3.79.87 \(\int \frac {216-112 x+4 x^2-x^3-2 x^4+x^5+6 x^4 \log (2-x)}{-2 x^4+x^5} \, dx\)

Optimal. Leaf size=36 \[ x+\frac {5+x+3 \left (-\frac {4 \left (-3+\frac {x}{2}\right )}{x}+x^2 \log ^2(2-x)\right )}{x^2} \]

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Rubi [A]  time = 0.27, antiderivative size = 25, normalized size of antiderivative = 0.69, number of steps used = 7, number of rules used = 5, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1593, 6742, 14, 2390, 2301} \begin {gather*} \frac {36}{x^3}-\frac {1}{x^2}+x+\frac {1}{x}+3 \log ^2(2-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(216 - 112*x + 4*x^2 - x^3 - 2*x^4 + x^5 + 6*x^4*Log[2 - x])/(-2*x^4 + x^5),x]

[Out]

36/x^3 - x^(-2) + x^(-1) + x + 3*Log[2 - x]^2

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 1593

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {216-112 x+4 x^2-x^3-2 x^4+x^5+6 x^4 \log (2-x)}{(-2+x) x^4} \, dx\\ &=\int \left (\frac {-108+2 x-x^2+x^4}{x^4}+\frac {6 \log (2-x)}{-2+x}\right ) \, dx\\ &=6 \int \frac {\log (2-x)}{-2+x} \, dx+\int \frac {-108+2 x-x^2+x^4}{x^4} \, dx\\ &=6 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,2-x\right )+\int \left (1-\frac {108}{x^4}+\frac {2}{x^3}-\frac {1}{x^2}\right ) \, dx\\ &=\frac {36}{x^3}-\frac {1}{x^2}+\frac {1}{x}+x+3 \log ^2(2-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 25, normalized size = 0.69 \begin {gather*} \frac {36}{x^3}-\frac {1}{x^2}+\frac {1}{x}+x+3 \log ^2(2-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(216 - 112*x + 4*x^2 - x^3 - 2*x^4 + x^5 + 6*x^4*Log[2 - x])/(-2*x^4 + x^5),x]

[Out]

36/x^3 - x^(-2) + x^(-1) + x + 3*Log[2 - x]^2

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fricas [A]  time = 0.53, size = 28, normalized size = 0.78 \begin {gather*} \frac {3 \, x^{3} \log \left (-x + 2\right )^{2} + x^{4} + x^{2} - x + 36}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^4*log(2-x)+x^5-2*x^4-x^3+4*x^2-112*x+216)/(x^5-2*x^4),x, algorithm="fricas")

[Out]

(3*x^3*log(-x + 2)^2 + x^4 + x^2 - x + 36)/x^3

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giac [A]  time = 0.16, size = 24, normalized size = 0.67 \begin {gather*} 3 \, \log \left (-x + 2\right )^{2} + x + \frac {x^{2} - x + 36}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^4*log(2-x)+x^5-2*x^4-x^3+4*x^2-112*x+216)/(x^5-2*x^4),x, algorithm="giac")

[Out]

3*log(-x + 2)^2 + x + (x^2 - x + 36)/x^3

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maple [A]  time = 0.24, size = 27, normalized size = 0.75

method	result	size
derivativedivides	\(-2+x -\frac {1}{x^{2}}+\frac {1}{x}+\frac {36}{x^{3}}+3 \ln \left (2-x \right )^{2}\)	\(27\)
default	\(-2+x -\frac {1}{x^{2}}+\frac {1}{x}+\frac {36}{x^{3}}+3 \ln \left (2-x \right )^{2}\)	\(27\)
risch	\(3 \ln \left (2-x \right )^{2}+\frac {x^{4}+x^{2}-x +36}{x^{3}}\)	\(27\)
norman	\(\frac {36+x^{2}+x^{4}-x +3 x^{3} \ln \left (2-x \right )^{2}}{x^{3}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^4*ln(2-x)+x^5-2*x^4-x^3+4*x^2-112*x+216)/(x^5-2*x^4),x,method=_RETURNVERBOSE)

[Out]

-2+x-1/x^2+1/x+36/x^3+3*ln(2-x)^2

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maxima [A]  time = 0.37, size = 40, normalized size = 1.11 \begin {gather*} 3 \, \log \left (-x + 2\right )^{2} + x - \frac {28 \, {\left (x + 1\right )}}{x^{2}} + \frac {2}{x} + \frac {9 \, {\left (3 \, x^{2} + 3 \, x + 4\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^4*log(2-x)+x^5-2*x^4-x^3+4*x^2-112*x+216)/(x^5-2*x^4),x, algorithm="maxima")

[Out]

3*log(-x + 2)^2 + x - 28*(x + 1)/x^2 + 2/x + 9*(3*x^2 + 3*x + 4)/x^3

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mupad [B]  time = 5.35, size = 24, normalized size = 0.67 \begin {gather*} x+3\,{\ln \left (2-x\right )}^2+\frac {x^2-x+36}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x^2 - 112*x - x^3 - 2*x^4 + x^5 + 6*x^4*log(2 - x) + 216)/(2*x^4 - x^5),x)

[Out]

x + 3*log(2 - x)^2 + (x^2 - x + 36)/x^3

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sympy [A]  time = 0.14, size = 19, normalized size = 0.53 \begin {gather*} x + 3 \log {\left (2 - x \right )}^{2} + \frac {x^{2} - x + 36}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x**4*ln(2-x)+x**5-2*x**4-x**3+4*x**2-112*x+216)/(x**5-2*x**4),x)

[Out]

x + 3*log(2 - x)**2 + (x**2 - x + 36)/x**3

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