3.77.25 \(\int \frac {25+4 x-2 x^2}{-1+x} \, dx\)

Optimal. Leaf size=16 \[ x \left (2-x+\frac {27 \log (-1+x)}{x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {683} \begin {gather*} -x^2+2 x+27 \log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(25 + 4*x - 2*x^2)/(-1 + x),x]

[Out]

2*x - x^2 + 27*Log[1 - x]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+\frac {27}{-1+x}-2 x\right ) \, dx\\ &=2 x-x^2+27 \log (1-x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 14, normalized size = 0.88 \begin {gather*} -(-1+x)^2+27 \log (-1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 + 4*x - 2*x^2)/(-1 + x),x]

[Out]

-(-1 + x)^2 + 27*Log[-1 + x]

________________________________________________________________________________________

fricas [A]  time = 1.08, size = 15, normalized size = 0.94 \begin {gather*} -x^{2} + 2 \, x + 27 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+4*x+25)/(x-1),x, algorithm="fricas")

[Out]

-x^2 + 2*x + 27*log(x - 1)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 16, normalized size = 1.00 \begin {gather*} -x^{2} + 2 \, x + 27 \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+4*x+25)/(x-1),x, algorithm="giac")

[Out]

-x^2 + 2*x + 27*log(abs(x - 1))

________________________________________________________________________________________

maple [A]  time = 0.21, size = 16, normalized size = 1.00

method	result	size
default	\(-x^{2}+2 x +27 \ln \left (x -1\right )\)	\(16\)
norman	\(-x^{2}+2 x +27 \ln \left (x -1\right )\)	\(16\)
risch	\(-x^{2}+2 x +27 \ln \left (x -1\right )\)	\(16\)
meijerg	\(27 \ln \left (1-x \right )-\frac {x \left (6+3 x \right )}{3}+4 x\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2+4*x+25)/(x-1),x,method=_RETURNVERBOSE)

[Out]

-x^2+2*x+27*ln(x-1)

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 15, normalized size = 0.94 \begin {gather*} -x^{2} + 2 \, x + 27 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+4*x+25)/(x-1),x, algorithm="maxima")

[Out]

-x^2 + 2*x + 27*log(x - 1)

________________________________________________________________________________________

mupad [B]  time = 5.29, size = 15, normalized size = 0.94 \begin {gather*} 2\,x+27\,\ln \left (x-1\right )-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - 2*x^2 + 25)/(x - 1),x)

[Out]

2*x + 27*log(x - 1) - x^2

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 12, normalized size = 0.75 \begin {gather*} - x^{2} + 2 x + 27 \log {\left (x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2+4*x+25)/(x-1),x)

[Out]

-x**2 + 2*x + 27*log(x - 1)

________________________________________________________________________________________