\(\int \frac {1}{x (\sqrt {a+b x}+\sqrt {c+b x})^2} \, dx\) [9]

Optimal result

Integrand size = 25, antiderivative size = 133 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {2 b x}{(a-c)^2}-\frac {2 \sqrt {a+b x} \sqrt {c+b x}}{(a-c)^2}-\frac {2 (a+c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{(a-c)^2}+\frac {4 \sqrt {a} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+b x}}\right )}{(a-c)^2}+\frac {(a+c) \log (x)}{(a-c)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {\log \left (\sqrt {a} \sqrt {c}+b x-\sqrt {a+b x} \sqrt {c+b x}\right )}{\left (\sqrt {a}+\sqrt {c}\right )^2}+\frac {2 \left (c+b x-\sqrt {a+b x} \sqrt {c+b x}\right )+\left (\sqrt {a}+\sqrt {c}\right )^2 \log \left (\sqrt {a} \sqrt {c}-b x+\sqrt {a+b x} \sqrt {c+b x}\right )}{(a-c)^2} \] Input:

Integrate[1/(x*(Sqrt[a + b*x] + Sqrt[c + b*x])^2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7240, 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.

\(\displaystyle \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {b x+c}\right )^2} \, dx\)

\(\Big \downarrow \) 7240

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

\(\Big \downarrow \) 2009

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

Input:

Int[1/(x*(Sqrt[a + b*x] + Sqrt[c + b*x])^2),x]
 

Output:

(2*b*x - 2*Sqrt[a + b*x]*Sqrt[c + b*x] - 2*(a + c)*ArcTanh[Sqrt[a + b*x]/S 
qrt[c + b*x]] + 4*Sqrt[a]*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a] 
*Sqrt[c + b*x])] + (a + c)*Log[x])/(a - c)^2
 

Defintions of rubi rules used

rule 2009

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

rule 7240

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.02 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.94

method	result	size
default	\(\frac {a \ln \left (x \right )}{\left (a -c \right )^{2}}+\frac {c \ln \left (x \right )}{\left (a -c \right )^{2}}+\frac {2 b x}{\left (a -c \right )^{2}}+\frac {\sqrt {b x +a}\, \sqrt {b x +c}\, \left (2 \ln \left (\frac {a b x +b c x +2 \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}+2 a c}{x}\right ) \operatorname {csgn}\left (b \right ) a c -2 \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \operatorname {csgn}\left (b \right )-\sqrt {a c}\, \ln \left (\frac {\left (2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \operatorname {csgn}\left (b \right )+2 b x +a +c \right ) \operatorname {csgn}\left (b \right )}{2}\right ) a -\sqrt {a c}\, \ln \left (\frac {\left (2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \operatorname {csgn}\left (b \right )+2 b x +a +c \right ) \operatorname {csgn}\left (b \right )}{2}\right ) c \right ) \operatorname {csgn}\left (b \right )}{\left (a -c \right )^{2} \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}}\)	\(258\)

Input:

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

Output:

1/(a-c)^2*a*ln(x)+1/(a-c)^2*c*ln(x)+2*b*x/(a-c)^2+1/(a-c)^2*(b*x+a)^(1/2)* 
(b*x+c)^(1/2)*(2*ln((a*b*x+b*c*x+2*(a*c)^(1/2)*(b^2*x^2+a*b*x+b*c*x+a*c)^( 
1/2)+2*a*c)/x)*csgn(b)*a*c-2*(a*c)^(1/2)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*c 
sgn(b)-(a*c)^(1/2)*ln(1/2*(2*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*csgn(b)+2*b*x 
+a+c)*csgn(b))*a-(a*c)^(1/2)*ln(1/2*(2*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*csg 
n(b)+2*b*x+a+c)*csgn(b))*c)*csgn(b)/(a*c)^(1/2)/(b^2*x^2+a*b*x+b*c*x+a*c)^ 
(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.18 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\left [\frac {2 \, b x + {\left (a + c\right )} \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right ) + {\left (a + c\right )} \log \left (x\right ) + 2 \, \sqrt {a c} \log \left (\frac {2 \, a^{2} c + 2 \, a c^{2} + 2 \, {\left (2 \, a c + \sqrt {a c} {\left (a + c\right )}\right )} \sqrt {b x + a} \sqrt {b x + c} + {\left (a^{2} b + 2 \, a b c + b c^{2}\right )} x + 2 \, {\left (2 \, a c + {\left (a b + b c\right )} x\right )} \sqrt {a c}}{x}\right ) - 2 \, \sqrt {b x + a} \sqrt {b x + c}}{a^{2} - 2 \, a c + c^{2}}, \frac {2 \, b x + {\left (a + c\right )} \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right ) + {\left (a + c\right )} \log \left (x\right ) - 4 \, \sqrt {-a c} \arctan \left (-\frac {\sqrt {-a c} b x - \sqrt {-a c} \sqrt {b x + a} \sqrt {b x + c}}{a c}\right ) - 2 \, \sqrt {b x + a} \sqrt {b x + c}}{a^{2} - 2 \, a c + c^{2}}\right ] \] Input:

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

Output:

[(2*b*x + (a + c)*log(-2*b*x + 2*sqrt(b*x + a)*sqrt(b*x + c) - a - c) + (a 
 + c)*log(x) + 2*sqrt(a*c)*log((2*a^2*c + 2*a*c^2 + 2*(2*a*c + sqrt(a*c)*( 
a + c))*sqrt(b*x + a)*sqrt(b*x + c) + (a^2*b + 2*a*b*c + b*c^2)*x + 2*(2*a 
*c + (a*b + b*c)*x)*sqrt(a*c))/x) - 2*sqrt(b*x + a)*sqrt(b*x + c))/(a^2 - 
2*a*c + c^2), (2*b*x + (a + c)*log(-2*b*x + 2*sqrt(b*x + a)*sqrt(b*x + c) 
- a - c) + (a + c)*log(x) - 4*sqrt(-a*c)*arctan(-(sqrt(-a*c)*b*x - sqrt(-a 
*c)*sqrt(b*x + a)*sqrt(b*x + c))/(a*c)) - 2*sqrt(b*x + a)*sqrt(b*x + c))/( 
a^2 - 2*a*c + c^2)]
 

Sympy [F]

\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\int \frac {1}{x \left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\int { \frac {1}{x {\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.46 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {4 \, a c \arctan \left (\frac {{\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - a - c}{2 \, \sqrt {-a c}}\right )}{{\left (a^{2} - 2 \, a c + c^{2}\right )} \sqrt {-a c}} - \frac {2 \, {\left (a^{2} - 2 \, a c + c^{2}\right )} \sqrt {b x + a} \sqrt {b x + c}}{a^{4} - 4 \, a^{3} c + 6 \, a^{2} c^{2} - 4 \, a c^{3} + c^{4}} + \frac {{\left (a + c\right )} \log \left ({\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2}\right )}{a^{2} - 2 \, a c + c^{2}} + \frac {{\left (a + c\right )} \log \left ({\left | b x \right |}\right )}{a^{2} - 2 \, a c + c^{2}} + \frac {2 \, {\left (b x + a\right )}}{a^{2} - 2 \, a c + c^{2}} \] Input:

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

Output:

4*a*c*arctan(1/2*((sqrt(b*x + a) - sqrt(b*x + c))^2 - a - c)/sqrt(-a*c))/( 
(a^2 - 2*a*c + c^2)*sqrt(-a*c)) - 2*(a^2 - 2*a*c + c^2)*sqrt(b*x + a)*sqrt 
(b*x + c)/(a^4 - 4*a^3*c + 6*a^2*c^2 - 4*a*c^3 + c^4) + (a + c)*log((sqrt( 
b*x + a) - sqrt(b*x + c))^2)/(a^2 - 2*a*c + c^2) + (a + c)*log(abs(b*x))/( 
a^2 - 2*a*c + c^2) + 2*(b*x + a)/(a^2 - 2*a*c + c^2)
 

Mupad [B] (verification not implemented)

Time = 31.33 (sec) , antiderivative size = 524, normalized size of antiderivative = 3.94 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {2\,b\,x}{{\left (a-c\right )}^2}-\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+b\,x}-\sqrt {c}}+1\right )\,\left (\frac {4\,c}{{\left (a-c\right )}^2}+\frac {2}{a-c}\right )-\frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (4\,a+4\,c\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^3\,\left (a^2-2\,a\,c+c^2\right )}+\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (4\,a+4\,c\right )}{\left (\sqrt {c+b\,x}-\sqrt {c}\right )\,\left (a^2-2\,a\,c+c^2\right )}-\frac {16\,\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^2\,\left (a^2-2\,a\,c+c^2\right )}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^4}-\frac {2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^2}+1}+\frac {2\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+b\,x}-\sqrt {c}}-1\right )\,\left (a+c\right )}{{\left (a-c\right )}^2}+\frac {\ln \left (x\right )\,\left (a+c\right )}{a^2-2\,a\,c+c^2}+\frac {2\,\sqrt {a}\,\sqrt {c}\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+b\,x}-\sqrt {c}}\right )}{{\left (a-c\right )}^2}-\frac {2\,\sqrt {a}\,\sqrt {c}\,\ln \left (\frac {a\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+b\,x}-\sqrt {c}}-\sqrt {a}\,\sqrt {c}+\frac {c\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+b\,x}-\sqrt {c}}-\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^2}\right )}{a^2-2\,a\,c+c^2} \] Input:

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

Output:

(2*b*x)/(a - c)^2 - log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^( 
1/2)) + 1)*((4*c)/(a - c)^2 + 2/(a - c)) - ((((a + b*x)^(1/2) - a^(1/2))^3 
*(4*a + 4*c))/(((c + b*x)^(1/2) - c^(1/2))^3*(a^2 - 2*a*c + c^2)) + (((a + 
 b*x)^(1/2) - a^(1/2))*(4*a + 4*c))/(((c + b*x)^(1/2) - c^(1/2))*(a^2 - 2* 
a*c + c^2)) - (16*a^(1/2)*c^(1/2)*((a + b*x)^(1/2) - a^(1/2))^2)/(((c + b* 
x)^(1/2) - c^(1/2))^2*(a^2 - 2*a*c + c^2)))/(((a + b*x)^(1/2) - a^(1/2))^4 
/((c + b*x)^(1/2) - c^(1/2))^4 - (2*((a + b*x)^(1/2) - a^(1/2))^2)/((c + b 
*x)^(1/2) - c^(1/2))^2 + 1) + (2*log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x 
)^(1/2) - c^(1/2)) - 1)*(a + c))/(a - c)^2 + (log(x)*(a + c))/(a^2 - 2*a*c 
 + c^2) + (2*a^(1/2)*c^(1/2)*log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1 
/2) - c^(1/2))))/(a - c)^2 - (2*a^(1/2)*c^(1/2)*log((a*((a + b*x)^(1/2) - 
a^(1/2)))/((c + b*x)^(1/2) - c^(1/2)) - a^(1/2)*c^(1/2) + (c*((a + b*x)^(1 
/2) - a^(1/2)))/((c + b*x)^(1/2) - c^(1/2)) - (a^(1/2)*c^(1/2)*((a + b*x)^ 
(1/2) - a^(1/2))^2)/((c + b*x)^(1/2) - c^(1/2))^2))/(a^2 - 2*a*c + c^2)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.59 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {-2 \sqrt {b x +a}\, \sqrt {b x +c}-2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +c}-\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right )-2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +c}+\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right )+2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {b x +a}\, \sqrt {b x +c}+2 \sqrt {c}\, \sqrt {a}+2 b x \right )+\mathrm {log}\left (\sqrt {b x +c}-\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) a +\mathrm {log}\left (\sqrt {b x +c}-\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) c +\mathrm {log}\left (\sqrt {b x +c}+\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) a +\mathrm {log}\left (\sqrt {b x +c}+\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) c +\mathrm {log}\left (2 \sqrt {b x +a}\, \sqrt {b x +c}+2 \sqrt {c}\, \sqrt {a}+2 b x \right ) a +\mathrm {log}\left (2 \sqrt {b x +a}\, \sqrt {b x +c}+2 \sqrt {c}\, \sqrt {a}+2 b x \right ) c -4 \,\mathrm {log}\left (\frac {\sqrt {b x +c}+\sqrt {b x +a}}{\sqrt {a -c}}\right ) a -4 \,\mathrm {log}\left (\frac {\sqrt {b x +c}+\sqrt {b x +a}}{\sqrt {a -c}}\right ) c +a +2 b x +c}{a^{2}-2 a c +c^{2}} \] Input:

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

Output:

( - 2*sqrt(a + b*x)*sqrt(b*x + c) - 2*sqrt(c)*sqrt(a)*log(sqrt(b*x + c) - 
sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x)) - 2*sqrt(c)*sqrt(a)*log(s 
qrt(b*x + c) + sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x)) + 2*sqrt(c 
)*sqrt(a)*log(2*sqrt(a + b*x)*sqrt(b*x + c) + 2*sqrt(c)*sqrt(a) + 2*b*x) + 
 log(sqrt(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a + 
log(sqrt(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*c + l 
og(sqrt(b*x + c) + sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a + lo 
g(sqrt(b*x + c) + sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*c + log 
(2*sqrt(a + b*x)*sqrt(b*x + c) + 2*sqrt(c)*sqrt(a) + 2*b*x)*a + log(2*sqrt 
(a + b*x)*sqrt(b*x + c) + 2*sqrt(c)*sqrt(a) + 2*b*x)*c - 4*log((sqrt(b*x + 
 c) + sqrt(a + b*x))/sqrt(a - c))*a - 4*log((sqrt(b*x + c) + sqrt(a + b*x) 
)/sqrt(a - c))*c + a + 2*b*x + c)/(a**2 - 2*a*c + c**2)