3.32.36 \(\int \frac {48-320 x-368 x^2-96 x^3-18 x^4-21 x^5-8 x^6-x^7+(-144-136 x-32 x^2) \log (2 x+x^2)}{18 x^4+21 x^5+8 x^6+x^7} \, dx\)

Optimal. Leaf size=26 \[ -x+\frac {8 \left (4 x+\log \left (2 x+x^2\right )\right )}{x^3 (3+x)} \]

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Rubi [B]  time = 1.12, antiderivative size = 143, normalized size of antiderivative = 5.50, number of steps used = 44, number of rules used = 16, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {6741, 6742, 44, 88, 72, 77, 2513, 2357, 2304, 2314, 31, 2418, 2395, 36, 29, 74} \begin {gather*} \frac {8 \log (x)}{3 x^3}+\frac {8 \log (x+2)}{3 x^3}-\frac {8 (\log (x)+\log (x+2)-\log (x (x+2)))}{(x+3) x^3}+\frac {32}{3 x^2}-\frac {8 \log (x)}{9 x^2}-\frac {8 \log (x+2)}{9 x^2}-x+\frac {32}{9 (x+3)}-\frac {32}{9 x}+\frac {8 x \log (x)}{81 (x+3)}-\frac {8 \log (x)}{81}-\frac {8 \log (x+2)}{27 (x+3)}+\frac {8 \log (x)}{27 x}+\frac {8 \log (x+2)}{27 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(48 - 320*x - 368*x^2 - 96*x^3 - 18*x^4 - 21*x^5 - 8*x^6 - x^7 + (-144 - 136*x - 32*x^2)*Log[2*x + x^2])/(
18*x^4 + 21*x^5 + 8*x^6 + x^7),x]

[Out]

32/(3*x^2) - 32/(9*x) - x + 32/(9*(3 + x)) - (8*Log[x])/81 + (8*Log[x])/(3*x^3) - (8*Log[x])/(9*x^2) + (8*Log[
x])/(27*x) + (8*x*Log[x])/(81*(3 + x)) + (8*Log[2 + x])/(3*x^3) - (8*Log[2 + x])/(9*x^2) + (8*Log[2 + x])/(27*
x) - (8*Log[2 + x])/(27*(3 + x)) - (8*(Log[x] + Log[2 + x] - Log[x*(2 + x)]))/(x^3*(3 + x))

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 74

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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 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 2395

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2513

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

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 {48-320 x-368 x^2-96 x^3-18 x^4-21 x^5-8 x^6-x^7+\left (-144-136 x-32 x^2\right ) \log \left (2 x+x^2\right )}{x^4 (2+x) (3+x)^2} \, dx\\ &=\int \left (-\frac {18}{(2+x) (3+x)^2}+\frac {48}{x^4 (2+x) (3+x)^2}-\frac {320}{x^3 (2+x) (3+x)^2}-\frac {368}{x^2 (2+x) (3+x)^2}-\frac {96}{x (2+x) (3+x)^2}-\frac {21 x}{(2+x) (3+x)^2}-\frac {8 x^2}{(2+x) (3+x)^2}-\frac {x^3}{(2+x) (3+x)^2}-\frac {8 (9+4 x) \log (x (2+x))}{x^4 (3+x)^2}\right ) \, dx\\ &=-\left (8 \int \frac {x^2}{(2+x) (3+x)^2} \, dx\right )-8 \int \frac {(9+4 x) \log (x (2+x))}{x^4 (3+x)^2} \, dx-18 \int \frac {1}{(2+x) (3+x)^2} \, dx-21 \int \frac {x}{(2+x) (3+x)^2} \, dx+48 \int \frac {1}{x^4 (2+x) (3+x)^2} \, dx-96 \int \frac {1}{x (2+x) (3+x)^2} \, dx-320 \int \frac {1}{x^3 (2+x) (3+x)^2} \, dx-368 \int \frac {1}{x^2 (2+x) (3+x)^2} \, dx-\int \frac {x^3}{(2+x) (3+x)^2} \, dx\\ &=-\left (8 \int \left (\frac {4}{2+x}-\frac {9}{(3+x)^2}-\frac {3}{3+x}\right ) \, dx\right )-8 \int \frac {(9+4 x) \log (x)}{x^4 (3+x)^2} \, dx-8 \int \frac {(9+4 x) \log (2+x)}{x^4 (3+x)^2} \, dx-18 \int \left (\frac {1}{-3-x}+\frac {1}{2+x}-\frac {1}{(3+x)^2}\right ) \, dx-21 \int \left (-\frac {2}{2+x}+\frac {3}{(3+x)^2}+\frac {2}{3+x}\right ) \, dx+48 \int \left (\frac {1}{18 x^4}-\frac {7}{108 x^3}+\frac {11}{216 x^2}-\frac {131}{3888 x}+\frac {1}{16 (2+x)}-\frac {1}{81 (3+x)^2}-\frac {7}{243 (3+x)}\right ) \, dx-96 \int \left (\frac {1}{18 x}-\frac {1}{2 (2+x)}+\frac {1}{3 (3+x)^2}+\frac {4}{9 (3+x)}\right ) \, dx-320 \int \left (\frac {1}{18 x^3}-\frac {7}{108 x^2}+\frac {11}{216 x}-\frac {1}{8 (2+x)}+\frac {1}{27 (3+x)^2}+\frac {2}{27 (3+x)}\right ) \, dx-368 \int \left (\frac {1}{18 x^2}-\frac {7}{108 x}+\frac {1}{4 (2+x)}-\frac {1}{9 (3+x)^2}-\frac {5}{27 (3+x)}\right ) \, dx+(8 (\log (x)+\log (2+x)-\log (x (2+x)))) \int \frac {9+4 x}{x^4 (3+x)^2} \, dx-\int \left (1-\frac {8}{2+x}+\frac {27}{(3+x)^2}\right ) \, dx\\ &=-\frac {8}{9 x^3}+\frac {94}{9 x^2}-\frac {74}{27 x}-x+\frac {32}{9 (3+x)}+\frac {49 \log (x)}{81}-\log (2+x)-\frac {8 (\log (x)+\log (2+x)-\log (x (2+x)))}{x^3 (3+x)}+\frac {32}{81} \log (3+x)-8 \int \left (\frac {\log (x)}{x^4}-\frac {2 \log (x)}{9 x^3}+\frac {\log (x)}{27 x^2}-\frac {\log (x)}{27 (3+x)^2}\right ) \, dx-8 \int \left (\frac {\log (2+x)}{x^4}-\frac {2 \log (2+x)}{9 x^3}+\frac {\log (2+x)}{27 x^2}-\frac {\log (2+x)}{27 (3+x)^2}\right ) \, dx\\ &=-\frac {8}{9 x^3}+\frac {94}{9 x^2}-\frac {74}{27 x}-x+\frac {32}{9 (3+x)}+\frac {49 \log (x)}{81}-\log (2+x)-\frac {8 (\log (x)+\log (2+x)-\log (x (2+x)))}{x^3 (3+x)}+\frac {32}{81} \log (3+x)-\frac {8}{27} \int \frac {\log (x)}{x^2} \, dx+\frac {8}{27} \int \frac {\log (x)}{(3+x)^2} \, dx-\frac {8}{27} \int \frac {\log (2+x)}{x^2} \, dx+\frac {8}{27} \int \frac {\log (2+x)}{(3+x)^2} \, dx+\frac {16}{9} \int \frac {\log (x)}{x^3} \, dx+\frac {16}{9} \int \frac {\log (2+x)}{x^3} \, dx-8 \int \frac {\log (x)}{x^4} \, dx-8 \int \frac {\log (2+x)}{x^4} \, dx\\ &=\frac {10}{x^2}-\frac {22}{9 x}-x+\frac {32}{9 (3+x)}+\frac {49 \log (x)}{81}+\frac {8 \log (x)}{3 x^3}-\frac {8 \log (x)}{9 x^2}+\frac {8 \log (x)}{27 x}+\frac {8 x \log (x)}{81 (3+x)}-\log (2+x)+\frac {8 \log (2+x)}{3 x^3}-\frac {8 \log (2+x)}{9 x^2}+\frac {8 \log (2+x)}{27 x}-\frac {8 \log (2+x)}{27 (3+x)}-\frac {8 (\log (x)+\log (2+x)-\log (x (2+x)))}{x^3 (3+x)}+\frac {32}{81} \log (3+x)-\frac {8}{81} \int \frac {1}{3+x} \, dx-\frac {8}{27} \int \frac {1}{x (2+x)} \, dx+\frac {8}{27} \int \frac {1}{(2+x) (3+x)} \, dx+\frac {8}{9} \int \frac {1}{x^2 (2+x)} \, dx-\frac {8}{3} \int \frac {1}{x^3 (2+x)} \, dx\\ &=\frac {10}{x^2}-\frac {22}{9 x}-x+\frac {32}{9 (3+x)}+\frac {49 \log (x)}{81}+\frac {8 \log (x)}{3 x^3}-\frac {8 \log (x)}{9 x^2}+\frac {8 \log (x)}{27 x}+\frac {8 x \log (x)}{81 (3+x)}-\log (2+x)+\frac {8 \log (2+x)}{3 x^3}-\frac {8 \log (2+x)}{9 x^2}+\frac {8 \log (2+x)}{27 x}-\frac {8 \log (2+x)}{27 (3+x)}-\frac {8 (\log (x)+\log (2+x)-\log (x (2+x)))}{x^3 (3+x)}+\frac {8}{27} \log (3+x)-\frac {4}{27} \int \frac {1}{x} \, dx+\frac {4}{27} \int \frac {1}{2+x} \, dx+\frac {8}{27} \int \frac {1}{2+x} \, dx-\frac {8}{27} \int \frac {1}{3+x} \, dx+\frac {8}{9} \int \left (\frac {1}{2 x^2}-\frac {1}{4 x}+\frac {1}{4 (2+x)}\right ) \, dx-\frac {8}{3} \int \left (\frac {1}{2 x^3}-\frac {1}{4 x^2}+\frac {1}{8 x}-\frac {1}{8 (2+x)}\right ) \, dx\\ &=\frac {32}{3 x^2}-\frac {32}{9 x}-x+\frac {32}{9 (3+x)}-\frac {8 \log (x)}{81}+\frac {8 \log (x)}{3 x^3}-\frac {8 \log (x)}{9 x^2}+\frac {8 \log (x)}{27 x}+\frac {8 x \log (x)}{81 (3+x)}+\frac {8 \log (2+x)}{3 x^3}-\frac {8 \log (2+x)}{9 x^2}+\frac {8 \log (2+x)}{27 x}-\frac {8 \log (2+x)}{27 (3+x)}-\frac {8 (\log (x)+\log (2+x)-\log (x (2+x)))}{x^3 (3+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 30, normalized size = 1.15 \begin {gather*} -\frac {-32 x+3 x^4+x^5-8 \log (x (2+x))}{x^3 (3+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(48 - 320*x - 368*x^2 - 96*x^3 - 18*x^4 - 21*x^5 - 8*x^6 - x^7 + (-144 - 136*x - 32*x^2)*Log[2*x + x
^2])/(18*x^4 + 21*x^5 + 8*x^6 + x^7),x]

[Out]

-((-32*x + 3*x^4 + x^5 - 8*Log[x*(2 + x)])/(x^3*(3 + x)))

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fricas [A]  time = 0.66, size = 35, normalized size = 1.35 \begin {gather*} -\frac {x^{5} + 3 \, x^{4} - 32 \, x - 8 \, \log \left (x^{2} + 2 \, x\right )}{x^{4} + 3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x^2-136*x-144)*log(x^2+2*x)-x^7-8*x^6-21*x^5-18*x^4-96*x^3-368*x^2-320*x+48)/(x^7+8*x^6+21*x^5
+18*x^4),x, algorithm="fricas")

[Out]

-(x^5 + 3*x^4 - 32*x - 8*log(x^2 + 2*x))/(x^4 + 3*x^3)

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giac [A]  time = 0.24, size = 48, normalized size = 1.85 \begin {gather*} -\frac {8}{27} \, {\left (\frac {1}{x + 3} - \frac {x^{2} - 3 \, x + 9}{x^{3}}\right )} \log \left (x^{2} + 2 \, x\right ) - x + \frac {32}{9 \, {\left (x + 3\right )}} - \frac {32 \, {\left (x - 3\right )}}{9 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x^2-136*x-144)*log(x^2+2*x)-x^7-8*x^6-21*x^5-18*x^4-96*x^3-368*x^2-320*x+48)/(x^7+8*x^6+21*x^5
+18*x^4),x, algorithm="giac")

[Out]

-8/27*(1/(x + 3) - (x^2 - 3*x + 9)/x^3)*log(x^2 + 2*x) - x + 32/9/(x + 3) - 32/9*(x - 3)/x^2

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maple [A]  time = 0.07, size = 40, normalized size = 1.54

method	result	size
risch	\(\frac {8 \ln \left (x^{2}+2 x \right )}{\left (3+x \right ) x^{3}}-\frac {x^{4}+3 x^{3}-32}{x^{2} \left (3+x \right )}\)	\(40\)
default	\(-x -\frac {8}{9 x^{3}}+\frac {94}{9 x^{2}}-\frac {74}{27 x}-\ln \relax (x )-\ln \left (2+x \right )+\frac {32}{9 \left (3+x \right )}+\frac {\frac {8}{3}+x^{4} \ln \left (x^{2}+2 x \right )-\frac {22 x^{3}}{27}+\frac {14 x}{9}-\frac {20 x^{2}}{9}+3 x^{3} \ln \left (x^{2}+2 x \right )+8 \ln \left (x^{2}+2 x \right )}{\left (3+x \right ) x^{3}}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-32*x^2-136*x-144)*ln(x^2+2*x)-x^7-8*x^6-21*x^5-18*x^4-96*x^3-368*x^2-320*x+48)/(x^7+8*x^6+21*x^5+18*x^4
),x,method=_RETURNVERBOSE)

[Out]

8/(3+x)/x^3*ln(x^2+2*x)-(x^4+3*x^3-32)/x^2/(3+x)

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maxima [B]  time = 0.62, size = 143, normalized size = 5.50 \begin {gather*} -x - \frac {66 \, x^{3} + 180 \, x^{2} - 81 \, {\left (x^{4} + 3 \, x^{3} + 8\right )} \log \left (x + 2\right ) + {\left (49 \, x^{4} + 147 \, x^{3} - 648\right )} \log \relax (x) - 126 \, x - 216}{81 \, {\left (x^{4} + 3 \, x^{3}\right )}} - \frac {2 \, {\left (25 \, x^{3} + 78 \, x^{2} - 51 \, x + 36\right )}}{27 \, {\left (x^{4} + 3 \, x^{3}\right )}} - \frac {80 \, {\left (x^{2} + 6 \, x - 3\right )}}{9 \, {\left (x^{3} + 3 \, x^{2}\right )}} - \frac {184 \, {\left (x - 3\right )}}{9 \, {\left (x^{2} + 3 \, x\right )}} + \frac {32}{x + 3} - \log \left (x + 2\right ) + \frac {49}{81} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x^2-136*x-144)*log(x^2+2*x)-x^7-8*x^6-21*x^5-18*x^4-96*x^3-368*x^2-320*x+48)/(x^7+8*x^6+21*x^5
+18*x^4),x, algorithm="maxima")

[Out]

-x - 1/81*(66*x^3 + 180*x^2 - 81*(x^4 + 3*x^3 + 8)*log(x + 2) + (49*x^4 + 147*x^3 - 648)*log(x) - 126*x - 216)
/(x^4 + 3*x^3) - 2/27*(25*x^3 + 78*x^2 - 51*x + 36)/(x^4 + 3*x^3) - 80/9*(x^2 + 6*x - 3)/(x^3 + 3*x^2) - 184/9
*(x - 3)/(x^2 + 3*x) + 32/(x + 3) - log(x + 2) + 49/81*log(x)

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mupad [B]  time = 2.07, size = 33, normalized size = 1.27 \begin {gather*} \frac {32\,x+8\,\ln \left (x^2+2\,x\right )-3\,x^4-x^5}{x^3\,\left (x+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(320*x + log(2*x + x^2)*(136*x + 32*x^2 + 144) + 368*x^2 + 96*x^3 + 18*x^4 + 21*x^5 + 8*x^6 + x^7 - 48)/(
18*x^4 + 21*x^5 + 8*x^6 + x^7),x)

[Out]

(32*x + 8*log(2*x + x^2) - 3*x^4 - x^5)/(x^3*(x + 3))

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sympy [A]  time = 0.22, size = 29, normalized size = 1.12 \begin {gather*} - x + \frac {8 \log {\left (x^{2} + 2 x \right )}}{x^{4} + 3 x^{3}} + \frac {32}{x^{3} + 3 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x**2-136*x-144)*ln(x**2+2*x)-x**7-8*x**6-21*x**5-18*x**4-96*x**3-368*x**2-320*x+48)/(x**7+8*x*
*6+21*x**5+18*x**4),x)

[Out]

-x + 8*log(x**2 + 2*x)/(x**4 + 3*x**3) + 32/(x**3 + 3*x**2)

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