3.40.19 \(\int \frac {-9-14 x+6 x^2+4 x^3}{17-9 x-7 x^2+2 x^3+x^4} \, dx\) [3919]

Optimal. Leaf size=17 \[ \log (4)+\log \left (-1+x-\left (-4+x+x^2\right )^2\right ) \]

[Out]

2*ln(2)+ln(-1-(x^2+x-4)^2+x)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.12, number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1601} \begin {gather*} \log \left (x^4+2 x^3-7 x^2-9 x+17\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - 14*x + 6*x^2 + 4*x^3)/(17 - 9*x - 7*x^2 + 2*x^3 + x^4),x]

[Out]

Log[17 - 9*x - 7*x^2 + 2*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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (17-9 x-7 x^2+2 x^3+x^4\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 19, normalized size = 1.12 \begin {gather*} \log \left (17-9 x-7 x^2+2 x^3+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 14*x + 6*x^2 + 4*x^3)/(17 - 9*x - 7*x^2 + 2*x^3 + x^4),x]

[Out]

Log[17 - 9*x - 7*x^2 + 2*x^3 + x^4]

________________________________________________________________________________________

Maple [A]
time = 0.22, size = 20, normalized size = 1.18

method	result	size
derivativedivides	\(\ln \left (x^{4}+2 x^{3}-7 x^{2}-9 x +17\right )\)	\(20\)
default	\(\ln \left (x^{4}+2 x^{3}-7 x^{2}-9 x +17\right )\)	\(20\)
norman	\(\ln \left (x^{4}+2 x^{3}-7 x^{2}-9 x +17\right )\)	\(20\)
risch	\(\ln \left (x^{4}+2 x^{3}-7 x^{2}-9 x +17\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3+6*x^2-14*x-9)/(x^4+2*x^3-7*x^2-9*x+17),x,method=_RETURNVERBOSE)

[Out]

ln(x^4+2*x^3-7*x^2-9*x+17)

________________________________________________________________________________________

Maxima [A]
time = 0.70, size = 19, normalized size = 1.12 \begin {gather*} \log \left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 9 \, x + 17\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+6*x^2-14*x-9)/(x^4+2*x^3-7*x^2-9*x+17),x, algorithm="maxima")

[Out]

log(x^4 + 2*x^3 - 7*x^2 - 9*x + 17)

________________________________________________________________________________________

Fricas [A]
time = 0.41, size = 19, normalized size = 1.12 \begin {gather*} \log \left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 9 \, x + 17\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+6*x^2-14*x-9)/(x^4+2*x^3-7*x^2-9*x+17),x, algorithm="fricas")

[Out]

log(x^4 + 2*x^3 - 7*x^2 - 9*x + 17)

________________________________________________________________________________________

Sympy [A]
time = 0.03, size = 19, normalized size = 1.12 \begin {gather*} \log {\left (x^{4} + 2 x^{3} - 7 x^{2} - 9 x + 17 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3+6*x**2-14*x-9)/(x**4+2*x**3-7*x**2-9*x+17),x)

[Out]

log(x**4 + 2*x**3 - 7*x**2 - 9*x + 17)

________________________________________________________________________________________

Giac [A]
time = 0.40, size = 20, normalized size = 1.18 \begin {gather*} \log \left ({\left | x^{4} + 2 \, x^{3} - 7 \, x^{2} - 9 \, x + 17 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+6*x^2-14*x-9)/(x^4+2*x^3-7*x^2-9*x+17),x, algorithm="giac")

[Out]

log(abs(x^4 + 2*x^3 - 7*x^2 - 9*x + 17))

________________________________________________________________________________________

Mupad [B]
time = 0.09, size = 19, normalized size = 1.12 \begin {gather*} \ln \left (x^4+2\,x^3-7\,x^2-9\,x+17\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(14*x - 6*x^2 - 4*x^3 + 9)/(2*x^3 - 7*x^2 - 9*x + x^4 + 17),x)

[Out]

log(2*x^3 - 7*x^2 - 9*x + x^4 + 17)

________________________________________________________________________________________