27*x^5 + 27*x^6 - 9*x^7 + x^8),x]

[Out]

((1 + x)*(-4*E*(-3 + x)*x^3 + (-3 + x)^2*x^5 + 4*E^2*(1 + x)))/((-3 + x)^2*x^4)

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fricas [B]  time = 0.45, size = 66, normalized size = 2.54 \begin {gather*} \frac {x^{8} - 5 \, x^{7} + 3 \, x^{6} + 9 \, x^{5} + 4 \, {\left (x^{2} + 2 \, x + 1\right )} e^{2} - 4 \, {\left (x^{5} - 2 \, x^{4} - 3 \, x^{3}\right )} e}{x^{6} - 6 \, x^{5} + 9 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^3-16*x^2+48*x+48)*exp(1)^2+(4*x^6-4*x^5-36*x^4+36*x^3)*exp(1)+2*x^9-17*x^8+45*x^7-27*x^6-27*
x^5)/(x^8-9*x^7+27*x^6-27*x^5),x, algorithm="fricas")

[Out]

(x^8 - 5*x^7 + 3*x^6 + 9*x^5 + 4*(x^2 + 2*x + 1)*e^2 - 4*(x^5 - 2*x^4 - 3*x^3)*e)/(x^6 - 6*x^5 + 9*x^4)

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giac [B]  time = 0.13, size = 67, normalized size = 2.58 \begin {gather*} x^{2} + x - \frac {16 \, {\left (10 \, x e^{2} + 81 \, x e - 42 \, e^{2} - 243 \, e\right )}}{243 \, {\left (x - 3\right )}^{2}} + \frac {4 \, {\left (40 \, x^{3} e^{2} + 81 \, x^{3} e + 72 \, x^{2} e^{2} + 72 \, x e^{2} + 27 \, e^{2}\right )}}{243 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^3-16*x^2+48*x+48)*exp(1)^2+(4*x^6-4*x^5-36*x^4+36*x^3)*exp(1)+2*x^9-17*x^8+45*x^7-27*x^6-27*
x^5)/(x^8-9*x^7+27*x^6-27*x^5),x, algorithm="giac")

[Out]

x^2 + x - 16/243*(10*x*e^2 + 81*x*e - 42*e^2 - 243*e)/(x - 3)^2 + 4/243*(40*x^3*e^2 + 81*x^3*e + 72*x^2*e^2 +
72*x*e^2 + 27*e^2)/x^4

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maple [B]  time = 0.08, size = 58, normalized size = 2.23

method	result	size
risch	\(x^{2}+x +\frac {-4 x^{5} {\mathrm e}+8 x^{4} {\mathrm e}+4 x^{2} {\mathrm e}^{2}+12 x^{3} {\mathrm e}+8 \,{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}}{x^{4} \left (x^{2}-6 x +9\right )}\)	\(58\)
default	\(x^{2}+x -\frac {-\frac {160 \,{\mathrm e}^{2}}{243}-\frac {4 \,{\mathrm e}}{3}}{x}+\frac {4 \,{\mathrm e}^{2}}{9 x^{4}}+\frac {32 \,{\mathrm e}^{2}}{27 x^{3}}+\frac {32 \,{\mathrm e}^{2}}{27 x^{2}}-\frac {\frac {160 \,{\mathrm e}^{2}}{243}+\frac {16 \,{\mathrm e}}{3}}{x -3}+\frac {64 \,{\mathrm e}^{2}}{81 \left (x -3\right )^{2}}\)	\(66\)
norman	\(\frac {x^{8}+\left (8 \,{\mathrm e}-27\right ) x^{4}+\left (-4 \,{\mathrm e}+27\right ) x^{5}-5 x^{7}+4 \,{\mathrm e}^{2}+8 \,{\mathrm e}^{2} x +4 x^{2} {\mathrm e}^{2}+12 x^{3} {\mathrm e}}{x^{4} \left (x -3\right )^{2}}\)	\(68\)
gosper	\(\frac {x^{8}-5 x^{7}-4 x^{5} {\mathrm e}+8 x^{4} {\mathrm e}+27 x^{5}+4 x^{2} {\mathrm e}^{2}+12 x^{3} {\mathrm e}-27 x^{4}+8 \,{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}}{x^{4} \left (x^{2}-6 x +9\right )}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^3-16*x^2+48*x+48)*exp(1)^2+(4*x^6-4*x^5-36*x^4+36*x^3)*exp(1)+2*x^9-17*x^8+45*x^7-27*x^6-27*x^5)/(
x^8-9*x^7+27*x^6-27*x^5),x,method=_RETURNVERBOSE)

[Out]

x^2+x+(-4*x^5*exp(1)+8*x^4*exp(1)+4*x^2*exp(2)+12*x^3*exp(1)+8*exp(2)*x+4*exp(2))/x^4/(x^2-6*x+9)

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maxima [B]  time = 0.36, size = 60, normalized size = 2.31 \begin {gather*} x^{2} + x - \frac {4 \, {\left (x^{5} e - 2 \, x^{4} e - 3 \, x^{3} e - x^{2} e^{2} - 2 \, x e^{2} - e^{2}\right )}}{x^{6} - 6 \, x^{5} + 9 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^3-16*x^2+48*x+48)*exp(1)^2+(4*x^6-4*x^5-36*x^4+36*x^3)*exp(1)+2*x^9-17*x^8+45*x^7-27*x^6-27*
x^5)/(x^8-9*x^7+27*x^6-27*x^5),x, algorithm="maxima")

[Out]

x^2 + x - 4*(x^5*e - 2*x^4*e - 3*x^3*e - x^2*e^2 - 2*x*e^2 - e^2)/(x^6 - 6*x^5 + 9*x^4)

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mupad [B]  time = 4.42, size = 49, normalized size = 1.88 \begin {gather*} \frac {\left (x+1\right )\,\left (x^7-6\,x^6+9\,x^5-4\,\mathrm {e}\,x^4+12\,\mathrm {e}\,x^3+4\,{\mathrm {e}}^2\,x+4\,{\mathrm {e}}^2\right )}{x^4\,{\left (x-3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2)*(48*x - 16*x^2 - 16*x^3 + 48) - 27*x^5 - 27*x^6 + 45*x^7 - 17*x^8 + 2*x^9 + exp(1)*(36*x^3 - 36*x
^4 - 4*x^5 + 4*x^6))/(27*x^5 - 27*x^6 + 9*x^7 - x^8),x)

[Out]

((x + 1)*(4*exp(2) + 4*x*exp(2) + 12*x^3*exp(1) - 4*x^4*exp(1) + 9*x^5 - 6*x^6 + x^7))/(x^4*(x - 3)^2)

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sympy [B]  time = 1.44, size = 63, normalized size = 2.42 \begin {gather*} x^{2} + x + \frac {- 4 e x^{5} + 8 e x^{4} + 12 e x^{3} + 4 x^{2} e^{2} + 8 x e^{2} + 4 e^{2}}{x^{6} - 6 x^{5} + 9 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**3-16*x**2+48*x+48)*exp(1)**2+(4*x**6-4*x**5-36*x**4+36*x**3)*exp(1)+2*x**9-17*x**8+45*x**7-
27*x**6-27*x**5)/(x**8-9*x**7+27*x**6-27*x**5),x)

[Out]

x**2 + x + (-4*E*x**5 + 8*E*x**4 + 12*E*x**3 + 4*x**2*exp(2) + 8*x*exp(2) + 4*exp(2))/(x**6 - 6*x**5 + 9*x**4)

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