3.90.84 \(\int \frac {-3-7 x^2+9 x^3-2 x^4+3 x^5+(3-9 x-16 x^2-12 x^3-9 x^4+3 x^5) \log (x)}{-x^2+3 x^3-3 x^4+x^5} \, dx\)

Optimal. Leaf size=25 \[ \frac {\left (1+x+\left (2+\frac {1}{x^2}\right ) x\right ) \left (3+x^2\right ) \log (x)}{(1-x)^2} \]

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Rubi [B]  time = 0.54, antiderivative size = 74, normalized size of antiderivative = 2.96, number of steps used = 19, number of rules used = 11, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.153, Rules used = {6741, 6742, 44, 37, 43, 2357, 2295, 2319, 2314, 31, 2304} \begin {gather*} -\frac {9 x^2}{2 (1-x)^2}-\frac {9}{1-x}+\frac {9}{2 (1-x)^2}-\frac {18 x \log (x)}{1-x}+3 x \log (x)+\frac {20 \log (x)}{(1-x)^2}-11 \log (x)+\frac {3 \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 - 7*x^2 + 9*x^3 - 2*x^4 + 3*x^5 + (3 - 9*x - 16*x^2 - 12*x^3 - 9*x^4 + 3*x^5)*Log[x])/(-x^2 + 3*x^3 -
3*x^4 + x^5),x]

[Out]

9/(2*(1 - x)^2) - 9/(1 - x) - (9*x^2)/(2*(1 - x)^2) - 11*Log[x] + (20*Log[x])/(1 - x)^2 + (3*Log[x])/x + 3*x*L
og[x] - (18*x*Log[x])/(1 - x)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2314

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

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 6741

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

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 {3+7 x^2-9 x^3+2 x^4-3 x^5-\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{(1-x)^3 x^2} \, dx\\ &=\int \left (-\frac {7}{(-1+x)^3}-\frac {3}{(-1+x)^3 x^2}+\frac {9 x}{(-1+x)^3}-\frac {2 x^2}{(-1+x)^3}+\frac {3 x^3}{(-1+x)^3}+\frac {\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{(-1+x)^3 x^2}\right ) \, dx\\ &=\frac {7}{2 (1-x)^2}-2 \int \frac {x^2}{(-1+x)^3} \, dx-3 \int \frac {1}{(-1+x)^3 x^2} \, dx+3 \int \frac {x^3}{(-1+x)^3} \, dx+9 \int \frac {x}{(-1+x)^3} \, dx+\int \frac {\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{(-1+x)^3 x^2} \, dx\\ &=\frac {7}{2 (1-x)^2}-\frac {9 x^2}{2 (1-x)^2}-2 \int \left (\frac {1}{(-1+x)^3}+\frac {2}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx+3 \int \left (1+\frac {1}{(-1+x)^3}+\frac {3}{(-1+x)^2}+\frac {3}{-1+x}\right ) \, dx-3 \int \left (\frac {1}{(-1+x)^3}-\frac {2}{(-1+x)^2}+\frac {3}{-1+x}-\frac {1}{x^2}-\frac {3}{x}\right ) \, dx+\int \left (3 \log (x)-\frac {40 \log (x)}{(-1+x)^3}-\frac {18 \log (x)}{(-1+x)^2}-\frac {3 \log (x)}{x^2}\right ) \, dx\\ &=\frac {9}{2 (1-x)^2}+\frac {11}{1-x}-\frac {3}{x}+3 x-\frac {9 x^2}{2 (1-x)^2}-2 \log (1-x)+9 \log (x)+3 \int \log (x) \, dx-3 \int \frac {\log (x)}{x^2} \, dx-18 \int \frac {\log (x)}{(-1+x)^2} \, dx-40 \int \frac {\log (x)}{(-1+x)^3} \, dx\\ &=\frac {9}{2 (1-x)^2}+\frac {11}{1-x}-\frac {9 x^2}{2 (1-x)^2}-2 \log (1-x)+9 \log (x)+\frac {20 \log (x)}{(1-x)^2}+\frac {3 \log (x)}{x}+3 x \log (x)-\frac {18 x \log (x)}{1-x}-18 \int \frac {1}{-1+x} \, dx-20 \int \frac {1}{(-1+x)^2 x} \, dx\\ &=\frac {9}{2 (1-x)^2}+\frac {11}{1-x}-\frac {9 x^2}{2 (1-x)^2}-20 \log (1-x)+9 \log (x)+\frac {20 \log (x)}{(1-x)^2}+\frac {3 \log (x)}{x}+3 x \log (x)-\frac {18 x \log (x)}{1-x}-20 \int \left (\frac {1}{1-x}+\frac {1}{(-1+x)^2}+\frac {1}{x}\right ) \, dx\\ &=\frac {9}{2 (1-x)^2}-\frac {9}{1-x}-\frac {9 x^2}{2 (1-x)^2}-11 \log (x)+\frac {20 \log (x)}{(1-x)^2}+\frac {3 \log (x)}{x}+3 x \log (x)-\frac {18 x \log (x)}{1-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 29, normalized size = 1.16 \begin {gather*} \frac {\left (3+3 x+10 x^2+x^3+3 x^4\right ) \log (x)}{(-1+x)^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 - 7*x^2 + 9*x^3 - 2*x^4 + 3*x^5 + (3 - 9*x - 16*x^2 - 12*x^3 - 9*x^4 + 3*x^5)*Log[x])/(-x^2 + 3*
x^3 - 3*x^4 + x^5),x]

[Out]

((3 + 3*x + 10*x^2 + x^3 + 3*x^4)*Log[x])/((-1 + x)^2*x)

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fricas [A]  time = 0.49, size = 33, normalized size = 1.32 \begin {gather*} \frac {{\left (3 \, x^{4} + x^{3} + 10 \, x^{2} + 3 \, x + 3\right )} \log \relax (x)}{x^{3} - 2 \, x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5-9*x^4-12*x^3-16*x^2-9*x+3)*log(x)+3*x^5-2*x^4+9*x^3-7*x^2-3)/(x^5-3*x^4+3*x^3-x^2),x, algori
thm="fricas")

[Out]

(3*x^4 + x^3 + 10*x^2 + 3*x + 3)*log(x)/(x^3 - 2*x^2 + x)

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giac [A]  time = 0.12, size = 34, normalized size = 1.36 \begin {gather*} {\left (3 \, x + \frac {2 \, {\left (9 \, x + 1\right )}}{x^{2} - 2 \, x + 1} + \frac {3}{x}\right )} \log \relax (x) + 7 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5-9*x^4-12*x^3-16*x^2-9*x+3)*log(x)+3*x^5-2*x^4+9*x^3-7*x^2-3)/(x^5-3*x^4+3*x^3-x^2),x, algori
thm="giac")

[Out]

(3*x + 2*(9*x + 1)/(x^2 - 2*x + 1) + 3/x)*log(x) + 7*log(x)

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maple [A]  time = 0.07, size = 39, normalized size = 1.56

method	result	size
norman	\(\frac {6 x^{3} \ln \relax (x )+8 x \ln \relax (x )+3 x^{4} \ln \relax (x )+3 \ln \relax (x )}{x \left (x -1\right )^{2}}-5 \ln \relax (x )\)	\(39\)
default	\(9 \ln \relax (x )+3 x \ln \relax (x )-\frac {20 \ln \relax (x ) x \left (x -2\right )}{\left (x -1\right )^{2}}+\frac {3 \ln \relax (x )}{x}+\frac {18 \ln \relax (x ) x}{x -1}\)	\(41\)
risch	\(\frac {\left (3 x^{4}-6 x^{3}+24 x^{2}-4 x +3\right ) \ln \relax (x )}{x \left (x^{2}-2 x +1\right )}+7 \ln \relax (x )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^5-9*x^4-12*x^3-16*x^2-9*x+3)*ln(x)+3*x^5-2*x^4+9*x^3-7*x^2-3)/(x^5-3*x^4+3*x^3-x^2),x,method=_RETURN
VERBOSE)

[Out]

(6*x^3*ln(x)+8*x*ln(x)+3*x^4*ln(x)+3*ln(x))/x/(x-1)^2-5*ln(x)

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maxima [B]  time = 0.41, size = 152, normalized size = 6.08 \begin {gather*} 3 \, x - \frac {3 \, x^{4} - 6 \, x^{3} - 20 \, x^{2} - {\left (3 \, x^{4} - 6 \, x^{3} + 24 \, x^{2} - 4 \, x + 3\right )} \log \relax (x) + 26 \, x - 3}{x^{3} - 2 \, x^{2} + x} - \frac {3 \, {\left (6 \, x^{2} - 9 \, x + 2\right )}}{2 \, {\left (x^{3} - 2 \, x^{2} + x\right )}} - \frac {3 \, {\left (6 \, x - 5\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {4 \, x - 3}{x^{2} - 2 \, x + 1} - \frac {9 \, {\left (2 \, x - 1\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {7}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + 7 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5-9*x^4-12*x^3-16*x^2-9*x+3)*log(x)+3*x^5-2*x^4+9*x^3-7*x^2-3)/(x^5-3*x^4+3*x^3-x^2),x, algori
thm="maxima")

[Out]

3*x - (3*x^4 - 6*x^3 - 20*x^2 - (3*x^4 - 6*x^3 + 24*x^2 - 4*x + 3)*log(x) + 26*x - 3)/(x^3 - 2*x^2 + x) - 3/2*
(6*x^2 - 9*x + 2)/(x^3 - 2*x^2 + x) - 3/2*(6*x - 5)/(x^2 - 2*x + 1) + (4*x - 3)/(x^2 - 2*x + 1) - 9/2*(2*x - 1
)/(x^2 - 2*x + 1) + 7/2/(x^2 - 2*x + 1) + 7*log(x)

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mupad [B]  time = 7.89, size = 29, normalized size = 1.16 \begin {gather*} \frac {\ln \relax (x)\,\left (3\,x^4+x^3+10\,x^2+3\,x+3\right )}{x\,{\left (x-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(9*x + 16*x^2 + 12*x^3 + 9*x^4 - 3*x^5 - 3) + 7*x^2 - 9*x^3 + 2*x^4 - 3*x^5 + 3)/(x^2 - 3*x^3 + 3*
x^4 - x^5),x)

[Out]

(log(x)*(3*x + 10*x^2 + x^3 + 3*x^4 + 3))/(x*(x - 1)^2)

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sympy [A]  time = 0.38, size = 37, normalized size = 1.48 \begin {gather*} 7 \log {\relax (x )} + \frac {\left (3 x^{4} - 6 x^{3} + 24 x^{2} - 4 x + 3\right ) \log {\relax (x )}}{x^{3} - 2 x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**5-9*x**4-12*x**3-16*x**2-9*x+3)*ln(x)+3*x**5-2*x**4+9*x**3-7*x**2-3)/(x**5-3*x**4+3*x**3-x**2
),x)

[Out]

7*log(x) + (3*x**4 - 6*x**3 + 24*x**2 - 4*x + 3)*log(x)/(x**3 - 2*x**2 + x)

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