\(\int \frac {\sqrt {x}}{-a+b x} \, dx\) [471]

Optimal result

Integrand size = 15, antiderivative size = 40 \[ \int \frac {\sqrt {x}}{-a+b x} \, dx=\frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \]

[Out]

-2*arctanh(b^(1/2)*x^(1/2)/a^(1/2))*a^(1/2)/b^(3/2)+2*x^(1/2)/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 65, 214} \[ \int \frac {\sqrt {x}}{-a+b x} \, dx=\frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \]

[In]

Int[Sqrt[x]/(-a + b*x),x]

[Out]

(2*Sqrt[x])/b - (2*Sqrt[a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(3/2)

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {x}}{b}+\frac {a \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{b} \\ & = \frac {2 \sqrt {x}}{b}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{b} \\ & = \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x}}{-a+b x} \, dx=\frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \]

[In]

Integrate[Sqrt[x]/(-a + b*x),x]

[Out]

(2*Sqrt[x])/b - (2*Sqrt[a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(3/2)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(\frac {2 \sqrt {x}}{b}-\frac {2 a \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\)	\(32\)
default	\(\frac {2 \sqrt {x}}{b}-\frac {2 a \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\)	\(32\)
risch	\(\frac {2 \sqrt {x}}{b}-\frac {2 a \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\)	\(32\)

[In]

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

[Out]

2*x^(1/2)/b-2*a/b/(a*b)^(1/2)*arctanh(b*x^(1/2)/(a*b)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.08 \[ \int \frac {\sqrt {x}}{-a+b x} \, dx=\left [\frac {\sqrt {\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {\frac {a}{b}} + a}{b x - a}\right ) + 2 \, \sqrt {x}}{b}, \frac {2 \, {\left (\sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {-\frac {a}{b}}}{a}\right ) + \sqrt {x}\right )}}{b}\right ] \]

[In]

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

[Out]

[(sqrt(a/b)*log((b*x - 2*b*sqrt(x)*sqrt(a/b) + a)/(b*x - a)) + 2*sqrt(x))/b, 2*(sqrt(-a/b)*arctan(b*sqrt(x)*sq
rt(-a/b)/a) + sqrt(x))/b]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (36) = 72\).

Time = 0.47 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.08 \[ \int \frac {\sqrt {x}}{-a+b x} \, dx=\begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 x^{\frac {3}{2}}}{3 a} & \text {for}\: b = 0 \\\frac {2 \sqrt {x}}{b} & \text {for}\: a = 0 \\\frac {a \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{b^{2} \sqrt {\frac {a}{b}}} - \frac {a \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{b^{2} \sqrt {\frac {a}{b}}} + \frac {2 \sqrt {x}}{b} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*sqrt(x), Eq(a, 0) & Eq(b, 0)), (-2*x**(3/2)/(3*a), Eq(b, 0)), (2*sqrt(x)/b, Eq(a, 0)), (a*log(s
qrt(x) - sqrt(a/b))/(b**2*sqrt(a/b)) - a*log(sqrt(x) + sqrt(a/b))/(b**2*sqrt(a/b)) + 2*sqrt(x)/b, True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {x}}{-a+b x} \, dx=\frac {a \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {2 \, \sqrt {x}}{b} \]

[In]

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

[Out]

a*log((b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) + sqrt(a*b)))/(sqrt(a*b)*b) + 2*sqrt(x)/b

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {x}}{-a+b x} \, dx=\frac {2 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} b} + \frac {2 \, \sqrt {x}}{b} \]

[In]

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

[Out]

2*a*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*b) + 2*sqrt(x)/b

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {x}}{-a+b x} \, dx=\frac {2\,\sqrt {x}}{b}-\frac {2\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \]

[In]

int(-x^(1/2)/(a - b*x),x)

[Out]

(2*x^(1/2))/b - (2*a^(1/2)*atanh((b^(1/2)*x^(1/2))/a^(1/2)))/b^(3/2)