\(\int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx\) [2715]

Optimal result

Integrand size = 35, antiderivative size = 248 \[ \int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx=-\frac {3 \sqrt {a x+\sqrt {-b+a x}}}{2 a}+\sqrt {-b+a x} \left (-\frac {2}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}}}{a}\right )-\frac {4 (-3+2 b) \arctan \left (\frac {1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-5+4 b}}\right )}{a \sqrt {-5+4 b}}+\frac {(-19+4 b) \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}\right )}{4 a}+\frac {2 \log \left (1-2 a x-\sqrt {a x+\sqrt {-b+a x}}+2 \sqrt {-b+a x} \sqrt {a x+\sqrt {-b+a x}}\right )}{a} \] Output:

-3/2*(a*x+(a*x-b)^(1/2))^(1/2)/a+(a*x-b)^(1/2)*(-2/a+(a*x+(a*x-b)^(1/2))^( 
1/2)/a)-4*(-3+2*b)*arctan((1-2*(a*x-b)^(1/2)+2*(a*x+(a*x-b)^(1/2))^(1/2))/ 
(-5+4*b)^(1/2))/a/(-5+4*b)^(1/2)+1/4*(-19+4*b)*ln(1+2*(a*x-b)^(1/2)-2*(a*x 
+(a*x-b)^(1/2))^(1/2))/a+2*ln(1-2*a*x-(a*x+(a*x-b)^(1/2))^(1/2)+2*(a*x-b)^ 
(1/2)*(a*x+(a*x-b)^(1/2))^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx=\frac {-8 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}} \left (-3+2 \sqrt {-b+a x}\right )-\frac {16 (-3+2 b) \arctan \left (\frac {1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-5+4 b}}\right )}{\sqrt {-5+4 b}}+(-19+4 b) \log \left (a \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}\right )\right )+8 \log \left (1-2 b+2 (b-a x)+\sqrt {a x+\sqrt {-b+a x}} \left (-1+2 \sqrt {-b+a x}\right )\right )}{4 a} \] Input:

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

Output:

(-8*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]]*(-3 + 2*Sqrt[-b + a*x]) 
- (16*(-3 + 2*b)*ArcTan[(1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x 
]])/Sqrt[-5 + 4*b]])/Sqrt[-5 + 4*b] + (-19 + 4*b)*Log[a*(-1 - 2*Sqrt[-b + 
a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]])] + 8*Log[1 - 2*b + 2*(b - a*x) + Sqrt 
[a*x + Sqrt[-b + a*x]]*(-1 + 2*Sqrt[-b + a*x])])/(4*a)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.68, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {7267, 7293, 2009}

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.

\(\displaystyle \int \frac {\sqrt {a x-b}}{\sqrt {\sqrt {a x-b}+a x}+1} \, dx\)

\(\Big \downarrow \) 7267

\(\displaystyle \frac {2 \int \frac {a x-b}{\sqrt {a x+\sqrt {a x-b}}+1}d\sqrt {a x-b}}{a}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {2 \int \left (\frac {\sqrt {a x+\sqrt {a x-b}} (1-b)}{a x+\sqrt {a x-b}-1}-\frac {\sqrt {a x-b} \sqrt {a x+\sqrt {a x-b}}}{a x+\sqrt {a x-b}-1}+\sqrt {a x+\sqrt {a x-b}}+\frac {b+\sqrt {a x-b}-1}{a x+\sqrt {a x-b}-1}-1\right )d\sqrt {a x-b}}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\text {arctanh}\left (\sqrt {\sqrt {a x-b}+a x}\right )+\frac {(3-2 b) \text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{\sqrt {5-4 b}}\right )}{\sqrt {5-4 b}}-\frac {1}{8} (1-4 b) \text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )+(1-b) \text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )-\frac {2 (1-b) \text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{\sqrt {5-4 b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {5-4 b}}-\frac {\text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{\sqrt {5-4 b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {5-4 b}}+\frac {1}{4} \sqrt {\sqrt {a x-b}+a x} \left (2 \sqrt {a x-b}+1\right )-\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}+\frac {1}{2} \log \left (-\sqrt {a x-b}-a x+1\right )\right )}{a}\)

Input:

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

Output:

(2*(-Sqrt[-b + a*x] - Sqrt[a*x + Sqrt[-b + a*x]] + (Sqrt[a*x + Sqrt[-b + a 
*x]]*(1 + 2*Sqrt[-b + a*x]))/4 + ArcTanh[Sqrt[a*x + Sqrt[-b + a*x]]] + ((3 
 - 2*b)*ArcTanh[(1 + 2*Sqrt[-b + a*x])/Sqrt[5 - 4*b]])/Sqrt[5 - 4*b] + Arc 
Tanh[(1 + 2*Sqrt[-b + a*x])/(2*Sqrt[a*x + Sqrt[-b + a*x]])]/2 - ((1 - 4*b) 
*ArcTanh[(1 + 2*Sqrt[-b + a*x])/(2*Sqrt[a*x + Sqrt[-b + a*x]])])/8 + (1 - 
b)*ArcTanh[(1 + 2*Sqrt[-b + a*x])/(2*Sqrt[a*x + Sqrt[-b + a*x]])] - ArcTan 
h[(1 + 2*Sqrt[-b + a*x])/(Sqrt[5 - 4*b]*Sqrt[a*x + Sqrt[-b + a*x]])]/Sqrt[ 
5 - 4*b] - (2*(1 - b)*ArcTanh[(1 + 2*Sqrt[-b + a*x])/(Sqrt[5 - 4*b]*Sqrt[a 
*x + Sqrt[-b + a*x]])])/Sqrt[5 - 4*b] + Log[1 - a*x - Sqrt[-b + a*x]]/2))/ 
a
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(621\) vs. \(2(208)=416\).

Time = 0.19 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.51

method	result	size
derivativedivides	\(\frac {\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{2}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{4}-\frac {2 \left (3-2 b +\sqrt {5-4 b}\right ) \left (\frac {\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}{2}-\frac {\sqrt {5-4 b}\, \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {2-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{2}\right )}{\sqrt {5-4 b}}+\frac {2 \left (-3+2 b +\sqrt {5-4 b}\right ) \left (-\frac {\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}{2}-\frac {\sqrt {5-4 b}\, \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {2+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{2}\right )}{\sqrt {5-4 b}}-2 \sqrt {a x -b}+\ln \left (a x +\sqrt {a x -b}-1\right )-\frac {4 \left (-b +\frac {3}{2}\right ) \arctan \left (\frac {2 \sqrt {a x -b}+1}{\sqrt {-5+4 b}}\right )}{\sqrt {-5+4 b}}}{a}\)	\(622\)
default	\(\frac {\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{2}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{4}-\frac {2 \left (3-2 b +\sqrt {5-4 b}\right ) \left (\frac {\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}{2}-\frac {\sqrt {5-4 b}\, \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {2-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{2}\right )}{\sqrt {5-4 b}}+\frac {2 \left (-3+2 b +\sqrt {5-4 b}\right ) \left (-\frac {\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}{2}-\frac {\sqrt {5-4 b}\, \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {2+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{2}\right )}{\sqrt {5-4 b}}-2 \sqrt {a x -b}+\ln \left (a x +\sqrt {a x -b}-1\right )-\frac {4 \left (-b +\frac {3}{2}\right ) \arctan \left (\frac {2 \sqrt {a x -b}+1}{\sqrt {-5+4 b}}\right )}{\sqrt {-5+4 b}}}{a}\)	\(622\)

Input:

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

Output:

2/a*(1/4*(2*(a*x-b)^(1/2)+1)*(a*x+(a*x-b)^(1/2))^(1/2)+1/8*(4*b-1)*ln(1/2+ 
(a*x-b)^(1/2)+(a*x+(a*x-b)^(1/2))^(1/2))-(3-2*b+(5-4*b)^(1/2))/(5-4*b)^(1/ 
2)*(1/2*(((a*x-b)^(1/2)+1/2+1/2*(5-4*b)^(1/2))^2-(5-4*b)^(1/2)*((a*x-b)^(1 
/2)+1/2+1/2*(5-4*b)^(1/2))+1)^(1/2)-1/4*(5-4*b)^(1/2)*ln((a*x-b)^(1/2)+1/2 
+(((a*x-b)^(1/2)+1/2+1/2*(5-4*b)^(1/2))^2-(5-4*b)^(1/2)*((a*x-b)^(1/2)+1/2 
+1/2*(5-4*b)^(1/2))+1)^(1/2))-1/2*arctanh(1/2*(2-(5-4*b)^(1/2)*((a*x-b)^(1 
/2)+1/2+1/2*(5-4*b)^(1/2)))/(((a*x-b)^(1/2)+1/2+1/2*(5-4*b)^(1/2))^2-(5-4* 
b)^(1/2)*((a*x-b)^(1/2)+1/2+1/2*(5-4*b)^(1/2))+1)^(1/2)))+(-3+2*b+(5-4*b)^ 
(1/2))/(5-4*b)^(1/2)*(-1/2*(((a*x-b)^(1/2)+1/2-1/2*(5-4*b)^(1/2))^2+(5-4*b 
)^(1/2)*((a*x-b)^(1/2)+1/2-1/2*(5-4*b)^(1/2))+1)^(1/2)-1/4*(5-4*b)^(1/2)*l 
n((a*x-b)^(1/2)+1/2+(((a*x-b)^(1/2)+1/2-1/2*(5-4*b)^(1/2))^2+(5-4*b)^(1/2) 
*((a*x-b)^(1/2)+1/2-1/2*(5-4*b)^(1/2))+1)^(1/2))+1/2*arctanh(1/2*(2+(5-4*b 
)^(1/2)*((a*x-b)^(1/2)+1/2-1/2*(5-4*b)^(1/2)))/(((a*x-b)^(1/2)+1/2-1/2*(5- 
4*b)^(1/2))^2+(5-4*b)^(1/2)*((a*x-b)^(1/2)+1/2-1/2*(5-4*b)^(1/2))+1)^(1/2) 
))-(a*x-b)^(1/2)+1/2*ln(a*x+(a*x-b)^(1/2)-1)-2*(-b+3/2)/(-5+4*b)^(1/2)*arc 
tan((2*(a*x-b)^(1/2)+1)/(-5+4*b)^(1/2)))
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (irrational residues)
 

Sympy [F]

\[ \int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx=\int \frac {\sqrt {a x - b}}{\sqrt {a x + \sqrt {a x - b}} + 1}\, dx \] Input:

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

Output:

Integral(sqrt(a*x - b)/(sqrt(a*x + sqrt(a*x - b)) + 1), x)
 

Maxima [F]

\[ \int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx=\int { \frac {\sqrt {a x - b}}{\sqrt {a x + \sqrt {a x - b}} + 1} \,d x } \] Input:

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

Output:

integrate(sqrt(a*x - b)/(sqrt(a*x + sqrt(a*x - b)) + 1), x)
 

Giac [A] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.59 \[ \int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx=\frac {1}{2} \, \sqrt {a x + \sqrt {a x - b}} {\left (\frac {2 \, \sqrt {a x - b}}{a} - \frac {3}{a}\right )} + \frac {{\left (4 \, b - 11\right )} \log \left ({\left | -2 \, \sqrt {a x - b} + 2 \, \sqrt {a x + \sqrt {a x - b}} - 1 \right |}\right )}{4 \, a} + \frac {2 \, {\left (2 \, b - 3\right )} \arctan \left (-\frac {2 \, \sqrt {a x - b} - 2 \, \sqrt {a x + \sqrt {a x - b}} + 3}{\sqrt {4 \, b - 5}}\right )}{a \sqrt {4 \, b - 5}} - \frac {2 \, {\left (2 \, b - 3\right )} \arctan \left (-\frac {2 \, \sqrt {a x - b} - 2 \, \sqrt {a x + \sqrt {a x - b}} - 1}{\sqrt {4 \, b - 5}}\right )}{a \sqrt {4 \, b - 5}} + \frac {2 \, {\left (2 \, b - 3\right )} \arctan \left (\frac {2 \, \sqrt {a x - b} + 1}{\sqrt {4 \, b - 5}}\right )}{a \sqrt {4 \, b - 5}} + \frac {\log \left (a x + \sqrt {a x - b} - 1\right )}{a} - \frac {\log \left ({\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} + b + 3 \, \sqrt {a x - b} - 3 \, \sqrt {a x + \sqrt {a x - b}} + 1\right )}{a} + \frac {\log \left ({\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} + b - \sqrt {a x - b} + \sqrt {a x + \sqrt {a x - b}} - 1\right )}{a} - \frac {2 \, \sqrt {a x - b}}{a} \] Input:

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

Output:

1/2*sqrt(a*x + sqrt(a*x - b))*(2*sqrt(a*x - b)/a - 3/a) + 1/4*(4*b - 11)*l 
og(abs(-2*sqrt(a*x - b) + 2*sqrt(a*x + sqrt(a*x - b)) - 1))/a + 2*(2*b - 3 
)*arctan(-(2*sqrt(a*x - b) - 2*sqrt(a*x + sqrt(a*x - b)) + 3)/sqrt(4*b - 5 
))/(a*sqrt(4*b - 5)) - 2*(2*b - 3)*arctan(-(2*sqrt(a*x - b) - 2*sqrt(a*x + 
 sqrt(a*x - b)) - 1)/sqrt(4*b - 5))/(a*sqrt(4*b - 5)) + 2*(2*b - 3)*arctan 
((2*sqrt(a*x - b) + 1)/sqrt(4*b - 5))/(a*sqrt(4*b - 5)) + log(a*x + sqrt(a 
*x - b) - 1)/a - log((sqrt(a*x - b) - sqrt(a*x + sqrt(a*x - b)))^2 + b + 3 
*sqrt(a*x - b) - 3*sqrt(a*x + sqrt(a*x - b)) + 1)/a + log((sqrt(a*x - b) - 
 sqrt(a*x + sqrt(a*x - b)))^2 + b - sqrt(a*x - b) + sqrt(a*x + sqrt(a*x - 
b)) - 1)/a - 2*sqrt(a*x - b)/a
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx=\int \frac {\sqrt {a\,x-b}}{\sqrt {a\,x+\sqrt {a\,x-b}}+1} \,d x \] Input:

int((a*x - b)^(1/2)/((a*x + (a*x - b)^(1/2))^(1/2) + 1),x)
 

Output:

int((a*x - b)^(1/2)/((a*x + (a*x - b)^(1/2))^(1/2) + 1), x)
 

Reduce [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 927, normalized size of antiderivative = 3.74 \[ \int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx =\text {Too large to display} \] Input:

int((a*x-b)^(1/2)/(1+(a*x+(a*x-b)^(1/2))^(1/2)),x)
 

Output:

(32*sqrt(4*b - 1)*sqrt( - 4*b + 1)*atan((8*sqrt(a*x - b)*sqrt(sqrt(a*x - b 
) + a*x)*b - 2*sqrt(a*x - b)*sqrt(sqrt(a*x - b) + a*x) + 4*sqrt(sqrt(a*x - 
 b) + a*x)*b - sqrt(sqrt(a*x - b) + a*x) + 8*sqrt(a*x - b)*b - 2*sqrt(a*x 
- b) + 8*a*b*x - 2*a*x)/(2*sqrt(4*b - 1)*sqrt(sqrt(a*x - b) + a*x)*sqrt( - 
 4*b + 1) + 2*sqrt(4*b - 1)*sqrt(a*x - b)*sqrt( - 4*b + 1) + sqrt(4*b - 1) 
*sqrt( - 4*b + 1)))*b - 40*sqrt(4*b - 1)*sqrt( - 4*b + 1)*atan((8*sqrt(a*x 
 - b)*sqrt(sqrt(a*x - b) + a*x)*b - 2*sqrt(a*x - b)*sqrt(sqrt(a*x - b) + a 
*x) + 4*sqrt(sqrt(a*x - b) + a*x)*b - sqrt(sqrt(a*x - b) + a*x) + 8*sqrt(a 
*x - b)*b - 2*sqrt(a*x - b) + 8*a*b*x - 2*a*x)/(2*sqrt(4*b - 1)*sqrt(sqrt( 
a*x - b) + a*x)*sqrt( - 4*b + 1) + 2*sqrt(4*b - 1)*sqrt(a*x - b)*sqrt( - 4 
*b + 1) + sqrt(4*b - 1)*sqrt( - 4*b + 1))) + 128*sqrt(4*b - 5)*atan((2*sqr 
t(sqrt(a*x - b) + a*x) + 2*sqrt(a*x - b) + 3)/sqrt(4*b - 5))*b**2 - 224*sq 
rt(4*b - 5)*atan((2*sqrt(sqrt(a*x - b) + a*x) + 2*sqrt(a*x - b) + 3)/sqrt( 
4*b - 5))*b + 48*sqrt(4*b - 5)*atan((2*sqrt(sqrt(a*x - b) + a*x) + 2*sqrt( 
a*x - b) + 3)/sqrt(4*b - 5)) + 64*sqrt(a*x - b)*sqrt(sqrt(a*x - b) + a*x)* 
b**2 - 96*sqrt(a*x - b)*sqrt(sqrt(a*x - b) + a*x)*b + 20*sqrt(a*x - b)*sqr 
t(sqrt(a*x - b) + a*x) - 96*sqrt(sqrt(a*x - b) + a*x)*b**2 + 144*sqrt(sqrt 
(a*x - b) + a*x)*b - 30*sqrt(sqrt(a*x - b) + a*x) - 128*sqrt(a*x - b)*b**2 
 + 192*sqrt(a*x - b)*b - 40*sqrt(a*x - b) + 64*log(4*sqrt(a*x - b) + 4*a*x 
 - 4)*b**2 - 96*log(4*sqrt(a*x - b) + 4*a*x - 4)*b + 20*log(4*sqrt(a*x ...