Integrand size = 25, antiderivative size = 141 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=-\frac {a+c}{(a-c)^2 x}+\frac {2 \sqrt {a+b x} \sqrt {c+b x}}{(a-c)^2 x}-\frac {4 b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{(a-c)^2}+\frac {2 b (a+c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+b x}}\right )}{\sqrt {a} (a-c)^2 \sqrt {c}}+\frac {2 b \log (x)}{(a-c)^2} \]
(-a-c)/(a-c)^2/x-4*b*arctanh((b*x+a)^(1/2)/(b*x+c)^(1/2))/(a-c)^2+2*b*ln(x )/(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-c)^2/a^(1/2)/c^(1/2)+2*(b*x+a)^(1/2)*(b*x+c)^(1/2)/(a-c)^2/x
Time = 0.52 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {\frac {2 b (a+c) \text {arctanh}\left (\frac {-b x+\sqrt {a+b x} \sqrt {c+b x}}{\sqrt {a} \sqrt {c}}\right )}{\sqrt {a} \sqrt {c}}-\frac {a+c-2 b x-2 \sqrt {a+b x} \sqrt {c+b x}-2 b x \log \left (b x \left (a+c+2 b x-2 \sqrt {a+b x} \sqrt {c+b x}\right )\right )}{x}}{(a-c)^2} \]
((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
Time = 0.40 (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 |
\(\displaystyle \frac {\frac {2 b (a+c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {b x+c}}\right )}{\sqrt {a} \sqrt {c}}-4 b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {b x+c}}\right )+\frac {2 \sqrt {a+b x} \sqrt {b x+c}}{x}-\frac {a+c}{x}+2 b \log (x)}{(a-c)^2}\) |
(-((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
3.5.10.3.1 Defintions of rubi rules used
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.94
method | result | size |
default | \(-\frac {a}{x \left (a -c \right )^{2}}-\frac {c}{x \left (a -c \right )^{2}}+\frac {2 b \ln \left (x \right )}{\left (a -c \right )^{2}}+\frac {\sqrt {b x +a}\, \sqrt {b x +c}\, \left (\operatorname {csgn}\left (b \right ) \ln \left (\frac {a b x +b c x +2 \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}+2 a c}{x}\right ) x a b +\operatorname {csgn}\left (b \right ) \ln \left (\frac {a b x +b c x +2 \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}+2 a c}{x}\right ) x b c -2 \ln \left (\frac {\left (2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \operatorname {csgn}\left (b \right )+2 b x +a +c \right ) \operatorname {csgn}\left (b \right )}{2}\right ) x b \sqrt {a c}+2 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \sqrt {a c}\right ) \operatorname {csgn}\left (b \right )}{\left (a -c \right )^{2} \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, x \sqrt {a c}}\) | \(274\) |
-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*csgn(b)*(b^2*x^2+a*b*x +b*c*x+a*c)^(1/2)*(a*c)^(1/2))*csgn(b)/(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)/x/( a*c)^(1/2)
Time = 0.31 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.60 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\left [\frac {2 \, a b c x \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right ) + 2 \, a b c x \log \left (x\right ) + 2 \, a b c x + {\left (a b + b c\right )} \sqrt {a c} x \log \left (\frac {2 \, a^{2} c + 2 \, a c^{2} + 2 \, {\left (2 \, a c + \sqrt {a c} {\left (a + c\right )}\right )} \sqrt {b x + a} \sqrt {b x + c} + {\left (a^{2} b + 2 \, a b c + b c^{2}\right )} x + 2 \, {\left (2 \, a c + {\left (a b + b c\right )} x\right )} \sqrt {a c}}{x}\right ) + 2 \, \sqrt {b x + a} \sqrt {b x + c} a c - a^{2} c - a c^{2}}{{\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3}\right )} x}, \frac {2 \, a b c x \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right ) + 2 \, a b c x \log \left (x\right ) + 2 \, a b c x - 2 \, {\left (a b + b c\right )} \sqrt {-a c} x \arctan \left (-\frac {\sqrt {-a c} b x - \sqrt {-a c} \sqrt {b x + a} \sqrt {b x + c}}{a c}\right ) + 2 \, \sqrt {b x + a} \sqrt {b x + c} a c - a^{2} c - a c^{2}}{{\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3}\right )} x}\right ] \]
[(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)]
\[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\int \frac {1}{x^{2} \left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{2}}\, dx \]
\[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\int { \frac {1}{x^{2} {\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (121) = 242\).
Time = 0.77 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.21 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {2 \, b \log \left ({\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2}\right )}{a^{2} - 2 \, a c + c^{2}} + \frac {2 \, b \log \left ({\left | b x \right |}\right )}{a^{2} - 2 \, a c + c^{2}} + \frac {2 \, {\left (a b + b c\right )} \arctan \left (\frac {{\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - a - c}{2 \, \sqrt {-a c}}\right )}{{\left (a^{2} - 2 \, a c + c^{2}\right )} \sqrt {-a c}} - \frac {4 \, {\left (a b {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} + b c {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - a^{2} b + 2 \, a b c - b c^{2}\right )}}{{\left ({\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{4} - 2 \, a {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - 2 \, c {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} + a^{2} - 2 \, a c + c^{2}\right )} {\left (a^{2} - 2 \, a c + c^{2}\right )}} - \frac {2 \, {\left (b x + a\right )} b - a b + b c}{{\left (a^{2} - 2 \, a c + c^{2}\right )} b x} \]
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)
Time = 44.38 (sec) , antiderivative size = 7637, normalized size of antiderivative = 54.16 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\text {Too large to display} \]
(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 - ...