Integrand size = 23, antiderivative size = 261 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {8 a^2 (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac {2 a (a+3 c) (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac {16 a (a+b x)^{5/2}}{5 b^2 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{5/2}}{5 b^2 (a-c)^3}+\frac {8 (a+b x)^{7/2}}{7 b^2 (a-c)^3}-\frac {8 c^2 (c+b x)^{3/2}}{3 b^2 (a-c)^3}+\frac {2 c (3 a+c) (c+b x)^{3/2}}{3 b^2 (a-c)^3}+\frac {16 c (c+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac {2 (3 a+c) (c+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac {8 (c+b x)^{7/2}}{7 b^2 (a-c)^3} \] Output:
8/3*a^2*(b*x+a)^(3/2)/b^2/(a-c)^3-2/3*a*(a+3*c)*(b*x+a)^(3/2)/b^2/(a-c)^3- 16/5*a*(b*x+a)^(5/2)/b^2/(a-c)^3+2/5*(a+3*c)*(b*x+a)^(5/2)/b^2/(a-c)^3+8/7 *(b*x+a)^(7/2)/b^2/(a-c)^3-8/3*c^2*(b*x+c)^(3/2)/b^2/(a-c)^3+2/3*c*(3*a+c) *(b*x+c)^(3/2)/b^2/(a-c)^3+16/5*c*(b*x+c)^(5/2)/b^2/(a-c)^3-2/5*(3*a+c)*(b *x+c)^(5/2)/b^2/(a-c)^3-8/7*(b*x+c)^(7/2)/b^2/(a-c)^3
Time = 0.84 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.36 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {2 \left ((c+b x)^{3/2} \left (-6 c^2+9 b c x-20 b^2 x^2+7 a (2 c-3 b x)\right )+(a+b x)^{3/2} \left (6 a^2-a (14 c+9 b x)+b x (21 c+20 b x)\right )\right )}{35 b^2 (a-c)^3} \] Input:
Integrate[x/(Sqrt[a + b*x] + Sqrt[c + b*x])^3,x]
Output:
(2*((c + b*x)^(3/2)*(-6*c^2 + 9*b*c*x - 20*b^2*x^2 + 7*a*(2*c - 3*b*x)) + (a + b*x)^(3/2)*(6*a^2 - a*(14*c + 9*b*x) + b*x*(21*c + 20*b*x))))/(35*b^2 *(a - c)^3)
Time = 0.71 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.76, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {b x+c}\right )^3} \, dx\) |
| \(\Big \downarrow \) 7240 |
| \(\displaystyle \frac {\int \left (4 b \sqrt {a+b x} x^2-4 b \sqrt {c+b x} x^2+(a+3 c) \sqrt {a+b x} x-(3 a+c) \sqrt {c+b x} x\right )dx}{(a-c)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {8 a^2 (a+b x)^{3/2}}{3 b^2}+\frac {2 (a+3 c) (a+b x)^{5/2}}{5 b^2}-\frac {2 a (a+3 c) (a+b x)^{3/2}}{3 b^2}-\frac {2 (3 a+c) (b x+c)^{5/2}}{5 b^2}+\frac {2 c (3 a+c) (b x+c)^{3/2}}{3 b^2}+\frac {8 (a+b x)^{7/2}}{7 b^2}-\frac {16 a (a+b x)^{5/2}}{5 b^2}-\frac {8 c^2 (b x+c)^{3/2}}{3 b^2}-\frac {8 (b x+c)^{7/2}}{7 b^2}+\frac {16 c (b x+c)^{5/2}}{5 b^2}}{(a-c)^3}\) |
Input:
Int[x/(Sqrt[a + b*x] + Sqrt[c + b*x])^3,x]
Output:
((8*a^2*(a + b*x)^(3/2))/(3*b^2) - (2*a*(a + 3*c)*(a + b*x)^(3/2))/(3*b^2) - (16*a*(a + b*x)^(5/2))/(5*b^2) + (2*(a + 3*c)*(a + b*x)^(5/2))/(5*b^2) + (8*(a + b*x)^(7/2))/(7*b^2) - (8*c^2*(c + b*x)^(3/2))/(3*b^2) + (2*c*(3* a + c)*(c + b*x)^(3/2))/(3*b^2) + (16*c*(c + b*x)^(5/2))/(5*b^2) - (2*(3*a + c)*(c + b*x)^(5/2))/(5*b^2) - (8*(c + b*x)^(7/2))/(7*b^2))/(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]
Time = 0.01 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.85
| method | result | size |
| default | \(\frac {2 a \left (\frac {\left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {a \left (b x +a \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b^{2}}+\frac {6 c \left (\frac {\left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {a \left (b x +a \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b^{2}}-\frac {6 a \left (\frac {\left (b x +c \right )^{\frac {5}{2}}}{5}-\frac {c \left (b x +c \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b^{2}}-\frac {2 c \left (\frac {\left (b x +c \right )^{\frac {5}{2}}}{5}-\frac {c \left (b x +c \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b^{2}}+\frac {\frac {8 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {16 a \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {8 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}}{\left (a -c \right )^{3} b^{2}}-\frac {8 \left (\frac {\left (b x +c \right )^{\frac {7}{2}}}{7}-\frac {2 c \left (b x +c \right )^{\frac {5}{2}}}{5}+\frac {c^{2} \left (b x +c \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b^{2}}\) | \(222\) |
Input:
int(x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
2/(a-c)^3*a/b^2*(1/5*(b*x+a)^(5/2)-1/3*a*(b*x+a)^(3/2))+6/(a-c)^3*c/b^2*(1 /5*(b*x+a)^(5/2)-1/3*a*(b*x+a)^(3/2))-6/(a-c)^3*a/b^2*(1/5*(b*x+c)^(5/2)-1 /3*c*(b*x+c)^(3/2))-2/(a-c)^3*c/b^2*(1/5*(b*x+c)^(5/2)-1/3*c*(b*x+c)^(3/2) )+8/(a-c)^3/b^2*(1/7*(b*x+a)^(7/2)-2/5*a*(b*x+a)^(5/2)+1/3*a^2*(b*x+a)^(3/ 2))-8/(a-c)^3/b^2*(1/7*(b*x+c)^(7/2)-2/5*c*(b*x+c)^(5/2)+1/3*c^2*(b*x+c)^( 3/2))
Time = 0.07 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.61 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {2 \, {\left ({\left (20 \, b^{3} x^{3} + 6 \, a^{3} - 14 \, a^{2} c + {\left (11 \, a b^{2} + 21 \, b^{2} c\right )} x^{2} - {\left (3 \, a^{2} b - 7 \, a b c\right )} x\right )} \sqrt {b x + a} - {\left (20 \, b^{3} x^{3} - 14 \, a c^{2} + 6 \, c^{3} + {\left (21 \, a b^{2} + 11 \, b^{2} c\right )} x^{2} + {\left (7 \, a b c - 3 \, b c^{2}\right )} x\right )} \sqrt {b x + c}\right )}}{35 \, {\left (a^{3} b^{2} - 3 \, a^{2} b^{2} c + 3 \, a b^{2} c^{2} - b^{2} c^{3}\right )}} \] Input:
integrate(x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="fricas")
Output:
2/35*((20*b^3*x^3 + 6*a^3 - 14*a^2*c + (11*a*b^2 + 21*b^2*c)*x^2 - (3*a^2* b - 7*a*b*c)*x)*sqrt(b*x + a) - (20*b^3*x^3 - 14*a*c^2 + 6*c^3 + (21*a*b^2 + 11*b^2*c)*x^2 + (7*a*b*c - 3*b*c^2)*x)*sqrt(b*x + c))/(a^3*b^2 - 3*a^2* b^2*c + 3*a*b^2*c^2 - b^2*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 942 vs. \(2 (240) = 480\).
Time = 0.94 (sec) , antiderivative size = 942, normalized size of antiderivative = 3.61 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\begin {cases} \frac {12 a^{2}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {54 a b x}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {44 a c}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {36 a \sqrt {a + b x} \sqrt {b x + c}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {40 b^{2} x^{2}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {54 b c x}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {30 b x \sqrt {a + b x} \sqrt {b x + c}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {12 c^{2}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {36 c \sqrt {a + b x} \sqrt {b x + c}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 \left (\sqrt {a} + \sqrt {c}\right )^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(x/((b*x+a)**(1/2)+(b*x+c)**(1/2))**3,x)
Output:
Piecewise((12*a**2/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 1 40*b**3*x*sqrt(a + b*x) + 140*b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b *x) + 35*b**2*c*sqrt(b*x + c)) + 54*a*b*x/(35*a*b**2*sqrt(a + b*x) + 105*a *b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140*b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 44*a*c/(35*a*b**2* sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140* b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 36*a*sqrt(a + b*x)*sqrt(b*x + c)/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2* sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140*b**3*x*sqrt(b*x + c) + 105* b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 40*b**2*x**2/(35*a*b**2* sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140* b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 54*b*c*x/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3 *x*sqrt(a + b*x) + 140*b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 3 5*b**2*c*sqrt(b*x + c)) + 30*b*x*sqrt(a + b*x)*sqrt(b*x + c)/(35*a*b**2*sq rt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140*b* *3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 12*c**2/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3*x* sqrt(a + b*x) + 140*b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b **2*c*sqrt(b*x + c)) + 36*c*sqrt(a + b*x)*sqrt(b*x + c)/(35*a*b**2*sqrt...
\[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\int { \frac {x}{{\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{3}} \,d x } \] Input:
integrate(x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="maxima")
Output:
integrate(x/(sqrt(b*x + a) + sqrt(b*x + c))^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (221) = 442\).
Time = 0.15 (sec) , antiderivative size = 866, normalized size of antiderivative = 3.32 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="giac")
Output:
-2/35*((((b*x + a)*(20*(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)*(b*x + a)/(a^9*b^4 - 9*a^8 *b^4*c + 36*a^7*b^4*c^2 - 84*a^6*b^4*c^3 + 126*a^5*b^4*c^4 - 126*a^4*b^4*c ^5 + 84*a^3*b^4*c^6 - 36*a^2*b^4*c^7 + 9*a*b^4*c^8 - b^4*c^9) - (39*a^7*b^ 3 - 245*a^6*b^3*c + 651*a^5*b^3*c^2 - 945*a^4*b^3*c^3 + 805*a^3*b^3*c^4 - 399*a^2*b^3*c^5 + 105*a*b^3*c^6 - 11*b^3*c^7)/(a^9*b^4 - 9*a^8*b^4*c + 36* a^7*b^4*c^2 - 84*a^6*b^4*c^3 + 126*a^5*b^4*c^4 - 126*a^4*b^4*c^5 + 84*a^3* b^4*c^6 - 36*a^2*b^4*c^7 + 9*a*b^4*c^8 - b^4*c^9)) + 3*(6*a^8*b^3 - 41*a^7 *b^3*c + 119*a^6*b^3*c^2 - 189*a^5*b^3*c^3 + 175*a^4*b^3*c^4 - 91*a^3*b^3* c^5 + 21*a^2*b^3*c^6 + a*b^3*c^7 - b^3*c^8)/(a^9*b^4 - 9*a^8*b^4*c + 36*a^ 7*b^4*c^2 - 84*a^6*b^4*c^3 + 126*a^5*b^4*c^4 - 126*a^4*b^4*c^5 + 84*a^3*b^ 4*c^6 - 36*a^2*b^4*c^7 + 9*a*b^4*c^8 - b^4*c^9))*(b*x + a) + (a^9*b^3 - 2* a^8*b^3*c - 20*a^7*b^3*c^2 + 112*a^6*b^3*c^3 - 266*a^5*b^3*c^4 + 364*a^4*b ^3*c^5 - 308*a^3*b^3*c^6 + 160*a^2*b^3*c^7 - 47*a*b^3*c^8 + 6*b^3*c^9)/(a^ 9*b^4 - 9*a^8*b^4*c + 36*a^7*b^4*c^2 - 84*a^6*b^4*c^3 + 126*a^5*b^4*c^4 - 126*a^4*b^4*c^5 + 84*a^3*b^4*c^6 - 36*a^2*b^4*c^7 + 9*a*b^4*c^8 - b^4*c^9) )*sqrt(b*x + c) - (20*(b*x + a)^(7/2) - 49*(b*x + a)^(5/2)*a + 35*(b*x + a )^(3/2)*a^2 + 21*(b*x + a)^(5/2)*c - 35*(b*x + a)^(3/2)*a*c)/(a^3*b - 3*a^ 2*b*c + 3*a*b*c^2 - b*c^3))/b
Time = 22.73 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.48 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {x^2\,\left (\frac {48\,b\,c}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {c+b\,x}}{5\,b}-\frac {x^2\,\left (\frac {48\,a\,b}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {a+b\,x}}{5\,b}-\frac {2\,a\,\left (\frac {2\,a\,\left (a+3\,c\right )}{{\left (a-c\right )}^3}+\frac {4\,a\,\left (\frac {48\,a\,b}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{5\,b}\right )\,\sqrt {a+b\,x}}{3\,b^2}+\frac {8\,b\,x^3\,\sqrt {a+b\,x}}{7\,{\left (a-c\right )}^3}+\frac {2\,c\,\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {4\,c\,\left (\frac {48\,b\,c}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{5\,b}\right )\,\sqrt {c+b\,x}}{3\,b^2}-\frac {8\,b\,x^3\,\sqrt {c+b\,x}}{7\,{\left (a-c\right )}^3}+\frac {x\,\left (\frac {2\,a\,\left (a+3\,c\right )}{{\left (a-c\right )}^3}+\frac {4\,a\,\left (\frac {48\,a\,b}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{5\,b}\right )\,\sqrt {a+b\,x}}{3\,b}-\frac {x\,\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {4\,c\,\left (\frac {48\,b\,c}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{5\,b}\right )\,\sqrt {c+b\,x}}{3\,b} \] Input:
int(x/((a + b*x)^(1/2) + (c + b*x)^(1/2))^3,x)
Output:
(x^2*((48*b*c)/(7*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3)*(c + b*x)^(1/2 ))/(5*b) - (x^2*((48*a*b)/(7*(a - c)^3) - (2*b*(5*a + 3*c))/(a - c)^3)*(a + b*x)^(1/2))/(5*b) - (2*a*((2*a*(a + 3*c))/(a - c)^3 + (4*a*((48*a*b)/(7* (a - c)^3) - (2*b*(5*a + 3*c))/(a - c)^3))/(5*b))*(a + b*x)^(1/2))/(3*b^2) + (8*b*x^3*(a + b*x)^(1/2))/(7*(a - c)^3) + (2*c*((2*c*(3*a + c))/(a - c) ^3 + (4*c*((48*b*c)/(7*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3))/(5*b))*( c + b*x)^(1/2))/(3*b^2) - (8*b*x^3*(c + b*x)^(1/2))/(7*(a - c)^3) + (x*((2 *a*(a + 3*c))/(a - c)^3 + (4*a*((48*a*b)/(7*(a - c)^3) - (2*b*(5*a + 3*c)) /(a - c)^3))/(5*b))*(a + b*x)^(1/2))/(3*b) - (x*((2*c*(3*a + c))/(a - c)^3 + (4*c*((48*b*c)/(7*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3))/(5*b))*(c + b*x)^(1/2))/(3*b)
Time = 0.15 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {-\frac {6 \sqrt {b x +c}\, a \,b^{2} x^{2}}{5}-\frac {2 \sqrt {b x +c}\, a b c x}{5}+\frac {4 \sqrt {b x +c}\, a \,c^{2}}{5}-\frac {8 \sqrt {b x +c}\, b^{3} x^{3}}{7}-\frac {22 \sqrt {b x +c}\, b^{2} c \,x^{2}}{35}+\frac {6 \sqrt {b x +c}\, b \,c^{2} x}{35}-\frac {12 \sqrt {b x +c}\, c^{3}}{35}+\frac {12 \sqrt {b x +a}\, a^{3}}{35}-\frac {6 \sqrt {b x +a}\, a^{2} b x}{35}-\frac {4 \sqrt {b x +a}\, a^{2} c}{5}+\frac {22 \sqrt {b x +a}\, a \,b^{2} x^{2}}{35}+\frac {2 \sqrt {b x +a}\, a b c x}{5}+\frac {8 \sqrt {b x +a}\, b^{3} x^{3}}{7}+\frac {6 \sqrt {b x +a}\, b^{2} c \,x^{2}}{5}}{b^{2} \left (a^{3}-3 a^{2} c +3 a \,c^{2}-c^{3}\right )} \] Input:
int(x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x)
Output:
(2*( - 21*sqrt(b*x + c)*a*b**2*x**2 - 7*sqrt(b*x + c)*a*b*c*x + 14*sqrt(b* x + c)*a*c**2 - 20*sqrt(b*x + c)*b**3*x**3 - 11*sqrt(b*x + c)*b**2*c*x**2 + 3*sqrt(b*x + c)*b*c**2*x - 6*sqrt(b*x + c)*c**3 + 6*sqrt(a + b*x)*a**3 - 3*sqrt(a + b*x)*a**2*b*x - 14*sqrt(a + b*x)*a**2*c + 11*sqrt(a + b*x)*a*b **2*x**2 + 7*sqrt(a + b*x)*a*b*c*x + 20*sqrt(a + b*x)*b**3*x**3 + 21*sqrt( a + b*x)*b**2*c*x**2))/(35*b**2*(a**3 - 3*a**2*c + 3*a*c**2 - c**3))