\(\int \frac {x}{(\sqrt {a+b x}+\sqrt {a+c x})^2} \, dx\) [34]

Optimal result

Integrand size = 23, antiderivative size = 135 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {(b+c) x}{(b-c)^2}-\frac {2 \sqrt {a+b x} \sqrt {a+c x}}{(b-c)^2}+\frac {4 a \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{(b-c)^2}-\frac {2 a (b+c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {b} \sqrt {a+c x}}\right )}{\sqrt {b} (b-c)^2 \sqrt {c}}+\frac {2 a \log (x)}{(b-c)^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.88 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {4 a \sqrt {c} (b+c) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {a+c x}}{\sqrt {c} \left (\sqrt {a-\frac {a b}{c}}-\sqrt {a+b x}\right )}\right )+\sqrt {b} \left (a (b+c)+c \left (b x+c x-2 \sqrt {a+b x} \sqrt {a+c x}\right )+8 a c \text {arctanh}\left (\frac {-a b-b c x+c \sqrt {a+c x} \left (\sqrt {a-\frac {a b}{c}}-\sqrt {a+b x}\right )}{a (b-2 c)-b c x+2 \sqrt {a-\frac {a b}{c}} c \sqrt {a+b x}+\sqrt {a-\frac {a b}{c}} c \sqrt {a+c x}-c \sqrt {a+b x} \sqrt {a+c x}}\right )\right )}{\sqrt {b} (b-c)^2 c} \] Input:

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

Output:

(4*a*Sqrt[c]*(b + c)*ArcTanh[(Sqrt[b]*Sqrt[a + c*x])/(Sqrt[c]*(Sqrt[a - (a 
*b)/c] - Sqrt[a + b*x]))] + Sqrt[b]*(a*(b + c) + c*(b*x + c*x - 2*Sqrt[a + 
 b*x]*Sqrt[a + c*x]) + 8*a*c*ArcTanh[(-(a*b) - b*c*x + c*Sqrt[a + c*x]*(Sq 
rt[a - (a*b)/c] - Sqrt[a + b*x]))/(a*(b - 2*c) - b*c*x + 2*Sqrt[a - (a*b)/ 
c]*c*Sqrt[a + b*x] + Sqrt[a - (a*b)/c]*c*Sqrt[a + c*x] - c*Sqrt[a + b*x]*S 
qrt[a + c*x])]))/(Sqrt[b]*(b - c)^2*c)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {7241, 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.

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

\(\Big \downarrow \) 7241

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 7241

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

Maple [C] (verified)

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

Time = 0.01 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.97

method	result	size
default	\(\frac {x b}{\left (b -c \right )^{2}}+\frac {x c}{\left (b -c \right )^{2}}+\frac {2 a \ln \left (x \right )}{\left (b -c \right )^{2}}-\frac {\sqrt {b x +a}\, \sqrt {c x +a}\, \left (\ln \left (\frac {2 b c x +2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \sqrt {b c}+a b +a c}{2 \sqrt {b c}}\right ) \operatorname {csgn}\left (a \right ) a b +\ln \left (\frac {2 b c x +2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \sqrt {b c}+a b +a c}{2 \sqrt {b c}}\right ) \operatorname {csgn}\left (a \right ) a c +2 \sqrt {b c}\, \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right )-2 \sqrt {b c}\, \ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) a \right ) \operatorname {csgn}\left (a \right )}{\left (b -c \right )^{2} \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \sqrt {b c}}\)	\(266\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.56 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\left [\frac {2 \, a b c \log \left (x\right ) - 2 \, a b c \log \left (-\frac {{\left (b + c\right )} x - 2 \, \sqrt {b x + a} \sqrt {c x + a} + 2 \, a}{x}\right ) - 2 \, \sqrt {b x + a} \sqrt {c x + a} b c + {\left (a b + a c\right )} \sqrt {b c} \log \left (a b^{2} + 2 \, a b c + a c^{2} + 2 \, {\left (2 \, b c - \sqrt {b c} {\left (b + c\right )}\right )} \sqrt {b x + a} \sqrt {c x + a} + 2 \, {\left (b^{2} c + b c^{2}\right )} x - 2 \, {\left (2 \, b c x + a b + a c\right )} \sqrt {b c}\right ) + {\left (b^{2} c + b c^{2}\right )} x}{b^{3} c - 2 \, b^{2} c^{2} + b c^{3}}, \frac {2 \, a b c \log \left (x\right ) - 2 \, a b c \log \left (-\frac {{\left (b + c\right )} x - 2 \, \sqrt {b x + a} \sqrt {c x + a} + 2 \, a}{x}\right ) - 2 \, \sqrt {b x + a} \sqrt {c x + a} b c + 2 \, {\left (a b + a c\right )} \sqrt {-b c} \arctan \left (\frac {\sqrt {-b c} \sqrt {b x + a} \sqrt {c x + a} - \sqrt {-b c} a}{b c x}\right ) + {\left (b^{2} c + b c^{2}\right )} x}{b^{3} c - 2 \, b^{2} c^{2} + b c^{3}}\right ] \] Input:

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

Output:

[(2*a*b*c*log(x) - 2*a*b*c*log(-((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a) 
 + 2*a)/x) - 2*sqrt(b*x + a)*sqrt(c*x + a)*b*c + (a*b + a*c)*sqrt(b*c)*log 
(a*b^2 + 2*a*b*c + a*c^2 + 2*(2*b*c - sqrt(b*c)*(b + c))*sqrt(b*x + a)*sqr 
t(c*x + a) + 2*(b^2*c + b*c^2)*x - 2*(2*b*c*x + a*b + a*c)*sqrt(b*c)) + (b 
^2*c + b*c^2)*x)/(b^3*c - 2*b^2*c^2 + b*c^3), (2*a*b*c*log(x) - 2*a*b*c*lo 
g(-((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a) + 2*a)/x) - 2*sqrt(b*x + a)* 
sqrt(c*x + a)*b*c + 2*(a*b + a*c)*sqrt(-b*c)*arctan((sqrt(-b*c)*sqrt(b*x + 
 a)*sqrt(c*x + a) - sqrt(-b*c)*a)/(b*c*x)) + (b^2*c + b*c^2)*x)/(b^3*c - 2 
*b^2*c^2 + b*c^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (115) = 230\).

Time = 0.57 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.66 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {\frac {a {\left (b + c\right )} {\left | b \right |} \log \left ({\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2}\right )}{{\left (b^{2} - 2 \, b c + c^{2}\right )} \sqrt {b c}} + \frac {2 \, a {\left | b \right |} \log \left ({\left | {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2} - {\left (b^{2} + b c + 2 \, \sqrt {b c} b\right )} a \right |}\right )}{b^{2} - 2 \, b c + c^{2}} - \frac {2 \, a {\left | b \right |} \log \left ({\left | {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2} - {\left (b^{2} + b c - 2 \, \sqrt {b c} b\right )} a \right |}\right )}{b^{2} - 2 \, b c + c^{2}} + \frac {2 \, a b \log \left ({\left | b x \right |}\right )}{b^{2} - 2 \, b c + c^{2}} - \frac {2 \, \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} {\left (b^{2} {\left | b \right |} - 2 \, b c {\left | b \right |} + c^{2} {\left | b \right |}\right )} \sqrt {b x + a}}{b^{5} - 4 \, b^{4} c + 6 \, b^{3} c^{2} - 4 \, b^{2} c^{3} + b c^{4}} + \frac {{\left (b x + a\right )} b + {\left (b x + a\right )} c}{b^{2} - 2 \, b c + c^{2}}}{b} \] Input:

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

Output:

(a*(b + c)*abs(b)*log((sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b* 
c - a*b*c))^2)/((b^2 - 2*b*c + c^2)*sqrt(b*c)) + 2*a*abs(b)*log(abs((sqrt( 
b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2 - (b^2 + b*c + 
 2*sqrt(b*c)*b)*a))/(b^2 - 2*b*c + c^2) - 2*a*abs(b)*log(abs((sqrt(b*c)*sq 
rt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2 - (b^2 + b*c - 2*sqrt 
(b*c)*b)*a))/(b^2 - 2*b*c + c^2) + 2*a*b*log(abs(b*x))/(b^2 - 2*b*c + c^2) 
 - 2*sqrt(a*b^2 + (b*x + a)*b*c - a*b*c)*(b^2*abs(b) - 2*b*c*abs(b) + c^2* 
abs(b))*sqrt(b*x + a)/(b^5 - 4*b^4*c + 6*b^3*c^2 - 4*b^2*c^3 + b*c^4) + (( 
b*x + a)*b + (b*x + a)*c)/(b^2 - 2*b*c + c^2))/b
 

Mupad [B] (verification not implemented)

Time = 40.50 (sec) , antiderivative size = 5098, normalized size of antiderivative = 37.76 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*a*log(x))/(b^2 - 2*b*c + c^2) - (((4*a*c^2 + 4*a*b*c)*((a + b*x)^(1/2) 
- a^(1/2))^3)/((a + c*x)^(1/2) - a^(1/2))^3 + ((4*a*b^2 + 4*a*b*c)*((a + b 
*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1/2)) - (16*a*b*c*((a + b*x)^( 
1/2) - a^(1/2))^2)/((a + c*x)^(1/2) - a^(1/2))^2)/(b^4 - 2*b^3*c + b^2*c^2 
 - (((a + b*x)^(1/2) - a^(1/2))^2*(2*b*c^3 + 2*b^3*c - 4*b^2*c^2))/((a + c 
*x)^(1/2) - a^(1/2))^2 + (((a + b*x)^(1/2) - a^(1/2))^4*(c^4 - 2*b*c^3 + b 
^2*c^2))/((a + c*x)^(1/2) - a^(1/2))^4) - (2*a*log((((a + b*x)^(1/2) - (a 
+ c*x)^(1/2))*(b - (c*((a + b*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1 
/2))))/((a + c*x)^(1/2) - a^(1/2))))/(b^2 - 2*b*c + c^2) + (2*a*log(((a + 
b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^(1/2))))/(b - c)^2 + (x*(b + c) 
)/(b - c)^2 + (a*atan(((a*(b*c)^(1/2)*(b + c)*((2*((a + b*x)^(1/2) - a^(1/ 
2))*(32*a^3*b^2*c^10 - 64*a^3*b^3*c^9 + 8*a^3*b^4*c^8 + 240*a^3*b^5*c^7 + 
8*a^3*b^6*c^6 - 64*a^3*b^7*c^5 + 32*a^3*b^8*c^4))/(((a + c*x)^(1/2) - a^(1 
/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2)) - (4*(4*a^3*b^4*c^8 + 44 
*a^3*b^5*c^7 + 44*a^3*b^6*c^6 + 4*a^3*b^7*c^5))/(b^4 - 4*b^3*c - 4*b*c^3 + 
 c^4 + 6*b^2*c^2) + (2*a*(b*c)^(1/2)*(b + c)*((4*(4*a^2*b^3*c^11 + 2*a^2*b 
^4*c^10 - 18*a^2*b^5*c^9 + 12*a^2*b^6*c^8 + 12*a^2*b^7*c^7 - 18*a^2*b^8*c^ 
6 + 2*a^2*b^9*c^5 + 4*a^2*b^10*c^4))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^ 
2*c^2) - (2*((a + b*x)^(1/2) - a^(1/2))*(16*a^2*b^2*c^12 - 32*a^2*b^3*c^11 
 + 36*a^2*b^4*c^10 - 64*a^2*b^5*c^9 + 88*a^2*b^6*c^8 - 64*a^2*b^7*c^7 +...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.43 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {-2 \sqrt {c x +a}\, \sqrt {b x +a}\, b c +\sqrt {c}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {c}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {c x +a}\right ) a b +\sqrt {c}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {c}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {c x +a}\right ) a c -\sqrt {c}\, \sqrt {b}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {c x +a}\right ) a b -\sqrt {c}\, \sqrt {b}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {c x +a}\right ) a c +4 \,\mathrm {log}\left (\sqrt {b x +a}\, b +\sqrt {c x +a}\, b \right ) a b c +a b c +a \,c^{2}+b^{2} c x +b \,c^{2} x}{b c \left (b^{2}-2 b c +c^{2}\right )} \] Input:

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

Output:

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