\(\int \frac {1}{x^2 (\sqrt {a+b x}+\sqrt {c+b x})^2} \, dx\) [10]

Optimal result

Integrand size = 25, antiderivative size = 141 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=-\frac {a+c}{(a-c)^2 x}+\frac {2 \sqrt {a+b x} \sqrt {c+b x}}{(a-c)^2 x}-\frac {4 b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{(a-c)^2}+\frac {2 b (a+c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+b x}}\right )}{\sqrt {a} (a-c)^2 \sqrt {c}}+\frac {2 b \log (x)}{(a-c)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {\frac {2 b (a+c) \text {arctanh}\left (\frac {-b x+\sqrt {a+b x} \sqrt {c+b x}}{\sqrt {a} \sqrt {c}}\right )}{\sqrt {a} \sqrt {c}}-\frac {a+c-2 b x-2 \sqrt {a+b x} \sqrt {c+b x}-2 b x \log \left (b x \left (a+c+2 b x-2 \sqrt {a+b x} \sqrt {c+b x}\right )\right )}{x}}{(a-c)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.81, 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.

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

\(\Big \downarrow \) 7240

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-((a + c)/x) + (2*Sqrt[a + b*x]*Sqrt[c + b*x])/x - 4*b*ArcTanh[Sqrt[a + b 
*x]/Sqrt[c + b*x]] + (2*b*(a + c)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a] 
*Sqrt[c + b*x])])/(Sqrt[a]*Sqrt[c]) + 2*b*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 = 274, normalized size of antiderivative = 1.94

method	result	size
default	\(-\frac {a}{x \left (a -c \right )^{2}}-\frac {c}{x \left (a -c \right )^{2}}+\frac {2 b \ln \left (x \right )}{\left (a -c \right )^{2}}+\frac {\sqrt {b x +a}\, \sqrt {b x +c}\, \left (\operatorname {csgn}\left (b \right ) \ln \left (\frac {a b x +b c x +2 \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}+2 a c}{x}\right ) x a b +\operatorname {csgn}\left (b \right ) \ln \left (\frac {a b x +b c x +2 \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}+2 a c}{x}\right ) x b c -2 \ln \left (\frac {\left (2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \operatorname {csgn}\left (b \right )+2 b x +a +c \right ) \operatorname {csgn}\left (b \right )}{2}\right ) x b \sqrt {a c}+2 \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \operatorname {csgn}\left (b \right )\right ) \operatorname {csgn}\left (b \right )}{\left (a -c \right )^{2} \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, x \sqrt {a c}}\)	\(274\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.60 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\left [\frac {2 \, a b c x \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right ) + 2 \, a b c x \log \left (x\right ) + 2 \, a b c x + {\left (a b + b c\right )} \sqrt {a c} x \log \left (\frac {2 \, a^{2} c + 2 \, a c^{2} + 2 \, {\left (2 \, a c + \sqrt {a c} {\left (a + c\right )}\right )} \sqrt {b x + a} \sqrt {b x + c} + {\left (a^{2} b + 2 \, a b c + b c^{2}\right )} x + 2 \, {\left (2 \, a c + {\left (a b + b c\right )} x\right )} \sqrt {a c}}{x}\right ) + 2 \, \sqrt {b x + a} \sqrt {b x + c} a c - a^{2} c - a c^{2}}{{\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3}\right )} x}, \frac {2 \, a b c x \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right ) + 2 \, a b c x \log \left (x\right ) + 2 \, a b c x - 2 \, {\left (a b + b c\right )} \sqrt {-a c} x \arctan \left (-\frac {\sqrt {-a c} b x - \sqrt {-a c} \sqrt {b x + a} \sqrt {b x + c}}{a c}\right ) + 2 \, \sqrt {b x + a} \sqrt {b x + c} a c - a^{2} c - a c^{2}}{{\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3}\right )} x}\right ] \] Input:

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

Output:

[(2*a*b*c*x*log(-2*b*x + 2*sqrt(b*x + a)*sqrt(b*x + c) - a - c) + 2*a*b*c* 
x*log(x) + 2*a*b*c*x + (a*b + b*c)*sqrt(a*c)*x*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)*sq 
rt(b*x + c)*a*c - a^2*c - a*c^2)/((a^3*c - 2*a^2*c^2 + a*c^3)*x), (2*a*b*c 
*x*log(-2*b*x + 2*sqrt(b*x + a)*sqrt(b*x + c) - a - c) + 2*a*b*c*x*log(x) 
+ 2*a*b*c*x - 2*(a*b + b*c)*sqrt(-a*c)*x*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* 
c - a^2*c - a*c^2)/((a^3*c - 2*a^2*c^2 + a*c^3)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (121) = 242\).

Time = 0.43 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.21 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {2 \, b \log \left ({\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2}\right )}{a^{2} - 2 \, a c + c^{2}} + \frac {2 \, b \log \left ({\left | b x \right |}\right )}{a^{2} - 2 \, a c + c^{2}} + \frac {2 \, {\left (a b + b c\right )} \arctan \left (\frac {{\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - a - c}{2 \, \sqrt {-a c}}\right )}{{\left (a^{2} - 2 \, a c + c^{2}\right )} \sqrt {-a c}} - \frac {4 \, {\left (a b {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} + b c {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - a^{2} b + 2 \, a b c - b c^{2}\right )}}{{\left ({\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{4} - 2 \, a {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - 2 \, c {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} + a^{2} - 2 \, a c + c^{2}\right )} {\left (a^{2} - 2 \, a c + c^{2}\right )}} - \frac {2 \, {\left (b x + a\right )} b - a b + b c}{{\left (a^{2} - 2 \, a c + c^{2}\right )} b x} \] Input:

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

Output:

2*b*log((sqrt(b*x + a) - sqrt(b*x + c))^2)/(a^2 - 2*a*c + c^2) + 2*b*log(a 
bs(b*x))/(a^2 - 2*a*c + c^2) + 2*(a*b + b*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)) - 4 
*(a*b*(sqrt(b*x + a) - sqrt(b*x + c))^2 + b*c*(sqrt(b*x + a) - sqrt(b*x + 
c))^2 - a^2*b + 2*a*b*c - b*c^2)/(((sqrt(b*x + a) - sqrt(b*x + c))^4 - 2*a 
*(sqrt(b*x + a) - sqrt(b*x + c))^2 - 2*c*(sqrt(b*x + a) - sqrt(b*x + c))^2 
 + a^2 - 2*a*c + c^2)*(a^2 - 2*a*c + c^2)) - (2*(b*x + a)*b - a*b + b*c)/( 
(a^2 - 2*a*c + c^2)*b*x)
 

Mupad [B] (verification not implemented)

Time = 50.16 (sec) , antiderivative size = 7637, normalized size of antiderivative = 54.16 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*b*log(x))/(a^2 - 2*a*c + c^2) - ((((a + b*x)^(1/2) - a^(1/2))^2*((a^2*b 
)/2 + (b*c^2)/2 - (3*a*b*c)/2))/(((c + b*x)^(1/2) - c^(1/2))^2*(a*c^3 + a^ 
3*c - 2*a^2*c^2)) - b/(2*(a^2 - 2*a*c + c^2)) + (a^(1/2)*c^(1/2)*((a*b)/2 
+ (b*c)/2)*((a + b*x)^(1/2) - a^(1/2)))/(((c + b*x)^(1/2) - c^(1/2))*(a*c^ 
3 + a^3*c - 2*a^2*c^2)))/(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c 
^(1/2)) + ((a + b*x)^(1/2) - a^(1/2))^3/((c + b*x)^(1/2) - c^(1/2))^3 - (( 
a + c)*((a + b*x)^(1/2) - a^(1/2))^2)/(a^(1/2)*c^(1/2)*((c + b*x)^(1/2) - 
c^(1/2))^2)) + (b*atan(((b*((4*(4*a^4*b^3*c^12 + 8*a^5*b^3*c^11 - 32*a^6*b 
^3*c^10 - 8*a^7*b^3*c^9 + 56*a^8*b^3*c^8 - 8*a^9*b^3*c^7 - 32*a^10*b^3*c^6 
 + 8*a^11*b^3*c^5 + 4*a^12*b^3*c^4))/(a^7*c^15 - 8*a^8*c^14 + 28*a^9*c^13 
- 56*a^10*c^12 + 70*a^11*c^11 - 56*a^12*c^10 + 28*a^13*c^9 - 8*a^14*c^8 + 
a^15*c^7) + (4*b*((4*b*((4*(16*a^6*b*c^14 - 4*a^5*b*c^15 + 12*a^7*b*c^13 - 
 192*a^8*b*c^12 + 504*a^9*b*c^11 - 672*a^10*b*c^10 + 504*a^11*b*c^9 - 192* 
a^12*b*c^8 + 12*a^13*b*c^7 + 16*a^14*b*c^6 - 4*a^15*b*c^5))/(a^7*c^15 - 8* 
a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^12*c^10 + 28*a 
^13*c^9 - 8*a^14*c^8 + a^15*c^7) + (4*b*((4*(a^(9/2)*c^(35/2) - 8*a^(11/2) 
*c^(33/2) + 27*a^(13/2)*c^(31/2) - 49*a^(15/2)*c^(29/2) + 50*a^(17/2)*c^(2 
7/2) - 27*a^(19/2)*c^(25/2) + 6*a^(21/2)*c^(23/2) + 6*a^(23/2)*c^(21/2) - 
27*a^(25/2)*c^(19/2) + 50*a^(27/2)*c^(17/2) - 49*a^(29/2)*c^(15/2) + 27*a^ 
(31/2)*c^(13/2) - 8*a^(33/2)*c^(11/2) + a^(35/2)*c^(9/2)))/(a^7*c^15 - ...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {2 \sqrt {b x +a}\, \sqrt {b x +c}\, a c -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +c}-\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) a b x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +c}-\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) b c x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +c}+\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) a b x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +c}+\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) b c x +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {b x +a}\, \sqrt {b x +c}+2 \sqrt {c}\, \sqrt {a}+2 b x \right ) a b x +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {b x +a}\, \sqrt {b x +c}+2 \sqrt {c}\, \sqrt {a}+2 b x \right ) b c x +2 \,\mathrm {log}\left (\sqrt {b x +c}-\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) a b c x +2 \,\mathrm {log}\left (\sqrt {b x +c}+\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) a b c x +2 \,\mathrm {log}\left (2 \sqrt {b x +a}\, \sqrt {b x +c}+2 \sqrt {c}\, \sqrt {a}+2 b x \right ) a b c x -8 \,\mathrm {log}\left (\frac {\sqrt {b x +c}+\sqrt {b x +a}}{\sqrt {a -c}}\right ) a b c x -a^{2} c -2 a b c x -a \,c^{2}}{a c x \left (a^{2}-2 a c +c^{2}\right )} \] Input:

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

Output:

(2*sqrt(a + b*x)*sqrt(b*x + c)*a*c - sqrt(c)*sqrt(a)*log(sqrt(b*x + c) - s 
qrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a*b*x - sqrt(c)*sqrt(a)*lo 
g(sqrt(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*b*c*x - 
 sqrt(c)*sqrt(a)*log(sqrt(b*x + c) + sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqr 
t(a + b*x))*a*b*x - sqrt(c)*sqrt(a)*log(sqrt(b*x + c) + sqrt(2*sqrt(c)*sqr 
t(a) + a + c) + sqrt(a + b*x))*b*c*x + sqrt(c)*sqrt(a)*log(2*sqrt(a + b*x) 
*sqrt(b*x + c) + 2*sqrt(c)*sqrt(a) + 2*b*x)*a*b*x + sqrt(c)*sqrt(a)*log(2* 
sqrt(a + b*x)*sqrt(b*x + c) + 2*sqrt(c)*sqrt(a) + 2*b*x)*b*c*x + 2*log(sqr 
t(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a*b*c*x + 2* 
log(sqrt(b*x + c) + sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a*b*c 
*x + 2*log(2*sqrt(a + b*x)*sqrt(b*x + c) + 2*sqrt(c)*sqrt(a) + 2*b*x)*a*b* 
c*x - 8*log((sqrt(b*x + c) + sqrt(a + b*x))/sqrt(a - c))*a*b*c*x - a**2*c 
- 2*a*b*c*x - a*c**2)/(a*c*x*(a**2 - 2*a*c + c**2))