\(\int \frac {-1200 x^5+300 x^6}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx\) [2238]

Optimal result

Integrand size = 39, antiderivative size = 17 \[ \int \frac {-1200 x^5+300 x^6}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx=\log \left (2-\frac {25 x^6}{4 (-8+3 x)^2}\right ) \]

[Out]

ln(2-25/4*x^6/(3*x-8)^2)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(17)=34\).

Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.59, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1607, 2125, 2099, 1601} \[ \int \frac {-1200 x^5+300 x^6}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx=\frac {3}{7} \log \left (-25 x^6+72 x^2-384 x+512\right )+\frac {4}{7} \log \left (75 x^7-200 x^6-216 x^3+1728 x^2-4608 x+4096\right )-\frac {18}{7} \log (8-3 x) \]

[In]

Int[(-1200*x^5 + 300*x^6)/(4096 - 4608*x + 1728*x^2 - 216*x^3 - 200*x^6 + 75*x^7),x]

[Out]

(-18*Log[8 - 3*x])/7 + (3*Log[512 - 384*x + 72*x^2 - 25*x^6])/7 + (4*Log[4096 - 4608*x + 1728*x^2 - 216*x^3 -
200*x^6 + 75*x^7])/7

Rule 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 1607

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 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 2125

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[Coeff[Pm, x, m]*(Log[Qn]/(n*Coef
f[Qn, x, n])), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x
], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5 (-1200+300 x)}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx \\ & = \frac {4}{7} \log \left (4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7\right )+\frac {1}{525} \int \frac {1382400-1036800 x+194400 x^2-270000 x^5}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx \\ & = \frac {4}{7} \log \left (4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7\right )+\frac {1}{525} \int \left (-\frac {4050}{-8+3 x}+\frac {1350 \left (64-24 x+25 x^5\right )}{-512+384 x-72 x^2+25 x^6}\right ) \, dx \\ & = -\frac {18}{7} \log (8-3 x)+\frac {4}{7} \log \left (4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7\right )+\frac {18}{7} \int \frac {64-24 x+25 x^5}{-512+384 x-72 x^2+25 x^6} \, dx \\ & = -\frac {18}{7} \log (8-3 x)+\frac {3}{7} \log \left (512-384 x+72 x^2-25 x^6\right )+\frac {4}{7} \log \left (4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {-1200 x^5+300 x^6}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx=300 \left (-\frac {1}{150} \log (8-3 x)+\frac {1}{300} \log \left (512-384 x+72 x^2-25 x^6\right )\right ) \]

[In]

Integrate[(-1200*x^5 + 300*x^6)/(4096 - 4608*x + 1728*x^2 - 216*x^3 - 200*x^6 + 75*x^7),x]

[Out]

300*(-1/150*Log[8 - 3*x] + Log[512 - 384*x + 72*x^2 - 25*x^6]/300)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29

method	result	size
parallelrisch	\(-2 \ln \left (x -\frac {8}{3}\right )+\ln \left (x^{6}-\frac {72}{25} x^{2}+\frac {384}{25} x -\frac {512}{25}\right )\)	\(22\)
default	\(-2 \ln \left (3 x -8\right )+\ln \left (25 x^{6}-72 x^{2}+384 x -512\right )\)	\(26\)
norman	\(-2 \ln \left (3 x -8\right )+\ln \left (25 x^{6}-72 x^{2}+384 x -512\right )\)	\(26\)
risch	\(-2 \ln \left (3 x -8\right )+\ln \left (25 x^{6}-72 x^{2}+384 x -512\right )\)	\(26\)

[In]

int((300*x^6-1200*x^5)/(75*x^7-200*x^6-216*x^3+1728*x^2-4608*x+4096),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x-8/3)+ln(x^6-72/25*x^2+384/25*x-512/25)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.47 \[ \int \frac {-1200 x^5+300 x^6}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx=\log \left (25 \, x^{6} - 72 \, x^{2} + 384 \, x - 512\right ) - 2 \, \log \left (3 \, x - 8\right ) \]

[In]

integrate((300*x^6-1200*x^5)/(75*x^7-200*x^6-216*x^3+1728*x^2-4608*x+4096),x, algorithm="fricas")

[Out]

log(25*x^6 - 72*x^2 + 384*x - 512) - 2*log(3*x - 8)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \frac {-1200 x^5+300 x^6}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx=- 2 \log {\left (3 x - 8 \right )} + \log {\left (25 x^{6} - 72 x^{2} + 384 x - 512 \right )} \]

[In]

integrate((300*x**6-1200*x**5)/(75*x**7-200*x**6-216*x**3+1728*x**2-4608*x+4096),x)

[Out]

-2*log(3*x - 8) + log(25*x**6 - 72*x**2 + 384*x - 512)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.47 \[ \int \frac {-1200 x^5+300 x^6}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx=\log \left (25 \, x^{6} - 72 \, x^{2} + 384 \, x - 512\right ) - 2 \, \log \left (3 \, x - 8\right ) \]

[In]

integrate((300*x^6-1200*x^5)/(75*x^7-200*x^6-216*x^3+1728*x^2-4608*x+4096),x, algorithm="maxima")

[Out]

log(25*x^6 - 72*x^2 + 384*x - 512) - 2*log(3*x - 8)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59 \[ \int \frac {-1200 x^5+300 x^6}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx=\log \left ({\left | 25 \, x^{6} - 72 \, x^{2} + 384 \, x - 512 \right |}\right ) - 2 \, \log \left ({\left | 3 \, x - 8 \right |}\right ) \]

[In]

integrate((300*x^6-1200*x^5)/(75*x^7-200*x^6-216*x^3+1728*x^2-4608*x+4096),x, algorithm="giac")

[Out]

log(abs(25*x^6 - 72*x^2 + 384*x - 512)) - 2*log(abs(3*x - 8))

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.35 \[ \int \frac {-1200 x^5+300 x^6}{4096-4608 x+1728 x^2-216 x^3-200 x^6+75 x^7} \, dx=\ln \left (25\,x^6-72\,x^2+384\,x-512\right )-2\,\ln \left (x-\frac {8}{3}\right ) \]

[In]

int((1200*x^5 - 300*x^6)/(4608*x - 1728*x^2 + 216*x^3 + 200*x^6 - 75*x^7 - 4096),x)

[Out]

log(384*x - 72*x^2 + 25*x^6 - 512) - 2*log(x - 8/3)