Integrand size = 25, antiderivative size = 142 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {2 a x}{(b-c)^2}+\frac {(b+c) x^2}{2 (b-c)^2}-\frac {a \sqrt {a+b x} \sqrt {a+c x}}{2 b (b-c) c}-\frac {(a+b x)^{3/2} \sqrt {a+c x}}{b (b-c)^2}+\frac {a^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {b} \sqrt {a+c x}}\right )}{2 b^{3/2} c^{3/2}} \]
2*a*x/(b-c)^2+1/2*(b+c)*x^2/(b-c)^2+1/2*a^2*arctanh(c^(1/2)*(b*x+a)^(1/2)/ b^(1/2)/(c*x+a)^(1/2))/b^(3/2)/c^(3/2)-(b*x+a)^(3/2)*(c*x+a)^(1/2)/b/(b-c) ^2-1/2*a*(b*x+a)^(1/2)*(c*x+a)^(1/2)/b/(b-c)/c
Time = 0.83 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.23 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=-\frac {a^2 b (b-3 c)-b c^2 x \left (b x+c x-2 \sqrt {a+b x} \sqrt {a+c x}\right )+a c \left (-4 b c x+b \sqrt {a+b x} \sqrt {a+c x}+c \sqrt {a+b x} \sqrt {a+c x}\right )}{2 b (b-c)^2 c^2}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {a+c x}}{\sqrt {c} \left (\sqrt {a-\frac {a b}{c}}-\sqrt {a+b x}\right )}\right )}{b^{3/2} c^{3/2}} \]
-1/2*(a^2*b*(b - 3*c) - b*c^2*x*(b*x + c*x - 2*Sqrt[a + b*x]*Sqrt[a + c*x] ) + a*c*(-4*b*c*x + b*Sqrt[a + b*x]*Sqrt[a + c*x] + c*Sqrt[a + b*x]*Sqrt[a + c*x]))/(b*(b - c)^2*c^2) - (a^2*ArcTanh[(Sqrt[b]*Sqrt[a + c*x])/(Sqrt[c ]*(Sqrt[a - (a*b)/c] - Sqrt[a + b*x]))])/(b^(3/2)*c^(3/2))
Time = 0.41 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.94, 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.
\(\displaystyle \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx\) |
\(\Big \downarrow \) 7241 |
\(\displaystyle \frac {\int \left (2 a+(b+c) x-2 \sqrt {a+b x} \sqrt {a+c x}\right )dx}{(b-c)^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {a^2 (b-c)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {b} \sqrt {a+c x}}\right )}{2 b^{3/2} c^{3/2}}-\frac {a (b-c) \sqrt {a+b x} \sqrt {a+c x}}{2 b c}-\frac {(a+b x)^{3/2} \sqrt {a+c x}}{b}+2 a x+\frac {1}{2} x^2 (b+c)}{(b-c)^2}\) |
(2*a*x + ((b + c)*x^2)/2 - (a*(b - c)*Sqrt[a + b*x]*Sqrt[a + c*x])/(2*b*c) - ((a + b*x)^(3/2)*Sqrt[a + c*x])/b + (a^2*(b - c)^2*ArcTanh[(Sqrt[c]*Sqr t[a + b*x])/(Sqrt[b]*Sqrt[a + c*x])])/(2*b^(3/2)*c^(3/2)))/(b - c)^2
3.5.33.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)* (x_)^(n_.)])^(m_), x_Symbol] :> Simp[(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]
Time = 0.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.32
method | result | size |
default | \(\frac {x^{2} b}{2 \left (b -c \right )^{2}}+\frac {x^{2} c}{2 \left (b -c \right )^{2}}+\frac {2 a x}{\left (b -c \right )^{2}}-\frac {2 \left (\frac {\sqrt {b x +a}\, \left (c x +a \right )^{\frac {3}{2}}}{2 c}-\frac {\left (a b -a c \right ) \left (\frac {\sqrt {c x +a}\, \sqrt {b x +a}}{b}-\frac {\left (-a b +a c \right ) \sqrt {\left (b x +a \right ) \left (c x +a \right )}\, \ln \left (\frac {\frac {1}{2} a b +\frac {1}{2} a c +b c x}{\sqrt {b c}}+\sqrt {b c \,x^{2}+\left (a b +a c \right ) x +a^{2}}\right )}{2 b \sqrt {c x +a}\, \sqrt {b x +a}\, \sqrt {b c}}\right )}{4 c}\right )}{\left (b -c \right )^{2}}\) | \(187\) |
1/2*x^2/(b-c)^2*b+1/2*x^2/(b-c)^2*c+2*a*x/(b-c)^2-2/(b-c)^2*(1/2/c*(b*x+a) ^(1/2)*(c*x+a)^(3/2)-1/4*(a*b-a*c)/c*(1/b*(c*x+a)^(1/2)*(b*x+a)^(1/2)-1/2* (-a*b+a*c)/b*((b*x+a)*(c*x+a))^(1/2)/(c*x+a)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a *b+1/2*a*c+b*c*x)/(b*c)^(1/2)+(b*c*x^2+(a*b+a*c)*x+a^2)^(1/2))/(b*c)^(1/2) ))
Time = 0.32 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.62 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\left [\frac {8 \, a b^{2} c^{2} x + 2 \, {\left (b^{3} c^{2} + b^{2} c^{3}\right )} x^{2} + {\left (a^{2} b^{2} - 2 \, a^{2} b c + a^{2} c^{2}\right )} \sqrt {b c} \log \left (a b^{2} + 2 \, a b c + a c^{2} + 2 \, {\left (2 \, b c + \sqrt {b c} {\left (b + c\right )}\right )} \sqrt {b x + a} \sqrt {c x + a} + 2 \, {\left (b^{2} c + b c^{2}\right )} x + 2 \, {\left (2 \, b c x + a b + a c\right )} \sqrt {b c}\right ) - 2 \, {\left (2 \, b^{2} c^{2} x + a b^{2} c + a b c^{2}\right )} \sqrt {b x + a} \sqrt {c x + a}}{4 \, {\left (b^{4} c^{2} - 2 \, b^{3} c^{3} + b^{2} c^{4}\right )}}, \frac {4 \, a b^{2} c^{2} x + {\left (b^{3} c^{2} + b^{2} c^{3}\right )} x^{2} - {\left (a^{2} b^{2} - 2 \, a^{2} b c + a^{2} c^{2}\right )} \sqrt {-b c} \arctan \left (\frac {\sqrt {-b c} \sqrt {b x + a} \sqrt {c x + a} - \sqrt {-b c} a}{b c x}\right ) - {\left (2 \, b^{2} c^{2} x + a b^{2} c + a b c^{2}\right )} \sqrt {b x + a} \sqrt {c x + a}}{2 \, {\left (b^{4} c^{2} - 2 \, b^{3} c^{3} + b^{2} c^{4}\right )}}\right ] \]
[1/4*(8*a*b^2*c^2*x + 2*(b^3*c^2 + b^2*c^3)*x^2 + (a^2*b^2 - 2*a^2*b*c + a ^2*c^2)*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)*sqrt(c*x + a) + 2*(b^2*c + b*c^2)*x + 2*(2*b*c*x + a*b + a*c)*sqrt(b*c)) - 2*(2*b^2*c^2*x + a*b^2*c + a*b*c^2)*sqrt(b*x + a)*sqrt( c*x + a))/(b^4*c^2 - 2*b^3*c^3 + b^2*c^4), 1/2*(4*a*b^2*c^2*x + (b^3*c^2 + b^2*c^3)*x^2 - (a^2*b^2 - 2*a^2*b*c + a^2*c^2)*sqrt(-b*c)*arctan((sqrt(-b *c)*sqrt(b*x + a)*sqrt(c*x + a) - sqrt(-b*c)*a)/(b*c*x)) - (2*b^2*c^2*x + a*b^2*c + a*b*c^2)*sqrt(b*x + a)*sqrt(c*x + a))/(b^4*c^2 - 2*b^3*c^3 + b^2 *c^4)]
\[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\int \frac {x^{2}}{\left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{2}}\, dx \]
\[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\int { \frac {x^{2}}{{\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (116) = 232\).
Time = 0.83 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.92 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=-\frac {1}{2} \, \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} \sqrt {b x + a} {\left (\frac {2 \, {\left (b^{4} c^{2} {\left | b \right |} - b^{3} c^{3} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c^{2} - 3 \, b^{8} c^{3} + 3 \, b^{7} c^{4} - b^{6} c^{5}} + \frac {a b^{5} c {\left | b \right |} - 2 \, a b^{4} c^{2} {\left | b \right |} + a b^{3} c^{3} {\left | b \right |}}{b^{9} c^{2} - 3 \, b^{8} c^{3} + 3 \, b^{7} c^{4} - b^{6} c^{5}}\right )} - \frac {a^{2} {\left | b \right |} \log \left ({\left | -\sqrt {b c} \sqrt {b x + a} + \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} \right |}\right )}{2 \, \sqrt {b c} b^{2} c} + \frac {{\left (b x + a\right )}^{2} b + 2 \, {\left (b x + a\right )} a b + {\left (b x + a\right )}^{2} c - 2 \, {\left (b x + a\right )} a c}{2 \, {\left (b^{4} - 2 \, b^{3} c + b^{2} c^{2}\right )}} \]
-1/2*sqrt(a*b^2 + (b*x + a)*b*c - a*b*c)*sqrt(b*x + a)*(2*(b^4*c^2*abs(b) - b^3*c^3*abs(b))*(b*x + a)/(b^9*c^2 - 3*b^8*c^3 + 3*b^7*c^4 - b^6*c^5) + (a*b^5*c*abs(b) - 2*a*b^4*c^2*abs(b) + a*b^3*c^3*abs(b))/(b^9*c^2 - 3*b^8* c^3 + 3*b^7*c^4 - b^6*c^5)) - 1/2*a^2*abs(b)*log(abs(-sqrt(b*c)*sqrt(b*x + a) + sqrt(a*b^2 + (b*x + a)*b*c - a*b*c)))/(sqrt(b*c)*b^2*c) + 1/2*((b*x + a)^2*b + 2*(b*x + a)*a*b + (b*x + a)^2*c - 2*(b*x + a)*a*c)/(b^4 - 2*b^3 *c + b^2*c^2)
Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {2\,a\,x}{{\left (b-c\right )}^2}+\frac {x^2\,\left (b+c\right )}{2\,{\left (b-c\right )}^2}-\frac {2\,\left (\frac {x}{2}+\frac {a\,b+a\,c}{4\,b\,c}\right )\,\sqrt {a+b\,x}\,\sqrt {a+c\,x}}{{\left (b-c\right )}^2}+\frac {\ln \left (a\,b+a\,c+2\,b\,c\,x+2\,\sqrt {b}\,\sqrt {c}\,\sqrt {a+b\,x}\,\sqrt {a+c\,x}\right )\,{\left (a\,b-a\,c\right )}^2}{4\,b^{3/2}\,c^{3/2}\,{\left (b-c\right )}^2} \]