3.2.7 \(\int \frac {x^2}{-1+x} \, dx\) [107]

Optimal. Leaf size=15 \[ x+\frac {x^2}{2}+\log (1-x) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \begin {gather*} \frac {x^2}{2}+x+\log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(-1 + x),x]

[Out]

x + x^2/2 + Log[1 - x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.07 \begin {gather*} -\frac {3}{2}+x+\frac {x^2}{2}+\log (-1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(-1 + x),x]

[Out]

-3/2 + x + x^2/2 + Log[-1 + x]

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Maple [A]
time = 0.07, size = 12, normalized size = 0.80

method	result	size
default	\(\frac {x^{2}}{2}+x +\ln \left (x -1\right )\)	\(12\)
norman	\(\frac {x^{2}}{2}+x +\ln \left (x -1\right )\)	\(12\)
risch	\(\frac {x^{2}}{2}+x +\ln \left (x -1\right )\)	\(12\)
parallelrisch	\(\frac {x^{2}}{2}+x +\ln \left (x -1\right )\)	\(12\)
meijerg	\(\frac {x \left (3 x +6\right )}{6}+\ln \left (1-x \right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x-1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.37, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{2} \, x^{2} + x + \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + x + log(x - 1)

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Fricas [A]
time = 0.58, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{2} \, x^{2} + x + \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + x + log(x - 1)

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Sympy [A]
time = 0.02, size = 10, normalized size = 0.67 \begin {gather*} \frac {x^{2}}{2} + x + \log {\left (x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 + x + log(x - 1)

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Giac [A]
time = 0.48, size = 12, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, x^{2} + x + \log \left ({\left | x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + x + log(abs(x - 1))

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Mupad [B]
time = 0.02, size = 11, normalized size = 0.73 \begin {gather*} x+\ln \left (x-1\right )+\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x - 1),x)

[Out]

x + log(x - 1) + x^2/2

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Chatgpt [F] Failed to verify
time = 1.00, size = 20, normalized size = 1.33 \begin {gather*} -\frac {x^{3}}{3}-\frac {x^{2}}{2}-x -\ln \left (x -1\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(x^2/(x-1),x)

[Out]

-1/3*x^3-1/2*x^2-x-ln(x-1)

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