3.261 \(\int \frac{3+2 x^2}{(-1+x)^2 x} \, dx\)

Optimal. Leaf size=22 \[ \frac{5}{1-x}-\log (1-x)+3 \log (x) \]

[Out]

5/(1 - x) - Log[1 - x] + 3*Log[x]

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Rubi [A]  time = 0.0133821, antiderivative size = 22, normalized size of antiderivative = 1., 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 = {894} \[ \frac{5}{1-x}-\log (1-x)+3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x^2)/((-1 + x)^2*x),x]

[Out]

5/(1 - x) - Log[1 - x] + 3*Log[x]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{3+2 x^2}{(-1+x)^2 x} \, dx &=\int \left (\frac{1}{1-x}+\frac{5}{(-1+x)^2}+\frac{3}{x}\right ) \, dx\\ &=\frac{5}{1-x}-\log (1-x)+3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0095116, size = 20, normalized size = 0.91 \[ -\frac{5}{x-1}-\log (1-x)+3 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x^2)/((-1 + x)^2*x),x]

[Out]

-5/(-1 + x) - Log[1 - x] + 3*Log[x]

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Maple [A]  time = 0.007, size = 19, normalized size = 0.9 \begin{align*} -5\, \left ( x-1 \right ) ^{-1}-\ln \left ( x-1 \right ) +3\,\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2+3)/(x-1)^2/x,x)

[Out]

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

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Maxima [A]  time = 0.986294, size = 24, normalized size = 1.09 \begin{align*} -\frac{5}{x - 1} - \log \left (x - 1\right ) + 3 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+3)/(-1+x)^2/x,x, algorithm="maxima")

[Out]

-5/(x - 1) - log(x - 1) + 3*log(x)

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Fricas [A]  time = 1.2272, size = 73, normalized size = 3.32 \begin{align*} -\frac{{\left (x - 1\right )} \log \left (x - 1\right ) - 3 \,{\left (x - 1\right )} \log \left (x\right ) + 5}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+3)/(-1+x)^2/x,x, algorithm="fricas")

[Out]

-((x - 1)*log(x - 1) - 3*(x - 1)*log(x) + 5)/(x - 1)

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Sympy [A]  time = 0.102487, size = 14, normalized size = 0.64 \begin{align*} 3 \log{\left (x \right )} - \log{\left (x - 1 \right )} - \frac{5}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2+3)/(-1+x)**2/x,x)

[Out]

3*log(x) - log(x - 1) - 5/(x - 1)

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Giac [A]  time = 1.17183, size = 38, normalized size = 1.73 \begin{align*} -\frac{5}{x - 1} + 2 \, \log \left ({\left | x - 1 \right |}\right ) + 3 \, \log \left ({\left | -\frac{1}{x - 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+3)/(-1+x)^2/x,x, algorithm="giac")

[Out]

-5/(x - 1) + 2*log(abs(x - 1)) + 3*log(abs(-1/(x - 1) - 1))