3.11.53 \(\int \frac {116-78 x+4 x^2}{-3+2 x} \, dx\)

Optimal. Leaf size=25 \[ 4 \left (6-x+4 \left (-2+\frac {x}{16}\right ) x+\log (3 (-3+2 x))\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 0.60, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {698} \begin {gather*} x^2-36 x+4 \log (3-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(116 - 78*x + 4*x^2)/(-3 + 2*x),x]

[Out]

-36*x + x^2 + 4*Log[3 - 2*x]

Rule 698

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] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-36+2 x+\frac {8}{-3+2 x}\right ) \, dx\\ &=-36 x+x^2+4 \log (3-2 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.96 \begin {gather*} 2 \left (\frac {207}{8}-18 x+\frac {x^2}{2}+2 \log (-3+2 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(116 - 78*x + 4*x^2)/(-3 + 2*x),x]

[Out]

2*(207/8 - 18*x + x^2/2 + 2*Log[-3 + 2*x])

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fricas [A]  time = 0.53, size = 15, normalized size = 0.60 \begin {gather*} x^{2} - 36 \, x + 4 \, \log \left (2 \, x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-78*x+116)/(2*x-3),x, algorithm="fricas")

[Out]

x^2 - 36*x + 4*log(2*x - 3)

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giac [A]  time = 0.34, size = 16, normalized size = 0.64 \begin {gather*} x^{2} - 36 \, x + 4 \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-78*x+116)/(2*x-3),x, algorithm="giac")

[Out]

x^2 - 36*x + 4*log(abs(2*x - 3))

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maple [A]  time = 0.40, size = 16, normalized size = 0.64

method	result	size
default	\(-36 x +x^{2}+4 \ln \left (2 x -3\right )\)	\(16\)
norman	\(-36 x +x^{2}+4 \ln \left (2 x -3\right )\)	\(16\)
risch	\(-36 x +x^{2}+4 \ln \left (2 x -3\right )\)	\(16\)
meijerg	\(4 \ln \left (1-\frac {2 x}{3}\right )+\frac {x \left (2 x +6\right )}{2}-39 x\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2-78*x+116)/(2*x-3),x,method=_RETURNVERBOSE)

[Out]

-36*x+x^2+4*ln(2*x-3)

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maxima [A]  time = 0.40, size = 15, normalized size = 0.60 \begin {gather*} x^{2} - 36 \, x + 4 \, \log \left (2 \, x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-78*x+116)/(2*x-3),x, algorithm="maxima")

[Out]

x^2 - 36*x + 4*log(2*x - 3)

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mupad [B]  time = 0.66, size = 13, normalized size = 0.52 \begin {gather*} 4\,\ln \left (x-\frac {3}{2}\right )-36\,x+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 - 78*x + 116)/(2*x - 3),x)

[Out]

4*log(x - 3/2) - 36*x + x^2

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sympy [A]  time = 0.07, size = 14, normalized size = 0.56 \begin {gather*} x^{2} - 36 x + 4 \log {\left (2 x - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2-78*x+116)/(2*x-3),x)

[Out]

x**2 - 36*x + 4*log(2*x - 3)

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