Integrand size = 21, antiderivative size = 164 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=-\frac {2 a \sqrt {a+b x}}{(b-c)^3 x^2}-\frac {(2 b+3 c) \sqrt {a+b x}}{(b-c)^3 x}+\frac {2 a \sqrt {a+c x}}{(b-c)^3 x^2}+\frac {(3 b+2 c) \sqrt {a+c x}}{(b-c)^3 x}-\frac {3 b c \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)^3}+\frac {3 b c \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)^3} \] Output:
-2*a*(b*x+a)^(1/2)/(b-c)^3/x^2-(2*b+3*c)*(b*x+a)^(1/2)/(b-c)^3/x+2*a*(c*x+ a)^(1/2)/(b-c)^3/x^2+(3*b+2*c)*(c*x+a)^(1/2)/(b-c)^3/x-3*b*c*arctanh((b*x+ a)^(1/2)/a^(1/2))/a^(1/2)/(b-c)^3+3*b*c*arctanh((c*x+a)^(1/2)/a^(1/2))/a^( 1/2)/(b-c)^3
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.33 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {-\frac {3 (b+3 c) \left (a+b x+b x \sqrt {1+\frac {b x}{a}} \text {arctanh}\left (\sqrt {1+\frac {b x}{a}}\right )\right )}{x \sqrt {a+b x}}+\frac {3 (3 b+c) \left (a+c x+c x \sqrt {1+\frac {c x}{a}} \text {arctanh}\left (\sqrt {1+\frac {c x}{a}}\right )\right )}{x \sqrt {a+c x}}-\frac {8 b^2 (a+b x)^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},3,\frac {5}{2},1+\frac {b x}{a}\right )}{a^2}+\frac {8 c^2 (a+c x)^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},3,\frac {5}{2},1+\frac {c x}{a}\right )}{a^2}}{3 (b-c)^3} \] Input:
Integrate[(Sqrt[a + b*x] + Sqrt[a + c*x])^(-3),x]
Output:
((-3*(b + 3*c)*(a + b*x + b*x*Sqrt[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]] ))/(x*Sqrt[a + b*x]) + (3*(3*b + c)*(a + c*x + c*x*Sqrt[1 + (c*x)/a]*ArcTa nh[Sqrt[1 + (c*x)/a]]))/(x*Sqrt[a + c*x]) - (8*b^2*(a + b*x)^(3/2)*Hyperge ometric2F1[3/2, 3, 5/2, 1 + (b*x)/a])/a^2 + (8*c^2*(a + c*x)^(3/2)*Hyperge ometric2F1[3/2, 3, 5/2, 1 + (c*x)/a])/a^2)/(3*(b - c)^3)
Time = 0.68 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx\) |
| \(\Big \downarrow \) 7241 |
| \(\displaystyle \frac {\int \left (\frac {4 \sqrt {a+b x} a}{x^3}-\frac {4 \sqrt {a+c x} a}{x^3}+\frac {(b+3 c) \sqrt {a+b x}}{x^2}-\frac {(3 b+c) \sqrt {a+c x}}{x^2}\right )dx}{(b-c)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {b (b+3 c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {c (3 b+c) \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {c^2 \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {(b+3 c) \sqrt {a+b x}}{x}+\frac {(3 b+c) \sqrt {a+c x}}{x}-\frac {2 a \sqrt {a+b x}}{x^2}-\frac {b \sqrt {a+b x}}{x}+\frac {2 a \sqrt {a+c x}}{x^2}+\frac {c \sqrt {a+c x}}{x}}{(b-c)^3}\) |
Input:
Int[(Sqrt[a + b*x] + Sqrt[a + c*x])^(-3),x]
Output:
((-2*a*Sqrt[a + b*x])/x^2 - (b*Sqrt[a + b*x])/x - ((b + 3*c)*Sqrt[a + b*x] )/x + (2*a*Sqrt[a + c*x])/x^2 + (c*Sqrt[a + c*x])/x + ((3*b + c)*Sqrt[a + c*x])/x + (b^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a] - (b*(b + 3*c)*ArcT anh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a] - (c^2*ArcTanh[Sqrt[a + c*x]/Sqrt[a]]) /Sqrt[a] + (c*(3*b + c)*ArcTanh[Sqrt[a + c*x]/Sqrt[a]])/Sqrt[a])/(b - c)^3
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]
Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs. \(2(144)=288\).
Time = 0.01 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.83
| method | result | size |
| default | \(\frac {2 b^{2} \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (b -c \right )^{3}}+\frac {8 a \,b^{2} \left (\frac {-\frac {\left (b x +a \right )^{\frac {3}{2}}}{8 a}-\frac {\sqrt {b x +a}}{8}}{x^{2} b^{2}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{\left (b -c \right )^{3}}-\frac {8 a \,c^{2} \left (\frac {-\frac {\left (c x +a \right )^{\frac {3}{2}}}{8 a}-\frac {\sqrt {c x +a}}{8}}{x^{2} c^{2}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{\left (b -c \right )^{3}}+\frac {6 c b \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (b -c \right )^{3}}-\frac {6 b c \left (-\frac {\sqrt {c x +a}}{2 x c}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (b -c \right )^{3}}-\frac {2 c^{2} \left (-\frac {\sqrt {c x +a}}{2 x c}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (b -c \right )^{3}}\) | \(300\) |
Input:
int(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
2/(b-c)^3*b^2*(-1/2*(b*x+a)^(1/2)/x/b-1/2/a^(1/2)*arctanh((b*x+a)^(1/2)/a^ (1/2)))+8/(b-c)^3*a*b^2*((-1/8/a*(b*x+a)^(3/2)-1/8*(b*x+a)^(1/2))/x^2/b^2+ 1/8/a^(3/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-8/(b-c)^3*a*c^2*((-1/8/a*(c*x+ a)^(3/2)-1/8*(c*x+a)^(1/2))/x^2/c^2+1/8/a^(3/2)*arctanh((c*x+a)^(1/2)/a^(1 /2)))+6/(b-c)^3*c*b*(-1/2*(b*x+a)^(1/2)/x/b-1/2/a^(1/2)*arctanh((b*x+a)^(1 /2)/a^(1/2)))-6/(b-c)^3*b*c*(-1/2*(c*x+a)^(1/2)/x/c-1/2/a^(1/2)*arctanh((c *x+a)^(1/2)/a^(1/2)))-2/(b-c)^3*c^2*(-1/2*(c*x+a)^(1/2)/x/c-1/2/a^(1/2)*ar ctanh((c*x+a)^(1/2)/a^(1/2)))
Time = 0.09 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\left [-\frac {3 \, \sqrt {a} b c x^{2} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 3 \, \sqrt {a} b c x^{2} \log \left (\frac {c x - 2 \, \sqrt {c x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (2 \, a^{2} + {\left (2 \, a b + 3 \, a c\right )} x\right )} \sqrt {b x + a} - 2 \, {\left (2 \, a^{2} + {\left (3 \, a b + 2 \, a c\right )} x\right )} \sqrt {c x + a}}{2 \, {\left (a b^{3} - 3 \, a b^{2} c + 3 \, a b c^{2} - a c^{3}\right )} x^{2}}, \frac {3 \, \sqrt {-a} b c x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) - 3 \, \sqrt {-a} b c x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x + a}}\right ) - {\left (2 \, a^{2} + {\left (2 \, a b + 3 \, a c\right )} x\right )} \sqrt {b x + a} + {\left (2 \, a^{2} + {\left (3 \, a b + 2 \, a c\right )} x\right )} \sqrt {c x + a}}{{\left (a b^{3} - 3 \, a b^{2} c + 3 \, a b c^{2} - a c^{3}\right )} x^{2}}\right ] \] Input:
integrate(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="fricas")
Output:
[-1/2*(3*sqrt(a)*b*c*x^2*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 3* sqrt(a)*b*c*x^2*log((c*x - 2*sqrt(c*x + a)*sqrt(a) + 2*a)/x) + 2*(2*a^2 + (2*a*b + 3*a*c)*x)*sqrt(b*x + a) - 2*(2*a^2 + (3*a*b + 2*a*c)*x)*sqrt(c*x + a))/((a*b^3 - 3*a*b^2*c + 3*a*b*c^2 - a*c^3)*x^2), (3*sqrt(-a)*b*c*x^2*a rctan(sqrt(-a)/sqrt(b*x + a)) - 3*sqrt(-a)*b*c*x^2*arctan(sqrt(-a)/sqrt(c* x + a)) - (2*a^2 + (2*a*b + 3*a*c)*x)*sqrt(b*x + a) + (2*a^2 + (3*a*b + 2* a*c)*x)*sqrt(c*x + a))/((a*b^3 - 3*a*b^2*c + 3*a*b*c^2 - a*c^3)*x^2)]
\[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\int \frac {1}{\left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{3}}\, dx \] Input:
integrate(1/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)
Output:
Integral((sqrt(a + b*x) + sqrt(a + c*x))**(-3), x)
\[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\int { \frac {1}{{\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}^{3}} \,d x } \] Input:
integrate(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="maxima")
Output:
integrate((sqrt(b*x + a) + sqrt(c*x + a))^(-3), x)
Leaf count of result is larger than twice the leaf count of optimal. 2766 vs. \(2 (144) = 288\).
Time = 17.36 (sec) , antiderivative size = 2766, normalized size of antiderivative = 16.87 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="giac")
Output:
3*b*c*arctan(sqrt(b*x + a)/sqrt(-a))/((b^3 - 3*b^2*c + 3*b*c^2 - c^3)*sqrt (-a)) - 2*(3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c ))*a^3*b^7*c*abs(b) - 7*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)* b*c - a*b*c))*a^3*b^6*c^2*abs(b) + 3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))*a^3*b^5*c^3*abs(b) + 3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))*a^3*b^4*c^4*abs(b) - 2*(sqrt(b*c)* sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))*a^3*b^3*c^5*abs(b) - 3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^3*a^2*b^ 5*c*abs(b) - 3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b *c))^3*a^2*b^3*c^3*abs(b) + 6*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^3*a^2*b^2*c^4*abs(b) - 3*(sqrt(b*c)*sqrt(b*x + a) - sq rt(a*b^2 + (b*x + a)*b*c - a*b*c))^5*a*b^3*c*abs(b) - 3*(sqrt(b*c)*sqrt(b* x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^5*a*b^2*c^2*abs(b) - 6*(sqrt (b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^5*a*b*c^3*abs(b ) + 3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^7*b* c*abs(b) + 2*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c ))^7*c^2*abs(b))/((a^2*b^4 - 2*a^2*b^3*c + a^2*b^2*c^2 - 2*(sqrt(b*c)*sqrt (b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2*a*b^2 - 2*(sqrt(b*c)*sq rt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2*a*b*c + (sqrt(b*c)*sq rt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^4)^2*(b^3 - 3*b^2*c ...
Time = 24.99 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {c^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{4\,\sqrt {a}\,{\left (b-c\right )}^3\,{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}-\frac {\left (\frac {\sqrt {a}\,b^2}{4\,\left (a\,b^3-3\,a\,b^2\,c+3\,a\,b\,c^2-a\,c^3\right )}-\frac {\sqrt {a}\,\left (b^2+c\,b\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\left (\sqrt {a+c\,x}-\sqrt {a}\right )\,\left (a\,b^3-3\,a\,b^2\,c+3\,a\,b\,c^2-a\,c^3\right )}\right )\,{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}+\frac {3\,b\,c\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )}{\sqrt {a}\,\left (b^3-3\,b^2\,c+3\,b\,c^2-c^3\right )}-\frac {c\,\left (b+c\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a}\,{\left (b-c\right )}^3\,\left (\sqrt {a+c\,x}-\sqrt {a}\right )} \] Input:
int(1/((a + b*x)^(1/2) + (a + c*x)^(1/2))^3,x)
Output:
(c^2*((a + b*x)^(1/2) - a^(1/2))^2)/(4*a^(1/2)*(b - c)^3*((a + c*x)^(1/2) - a^(1/2))^2) - (((a^(1/2)*b^2)/(4*(a*b^3 - a*c^3 + 3*a*b*c^2 - 3*a*b^2*c) ) - (a^(1/2)*(b*c + b^2)*((a + b*x)^(1/2) - a^(1/2)))/(((a + c*x)^(1/2) - a^(1/2))*(a*b^3 - a*c^3 + 3*a*b*c^2 - 3*a*b^2*c)))*((a + c*x)^(1/2) - a^(1 /2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (3*b*c*log(((a + b*x)^(1/2) - a^(1 /2))/((a + c*x)^(1/2) - a^(1/2))))/(a^(1/2)*(3*b*c^2 - 3*b^2*c + b^3 - c^3 )) - (c*(b + c)*((a + b*x)^(1/2) - a^(1/2)))/(a^(1/2)*(b - c)^3*((a + c*x) ^(1/2) - a^(1/2)))
Time = 0.78 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {6 \sqrt {b}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {b}-b -c}\, \mathit {atan} \left (\frac {\sqrt {c}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {c x +a}}{\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {b}-b -c}}\right ) b c \,x^{2}+6 \sqrt {c}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {b}-b -c}\, \mathit {atan} \left (\frac {\sqrt {c}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {c x +a}}{\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {b}-b -c}}\right ) b c \,x^{2}-2 \sqrt {b x +a}\, a^{2} b +2 \sqrt {b x +a}\, a^{2} c -2 \sqrt {b x +a}\, a \,b^{2} x -\sqrt {b x +a}\, a b c x +3 \sqrt {b x +a}\, a \,c^{2} x +2 \sqrt {c x +a}\, a^{2} b -2 \sqrt {c x +a}\, a^{2} c +3 \sqrt {c x +a}\, a \,b^{2} x -\sqrt {c x +a}\, a b c x -2 \sqrt {c x +a}\, a \,c^{2} x}{a \,x^{2} \left (b^{4}-4 b^{3} c +6 b^{2} c^{2}-4 b \,c^{3}+c^{4}\right )} \] Input:
int(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x)
Output:
(6*sqrt(b)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(b) - b - c)*atan((sqrt(c)*sqrt(a + b*x) + sqrt(b)*sqrt(a + c*x))/(sqrt(a)*sqrt(2*sqrt(c)*sqrt(b) - b - c)))*b *c*x**2 + 6*sqrt(c)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(b) - b - c)*atan((sqrt(c)* sqrt(a + b*x) + sqrt(b)*sqrt(a + c*x))/(sqrt(a)*sqrt(2*sqrt(c)*sqrt(b) - b - c)))*b*c*x**2 - 2*sqrt(a + b*x)*a**2*b + 2*sqrt(a + b*x)*a**2*c - 2*sqr t(a + b*x)*a*b**2*x - sqrt(a + b*x)*a*b*c*x + 3*sqrt(a + b*x)*a*c**2*x + 2 *sqrt(a + c*x)*a**2*b - 2*sqrt(a + c*x)*a**2*c + 3*sqrt(a + c*x)*a*b**2*x - sqrt(a + c*x)*a*b*c*x - 2*sqrt(a + c*x)*a*c**2*x)/(a*x**2*(b**4 - 4*b**3 *c + 6*b**2*c**2 - 4*b*c**3 + c**4))