\(\int \frac {-5-15 x+30 x^2+14 x^3+x^4}{10-15 x+10 x^3+x^4} \, dx\) [5845]

Optimal result

Integrand size = 34, antiderivative size = 20 \[ \int \frac {-5-15 x+30 x^2+14 x^3+x^4}{10-15 x+10 x^3+x^4} \, dx=x+\log \left (-2+3 x-\left (2+\frac {x}{5}\right ) x^3\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6874, 1601} \[ \int \frac {-5-15 x+30 x^2+14 x^3+x^4}{10-15 x+10 x^3+x^4} \, dx=\log \left (x^4+10 x^3-15 x+10\right )+x \]

[In]

Int[(-5 - 15*x + 30*x^2 + 14*x^3 + x^4)/(10 - 15*x + 10*x^3 + x^4),x]

[Out]

x + Log[10 - 15*x + 10*x^3 + x^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 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (1-\frac {15-30 x^2-4 x^3}{10-15 x+10 x^3+x^4}\right ) \, dx \\ & = x-\int \frac {15-30 x^2-4 x^3}{10-15 x+10 x^3+x^4} \, dx \\ & = x+\log \left (10-15 x+10 x^3+x^4\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {-5-15 x+30 x^2+14 x^3+x^4}{10-15 x+10 x^3+x^4} \, dx=x+\log \left (10-15 x+10 x^3+x^4\right ) \]

[In]

Integrate[(-5 - 15*x + 30*x^2 + 14*x^3 + x^4)/(10 - 15*x + 10*x^3 + x^4),x]

[Out]

x + Log[10 - 15*x + 10*x^3 + x^4]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85

method	result	size
default	\(x +\ln \left (x^{4}+10 x^{3}-15 x +10\right )\)	\(17\)
norman	\(x +\ln \left (x^{4}+10 x^{3}-15 x +10\right )\)	\(17\)
risch	\(x +\ln \left (x^{4}+10 x^{3}-15 x +10\right )\)	\(17\)
parallelrisch	\(x +\ln \left (x^{4}+10 x^{3}-15 x +10\right )\)	\(17\)

[In]

int((x^4+14*x^3+30*x^2-15*x-5)/(x^4+10*x^3-15*x+10),x,method=_RETURNVERBOSE)

[Out]

x+ln(x^4+10*x^3-15*x+10)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {-5-15 x+30 x^2+14 x^3+x^4}{10-15 x+10 x^3+x^4} \, dx=x + \log \left (x^{4} + 10 \, x^{3} - 15 \, x + 10\right ) \]

[In]

integrate((x^4+14*x^3+30*x^2-15*x-5)/(x^4+10*x^3-15*x+10),x, algorithm="fricas")

[Out]

x + log(x^4 + 10*x^3 - 15*x + 10)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {-5-15 x+30 x^2+14 x^3+x^4}{10-15 x+10 x^3+x^4} \, dx=x + \log {\left (x^{4} + 10 x^{3} - 15 x + 10 \right )} \]

[In]

integrate((x**4+14*x**3+30*x**2-15*x-5)/(x**4+10*x**3-15*x+10),x)

[Out]

x + log(x**4 + 10*x**3 - 15*x + 10)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {-5-15 x+30 x^2+14 x^3+x^4}{10-15 x+10 x^3+x^4} \, dx=x + \log \left (x^{4} + 10 \, x^{3} - 15 \, x + 10\right ) \]

[In]

integrate((x^4+14*x^3+30*x^2-15*x-5)/(x^4+10*x^3-15*x+10),x, algorithm="maxima")

[Out]

x + log(x^4 + 10*x^3 - 15*x + 10)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-5-15 x+30 x^2+14 x^3+x^4}{10-15 x+10 x^3+x^4} \, dx=x + \log \left ({\left | x^{4} + 10 \, x^{3} - 15 \, x + 10 \right |}\right ) \]

[In]

integrate((x^4+14*x^3+30*x^2-15*x-5)/(x^4+10*x^3-15*x+10),x, algorithm="giac")

[Out]

x + log(abs(x^4 + 10*x^3 - 15*x + 10))

Mupad [B] (verification not implemented)

Time = 12.11 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {-5-15 x+30 x^2+14 x^3+x^4}{10-15 x+10 x^3+x^4} \, dx=x+\ln \left (x^4+10\,x^3-15\,x+10\right ) \]

[In]

int((30*x^2 - 15*x + 14*x^3 + x^4 - 5)/(10*x^3 - 15*x + x^4 + 10),x)

[Out]

x + log(10*x^3 - 15*x + x^4 + 10)