### 3.233 $$\int \frac{1+\frac{1}{\sqrt {x}}}{-1+\frac{1}{\sqrt {x}}} \, dx$$

Optimal. Leaf size=30 $-3 x^{2/3}-x-6 \sqrt {x}-6 \log \left (1-\sqrt {x}\right )$

[Out]

-6*x^(1/3) - 3*x^(2/3) - x - 6*Log[1 - x^(1/3)]

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Rubi [A]  time = 0.0170899, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.176, Rules used = {374, 376, 77} $-3 x^{2/3}-x-6 \sqrt {x}-6 \log \left (1-\sqrt {x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + x^(-1/3))/(-1 + x^(-1/3)),x]

[Out]

-6*x^(1/3) - 3*x^(2/3) - x - 6*Log[1 - x^(1/3)]

Rule 374

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[x^(n*(p + q))*(b + a/x^n)^
p*(d + c/x^n)^q, x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[p, q] && NegQ[n]

Rule 376

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Dis
t[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}
, x] && NeQ[b*c - a*d, 0] && FractionQ[n]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{1+\frac{1}{\sqrt {x}}}{-1+\frac{1}{\sqrt {x}}} \, dx &=\int \frac{1+\sqrt {x}}{1-\sqrt {x}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x^2 (1+x)}{1-x} \, dx,x,\sqrt {x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-2-\frac{2}{-1+x}-2 x-x^2\right ) \, dx,x,\sqrt {x}\right )\\ &=-6 \sqrt {x}-3 x^{2/3}-x-6 \log \left (1-\sqrt {x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0128185, size = 30, normalized size = 1. $-3 x^{2/3}-x-6 \sqrt {x}-6 \log \left (1-\sqrt {x}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-6*x^(1/3) - 3*x^(2/3) - x - 6*Log[1 - x^(1/3)]

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Maple [A]  time = 0.003, size = 23, normalized size = 0.8 \begin{align*} -x-3\,{x}^{2/3}-6\,\sqrt {x}-6\,\ln \left ( -1+\sqrt {x} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x-3*x^(2/3)-6*x^(1/3)-6*ln(-1+x^(1/3))

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Maxima [A]  time = 0.944397, size = 30, normalized size = 1. \begin{align*} -x - 3 \, x^{\frac{2}{3}} - 6 \, x^{\frac{1}{3}} - 6 \, \log \left (x^{\frac{1}{3}} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - 3*x^(2/3) - 6*x^(1/3) - 6*log(x^(1/3) - 1)

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Fricas [A]  time = 2.12879, size = 66, normalized size = 2.2 \begin{align*} -x - 3 \, x^{\frac{2}{3}} - 6 \, x^{\frac{1}{3}} - 6 \, \log \left (x^{\frac{1}{3}} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - 3*x^(2/3) - 6*x^(1/3) - 6*log(x^(1/3) - 1)

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Sympy [A]  time = 0.150008, size = 26, normalized size = 0.87 \begin{align*} - 3 x^{\frac{2}{3}} - 6 \sqrt {x} - x - 6 \log{\left (\sqrt {x} - 1 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x**(2/3) - 6*x**(1/3) - x - 6*log(x**(1/3) - 1)

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Giac [A]  time = 1.05281, size = 31, normalized size = 1.03 \begin{align*} -x - 3 \, x^{\frac{2}{3}} - 6 \, x^{\frac{1}{3}} - 6 \, \log \left ({\left | x^{\frac{1}{3}} - 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - 3*x^(2/3) - 6*x^(1/3) - 6*log(abs(x^(1/3) - 1))