Integrand size = 17, antiderivative size = 102 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {(c+d x) \sqrt {\frac {b}{d}-\frac {b c-a d}{d (c+d x)}}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {b}{d}-\frac {b c-a d}{d (c+d x)}}}{\sqrt {b}}\right )}{\sqrt {b} d^{3/2}} \] Output:
(d*x+c)*(b/d-(-a*d+b*c)/d/(d*x+c))^(1/2)/d-(-a*d+b*c)*arctanh(d^(1/2)*(b/d -(-a*d+b*c)/d/(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d^(3/2)
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {\sqrt {\frac {a+b x}{c+d x}} \left (\sqrt {d} (c+d x)+\frac {(-b c+a d) \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {a+b x}}\right )}{d^{3/2}} \] Input:
Integrate[Sqrt[(a + b*x)/(c + d*x)],x]
Output:
(Sqrt[(a + b*x)/(c + d*x)]*(Sqrt[d]*(c + d*x) + ((-(b*c) + a*d)*Sqrt[c + d *x]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(Sqrt[b]*Sqr t[a + b*x])))/d^(3/2)
Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2051, 252, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {\frac {a+b x}{c+d x}} \, dx\) |
\(\Big \downarrow \) 2051 |
\(\displaystyle 2 (b c-a d) \int \frac {a+b x}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\sqrt {\frac {a+b x}{c+d x}}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle 2 (b c-a d) \left (\frac {\sqrt {\frac {a+b x}{c+d x}}}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\sqrt {\frac {a+b x}{c+d x}}}{2 d}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle 2 (b c-a d) \left (\frac {\sqrt {\frac {a+b x}{c+d x}}}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{2 \sqrt {b} d^{3/2}}\right )\) |
Input:
Int[Sqrt[(a + b*x)/(c + d*x)],x]
Output:
2*(b*c - a*d)*(Sqrt[(a + b*x)/(c + d*x)]/(2*d*(b - (d*(a + b*x))/(c + d*x) )) - ArcTanh[(Sqrt[d]*Sqrt[(a + b*x)/(c + d*x)])/Sqrt[b]]/(2*Sqrt[b]*d^(3/ 2)))
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]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_ Symbol] :> With[{q = Denominator[p]}, Simp[q*e*((b*c - a*d)/n) Subst[Int[ x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^(1/n - 1)/(b*e - d*x^q)^(1/n + 1)), x], x, (e*((a + b*x^n)/(c + d*x^n)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[p] && IntegerQ[1/n]
Time = 0.12 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.49
method | result | size |
default | \(\frac {\sqrt {\frac {b x +a}{d x +c}}\, \left (d x +c \right ) \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a d -\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d \sqrt {b d}}\) | \(152\) |
Input:
int(((b*x+a)/(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*((b*x+a)/(d*x+c))^(1/2)*(d*x+c)*(ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^( 1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*d-ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x +c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b*c+2*((b*x+a)*(d*x+c))^(1/2) *(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/d/(b*d)^(1/2)
Time = 0.11 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.76 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\left [-\frac {{\left (b c - a d\right )} \sqrt {b d} \log \left (2 \, b d x + b c + a d + 2 \, \sqrt {b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}\right ) - 2 \, {\left (b d^{2} x + b c d\right )} \sqrt {\frac {b x + a}{d x + c}}}{2 \, b d^{2}}, \frac {{\left (b c - a d\right )} \sqrt {-b d} \arctan \left (\frac {\sqrt {-b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d x + a d}\right ) + {\left (b d^{2} x + b c d\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d^{2}}\right ] \] Input:
integrate(((b*x+a)/(d*x+c))^(1/2),x, algorithm="fricas")
Output:
[-1/2*((b*c - a*d)*sqrt(b*d)*log(2*b*d*x + b*c + a*d + 2*sqrt(b*d)*(d*x + c)*sqrt((b*x + a)/(d*x + c))) - 2*(b*d^2*x + b*c*d)*sqrt((b*x + a)/(d*x + c)))/(b*d^2), ((b*c - a*d)*sqrt(-b*d)*arctan(sqrt(-b*d)*(d*x + c)*sqrt((b* x + a)/(d*x + c))/(b*d*x + a*d)) + (b*d^2*x + b*c*d)*sqrt((b*x + a)/(d*x + c)))/(b*d^2)]
\[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\int \sqrt {\frac {a + b x}{c + d x}}\, dx \] Input:
integrate(((b*x+a)/(d*x+c))**(1/2),x)
Output:
Integral(sqrt((a + b*x)/(c + d*x)), x)
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.16 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {{\left (b c - a d\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d - \frac {{\left (b x + a\right )} d^{2}}{d x + c}} + \frac {{\left (b c - a d\right )} \log \left (\frac {d \sqrt {\frac {b x + a}{d x + c}} - \sqrt {b d}}{d \sqrt {\frac {b x + a}{d x + c}} + \sqrt {b d}}\right )}{2 \, \sqrt {b d} d} \] Input:
integrate(((b*x+a)/(d*x+c))^(1/2),x, algorithm="maxima")
Output:
(b*c - a*d)*sqrt((b*x + a)/(d*x + c))/(b*d - (b*x + a)*d^2/(d*x + c)) + 1/ 2*(b*c - a*d)*log((d*sqrt((b*x + a)/(d*x + c)) - sqrt(b*d))/(d*sqrt((b*x + a)/(d*x + c)) + sqrt(b*d)))/(sqrt(b*d)*d)
Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {{\left (b c \mathrm {sgn}\left (d x + c\right ) - a d \mathrm {sgn}\left (d x + c\right )\right )} \log \left ({\left | -b c - a d - 2 \, \sqrt {b d} {\left (\sqrt {b d} x - \sqrt {b d x^{2} + b c x + a d x + a c}\right )} \right |}\right )}{2 \, \sqrt {b d} d} + \frac {\sqrt {b d x^{2} + b c x + a d x + a c} \mathrm {sgn}\left (d x + c\right )}{d} \] Input:
integrate(((b*x+a)/(d*x+c))^(1/2),x, algorithm="giac")
Output:
1/2*(b*c*sgn(d*x + c) - a*d*sgn(d*x + c))*log(abs(-b*c - a*d - 2*sqrt(b*d) *(sqrt(b*d)*x - sqrt(b*d*x^2 + b*c*x + a*d*x + a*c))))/(sqrt(b*d)*d) + sqr t(b*d*x^2 + b*c*x + a*d*x + a*c)*sgn(d*x + c)/d
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.88 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {\frac {a+b\,x}{c+d\,x}}}{\sqrt {b}}\right )\,\left (a\,d-b\,c\right )}{\sqrt {b}\,d^{3/2}}+\frac {\left (a\,d-b\,c\right )\,\sqrt {\frac {a+b\,x}{c+d\,x}}}{b\,d\,\left (\frac {d\,\left (a+b\,x\right )}{b\,\left (c+d\,x\right )}-1\right )} \] Input:
int(((a + b*x)/(c + d*x))^(1/2),x)
Output:
(atanh((d^(1/2)*((a + b*x)/(c + d*x))^(1/2))/b^(1/2))*(a*d - b*c))/(b^(1/2 )*d^(3/2)) + ((a*d - b*c)*((a + b*x)/(c + d*x))^(1/2))/(b*d*((d*(a + b*x)) /(b*(c + d*x)) - 1))
Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, b d +\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a d -\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b c}{b \,d^{2}} \] Input:
int(((b*x+a)/(d*x+c))^(1/2),x)
Output:
(sqrt(c + d*x)*sqrt(a + b*x)*b*d + sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b *x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*d - sqrt(d)*sqrt(b)*log((s qrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b*c)/(b*d** 2)