3.3.5 \(\int \frac {-33750 x^2-101250 x^3+379586250 x^4-33750 x^5+(6+30 x-67440 x^2-134940 x^3-67470 x^4+6 x^5) \log (x)}{x+5 x^2+10 x^3+10 x^4+5 x^5+x^6} \, dx\)

Optimal. Leaf size=19 \[ 3 \left (\frac {5625 x^2}{(1+x)^2}-\log (x)\right )^2 \]

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Rubi [B]  time = 0.43, antiderivative size = 58, normalized size of antiderivative = 3.05, number of steps used = 14, number of rules used = 10, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6688, 12, 6742, 1620, 2357, 2301, 2319, 44, 2314, 31} \begin {gather*} -\frac {379687500}{x+1}+\frac {569531250}{(x+1)^2}-\frac {379687500}{(x+1)^3}+\frac {94921875}{(x+1)^4}+3 \log ^2(x)-\frac {67500 x \log (x)}{x+1}-\frac {33750 \log (x)}{(x+1)^2}+33750 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-33750*x^2 - 101250*x^3 + 379586250*x^4 - 33750*x^5 + (6 + 30*x - 67440*x^2 - 134940*x^3 - 67470*x^4 + 6*
x^5)*Log[x])/(x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6),x]

[Out]

94921875/(1 + x)^4 - 379687500/(1 + x)^3 + 569531250/(1 + x)^2 - 379687500/(1 + x) + 33750*Log[x] - (33750*Log
[x])/(1 + x)^2 - (67500*x*Log[x])/(1 + x) + 3*Log[x]^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

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

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 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

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 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 6688

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

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 {6 \left (1+3 x-11247 x^2+x^3\right ) \left (-5625 x^2+(1+x)^2 \log (x)\right )}{x (1+x)^5} \, dx\\ &=6 \int \frac {\left (1+3 x-11247 x^2+x^3\right ) \left (-5625 x^2+(1+x)^2 \log (x)\right )}{x (1+x)^5} \, dx\\ &=6 \int \left (-\frac {5625 x \left (1+3 x-11247 x^2+x^3\right )}{(1+x)^5}+\frac {\left (1+3 x-11247 x^2+x^3\right ) \log (x)}{x (1+x)^3}\right ) \, dx\\ &=6 \int \frac {\left (1+3 x-11247 x^2+x^3\right ) \log (x)}{x (1+x)^3} \, dx-33750 \int \frac {x \left (1+3 x-11247 x^2+x^3\right )}{(1+x)^5} \, dx\\ &=6 \int \left (\frac {\log (x)}{x}+\frac {11250 \log (x)}{(1+x)^3}-\frac {11250 \log (x)}{(1+x)^2}\right ) \, dx-33750 \int \left (\frac {11250}{(1+x)^5}-\frac {33750}{(1+x)^4}+\frac {33750}{(1+x)^3}-\frac {11251}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379721250}{1+x}-33750 \log (1+x)+6 \int \frac {\log (x)}{x} \, dx+67500 \int \frac {\log (x)}{(1+x)^3} \, dx-67500 \int \frac {\log (x)}{(1+x)^2} \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379721250}{1+x}-\frac {33750 \log (x)}{(1+x)^2}-\frac {67500 x \log (x)}{1+x}+3 \log ^2(x)-33750 \log (1+x)+33750 \int \frac {1}{x (1+x)^2} \, dx+67500 \int \frac {1}{1+x} \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379721250}{1+x}-\frac {33750 \log (x)}{(1+x)^2}-\frac {67500 x \log (x)}{1+x}+3 \log ^2(x)+33750 \log (1+x)+33750 \int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^2}\right ) \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379687500}{1+x}+33750 \log (x)-\frac {33750 \log (x)}{(1+x)^2}-\frac {67500 x \log (x)}{1+x}+3 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 41, normalized size = 2.16 \begin {gather*} 3 \left (-\frac {31640625 \left (1+4 x+6 x^2+4 x^3\right )}{(1+x)^4}-\frac {11250 x^2 \log (x)}{(1+x)^2}+\log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-33750*x^2 - 101250*x^3 + 379586250*x^4 - 33750*x^5 + (6 + 30*x - 67440*x^2 - 134940*x^3 - 67470*x^
4 + 6*x^5)*Log[x])/(x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6),x]

[Out]

3*((-31640625*(1 + 4*x + 6*x^2 + 4*x^3))/(1 + x)^4 - (11250*x^2*Log[x])/(1 + x)^2 + Log[x]^2)

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fricas [B]  time = 0.57, size = 77, normalized size = 4.05 \begin {gather*} -\frac {3 \, {\left (126562500 \, x^{3} - {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )} \log \relax (x)^{2} + 189843750 \, x^{2} + 11250 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \relax (x) + 126562500 \, x + 31640625\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5-67470*x^4-134940*x^3-67440*x^2+30*x+6)*log(x)-33750*x^5+379586250*x^4-101250*x^3-33750*x^2)/
(x^6+5*x^5+10*x^4+10*x^3+5*x^2+x),x, algorithm="fricas")

[Out]

-3*(126562500*x^3 - (x^4 + 4*x^3 + 6*x^2 + 4*x + 1)*log(x)^2 + 189843750*x^2 + 11250*(x^4 + 2*x^3 + x^2)*log(x
) + 126562500*x + 31640625)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {6 \, {\left (5625 \, x^{5} - 63264375 \, x^{4} + 16875 \, x^{3} + 5625 \, x^{2} - {\left (x^{5} - 11245 \, x^{4} - 22490 \, x^{3} - 11240 \, x^{2} + 5 \, x + 1\right )} \log \relax (x)\right )}}{x^{6} + 5 \, x^{5} + 10 \, x^{4} + 10 \, x^{3} + 5 \, x^{2} + x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5-67470*x^4-134940*x^3-67440*x^2+30*x+6)*log(x)-33750*x^5+379586250*x^4-101250*x^3-33750*x^2)/
(x^6+5*x^5+10*x^4+10*x^3+5*x^2+x),x, algorithm="giac")

[Out]

integrate(-6*(5625*x^5 - 63264375*x^4 + 16875*x^3 + 5625*x^2 - (x^5 - 11245*x^4 - 22490*x^3 - 11240*x^2 + 5*x
+ 1)*log(x))/(x^6 + 5*x^5 + 10*x^4 + 10*x^3 + 5*x^2 + x), x)

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maple [B]  time = 0.07, size = 59, normalized size = 3.11

method	result	size
default	\(\frac {569531250}{\left (x +1\right )^{2}}+\frac {94921875}{\left (x +1\right )^{4}}-\frac {379687500}{x +1}-\frac {379687500}{\left (x +1\right )^{3}}+3 \ln \relax (x )^{2}+\frac {33750 \ln \relax (x ) x \left (2+x \right )}{\left (x +1\right )^{2}}-\frac {67500 \ln \relax (x ) x}{x +1}\)	\(59\)
norman	\(\frac {5625 \ln \relax (x )[phone] x^{4}-45000 x^{3} \ln \relax (x )-28125 x^{4} \ln \relax (x )+22500 x \ln \relax (x )+3 \ln \relax (x )^{2}+12 x \ln \relax (x )^{2}+18 x^{2} \ln \relax (x )^{2}+12 x^{3} \ln \relax (x )^{2}+3 x^{4} \ln \relax (x )^{2}}{\left (x +1\right )^{4}}-5625 \ln \relax (x )\)	\(81\)
risch	\(3 \ln \relax (x )^{2}+\frac {33750 \left (2 x +1\right ) \ln \relax (x )}{x^{2}+2 x +1}-\frac {16875 \left (2 x^{4} \ln \relax (x )+8 x^{3} \ln \relax (x )+12 x^{2} \ln \relax (x )+22500 x^{3}+8 x \ln \relax (x )+33750 x^{2}+2 \ln \relax (x )+22500 x +5625\right )}{\left (x^{2}+2 x +1\right )^{2}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^5-67470*x^4-134940*x^3-67440*x^2+30*x+6)*ln(x)-33750*x^5+379586250*x^4-101250*x^3-33750*x^2)/(x^6+5*
x^5+10*x^4+10*x^3+5*x^2+x),x,method=_RETURNVERBOSE)

[Out]

569531250/(x+1)^2+94921875/(x+1)^4-379687500/(x+1)-379687500/(x+1)^3+3*ln(x)^2+33750*ln(x)*x*(2+x)/(x+1)^2-675
00*ln(x)*x/(x+1)

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maxima [B]  time = 0.78, size = 172, normalized size = 9.05 \begin {gather*} -\frac {5625 \, {\left (48 \, x^{3} + 108 \, x^{2} + 88 \, x + 25\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} - \frac {189793125 \, {\left (4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} + \frac {16875 \, {\left (6 \, x^{2} + 4 \, x + 1\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} - \frac {3 \, {\left (11250 \, x^{2} \log \relax (x) - {\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} - 11250 \, x - 11250\right )}}{x^{2} + 2 \, x + 1} + \frac {5625 \, {\left (4 \, x + 1\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5-67470*x^4-134940*x^3-67440*x^2+30*x+6)*log(x)-33750*x^5+379586250*x^4-101250*x^3-33750*x^2)/
(x^6+5*x^5+10*x^4+10*x^3+5*x^2+x),x, algorithm="maxima")

[Out]

-5625/2*(48*x^3 + 108*x^2 + 88*x + 25)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) - 189793125/2*(4*x^3 + 6*x^2 + 4*x + 1)
/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 16875/2*(6*x^2 + 4*x + 1)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) - 3*(11250*x^2*lo
g(x) - (x^2 + 2*x + 1)*log(x)^2 - 11250*x - 11250)/(x^2 + 2*x + 1) + 5625/2*(4*x + 1)/(x^4 + 4*x^3 + 6*x^2 + 4
*x + 1)

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mupad [B]  time = 0.52, size = 28, normalized size = 1.47 \begin {gather*} \frac {3\,{\left (\ln \relax (x)+x^2\,\ln \relax (x)+2\,x\,\ln \relax (x)-5625\,x^2\right )}^2}{{\left (x+1\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(33750*x^2 - log(x)*(30*x - 67440*x^2 - 134940*x^3 - 67470*x^4 + 6*x^5 + 6) + 101250*x^3 - 379586250*x^4
+ 33750*x^5)/(x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6),x)

[Out]

(3*(log(x) + x^2*log(x) + 2*x*log(x) - 5625*x^2)^2)/(x + 1)^4

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sympy [B]  time = 0.25, size = 61, normalized size = 3.21 \begin {gather*} \frac {\left (67500 x + 33750\right ) \log {\relax (x )}}{x^{2} + 2 x + 1} - \frac {379687500 x^{3} + 569531250 x^{2} + 379687500 x + 94921875}{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1} + 3 \log {\relax (x )}^{2} - 33750 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**5-67470*x**4-134940*x**3-67440*x**2+30*x+6)*ln(x)-33750*x**5+379586250*x**4-101250*x**3-33750
*x**2)/(x**6+5*x**5+10*x**4+10*x**3+5*x**2+x),x)

[Out]

(67500*x + 33750)*log(x)/(x**2 + 2*x + 1) - (379687500*x**3 + 569531250*x**2 + 379687500*x + 94921875)/(x**4 +
4*x**3 + 6*x**2 + 4*x + 1) + 3*log(x)**2 - 33750*log(x)

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