Integrand size = 21, antiderivative size = 64 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {(a-c)^2}{10 b \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^5}-\frac {1}{2 b \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \] Output:
1/10*(a-c)^2/b/((b*x+a)^(1/2)+(b*x+c)^(1/2))^5-1/2/b/((b*x+a)^(1/2)+(b*x+c )^(1/2))
Time = 0.64 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {2 \left ((-5 a+c-4 b x) (c+b x)^{3/2}+(a+b x)^{3/2} (-a+5 c+4 b x)\right )}{5 b (a-c)^3} \] Input:
Integrate[(Sqrt[a + b*x] + Sqrt[c + b*x])^(-3),x]
Output:
(2*((-5*a + c - 4*b*x)*(c + b*x)^(3/2) + (a + b*x)^(3/2)*(-a + 5*c + 4*b*x )))/(5*b*(a - c)^3)
Time = 0.50 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
\(\displaystyle \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {b x+c}\right )^3} \, dx\) |
\(\Big \downarrow \) 7240 |
\(\displaystyle \frac {\int \left (-\sqrt {c+b x} (3 a+c)+(a+3 c) \sqrt {a+b x}+4 b x \sqrt {a+b x}-4 b x \sqrt {c+b x}\right )dx}{(a-c)^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {2 (a+3 c) (a+b x)^{3/2}}{3 b}-\frac {2 (3 a+c) (b x+c)^{3/2}}{3 b}+\frac {8 (a+b x)^{5/2}}{5 b}-\frac {8 a (a+b x)^{3/2}}{3 b}-\frac {8 (b x+c)^{5/2}}{5 b}+\frac {8 c (b x+c)^{3/2}}{3 b}}{(a-c)^3}\) |
Input:
Int[(Sqrt[a + b*x] + Sqrt[c + b*x])^(-3),x]
Output:
((-8*a*(a + b*x)^(3/2))/(3*b) + (2*(a + 3*c)*(a + b*x)^(3/2))/(3*b) + (8*( a + b*x)^(5/2))/(5*b) + (8*c*(c + b*x)^(3/2))/(3*b) - (2*(3*a + c)*(c + b* x)^(3/2))/(3*b) - (8*(c + b*x)^(5/2))/(5*b))/(a - c)^3
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]
Leaf count of result is larger than twice the leaf count of optimal. \(145\) vs. \(2(52)=104\).
Time = 0.01 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.28
method | result | size |
default | \(\frac {2 a \left (b x +a \right )^{\frac {3}{2}}}{3 \left (a -c \right )^{3} b}+\frac {2 c \left (b x +a \right )^{\frac {3}{2}}}{\left (a -c \right )^{3} b}-\frac {2 a \left (b x +c \right )^{\frac {3}{2}}}{\left (a -c \right )^{3} b}-\frac {2 c \left (b x +c \right )^{\frac {3}{2}}}{3 \left (a -c \right )^{3} b}+\frac {\frac {8 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {8 a \left (b x +a \right )^{\frac {3}{2}}}{3}}{\left (a -c \right )^{3} b}-\frac {8 \left (\frac {\left (b x +c \right )^{\frac {5}{2}}}{5}-\frac {c \left (b x +c \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b}\) | \(146\) |
Input:
int(1/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
2/3/(a-c)^3*a*(b*x+a)^(3/2)/b+2/(a-c)^3*c*(b*x+a)^(3/2)/b-2/(a-c)^3*a*(b*x +c)^(3/2)/b-2/3/(a-c)^3*c*(b*x+c)^(3/2)/b+8/(a-c)^3/b*(1/5*(b*x+a)^(5/2)-1 /3*a*(b*x+a)^(3/2))-8/(a-c)^3/b*(1/5*(b*x+c)^(5/2)-1/3*c*(b*x+c)^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (52) = 104\).
Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.66 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {2 \, {\left ({\left (4 \, b^{2} x^{2} - a^{2} + 5 \, a c + {\left (3 \, a b + 5 \, b c\right )} x\right )} \sqrt {b x + a} - {\left (4 \, b^{2} x^{2} + 5 \, a c - c^{2} + {\left (5 \, a b + 3 \, b c\right )} x\right )} \sqrt {b x + c}\right )}}{5 \, {\left (a^{3} b - 3 \, a^{2} b c + 3 \, a b c^{2} - b c^{3}\right )}} \] Input:
integrate(1/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="fricas")
Output:
2/5*((4*b^2*x^2 - a^2 + 5*a*c + (3*a*b + 5*b*c)*x)*sqrt(b*x + a) - (4*b^2* x^2 + 5*a*c - c^2 + (5*a*b + 3*b*c)*x)*sqrt(b*x + c))/(a^3*b - 3*a^2*b*c + 3*a*b*c^2 - b*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (48) = 96\).
Time = 0.90 (sec) , antiderivative size = 384, normalized size of antiderivative = 6.00 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\begin {cases} - \frac {2 a}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} - \frac {4 b x}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} - \frac {2 c}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} - \frac {6 \sqrt {a + b x} \sqrt {b x + c}}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} & \text {for}\: b \neq 0 \\\frac {x}{\left (\sqrt {a} + \sqrt {c}\right )^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(1/((b*x+a)**(1/2)+(b*x+c)**(1/2))**3,x)
Output:
Piecewise((-2*a/(5*a*b*sqrt(a + b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*x*sq rt(a + b*x) + 20*b**2*x*sqrt(b*x + c) + 15*b*c*sqrt(a + b*x) + 5*b*c*sqrt( b*x + c)) - 4*b*x/(5*a*b*sqrt(a + b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*x* sqrt(a + b*x) + 20*b**2*x*sqrt(b*x + c) + 15*b*c*sqrt(a + b*x) + 5*b*c*sqr t(b*x + c)) - 2*c/(5*a*b*sqrt(a + b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*x* sqrt(a + b*x) + 20*b**2*x*sqrt(b*x + c) + 15*b*c*sqrt(a + b*x) + 5*b*c*sqr t(b*x + c)) - 6*sqrt(a + b*x)*sqrt(b*x + c)/(5*a*b*sqrt(a + b*x) + 15*a*b* sqrt(b*x + c) + 20*b**2*x*sqrt(a + b*x) + 20*b**2*x*sqrt(b*x + c) + 15*b*c *sqrt(a + b*x) + 5*b*c*sqrt(b*x + c)), Ne(b, 0)), (x/(sqrt(a) + sqrt(c))** 3, True))
\[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\int { \frac {1}{{\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{3}} \,d x } \] Input:
integrate(1/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="maxima")
Output:
integrate((sqrt(b*x + a) + sqrt(b*x + c))^(-3), x)
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (52) = 104\).
Time = 0.14 (sec) , antiderivative size = 427, normalized size of antiderivative = 6.67 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=-\frac {2}{5} \, {\left ({\left (b x + a\right )} {\left (\frac {4 \, {\left (a^{3} b^{2} - 3 \, a^{2} b^{2} c + 3 \, a b^{2} c^{2} - b^{2} c^{3}\right )} {\left (b x + a\right )}}{a^{6} b^{3} - 6 \, a^{5} b^{3} c + 15 \, a^{4} b^{3} c^{2} - 20 \, a^{3} b^{3} c^{3} + 15 \, a^{2} b^{3} c^{4} - 6 \, a b^{3} c^{5} + b^{3} c^{6}} - \frac {3 \, {\left (a^{4} b^{2} - 4 \, a^{3} b^{2} c + 6 \, a^{2} b^{2} c^{2} - 4 \, a b^{2} c^{3} + b^{2} c^{4}\right )}}{a^{6} b^{3} - 6 \, a^{5} b^{3} c + 15 \, a^{4} b^{3} c^{2} - 20 \, a^{3} b^{3} c^{3} + 15 \, a^{2} b^{3} c^{4} - 6 \, a b^{3} c^{5} + b^{3} c^{6}}\right )} - \frac {a^{5} b^{2} - 5 \, a^{4} b^{2} c + 10 \, a^{3} b^{2} c^{2} - 10 \, a^{2} b^{2} c^{3} + 5 \, a b^{2} c^{4} - b^{2} c^{5}}{a^{6} b^{3} - 6 \, a^{5} b^{3} c + 15 \, a^{4} b^{3} c^{2} - 20 \, a^{3} b^{3} c^{3} + 15 \, a^{2} b^{3} c^{4} - 6 \, a b^{3} c^{5} + b^{3} c^{6}}\right )} \sqrt {b x + c} + \frac {2 \, {\left (4 \, {\left (b x + a\right )}^{\frac {5}{2}} - 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 5 \, {\left (b x + a\right )}^{\frac {3}{2}} c\right )}}{5 \, {\left (a^{3} b - 3 \, a^{2} b c + 3 \, a b c^{2} - b c^{3}\right )}} \] Input:
integrate(1/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="giac")
Output:
-2/5*((b*x + a)*(4*(a^3*b^2 - 3*a^2*b^2*c + 3*a*b^2*c^2 - b^2*c^3)*(b*x + a)/(a^6*b^3 - 6*a^5*b^3*c + 15*a^4*b^3*c^2 - 20*a^3*b^3*c^3 + 15*a^2*b^3*c ^4 - 6*a*b^3*c^5 + b^3*c^6) - 3*(a^4*b^2 - 4*a^3*b^2*c + 6*a^2*b^2*c^2 - 4 *a*b^2*c^3 + b^2*c^4)/(a^6*b^3 - 6*a^5*b^3*c + 15*a^4*b^3*c^2 - 20*a^3*b^3 *c^3 + 15*a^2*b^3*c^4 - 6*a*b^3*c^5 + b^3*c^6)) - (a^5*b^2 - 5*a^4*b^2*c + 10*a^3*b^2*c^2 - 10*a^2*b^2*c^3 + 5*a*b^2*c^4 - b^2*c^5)/(a^6*b^3 - 6*a^5 *b^3*c + 15*a^4*b^3*c^2 - 20*a^3*b^3*c^3 + 15*a^2*b^3*c^4 - 6*a*b^3*c^5 + b^3*c^6))*sqrt(b*x + c) + 2/5*(4*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a + 5 *(b*x + a)^(3/2)*c)/(a^3*b - 3*a^2*b*c + 3*a*b*c^2 - b*c^3)
Time = 22.66 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.94 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {\left (\frac {2\,\left (a^2+3\,c\,a\right )}{{\left (a-c\right )}^3}+\frac {2\,a\,\left (\frac {32\,a\,b}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{3\,b}\right )\,\sqrt {a+b\,x}}{b}-\frac {\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {2\,c\,\left (\frac {32\,b\,c}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{3\,b}\right )\,\sqrt {c+b\,x}}{b}+\frac {8\,b\,x^2\,\sqrt {a+b\,x}}{5\,{\left (a-c\right )}^3}-\frac {8\,b\,x^2\,\sqrt {c+b\,x}}{5\,{\left (a-c\right )}^3}-\frac {x\,\left (\frac {32\,a\,b}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {a+b\,x}}{3\,b}+\frac {x\,\left (\frac {32\,b\,c}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {c+b\,x}}{3\,b} \] Input:
int(1/((a + b*x)^(1/2) + (c + b*x)^(1/2))^3,x)
Output:
(((2*(3*a*c + a^2))/(a - c)^3 + (2*a*((32*a*b)/(5*(a - c)^3) - (2*b*(5*a + 3*c))/(a - c)^3))/(3*b))*(a + b*x)^(1/2))/b - (((2*c*(3*a + c))/(a - c)^3 + (2*c*((32*b*c)/(5*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3))/(3*b))*(c + b*x)^(1/2))/b + (8*b*x^2*(a + b*x)^(1/2))/(5*(a - c)^3) - (8*b*x^2*(c + b*x)^(1/2))/(5*(a - c)^3) - (x*((32*a*b)/(5*(a - c)^3) - (2*b*(5*a + 3*c)) /(a - c)^3)*(a + b*x)^(1/2))/(3*b) + (x*((32*b*c)/(5*(a - c)^3) - (2*b*(3* a + 5*c))/(a - c)^3)*(c + b*x)^(1/2))/(3*b)
Time = 0.17 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.22 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {-2 \sqrt {b x +c}\, a b x -2 \sqrt {b x +c}\, a c -\frac {8 \sqrt {b x +c}\, b^{2} x^{2}}{5}-\frac {6 \sqrt {b x +c}\, b c x}{5}+\frac {2 \sqrt {b x +c}\, c^{2}}{5}-\frac {2 \sqrt {b x +a}\, a^{2}}{5}+\frac {6 \sqrt {b x +a}\, a b x}{5}+2 \sqrt {b x +a}\, a c +\frac {8 \sqrt {b x +a}\, b^{2} x^{2}}{5}+2 \sqrt {b x +a}\, b c x}{b \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )} \] Input:
int(1/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x)
Output:
(2*( - 5*sqrt(b*x + c)*a*b*x - 5*sqrt(b*x + c)*a*c - 4*sqrt(b*x + c)*b**2* x**2 - 3*sqrt(b*x + c)*b*c*x + sqrt(b*x + c)*c**2 - sqrt(a + b*x)*a**2 + 3 *sqrt(a + b*x)*a*b*x + 5*sqrt(a + b*x)*a*c + 4*sqrt(a + b*x)*b**2*x**2 + 5 *sqrt(a + b*x)*b*c*x))/(5*b*(a**3 - 3*a**2*c + 3*a*c**2 - c**3))