3.88.29 \(\int \frac {-4+28 x-5 x^2}{-28+4 x-7 x^2+x^3} \, dx\)

Optimal. Leaf size=19 \[ -\log \left (\frac {4}{625} (7-x) \left (4+x^2\right )^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 17, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2074, 260} \begin {gather*} -2 \log \left (x^2+4\right )-\log (7-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + 28*x - 5*x^2)/(-28 + 4*x - 7*x^2 + x^3),x]

[Out]

-Log[7 - x] - 2*Log[4 + x^2]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2074

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 {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{7-x}-\frac {4 x}{4+x^2}\right ) \, dx\\ &=-\log (7-x)-4 \int \frac {x}{4+x^2} \, dx\\ &=-\log (7-x)-2 \log \left (4+x^2\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 17, normalized size = 0.89 \begin {gather*} -\log (7-x)-2 \log \left (4+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 28*x - 5*x^2)/(-28 + 4*x - 7*x^2 + x^3),x]

[Out]

-Log[7 - x] - 2*Log[4 + x^2]

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 15, normalized size = 0.79 \begin {gather*} -2 \, \log \left (x^{2} + 4\right ) - \log \left (x - 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^2+28*x-4)/(x^3-7*x^2+4*x-28),x, algorithm="fricas")

[Out]

-2*log(x^2 + 4) - log(x - 7)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 16, normalized size = 0.84 \begin {gather*} -2 \, \log \left (x^{2} + 4\right ) - \log \left ({\left | x - 7 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^2+28*x-4)/(x^3-7*x^2+4*x-28),x, algorithm="giac")

[Out]

-2*log(x^2 + 4) - log(abs(x - 7))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 16, normalized size = 0.84

method	result	size
default	\(-2 \ln \left (x^{2}+4\right )-\ln \left (x -7\right )\)	\(16\)
norman	\(-2 \ln \left (x^{2}+4\right )-\ln \left (x -7\right )\)	\(16\)
risch	\(-2 \ln \left (x^{2}+4\right )-\ln \left (x -7\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5*x^2+28*x-4)/(x^3-7*x^2+4*x-28),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x^2+4)-ln(x-7)

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 15, normalized size = 0.79 \begin {gather*} -2 \, \log \left (x^{2} + 4\right ) - \log \left (x - 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^2+28*x-4)/(x^3-7*x^2+4*x-28),x, algorithm="maxima")

[Out]

-2*log(x^2 + 4) - log(x - 7)

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 16, normalized size = 0.84 \begin {gather*} -\ln \left (\left (16\,x-112\right )\,{\left (x^2+4\right )}^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x^2 - 28*x + 4)/(4*x - 7*x^2 + x^3 - 28),x)

[Out]

-log((16*x - 112)*(x^2 + 4)^2)

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 14, normalized size = 0.74 \begin {gather*} - \log {\left (x - 7 \right )} - 2 \log {\left (x^{2} + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x**2+28*x-4)/(x**3-7*x**2+4*x-28),x)

[Out]

-log(x - 7) - 2*log(x**2 + 4)

________________________________________________________________________________________