\(\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\) [572]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, 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 = {65, 223, 212} \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}} \]

[In]

Int[1/(Sqrt[x]*Sqrt[a + b*x]),x]

[Out]

(2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/Sqrt[b]

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx=-\frac {2 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {b}} \]

[In]

Integrate[1/(Sqrt[x]*Sqrt[a + b*x]),x]

[Out]

(-2*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])/Sqrt[b]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(20)=40\).

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71

method	result	size
default	\(\frac {\sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {b}}\)	\(48\)

[In]

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

[Out]

(x*(b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(1/2)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))/b^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx=\left [\frac {\log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right )}{\sqrt {b}}, -\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right )}{b}\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.98 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx=\frac {2 \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} \]

[In]

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

[Out]

2*asinh(sqrt(b)*sqrt(x)/sqrt(a))/sqrt(b)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).

Time = 0.55 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx=-\frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{\sqrt {b}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 77.66 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {b} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{{\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {-b}\,\sqrt {x}}\right )}{\sqrt {-b}} \]

[In]

int(1/(x^(1/2)*(a + b*x)^(1/2)),x)

[Out]

-(4*atan(((a + b*x)^(1/2) - a^(1/2))/((-b)^(1/2)*x^(1/2))))/(-b)^(1/2)