\(\int \frac {1}{x (\sqrt {a+b x}+\sqrt {c+b x})} \, dx\) [4]

Optimal result

Integrand size = 25, antiderivative size = 97 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\frac {2 \sqrt {a+b x}}{a-c}-\frac {2 \sqrt {c+b x}}{a-c}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a-c}+\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )}{a-c} \] Output:

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

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.65 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\frac {2 \left (\sqrt {a+b x}-\sqrt {c+b x}-\sqrt {-\left (\sqrt {a}-\sqrt {c}\right )^2} \arctan \left (\frac {\sqrt {a+b x}-\sqrt {c+b x}}{\sqrt {-\left (\sqrt {a}-\sqrt {c}\right )^2}}\right )-\sqrt {-\left (\sqrt {a}+\sqrt {c}\right )^2} \arctan \left (\frac {\sqrt {a+b x}-\sqrt {c+b x}}{\sqrt {-\left (\sqrt {a}+\sqrt {c}\right )^2}}\right )\right )}{a-c} \] Input:

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

Output:

(2*(Sqrt[a + b*x] - Sqrt[c + b*x] - Sqrt[-(Sqrt[a] - Sqrt[c])^2]*ArcTan[(S 
qrt[a + b*x] - Sqrt[c + b*x])/Sqrt[-(Sqrt[a] - Sqrt[c])^2]] - Sqrt[-(Sqrt[ 
a] + Sqrt[c])^2]*ArcTan[(Sqrt[a + b*x] - Sqrt[c + b*x])/Sqrt[-(Sqrt[a] + S 
qrt[c])^2]]))/(a - c)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2529, 60, 73, 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.

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

\(\Big \downarrow \) 2529

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \sqrt {a+b x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a-c}-\frac {2 \sqrt {b x+c}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )}{a-c}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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]
 

rule 2529

Int[(u_)/((e_.)*Sqrt[(a_.) + (b_.)*(x_)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)]), 
 x_Symbol] :> Simp[-d/(e*(b*c - a*d))   Int[u*Sqrt[a + b*x], x], x] + Simp[ 
b/(f*(b*c - a*d))   Int[u*Sqrt[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f} 
, x] && NeQ[b*c - a*d, 0] && EqQ[b*e^2 - d*f^2, 0]
 

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {2 \sqrt {b x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a -c}-\frac {2 \sqrt {b x +c}-2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )}{a -c}\)	\(73\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.15 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\left [-\frac {\sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + \sqrt {c} \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, \sqrt {b x + a} + 2 \, \sqrt {b x + c}}{a - c}, -\frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {b x + c}}\right ) + \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, \sqrt {b x + a} + 2 \, \sqrt {b x + c}}{a - c}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) - \sqrt {c} \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, \sqrt {b x + a} - 2 \, \sqrt {b x + c}}{a - c}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) - \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {b x + c}}\right ) + \sqrt {b x + a} - \sqrt {b x + c}\right )}}{a - c}\right ] \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\int \frac {1}{x \left (\sqrt {a + b x} + \sqrt {b x + c}\right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\int { \frac {1}{x {\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1016 vs. \(2 (81) = 162\).

Time = 0.28 (sec) , antiderivative size = 1016, normalized size of antiderivative = 10.47 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

2*a*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*(a - c)) - 2*(a^4*c - a^3*c^2 
 - a^2*c^3 + a*c^4 + 2*(a*c^2 + sqrt(a*c)*c^2)*(a - c)^2*sgn(-a + c) - 2*( 
a*c^2 + sqrt(a*c)*a*c)*(a - c)^2 + (a^2*c^2 - 2*a*c^3 + c^4 - (a^2*c - 2*a 
*c^2 + c^3)*sqrt(a*c))*abs(-a + c)*sgn(-a + c) - (a^3*c - 2*a^2*c^2 + a*c^ 
3 + (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*abs(-a + c) - (a^4*c - a^3*c^2 - a^ 
2*c^3 + a*c^4 + (a^3*c - a^2*c^2 - a*c^3 + c^4)*sqrt(a*c))*sgn(-a + c) + ( 
a^4 - a^3*c - a^2*c^2 + a*c^3)*sqrt(a*c))*arctan(-(sqrt(b*x + a) - sqrt(b* 
x + c))/sqrt(-(a^2 - c^2 + sqrt((a^2 - c^2)^2 - (a^3 - 3*a^2*c + 3*a*c^2 - 
 c^3)*(a - c)))/(a - c)))/((sqrt(-a)*a^4 - a^4*sqrt(-c) - 4*sqrt(-a)*a^3*c 
 + 4*a^3*sqrt(-c)*c + 6*sqrt(-a)*a^2*c^2 - 6*a^2*sqrt(-c)*c^2 - 4*sqrt(-a) 
*a*c^3 + 4*a*sqrt(-c)*c^3 + sqrt(-a)*c^4 - sqrt(-c)*c^4)*abs(-a + c)) + 2* 
(a^4*c - a^3*c^2 - a^2*c^3 + a*c^4 - 2*(a*c^2 + sqrt(a*c)*c^2)*(a - c)^2*s 
gn(-a + c) - 2*(a*c^2 - sqrt(a*c)*a*c)*(a - c)^2 + (a^2*c^2 - 2*a*c^3 + c^ 
4 - (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*abs(-a + c)*sgn(-a + c) + (a^3*c - 
2*a^2*c^2 + a*c^3 + (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*abs(-a + c) + (a^4* 
c - a^3*c^2 - a^2*c^3 + a*c^4 - (a^3*c - a^2*c^2 - a*c^3 + c^4)*sqrt(a*c)) 
*sgn(-a + c) + (a^4 - a^3*c - a^2*c^2 + a*c^3)*sqrt(a*c))*arctan(-(sqrt(b* 
x + a) - sqrt(b*x + c))/sqrt(-(a^2 - c^2 - sqrt((a^2 - c^2)^2 - (a^3 - 3*a 
^2*c + 3*a*c^2 - c^3)*(a - c)))/(a - c)))/((sqrt(-a)*a^4 - a^4*sqrt(-c) - 
4*sqrt(-a)*a^3*c + 4*a^3*sqrt(-c)*c + 6*sqrt(-a)*a^2*c^2 - 6*a^2*sqrt(-...
 

Mupad [B] (verification not implemented)

Time = 38.69 (sec) , antiderivative size = 2983, normalized size of antiderivative = 30.75 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((a^2*c^(5/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^3*c^(3/2)*(a*c 
^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^(7/2)*c*(a*c^3 + a^3*c - 2*a^2*c^2)^( 
1/2)*2i + a^(5/2)*c^2*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a*c^3*(a + b* 
x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^3*c*(a + b*x)^(1/2)*(a*c 
^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(3/2)*c^(5/2)*(a + b*x)^(1/2)*(a*c^3 
+ a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2)*c^(3/2)*(a + b*x)^(1/2)*(a*c^3 + a 
^3*c - 2*a^2*c^2)^(1/2)*2i - a^2*c^2*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^ 
2*c^2)^(1/2)*4i - a^(3/2)*c^(5/2)*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c 
^2)^(1/2)*2i - a^(5/2)*c^(3/2)*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2) 
^(1/2)*2i)/(2*a^5*c^(3/2) - 4*a^4*c^(5/2) + 2*a^(5/2)*c^4 + 2*a^3*c^(7/2) 
- 4*a^(7/2)*c^3 + 2*a^(9/2)*c^2 - 2*a^2*c^4*(a + b*x)^(1/2) + 4*a^3*c^3*(a 
 + b*x)^(1/2) - 2*a^4*c^2*(a + b*x)^(1/2) - 2*a^(3/2)*c^(9/2)*(a + b*x)^(1 
/2) + 2*a^(5/2)*c^(7/2)*(a + b*x)^(1/2) + 2*a^(7/2)*c^(5/2)*(a + b*x)^(1/2 
) - 2*a^(9/2)*c^(3/2)*(a + b*x)^(1/2) + 2*a^2*c^4*(c + b*x)^(1/2) - 4*a^3* 
c^3*(c + b*x)^(1/2) + 2*a^4*c^2*(c + b*x)^(1/2)))*(a + b*x)^(1/2)*(a*c^3 + 
 a^3*c - 2*a^2*c^2)^(1/2)*2i - 4*a^(3/2)*c - 8*a*c^(3/2) + atan((a^2*c^(5/ 
2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^3*c^(3/2)*(a*c^3 + a^3*c - 2*a 
^2*c^2)^(1/2)*2i - a^(7/2)*c*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2 
)*c^2*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a*c^3*(a + b*x)^(1/2)*(a*c^3 
+ a^3*c - 2*a^2*c^2)^(1/2)*2i + a^3*c*(a + b*x)^(1/2)*(a*c^3 + a^3*c - ...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 677, normalized size of antiderivative = 6.98 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(c)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - a - c)*atan((sqrt(b*x + c) 
+ sqrt(a + b*x))/sqrt(2*sqrt(c)*sqrt(a) - a - c))*a - 2*sqrt(c)*sqrt(a)*sq 
rt(2*sqrt(c)*sqrt(a) - a - c)*atan((sqrt(b*x + c) + sqrt(a + b*x))/sqrt(2* 
sqrt(c)*sqrt(a) - a - c))*c - 4*sqrt(2*sqrt(c)*sqrt(a) - a - c)*atan((sqrt 
(b*x + c) + sqrt(a + b*x))/sqrt(2*sqrt(c)*sqrt(a) - a - c))*a*c - 2*sqrt(a 
)*sqrt(a - c)*sqrt( - a + c)*atan((sqrt(a + b*x)*sqrt(b*x + c)*a - sqrt(a 
+ b*x)*sqrt(b*x + c)*c + a**2 + a*b*x - a*c - b*c*x)/(sqrt(a)*sqrt(a - c)* 
sqrt(b*x + c)*sqrt( - a + c) + sqrt(a)*sqrt(a + b*x)*sqrt(a - c)*sqrt( - a 
 + c)))*a + 2*sqrt(a)*sqrt(a - c)*sqrt( - a + c)*atan((sqrt(a + b*x)*sqrt( 
b*x + c)*a - sqrt(a + b*x)*sqrt(b*x + c)*c + a**2 + a*b*x - a*c - b*c*x)/( 
sqrt(a)*sqrt(a - c)*sqrt(b*x + c)*sqrt( - a + c) + sqrt(a)*sqrt(a + b*x)*s 
qrt(a - c)*sqrt( - a + c)))*c - 2*sqrt(b*x + c)*a**2 + 2*sqrt(b*x + c)*a*c 
 - sqrt(c)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + a + c)*log(sqrt(b*x + c) - sqr 
t(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a - sqrt(c)*sqrt(a)*sqrt(2*s 
qrt(c)*sqrt(a) + a + c)*log(sqrt(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a + c 
) + sqrt(a + b*x))*c + sqrt(c)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + a + c)*log 
(sqrt(b*x + c) + sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a + sqrt 
(c)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + a + c)*log(sqrt(b*x + c) + sqrt(2*sqr 
t(c)*sqrt(a) + a + c) + sqrt(a + b*x))*c + 2*sqrt(2*sqrt(c)*sqrt(a) + a + 
c)*log(sqrt(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))...