\(\int \frac {-16+30 x+(-25 x^3-10 x^4) \log (4)}{-8 x+30 x^2-25 x^3+(25 x^4+5 x^5) \log (4)} \, dx\) [8723]

Optimal result

Integrand size = 50, antiderivative size = 28 \[ \int \frac {-16+30 x+\left (-25 x^3-10 x^4\right ) \log (4)}{-8 x+30 x^2-25 x^3+\left (25 x^4+5 x^5\right ) \log (4)} \, dx=1-\log \left (-\left (5-\frac {3}{x}\right )^2+\frac {1}{x^2}+5 x (5+x) \log (4)\right ) \]

[Out]

1-ln(10*ln(2)*(5+x)*x+1/x^2-(5-3/x)^2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2099, 1601} \[ \int \frac {-16+30 x+\left (-25 x^3-10 x^4\right ) \log (4)}{-8 x+30 x^2-25 x^3+\left (25 x^4+5 x^5\right ) \log (4)} \, dx=2 \log (x)-\log \left (-5 x^4 \log (4)-25 x^3 \log (4)+25 x^2-30 x+8\right ) \]

[In]

Int[(-16 + 30*x + (-25*x^3 - 10*x^4)*Log[4])/(-8*x + 30*x^2 - 25*x^3 + (25*x^4 + 5*x^5)*Log[4]),x]

[Out]

2*Log[x] - Log[8 - 30*x + 25*x^2 - 25*x^3*Log[4] - 5*x^4*Log[4]]

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 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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{x}-\frac {5 \left (6-10 x+15 x^2 \log (4)+4 x^3 \log (4)\right )}{-8+30 x-25 x^2+25 x^3 \log (4)+5 x^4 \log (4)}\right ) \, dx \\ & = 2 \log (x)-5 \int \frac {6-10 x+15 x^2 \log (4)+4 x^3 \log (4)}{-8+30 x-25 x^2+25 x^3 \log (4)+5 x^4 \log (4)} \, dx \\ & = 2 \log (x)-\log \left (8-30 x+25 x^2-25 x^3 \log (4)-5 x^4 \log (4)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-16+30 x+\left (-25 x^3-10 x^4\right ) \log (4)}{-8 x+30 x^2-25 x^3+\left (25 x^4+5 x^5\right ) \log (4)} \, dx=2 \log (x)-\log \left (8-30 x+25 x^2-25 x^3 \log (4)-5 x^4 \log (4)\right ) \]

[In]

Integrate[(-16 + 30*x + (-25*x^3 - 10*x^4)*Log[4])/(-8*x + 30*x^2 - 25*x^3 + (25*x^4 + 5*x^5)*Log[4]),x]

[Out]

2*Log[x] - Log[8 - 30*x + 25*x^2 - 25*x^3*Log[4] - 5*x^4*Log[4]]

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18

method	result	size
default	\(-\ln \left (10 x^{4} \ln \left (2\right )+50 x^{3} \ln \left (2\right )-25 x^{2}+30 x -8\right )+2 \ln \left (x \right )\)	\(33\)
norman	\(-\ln \left (10 x^{4} \ln \left (2\right )+50 x^{3} \ln \left (2\right )-25 x^{2}+30 x -8\right )+2 \ln \left (x \right )\)	\(33\)
risch	\(2 \ln \left (-x \right )-\ln \left (10 x^{4} \ln \left (2\right )+50 x^{3} \ln \left (2\right )-25 x^{2}+30 x -8\right )\)	\(35\)
parallelrisch	\(2 \ln \left (x \right )-\ln \left (\frac {10 x^{4} \ln \left (2\right )+50 x^{3} \ln \left (2\right )-25 x^{2}+30 x -8}{10 \ln \left (2\right )}\right )\)	\(39\)

[In]

int((2*(-10*x^4-25*x^3)*ln(2)+30*x-16)/(2*(5*x^5+25*x^4)*ln(2)-25*x^3+30*x^2-8*x),x,method=_RETURNVERBOSE)

[Out]

-ln(10*x^4*ln(2)+50*x^3*ln(2)-25*x^2+30*x-8)+2*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {-16+30 x+\left (-25 x^3-10 x^4\right ) \log (4)}{-8 x+30 x^2-25 x^3+\left (25 x^4+5 x^5\right ) \log (4)} \, dx=-\log \left (-25 \, x^{2} + 10 \, {\left (x^{4} + 5 \, x^{3}\right )} \log \left (2\right ) + 30 \, x - 8\right ) + 2 \, \log \left (x\right ) \]

[In]

integrate((2*(-10*x^4-25*x^3)*log(2)+30*x-16)/(2*(5*x^5+25*x^4)*log(2)-25*x^3+30*x^2-8*x),x, algorithm="fricas
")

[Out]

-log(-25*x^2 + 10*(x^4 + 5*x^3)*log(2) + 30*x - 8) + 2*log(x)

Sympy [A] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {-16+30 x+\left (-25 x^3-10 x^4\right ) \log (4)}{-8 x+30 x^2-25 x^3+\left (25 x^4+5 x^5\right ) \log (4)} \, dx=2 \log {\left (x \right )} - \log {\left (x^{4} + 5 x^{3} - \frac {5 x^{2}}{2 \log {\left (2 \right )}} + \frac {3 x}{\log {\left (2 \right )}} - \frac {4}{5 \log {\left (2 \right )}} \right )} \]

[In]

integrate((2*(-10*x**4-25*x**3)*ln(2)+30*x-16)/(2*(5*x**5+25*x**4)*ln(2)-25*x**3+30*x**2-8*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-16+30 x+\left (-25 x^3-10 x^4\right ) \log (4)}{-8 x+30 x^2-25 x^3+\left (25 x^4+5 x^5\right ) \log (4)} \, dx=-\log \left (10 \, x^{4} \log \left (2\right ) + 50 \, x^{3} \log \left (2\right ) - 25 \, x^{2} + 30 \, x - 8\right ) + 2 \, \log \left (x\right ) \]

[In]

integrate((2*(-10*x^4-25*x^3)*log(2)+30*x-16)/(2*(5*x^5+25*x^4)*log(2)-25*x^3+30*x^2-8*x),x, algorithm="maxima
")

[Out]

-log(10*x^4*log(2) + 50*x^3*log(2) - 25*x^2 + 30*x - 8) + 2*log(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-16+30 x+\left (-25 x^3-10 x^4\right ) \log (4)}{-8 x+30 x^2-25 x^3+\left (25 x^4+5 x^5\right ) \log (4)} \, dx=-\log \left ({\left | 10 \, x^{4} \log \left (2\right ) + 50 \, x^{3} \log \left (2\right ) - 25 \, x^{2} + 30 \, x - 8 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate((2*(-10*x^4-25*x^3)*log(2)+30*x-16)/(2*(5*x^5+25*x^4)*log(2)-25*x^3+30*x^2-8*x),x, algorithm="giac")

[Out]

-log(abs(10*x^4*log(2) + 50*x^3*log(2) - 25*x^2 + 30*x - 8)) + 2*log(abs(x))

Mupad [B] (verification not implemented)

Time = 13.50 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-16+30 x+\left (-25 x^3-10 x^4\right ) \log (4)}{-8 x+30 x^2-25 x^3+\left (25 x^4+5 x^5\right ) \log (4)} \, dx=2\,\ln \left (x\right )-\ln \left (30\,\ln \left (2\right )\,x^4+150\,\ln \left (2\right )\,x^3-75\,x^2+90\,x-24\right ) \]

[In]

int((2*log(2)*(25*x^3 + 10*x^4) - 30*x + 16)/(8*x - 2*log(2)*(25*x^4 + 5*x^5) - 30*x^2 + 25*x^3),x)

[Out]

2*log(x) - log(90*x + 150*x^3*log(2) + 30*x^4*log(2) - 75*x^2 - 24)