\(\int \frac {1260 x^2+16 x^3+(-252 x^2-3 x^3) \log (84+x)}{2100+25 x+(-840-10 x) \log (84+x)+(84+x) \log ^2(84+x)} \, dx\) [6624]

Optimal result

Integrand size = 55, antiderivative size = 16 \[ \int \frac {1260 x^2+16 x^3+\left (-252 x^2-3 x^3\right ) \log (84+x)}{2100+25 x+(-840-10 x) \log (84+x)+(84+x) \log ^2(84+x)} \, dx=\log (4)-\frac {x^3}{-5+\log (84+x)} \]

[Out]

2*ln(2)-x^3/(ln(x+84)-5)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(16)=32\).

Time = 0.47 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.81, number of steps used = 28, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.255, Rules used = {6820, 6874, 2458, 2395, 2334, 2336, 2209, 2339, 30, 2343, 2346, 2446, 2436, 2437} \[ \int \frac {1260 x^2+16 x^3+\left (-252 x^2-3 x^3\right ) \log (84+x)}{2100+25 x+(-840-10 x) \log (84+x)+(84+x) \log ^2(84+x)} \, dx=\frac {(x+84)^3}{5-\log (x+84)}-\frac {252 (x+84)^2}{5-\log (x+84)}+\frac {21168 (x+84)}{5-\log (x+84)}-\frac {592704}{5-\log (x+84)} \]

[In]

Int[(1260*x^2 + 16*x^3 + (-252*x^2 - 3*x^3)*Log[84 + x])/(2100 + 25*x + (-840 - 10*x)*Log[84 + x] + (84 + x)*L
og[84 + x]^2),x]

[Out]

-592704/(5 - Log[84 + x]) + (21168*(84 + x))/(5 - Log[84 + x]) - (252*(84 + x)^2)/(5 - Log[84 + x]) + (84 + x)
^3/(5 - Log[84 + x])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2209

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

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2336

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2343

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

Rule 2346

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

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 2446

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

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 (4 (315+4 x)-3 (84+x) \log (84+x))}{(84+x) (5-\log (84+x))^2} \, dx \\ & = \int \left (\frac {x^3}{(84+x) (-5+\log (84+x))^2}-\frac {3 x^2}{-5+\log (84+x)}\right ) \, dx \\ & = -\left (3 \int \frac {x^2}{-5+\log (84+x)} \, dx\right )+\int \frac {x^3}{(84+x) (-5+\log (84+x))^2} \, dx \\ & = -\left (3 \int \left (\frac {7056}{-5+\log (84+x)}-\frac {168 (84+x)}{-5+\log (84+x)}+\frac {(84+x)^2}{-5+\log (84+x)}\right ) \, dx\right )+\text {Subst}\left (\int \frac {(-84+x)^3}{x (-5+\log (x))^2} \, dx,x,84+x\right ) \\ & = -\left (3 \int \frac {(84+x)^2}{-5+\log (84+x)} \, dx\right )+504 \int \frac {84+x}{-5+\log (84+x)} \, dx-21168 \int \frac {1}{-5+\log (84+x)} \, dx+\text {Subst}\left (\int \left (\frac {21168}{(-5+\log (x))^2}-\frac {592704}{x (-5+\log (x))^2}-\frac {252 x}{(-5+\log (x))^2}+\frac {x^2}{(-5+\log (x))^2}\right ) \, dx,x,84+x\right ) \\ & = -\left (3 \text {Subst}\left (\int \frac {x^2}{-5+\log (x)} \, dx,x,84+x\right )\right )-252 \text {Subst}\left (\int \frac {x}{(-5+\log (x))^2} \, dx,x,84+x\right )+504 \text {Subst}\left (\int \frac {x}{-5+\log (x)} \, dx,x,84+x\right )+21168 \text {Subst}\left (\int \frac {1}{(-5+\log (x))^2} \, dx,x,84+x\right )-21168 \text {Subst}\left (\int \frac {1}{-5+\log (x)} \, dx,x,84+x\right )-592704 \text {Subst}\left (\int \frac {1}{x (-5+\log (x))^2} \, dx,x,84+x\right )+\text {Subst}\left (\int \frac {x^2}{(-5+\log (x))^2} \, dx,x,84+x\right ) \\ & = \frac {21168 (84+x)}{5-\log (84+x)}-\frac {252 (84+x)^2}{5-\log (84+x)}+\frac {(84+x)^3}{5-\log (84+x)}-3 \text {Subst}\left (\int \frac {e^{3 x}}{-5+x} \, dx,x,\log (84+x)\right )+3 \text {Subst}\left (\int \frac {x^2}{-5+\log (x)} \, dx,x,84+x\right )+504 \text {Subst}\left (\int \frac {e^{2 x}}{-5+x} \, dx,x,\log (84+x)\right )-504 \text {Subst}\left (\int \frac {x}{-5+\log (x)} \, dx,x,84+x\right )-21168 \text {Subst}\left (\int \frac {e^x}{-5+x} \, dx,x,\log (84+x)\right )+21168 \text {Subst}\left (\int \frac {1}{-5+\log (x)} \, dx,x,84+x\right )-592704 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,-5+\log (84+x)\right ) \\ & = -3 e^{15} \text {Ei}(-3 (5-\log (84+x)))+504 e^{10} \text {Ei}(-2 (5-\log (84+x)))-21168 e^5 \text {Ei}(-5+\log (84+x))-\frac {592704}{5-\log (84+x)}+\frac {21168 (84+x)}{5-\log (84+x)}-\frac {252 (84+x)^2}{5-\log (84+x)}+\frac {(84+x)^3}{5-\log (84+x)}+3 \text {Subst}\left (\int \frac {e^{3 x}}{-5+x} \, dx,x,\log (84+x)\right )-504 \text {Subst}\left (\int \frac {e^{2 x}}{-5+x} \, dx,x,\log (84+x)\right )+21168 \text {Subst}\left (\int \frac {e^x}{-5+x} \, dx,x,\log (84+x)\right ) \\ & = -\frac {592704}{5-\log (84+x)}+\frac {21168 (84+x)}{5-\log (84+x)}-\frac {252 (84+x)^2}{5-\log (84+x)}+\frac {(84+x)^3}{5-\log (84+x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1260 x^2+16 x^3+\left (-252 x^2-3 x^3\right ) \log (84+x)}{2100+25 x+(-840-10 x) \log (84+x)+(84+x) \log ^2(84+x)} \, dx=-\frac {x^3}{-5+\log (84+x)} \]

[In]

Integrate[(1260*x^2 + 16*x^3 + (-252*x^2 - 3*x^3)*Log[84 + x])/(2100 + 25*x + (-840 - 10*x)*Log[84 + x] + (84
+ x)*Log[84 + x]^2),x]

[Out]

-(x^3/(-5 + Log[84 + x]))

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

method	result	size
norman	\(-\frac {x^{3}}{\ln \left (x +84\right )-5}\)	\(14\)
risch	\(-\frac {x^{3}}{\ln \left (x +84\right )-5}\)	\(14\)
parallelrisch	\(-\frac {x^{3}}{\ln \left (x +84\right )-5}\)	\(14\)
derivativedivides	\(\frac {592704}{\ln \left (x +84\right )-5}-\frac {21168 \left (x +84\right )}{\ln \left (x +84\right )-5}+\frac {252 \left (x +84\right )^{2}}{\ln \left (x +84\right )-5}-\frac {\left (x +84\right )^{3}}{\ln \left (x +84\right )-5}\)	\(55\)
default	\(\frac {592704}{\ln \left (x +84\right )-5}-\frac {21168 \left (x +84\right )}{\ln \left (x +84\right )-5}+\frac {252 \left (x +84\right )^{2}}{\ln \left (x +84\right )-5}-\frac {\left (x +84\right )^{3}}{\ln \left (x +84\right )-5}\)	\(55\)

[In]

int(((-3*x^3-252*x^2)*ln(x+84)+16*x^3+1260*x^2)/((x+84)*ln(x+84)^2+(-10*x-840)*ln(x+84)+25*x+2100),x,method=_R
ETURNVERBOSE)

[Out]

-x^3/(ln(x+84)-5)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1260 x^2+16 x^3+\left (-252 x^2-3 x^3\right ) \log (84+x)}{2100+25 x+(-840-10 x) \log (84+x)+(84+x) \log ^2(84+x)} \, dx=-\frac {x^{3}}{\log \left (x + 84\right ) - 5} \]

[In]

integrate(((-3*x^3-252*x^2)*log(x+84)+16*x^3+1260*x^2)/((x+84)*log(x+84)^2+(-10*x-840)*log(x+84)+25*x+2100),x,
algorithm="fricas")

[Out]

-x^3/(log(x + 84) - 5)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {1260 x^2+16 x^3+\left (-252 x^2-3 x^3\right ) \log (84+x)}{2100+25 x+(-840-10 x) \log (84+x)+(84+x) \log ^2(84+x)} \, dx=- \frac {x^{3}}{\log {\left (x + 84 \right )} - 5} \]

[In]

integrate(((-3*x**3-252*x**2)*ln(x+84)+16*x**3+1260*x**2)/((x+84)*ln(x+84)**2+(-10*x-840)*ln(x+84)+25*x+2100),
x)

[Out]

-x**3/(log(x + 84) - 5)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1260 x^2+16 x^3+\left (-252 x^2-3 x^3\right ) \log (84+x)}{2100+25 x+(-840-10 x) \log (84+x)+(84+x) \log ^2(84+x)} \, dx=-\frac {x^{3}}{\log \left (x + 84\right ) - 5} \]

[In]

integrate(((-3*x^3-252*x^2)*log(x+84)+16*x^3+1260*x^2)/((x+84)*log(x+84)^2+(-10*x-840)*log(x+84)+25*x+2100),x,
algorithm="maxima")

[Out]

-x^3/(log(x + 84) - 5)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1260 x^2+16 x^3+\left (-252 x^2-3 x^3\right ) \log (84+x)}{2100+25 x+(-840-10 x) \log (84+x)+(84+x) \log ^2(84+x)} \, dx=-\frac {x^{3}}{\log \left (x + 84\right ) - 5} \]

[In]

integrate(((-3*x^3-252*x^2)*log(x+84)+16*x^3+1260*x^2)/((x+84)*log(x+84)^2+(-10*x-840)*log(x+84)+25*x+2100),x,
algorithm="giac")

[Out]

-x^3/(log(x + 84) - 5)

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1260 x^2+16 x^3+\left (-252 x^2-3 x^3\right ) \log (84+x)}{2100+25 x+(-840-10 x) \log (84+x)+(84+x) \log ^2(84+x)} \, dx=-\frac {x^3}{\ln \left (x+84\right )-5} \]

[In]

int((1260*x^2 - log(x + 84)*(252*x^2 + 3*x^3) + 16*x^3)/(25*x - log(x + 84)*(10*x + 840) + log(x + 84)^2*(x +
84) + 2100),x)

[Out]

-x^3/(log(x + 84) - 5)