3.82.90 \(\int \frac {e^x (-8 x^3+4 x^4+2 x^5)}{27-54 x+27 x^2} \, dx\)

Optimal. Leaf size=15 \[ \frac {2 e^x x^4}{27 (-1+x)} \]

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Rubi [B]  time = 0.20, antiderivative size = 50, normalized size of antiderivative = 3.33, number of steps used = 18, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {27, 12, 1594, 2199, 2194, 2177, 2178, 2176} \begin {gather*} \frac {2 e^x x^3}{27}+\frac {2 e^x x^2}{27}+\frac {2 e^x x}{27}+\frac {2 e^x}{27}-\frac {2 e^x}{27 (1-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(-8*x^3 + 4*x^4 + 2*x^5))/(27 - 54*x + 27*x^2),x]

[Out]

(2*E^x)/27 - (2*E^x)/(27*(1 - x)) + (2*E^x*x)/27 + (2*E^x*x^2)/27 + (2*E^x*x^3)/27

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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - 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 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), 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] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$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 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-8 x^3+4 x^4+2 x^5\right )}{27 (-1+x)^2} \, dx\\ &=\frac {1}{27} \int \frac {e^x \left (-8 x^3+4 x^4+2 x^5\right )}{(-1+x)^2} \, dx\\ &=\frac {1}{27} \int \frac {e^x x^3 \left (-8+4 x+2 x^2\right )}{(-1+x)^2} \, dx\\ &=\frac {1}{27} \int \left (4 e^x-\frac {2 e^x}{(-1+x)^2}+\frac {2 e^x}{-1+x}+6 e^x x+8 e^x x^2+2 e^x x^3\right ) \, dx\\ &=-\left (\frac {2}{27} \int \frac {e^x}{(-1+x)^2} \, dx\right )+\frac {2}{27} \int \frac {e^x}{-1+x} \, dx+\frac {2}{27} \int e^x x^3 \, dx+\frac {4 \int e^x \, dx}{27}+\frac {2}{9} \int e^x x \, dx+\frac {8}{27} \int e^x x^2 \, dx\\ &=\frac {4 e^x}{27}-\frac {2 e^x}{27 (1-x)}+\frac {2 e^x x}{9}+\frac {8 e^x x^2}{27}+\frac {2 e^x x^3}{27}+\frac {2}{27} e \text {Ei}(-1+x)-\frac {2}{27} \int \frac {e^x}{-1+x} \, dx-\frac {2 \int e^x \, dx}{9}-\frac {2}{9} \int e^x x^2 \, dx-\frac {16}{27} \int e^x x \, dx\\ &=-\frac {2 e^x}{27}-\frac {2 e^x}{27 (1-x)}-\frac {10 e^x x}{27}+\frac {2 e^x x^2}{27}+\frac {2 e^x x^3}{27}+\frac {4}{9} \int e^x x \, dx+\frac {16 \int e^x \, dx}{27}\\ &=\frac {14 e^x}{27}-\frac {2 e^x}{27 (1-x)}+\frac {2 e^x x}{27}+\frac {2 e^x x^2}{27}+\frac {2 e^x x^3}{27}-\frac {4 \int e^x \, dx}{9}\\ &=\frac {2 e^x}{27}-\frac {2 e^x}{27 (1-x)}+\frac {2 e^x x}{27}+\frac {2 e^x x^2}{27}+\frac {2 e^x x^3}{27}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 15, normalized size = 1.00 \begin {gather*} \frac {2 e^x x^4}{27 (-1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-8*x^3 + 4*x^4 + 2*x^5))/(27 - 54*x + 27*x^2),x]

[Out]

(2*E^x*x^4)/(27*(-1 + x))

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fricas [A]  time = 0.60, size = 12, normalized size = 0.80 \begin {gather*} \frac {2 \, x^{4} e^{x}}{27 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+4*x^4-8*x^3)*exp(x)/(27*x^2-54*x+27),x, algorithm="fricas")

[Out]

2/27*x^4*e^x/(x - 1)

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giac [A]  time = 0.18, size = 12, normalized size = 0.80 \begin {gather*} \frac {2 \, x^{4} e^{x}}{27 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+4*x^4-8*x^3)*exp(x)/(27*x^2-54*x+27),x, algorithm="giac")

[Out]

2/27*x^4*e^x/(x - 1)

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maple [A]  time = 0.38, size = 13, normalized size = 0.87

method	result	size
gosper	\(\frac {2 x^{4} {\mathrm e}^{x}}{27 \left (x -1\right )}\)	\(13\)
norman	\(\frac {2 x^{4} {\mathrm e}^{x}}{27 \left (x -1\right )}\)	\(13\)
risch	\(\frac {2 x^{4} {\mathrm e}^{x}}{27 \left (x -1\right )}\)	\(13\)
default	\(\frac {2 \,{\mathrm e}^{x} x^{3}}{27}+\frac {2 \,{\mathrm e}^{x} x^{2}}{27}+\frac {2 \,{\mathrm e}^{x} x}{27}+\frac {2 \,{\mathrm e}^{x}}{27}+\frac {2 \,{\mathrm e}^{x}}{27 \left (x -1\right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^5+4*x^4-8*x^3)*exp(x)/(27*x^2-54*x+27),x,method=_RETURNVERBOSE)

[Out]

2/27*x^4/(x-1)*exp(x)

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maxima [A]  time = 0.41, size = 12, normalized size = 0.80 \begin {gather*} \frac {2 \, x^{4} e^{x}}{27 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+4*x^4-8*x^3)*exp(x)/(27*x^2-54*x+27),x, algorithm="maxima")

[Out]

2/27*x^4*e^x/(x - 1)

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mupad [B]  time = 5.01, size = 14, normalized size = 0.93 \begin {gather*} \frac {2\,x^4\,{\mathrm {e}}^x}{27\,\left (x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(4*x^4 - 8*x^3 + 2*x^5))/(27*x^2 - 54*x + 27),x)

[Out]

(2*x^4*exp(x))/(27*(x - 1))

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sympy [A]  time = 0.11, size = 12, normalized size = 0.80 \begin {gather*} \frac {2 x^{4} e^{x}}{27 x - 27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**5+4*x**4-8*x**3)*exp(x)/(27*x**2-54*x+27),x)

[Out]

2*x**4*exp(x)/(27*x - 27)

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