Integrand size = 25, antiderivative size = 123 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=-\frac {a}{(b-c)^2 x^2}-\frac {b+c}{(b-c)^2 x}+\frac {\sqrt {a+b x} \sqrt {a+c x}}{2 a (b-c) x}+\frac {\sqrt {a+b x} (a+c x)^{3/2}}{a (b-c)^2 x^2}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{2 a} \] Output:
-a/(b-c)^2/x^2-(b+c)/(b-c)^2/x+1/2*(b*x+a)^(1/2)*(c*x+a)^(1/2)/a/(b-c)/x+( b*x+a)^(1/2)*(c*x+a)^(3/2)/a/(b-c)^2/x^2-1/2*arctanh((b*x+a)^(1/2)/(c*x+a) ^(1/2))/a
Time = 1.37 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {-2 a^2+(b+c) x \sqrt {a+b x} \sqrt {a+c x}+2 a \left (-b x-c x+\sqrt {a+b x} \sqrt {a+c x}\right )-(b-c)^2 x^2 \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a+b x}}\right )}{2 a (b-c)^2 x^2} \] Input:
Integrate[1/(x*(Sqrt[a + b*x] + Sqrt[a + c*x])^2),x]
Output:
(-2*a^2 + (b + c)*x*Sqrt[a + b*x]*Sqrt[a + c*x] + 2*a*(-(b*x) - c*x + Sqrt [a + b*x]*Sqrt[a + c*x]) - (b - c)^2*x^2*ArcTanh[Sqrt[a + c*x]/Sqrt[a + b* x]])/(2*a*(b - c)^2*x^2)
Time = 0.66 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx\) |
| \(\Big \downarrow \) 7241 |
| \(\displaystyle \frac {\int \left (\frac {2 a}{x^3}-\frac {2 \sqrt {a+b x} \sqrt {a+c x}}{x^3}+\frac {b+c}{x^2}\right )dx}{(b-c)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {(b-c)^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{2 a}+\frac {\sqrt {a+b x} (a+c x)^{3/2}}{a x^2}+\frac {(b-c) \sqrt {a+b x} \sqrt {a+c x}}{2 a x}-\frac {a}{x^2}-\frac {b+c}{x}}{(b-c)^2}\) |
Input:
Int[1/(x*(Sqrt[a + b*x] + Sqrt[a + c*x])^2),x]
Output:
(-(a/x^2) - (b + c)/x + ((b - c)*Sqrt[a + b*x]*Sqrt[a + c*x])/(2*a*x) + (S qrt[a + b*x]*(a + c*x)^(3/2))/(a*x^2) - ((b - c)^2*ArcTanh[Sqrt[a + b*x]/S qrt[a + c*x]])/(2*a))/(b - c)^2
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.02 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.53
| method | result | size |
| default | \(-\frac {b}{x \left (b -c \right )^{2}}-\frac {c}{x \left (b -c \right )^{2}}-\frac {a}{\left (b -c \right )^{2} x^{2}}-\frac {\sqrt {b x +a}\, \sqrt {c x +a}\, \left (\ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{2} b^{2}-2 \ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{2} b c +\ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{2} c^{2}-2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right ) x b -2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right ) x c -4 \,\operatorname {csgn}\left (a \right ) a \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\right ) \operatorname {csgn}\left (a \right )}{4 \left (b -c \right )^{2} a \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x^{2}}\) | \(311\) |
Input:
int(1/x/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
-1/x/(b-c)^2*b-1/x/(b-c)^2*c-a/(b-c)^2/x^2-1/4/(b-c)^2*(b*x+a)^(1/2)*(c*x+ a)^(1/2)/a*(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 )*x^2*b^2-2*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 )*x^2*b*c+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)* x^2*c^2-2*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*csgn(a)*x*b-2*(b*c*x^2+a*b*x+a*c *x+a^2)^(1/2)*csgn(a)*x*c-4*csgn(a)*a*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2))*csg n(a)/(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)/x^2
Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {4 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} x^{2} \log \left (-\frac {{\left (b + c\right )} x - 2 \, \sqrt {b x + a} \sqrt {c x + a} + 2 \, a}{x}\right ) + {\left (b^{2} + 2 \, b c + c^{2}\right )} x^{2} + 8 \, {\left ({\left (b + c\right )} x + 2 \, a\right )} \sqrt {b x + a} \sqrt {c x + a} - 16 \, a^{2} - 16 \, {\left (a b + a c\right )} x}{16 \, {\left (a b^{2} - 2 \, a b c + a c^{2}\right )} x^{2}} \] Input:
integrate(1/x/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x, algorithm="fricas")
Output:
1/16*(4*(b^2 - 2*b*c + c^2)*x^2*log(-((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a) + 2*a)/x) + (b^2 + 2*b*c + c^2)*x^2 + 8*((b + c)*x + 2*a)*sqrt(b*x + a)*sqrt(c*x + a) - 16*a^2 - 16*(a*b + a*c)*x)/((a*b^2 - 2*a*b*c + a*c^2)* x^2)
\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\int \frac {1}{x \left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{2}}\, dx \] Input:
integrate(1/x/((b*x+a)**(1/2)+(c*x+a)**(1/2))**2,x)
Output:
Integral(1/(x*(sqrt(a + b*x) + sqrt(a + c*x))**2), x)
\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\int { \frac {1}{x {\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}^{2}} \,d x } \] Input:
integrate(1/x/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x, algorithm="maxima")
Output:
integrate(1/(x*(sqrt(b*x + a) + sqrt(c*x + a))^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (107) = 214\).
Time = 2.57 (sec) , antiderivative size = 532, normalized size of antiderivative = 4.33 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=-\frac {\sqrt {b c} {\left | b \right |} \arctan \left (-\frac {a b^{2} + a b c - {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2}}{2 \, \sqrt {-b c} a b}\right )}{2 \, \sqrt {-b c} a b} - \frac {{\left (b^{2} + 6 \, b c + c^{2}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{6} {\left | b \right |} - {\left (3 \, b^{4} + 5 \, b^{3} c + 5 \, b^{2} c^{2} + 3 \, b c^{3}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{4} a {\left | b \right |} + {\left (3 \, b^{6} - 4 \, b^{5} c + 2 \, b^{4} c^{2} - 4 \, b^{3} c^{3} + 3 \, b^{2} c^{4}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2} a^{2} {\left | b \right |} - {\left (b^{8} - 3 \, b^{7} c + 2 \, b^{6} c^{2} + 2 \, b^{5} c^{3} - 3 \, b^{4} c^{4} + b^{3} c^{5}\right )} \sqrt {b c} a^{3} {\left | b \right |}}{{\left ({\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{4} - 2 \, {\left (b^{2} + b c\right )} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2} a + {\left (b^{4} - 2 \, b^{3} c + b^{2} c^{2}\right )} a^{2}\right )}^{2} {\left (b^{2} - 2 \, b c + c^{2}\right )}} - \frac {{\left (b x + a\right )} b^{2} + {\left (b x + a\right )} b c - a b c}{{\left (b^{2} - 2 \, b c + c^{2}\right )} b^{2} x^{2}} \] Input:
integrate(1/x/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x, algorithm="giac")
Output:
-1/2*sqrt(b*c)*abs(b)*arctan(-1/2*(a*b^2 + a*b*c - (sqrt(b*c)*sqrt(b*x + a ) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2)/(sqrt(-b*c)*a*b))/(sqrt(-b*c)* a*b) - ((b^2 + 6*b*c + c^2)*sqrt(b*c)*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^ 2 + (b*x + a)*b*c - a*b*c))^6*abs(b) - (3*b^4 + 5*b^3*c + 5*b^2*c^2 + 3*b* c^3)*sqrt(b*c)*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b *c))^4*a*abs(b) + (3*b^6 - 4*b^5*c + 2*b^4*c^2 - 4*b^3*c^3 + 3*b^2*c^4)*sq rt(b*c)*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2* a^2*abs(b) - (b^8 - 3*b^7*c + 2*b^6*c^2 + 2*b^5*c^3 - 3*b^4*c^4 + b^3*c^5) *sqrt(b*c)*a^3*abs(b))/(((sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a) *b*c - a*b*c))^4 - 2*(b^2 + b*c)*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + ( b*x + a)*b*c - a*b*c))^2*a + (b^4 - 2*b^3*c + b^2*c^2)*a^2)^2*(b^2 - 2*b*c + c^2)) - ((b*x + a)*b^2 + (b*x + a)*b*c - a*b*c)/((b^2 - 2*b*c + c^2)*b^ 2*x^2)
Time = 36.21 (sec) , antiderivative size = 787, normalized size of antiderivative = 6.40 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/(x*((a + b*x)^(1/2) + (a + c*x)^(1/2))^2),x)
Output:
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)))/(4*a) - (b^ 4/2 + (((a + b*x)^(1/2) - a^(1/2))^4*(4*b*c^3 + 4*b^3*c - b^4/2 - c^4/2 + (3*b^2*c^2)/2))/((a + c*x)^(1/2) - a^(1/2))^4 - ((2*b^3*c + 2*b^4)*((a + b *x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1/2)) - ((b*c^3 + b^2*c^2)*((a + b*x)^(1/2) - a^(1/2))^5)/((a + c*x)^(1/2) - a^(1/2))^5 + (((a + b*x)^(1 /2) - a^(1/2))^2*(6*b^3*c + (5*b^4)/2 + (5*b^2*c^2)/2))/((a + c*x)^(1/2) - a^(1/2))^2 - (((a + b*x)^(1/2) - a^(1/2))^3*(b*c^3 + 6*b^3*c + b^4 + 6*b^ 2*c^2))/((a + c*x)^(1/2) - a^(1/2))^3)/((((a + b*x)^(1/2) - a^(1/2))^4*(8* a*b^4 + 8*a*c^4 - 48*a*b^2*c^2 + 16*a*b*c^3 + 16*a*b^3*c))/((a + c*x)^(1/2 ) - a^(1/2))^4 - (((a + b*x)^(1/2) - a^(1/2))^3*(16*a*b^4 - 16*a*b^2*c^2 + 16*a*b*c^3 - 16*a*b^3*c))/((a + c*x)^(1/2) - a^(1/2))^3 - (((a + b*x)^(1/ 2) - a^(1/2))^5*(16*a*c^4 - 16*a*b^2*c^2 - 16*a*b*c^3 + 16*a*b^3*c))/((a + c*x)^(1/2) - a^(1/2))^5 + (((a + b*x)^(1/2) - a^(1/2))^2*(8*a*b^4 + 8*a*b ^2*c^2 - 16*a*b^3*c))/((a + c*x)^(1/2) - a^(1/2))^2 + (((a + b*x)^(1/2) - a^(1/2))^6*(8*a*c^4 + 8*a*b^2*c^2 - 16*a*b*c^3))/((a + c*x)^(1/2) - a^(1/2 ))^6) - log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^(1/2)))/(4*a) - (a + x*(b + c))/(x^2*(b^2 - 2*b*c + c^2)) - (c^2*((a + b*x)^(1/2) - a^( 1/2))^2)/(16*a*(b - c)^2*((a + c*x)^(1/2) - a^(1/2))^2) + (c*(b + c)*((a + b*x)^(1/2) - a^(1/2)))/(8*a*(b - c)^2*((a + c*x)^(1/2) - a^(1/2)))
Time = 0.16 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.04 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {4 \sqrt {c x +a}\, \sqrt {b x +a}\, a +2 \sqrt {c x +a}\, \sqrt {b x +a}\, b x +2 \sqrt {c x +a}\, \sqrt {b x +a}\, c x +\mathrm {log}\left (2 \sqrt {c x +a}\, \sqrt {b x +a}\, b -2 a b -b^{2} x -b c x \right ) b^{2} x^{2}-2 \,\mathrm {log}\left (2 \sqrt {c x +a}\, \sqrt {b x +a}\, b -2 a b -b^{2} x -b c x \right ) b c \,x^{2}+\mathrm {log}\left (2 \sqrt {c x +a}\, \sqrt {b x +a}\, b -2 a b -b^{2} x -b c x \right ) c^{2} x^{2}-\mathrm {log}\left (b x \right ) b^{2} x^{2}+2 \,\mathrm {log}\left (b x \right ) b c \,x^{2}-\mathrm {log}\left (b x \right ) c^{2} x^{2}-4 a^{2}-4 a b x -4 a c x -2 b^{2} x^{2}-2 b c \,x^{2}}{4 a \,x^{2} \left (b^{2}-2 b c +c^{2}\right )} \] Input:
int(1/x/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x)
Output:
(4*sqrt(a + c*x)*sqrt(a + b*x)*a + 2*sqrt(a + c*x)*sqrt(a + b*x)*b*x + 2*s qrt(a + c*x)*sqrt(a + b*x)*c*x + log(2*sqrt(a + c*x)*sqrt(a + b*x)*b - 2*a *b - b**2*x - b*c*x)*b**2*x**2 - 2*log(2*sqrt(a + c*x)*sqrt(a + b*x)*b - 2 *a*b - b**2*x - b*c*x)*b*c*x**2 + log(2*sqrt(a + c*x)*sqrt(a + b*x)*b - 2* a*b - b**2*x - b*c*x)*c**2*x**2 - log(b*x)*b**2*x**2 + 2*log(b*x)*b*c*x**2 - log(b*x)*c**2*x**2 - 4*a**2 - 4*a*b*x - 4*a*c*x - 2*b**2*x**2 - 2*b*c*x **2)/(4*a*x**2*(b**2 - 2*b*c + c**2))