3.18.58 \(\int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx\) [1758]

3.18.58.1 Optimal result

Integrand size = 28, antiderivative size = 118 \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\frac {\left (-3+2 b \sqrt {b+a x}\right ) \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}}{2 a b^2}+\frac {\left (-3-4 b^3\right ) \log \left (-1-2 b \sqrt {b+a x}+2 \sqrt {b} \sqrt {-b^2+\sqrt {b+a x}+b (b+a x)}\right )}{4 a b^{5/2}} \]

output

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

3.18.58.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\frac {2 \sqrt {b} \sqrt {a b x+\sqrt {b+a x}} \left (-3+2 b \sqrt {b+a x}\right )-\left (3+4 b^3\right ) \log \left (a b^2 \left (1+2 b \sqrt {b+a x}-2 \sqrt {b} \sqrt {a b x+\sqrt {b+a x}}\right )\right )}{4 a b^{5/2}} \]

input

Integrate[Sqrt[b + a*x]/Sqrt[a*b*x + Sqrt[b + a*x]],x]
 

output

(2*Sqrt[b]*Sqrt[a*b*x + Sqrt[b + a*x]]*(-3 + 2*b*Sqrt[b + a*x]) - (3 + 4*b 
^3)*Log[a*b^2*(1 + 2*b*Sqrt[b + a*x] - 2*Sqrt[b]*Sqrt[a*b*x + Sqrt[b + a*x 
]])])/(4*a*b^(5/2))
 

3.18.58.3 Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.30, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {7267, 1166, 27, 1160, 1092, 219}

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.

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 1166

\(\displaystyle \frac {2 \left (\frac {\int \frac {2 b^2-3 \sqrt {b+a x}}{2 \sqrt {-b^2+(b+a x) b+\sqrt {b+a x}}}d\sqrt {b+a x}}{2 b}+\frac {\sqrt {a x+b} \sqrt {b (a x+b)+\sqrt {a x+b}-b^2}}{2 b}\right )}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {\int \frac {2 b^2-3 \sqrt {b+a x}}{\sqrt {-b^2+(b+a x) b+\sqrt {b+a x}}}d\sqrt {b+a x}}{4 b}+\frac {\sqrt {a x+b} \sqrt {b (a x+b)+\sqrt {a x+b}-b^2}}{2 b}\right )}{a}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {2 \left (\frac {\frac {\left (4 b^3+3\right ) \int \frac {1}{\sqrt {-b^2+(b+a x) b+\sqrt {b+a x}}}d\sqrt {b+a x}}{2 b}-\frac {3 \sqrt {b (a x+b)+\sqrt {a x+b}-b^2}}{b}}{4 b}+\frac {\sqrt {a x+b} \sqrt {b (a x+b)+\sqrt {a x+b}-b^2}}{2 b}\right )}{a}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {2 \left (\frac {\frac {\left (4 b^3+3\right ) \int \frac {1}{3 b-a x}d\frac {2 \sqrt {b+a x} b+1}{\sqrt {-b^2+(b+a x) b+\sqrt {b+a x}}}}{b}-\frac {3 \sqrt {b (a x+b)+\sqrt {a x+b}-b^2}}{b}}{4 b}+\frac {\sqrt {a x+b} \sqrt {b (a x+b)+\sqrt {a x+b}-b^2}}{2 b}\right )}{a}\)

\(\Big \downarrow \) 219

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

input

Int[Sqrt[b + a*x]/Sqrt[a*b*x + Sqrt[b + a*x]],x]
 

output

(2*((Sqrt[b + a*x]*Sqrt[-b^2 + Sqrt[b + a*x] + b*(b + a*x)])/(2*b) + ((-3* 
Sqrt[-b^2 + Sqrt[b + a*x] + b*(b + a*x)])/b + ((3 + 4*b^3)*ArcTanh[(1 + 2* 
b*Sqrt[b + a*x])/(2*Sqrt[b]*Sqrt[-b^2 + Sqrt[b + a*x] + b*(b + a*x)])])/(2 
*b^(3/2)))/(4*b)))/a
 

3.18.58.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

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]]
 

3.18.58.4 Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(\frac {\frac {\sqrt {a x +b}\, \sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}}{b}-\frac {3 \left (\frac {\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}}{b}-\frac {\ln \left (\frac {\frac {1}{2}+b \sqrt {a x +b}}{\sqrt {b}}+\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}\right )}{2 b^{\frac {3}{2}}}\right )}{2 b}+\sqrt {b}\, \ln \left (\frac {\frac {1}{2}+b \sqrt {a x +b}}{\sqrt {b}}+\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}\right )}{a}\)	\(161\)
default	\(\frac {\frac {\sqrt {a x +b}\, \sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}}{b}-\frac {3 \left (\frac {\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}}{b}-\frac {\ln \left (\frac {\frac {1}{2}+b \sqrt {a x +b}}{\sqrt {b}}+\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}\right )}{2 b^{\frac {3}{2}}}\right )}{2 b}+\sqrt {b}\, \ln \left (\frac {\frac {1}{2}+b \sqrt {a x +b}}{\sqrt {b}}+\sqrt {-b^{2}+\sqrt {a x +b}+b \left (a x +b \right )}\right )}{a}\)	\(161\)

input

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

output

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

3.18.58.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.18.58.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (99) = 198\).

Time = 0.64 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.92 \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\begin {cases} \frac {2 \left (\begin {cases} \left (\frac {b}{2} + \frac {3}{8 b^{2}}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {- b^{2} + b \left (a x + b\right ) + \sqrt {a x + b}} + 2 b \sqrt {a x + b} + 1 \right )}}{\sqrt {b}} & \text {for}\: b^{2} + \frac {1}{4 b} \neq 0 \\\frac {\left (\sqrt {a x + b} + \frac {1}{2 b}\right ) \log {\left (\sqrt {a x + b} + \frac {1}{2 b} \right )}}{\sqrt {b \left (\sqrt {a x + b} + \frac {1}{2 b}\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \left (\frac {\sqrt {a x + b}}{2 b} - \frac {3}{4 b^{2}}\right ) \sqrt {- b^{2} + b \left (a x + b\right ) + \sqrt {a x + b}} & \text {for}\: b \neq 0 \\2 b^{4} \sqrt {- b^{2} + \sqrt {a x + b}} + \frac {4 b^{2} \left (- b^{2} + \sqrt {a x + b}\right )^{\frac {3}{2}}}{3} + \frac {2 \left (- b^{2} + \sqrt {a x + b}\right )^{\frac {5}{2}}}{5} & \text {otherwise} \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\\sqrt [4]{b} x & \text {otherwise} \end {cases} \]

input

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

output

Piecewise((2*Piecewise(((b/2 + 3/(8*b**2))*Piecewise((log(2*sqrt(b)*sqrt(- 
b**2 + b*(a*x + b) + sqrt(a*x + b)) + 2*b*sqrt(a*x + b) + 1)/sqrt(b), Ne(b 
**2 + 1/(4*b), 0)), ((sqrt(a*x + b) + 1/(2*b))*log(sqrt(a*x + b) + 1/(2*b) 
)/sqrt(b*(sqrt(a*x + b) + 1/(2*b))**2), True)) + (sqrt(a*x + b)/(2*b) - 3/ 
(4*b**2))*sqrt(-b**2 + b*(a*x + b) + sqrt(a*x + b)), Ne(b, 0)), (2*b**4*sq 
rt(-b**2 + sqrt(a*x + b)) + 4*b**2*(-b**2 + sqrt(a*x + b))**(3/2)/3 + 2*(- 
b**2 + sqrt(a*x + b))**(5/2)/5, True))/a, Ne(a, 0)), (b**(1/4)*x, True))
 

3.18.58.7 Maxima [F]

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

input

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

output

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

3.18.58.8 Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {b+a x}}{\sqrt {a b x+\sqrt {b+a x}}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.18.58.9 Mupad [F(-1)]

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

input

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

output

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