Integrand size = 25, antiderivative size = 163 \[ \int \frac {x^3}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}+\frac {2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac {8 a (a+c x)^{3/2}}{3 (b-c)^3 c}+\frac {2 a (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c^2}-\frac {2 (3 b+c) (a+c x)^{5/2}}{5 (b-c)^3 c^2} \] Output:
8/3*a*(b*x+a)^(3/2)/b/(b-c)^3-2/3*a*(b+3*c)*(b*x+a)^(3/2)/b^2/(b-c)^3+2/5* (b+3*c)*(b*x+a)^(5/2)/b^2/(b-c)^3-8/3*a*(c*x+a)^(3/2)/(b-c)^3/c+2/3*a*(3*b +c)*(c*x+a)^(3/2)/(b-c)^3/c^2-2/5*(3*b+c)*(c*x+a)^(5/2)/(b-c)^3/c^2
Time = 1.88 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {2 \sqrt {a-\frac {a b}{c}+\frac {b (a+c x)}{c}} \left (a^2 b^3-4 a^2 b^2 c+5 a^2 b c^2-2 a^2 c^3-2 a b^3 (a+c x)+a b^2 c (a+c x)+a b c^2 (a+c x)+b^3 (a+c x)^2+3 b^2 c (a+c x)^2\right )}{5 b^2 (b-c)^3 c^2}+\frac {2 \left (5 a b (a+c x)^{3/2}-5 a c (a+c x)^{3/2}-3 b (a+c x)^{5/2}-c (a+c x)^{5/2}\right )}{5 (b-c)^3 c^2} \] Input:
Integrate[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]
Output:
(2*Sqrt[a - (a*b)/c + (b*(a + c*x))/c]*(a^2*b^3 - 4*a^2*b^2*c + 5*a^2*b*c^ 2 - 2*a^2*c^3 - 2*a*b^3*(a + c*x) + a*b^2*c*(a + c*x) + a*b*c^2*(a + c*x) + b^3*(a + c*x)^2 + 3*b^2*c*(a + c*x)^2))/(5*b^2*(b - c)^3*c^2) + (2*(5*a* b*(a + c*x)^(3/2) - 5*a*c*(a + c*x)^(3/2) - 3*b*(a + c*x)^(5/2) - c*(a + c *x)^(5/2)))/(5*(b - c)^3*c^2)
Time = 0.65 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.79, 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.
| \(\displaystyle \int \frac {x^3}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx\) |
| \(\Big \downarrow \) 7241 |
| \(\displaystyle \frac {\int \left (4 \sqrt {a+b x} a-4 \sqrt {a+c x} a+(b+3 c) x \sqrt {a+b x}-(3 b+c) x \sqrt {a+c x}\right )dx}{(b-c)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 (b+3 c) (a+b x)^{5/2}}{5 b^2}-\frac {2 a (b+3 c) (a+b x)^{3/2}}{3 b^2}-\frac {2 (3 b+c) (a+c x)^{5/2}}{5 c^2}+\frac {2 a (3 b+c) (a+c x)^{3/2}}{3 c^2}+\frac {8 a (a+b x)^{3/2}}{3 b}-\frac {8 a (a+c x)^{3/2}}{3 c}}{(b-c)^3}\) |
Input:
Int[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]
Output:
((8*a*(a + b*x)^(3/2))/(3*b) - (2*a*(b + 3*c)*(a + b*x)^(3/2))/(3*b^2) + ( 2*(b + 3*c)*(a + b*x)^(5/2))/(5*b^2) - (8*a*(a + c*x)^(3/2))/(3*c) + (2*a* (3*b + c)*(a + c*x)^(3/2))/(3*c^2) - (2*(3*b + c)*(a + c*x)^(5/2))/(5*c^2) )/(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]
Time = 0.01 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a \left (b x +a \right )^{\frac {3}{2}}}{3}}{\left (b -c \right )^{3} b}+\frac {8 a \left (b x +a \right )^{\frac {3}{2}}}{3 b \left (b -c \right )^{3}}-\frac {8 a \left (c x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right )^{3} c}+\frac {6 c \left (\frac {\left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {a \left (b x +a \right )^{\frac {3}{2}}}{3}\right )}{\left (b -c \right )^{3} b^{2}}-\frac {6 b \left (\frac {\left (c x +a \right )^{\frac {5}{2}}}{5}-\frac {a \left (c x +a \right )^{\frac {3}{2}}}{3}\right )}{\left (b -c \right )^{3} c^{2}}-\frac {2 \left (\frac {\left (c x +a \right )^{\frac {5}{2}}}{5}-\frac {a \left (c x +a \right )^{\frac {3}{2}}}{3}\right )}{\left (b -c \right )^{3} c}\) | \(172\) |
Input:
int(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
2/(b-c)^3/b*(1/5*(b*x+a)^(5/2)-1/3*a*(b*x+a)^(3/2))+8/3*a*(b*x+a)^(3/2)/b/ (b-c)^3-8/3*a*(c*x+a)^(3/2)/(b-c)^3/c+6/(b-c)^3*c/b^2*(1/5*(b*x+a)^(5/2)-1 /3*a*(b*x+a)^(3/2))-6/(b-c)^3*b/c^2*(1/5*(c*x+a)^(5/2)-1/3*a*(c*x+a)^(3/2) )-2/(b-c)^3/c*(1/5*(c*x+a)^(5/2)-1/3*a*(c*x+a)^(3/2))
Time = 0.07 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {2 \, {\left ({\left (6 \, a^{2} b c^{2} - 2 \, a^{2} c^{3} + {\left (b^{3} c^{2} + 3 \, b^{2} c^{3}\right )} x^{2} + {\left (7 \, a b^{2} c^{2} + a b c^{3}\right )} x\right )} \sqrt {b x + a} + {\left (2 \, a^{2} b^{3} - 6 \, a^{2} b^{2} c - {\left (3 \, b^{3} c^{2} + b^{2} c^{3}\right )} x^{2} - {\left (a b^{3} c + 7 \, a b^{2} c^{2}\right )} x\right )} \sqrt {c x + a}\right )}}{5 \, {\left (b^{5} c^{2} - 3 \, b^{4} c^{3} + 3 \, b^{3} c^{4} - b^{2} c^{5}\right )}} \] Input:
integrate(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="fricas")
Output:
2/5*((6*a^2*b*c^2 - 2*a^2*c^3 + (b^3*c^2 + 3*b^2*c^3)*x^2 + (7*a*b^2*c^2 + a*b*c^3)*x)*sqrt(b*x + a) + (2*a^2*b^3 - 6*a^2*b^2*c - (3*b^3*c^2 + b^2*c ^3)*x^2 - (a*b^3*c + 7*a*b^2*c^2)*x)*sqrt(c*x + a))/(b^5*c^2 - 3*b^4*c^3 + 3*b^3*c^4 - b^2*c^5)
\[ \int \frac {x^3}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\int \frac {x^{3}}{\left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{3}}\, dx \] Input:
integrate(x**3/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)
Output:
Integral(x**3/(sqrt(a + b*x) + sqrt(a + c*x))**3, x)
\[ \int \frac {x^3}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\int { \frac {x^{3}}{{\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}^{3}} \,d x } \] Input:
integrate(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="maxima")
Output:
integrate(x^3/(sqrt(b*x + a) + sqrt(c*x + a))^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (139) = 278\).
Time = 0.74 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.94 \[ \int \frac {x^3}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=-\frac {2}{5} \, \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} {\left ({\left (b x + a\right )} {\left (\frac {{\left (3 \, b^{12} c^{3} {\left | b \right |} - 8 \, b^{11} c^{4} {\left | b \right |} + 6 \, b^{10} c^{5} {\left | b \right |} - b^{8} c^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{18} c^{3} - 6 \, b^{17} c^{4} + 15 \, b^{16} c^{5} - 20 \, b^{15} c^{6} + 15 \, b^{14} c^{7} - 6 \, b^{13} c^{8} + b^{12} c^{9}} + \frac {a b^{13} c^{2} {\left | b \right |} - 2 \, a b^{12} c^{3} {\left | b \right |} - 2 \, a b^{11} c^{4} {\left | b \right |} + 8 \, a b^{10} c^{5} {\left | b \right |} - 7 \, a b^{9} c^{6} {\left | b \right |} + 2 \, a b^{8} c^{7} {\left | b \right |}}{b^{18} c^{3} - 6 \, b^{17} c^{4} + 15 \, b^{16} c^{5} - 20 \, b^{15} c^{6} + 15 \, b^{14} c^{7} - 6 \, b^{13} c^{8} + b^{12} c^{9}}\right )} - \frac {2 \, a^{2} b^{14} c {\left | b \right |} - 11 \, a^{2} b^{13} c^{2} {\left | b \right |} + 25 \, a^{2} b^{12} c^{3} {\left | b \right |} - 30 \, a^{2} b^{11} c^{4} {\left | b \right |} + 20 \, a^{2} b^{10} c^{5} {\left | b \right |} - 7 \, a^{2} b^{9} c^{6} {\left | b \right |} + a^{2} b^{8} c^{7} {\left | b \right |}}{b^{18} c^{3} - 6 \, b^{17} c^{4} + 15 \, b^{16} c^{5} - 20 \, b^{15} c^{6} + 15 \, b^{14} c^{7} - 6 \, b^{13} c^{8} + b^{12} c^{9}}\right )} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {5}{2}} b + 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a b + 3 \, {\left (b x + a\right )}^{\frac {5}{2}} c - 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a c\right )}}{5 \, {\left (b^{5} - 3 \, b^{4} c + 3 \, b^{3} c^{2} - b^{2} c^{3}\right )}} \] Input:
integrate(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="giac")
Output:
-2/5*sqrt(a*b^2 + (b*x + a)*b*c - a*b*c)*((b*x + a)*((3*b^12*c^3*abs(b) - 8*b^11*c^4*abs(b) + 6*b^10*c^5*abs(b) - b^8*c^7*abs(b))*(b*x + a)/(b^18*c^ 3 - 6*b^17*c^4 + 15*b^16*c^5 - 20*b^15*c^6 + 15*b^14*c^7 - 6*b^13*c^8 + b^ 12*c^9) + (a*b^13*c^2*abs(b) - 2*a*b^12*c^3*abs(b) - 2*a*b^11*c^4*abs(b) + 8*a*b^10*c^5*abs(b) - 7*a*b^9*c^6*abs(b) + 2*a*b^8*c^7*abs(b))/(b^18*c^3 - 6*b^17*c^4 + 15*b^16*c^5 - 20*b^15*c^6 + 15*b^14*c^7 - 6*b^13*c^8 + b^12 *c^9)) - (2*a^2*b^14*c*abs(b) - 11*a^2*b^13*c^2*abs(b) + 25*a^2*b^12*c^3*a bs(b) - 30*a^2*b^11*c^4*abs(b) + 20*a^2*b^10*c^5*abs(b) - 7*a^2*b^9*c^6*ab s(b) + a^2*b^8*c^7*abs(b))/(b^18*c^3 - 6*b^17*c^4 + 15*b^16*c^5 - 20*b^15* c^6 + 15*b^14*c^7 - 6*b^13*c^8 + b^12*c^9)) + 2/5*((b*x + a)^(5/2)*b + 5*( b*x + a)^(3/2)*a*b + 3*(b*x + a)^(5/2)*c - 5*(b*x + a)^(3/2)*a*c)/(b^5 - 3 *b^4*c + 3*b^3*c^2 - b^2*c^3)
Time = 22.69 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.64 \[ \int \frac {x^3}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {\left (\frac {8\,a^2}{{\left (b-c\right )}^3}+\frac {2\,a\,\left (\frac {8\,a\,\left (b+3\,c\right )}{5\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (5\,b+3\,c\right )}{{\left (b-c\right )}^3}\right )}{3\,b}\right )\,\sqrt {a+b\,x}}{b}-\frac {\left (\frac {8\,a^2}{{\left (b-c\right )}^3}+\frac {2\,a\,\left (\frac {8\,a\,\left (3\,b+c\right )}{5\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (3\,b+5\,c\right )}{{\left (b-c\right )}^3}\right )}{3\,c}\right )\,\sqrt {a+c\,x}}{c}-\frac {x\,\left (\frac {8\,a\,\left (b+3\,c\right )}{5\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (5\,b+3\,c\right )}{{\left (b-c\right )}^3}\right )\,\sqrt {a+b\,x}}{3\,b}+\frac {x\,\left (\frac {8\,a\,\left (3\,b+c\right )}{5\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (3\,b+5\,c\right )}{{\left (b-c\right )}^3}\right )\,\sqrt {a+c\,x}}{3\,c}+\frac {2\,x^2\,\left (b+3\,c\right )\,\sqrt {a+b\,x}}{5\,{\left (b-c\right )}^3}-\frac {2\,x^2\,\left (3\,b+c\right )\,\sqrt {a+c\,x}}{5\,{\left (b-c\right )}^3} \] Input:
int(x^3/((a + b*x)^(1/2) + (a + c*x)^(1/2))^3,x)
Output:
(((8*a^2)/(b - c)^3 + (2*a*((8*a*(b + 3*c))/(5*(b - c)^3) - (2*a*(5*b + 3* c))/(b - c)^3))/(3*b))*(a + b*x)^(1/2))/b - (((8*a^2)/(b - c)^3 + (2*a*((8 *a*(3*b + c))/(5*(b - c)^3) - (2*a*(3*b + 5*c))/(b - c)^3))/(3*c))*(a + c* x)^(1/2))/c - (x*((8*a*(b + 3*c))/(5*(b - c)^3) - (2*a*(5*b + 3*c))/(b - c )^3)*(a + b*x)^(1/2))/(3*b) + (x*((8*a*(3*b + c))/(5*(b - c)^3) - (2*a*(3* b + 5*c))/(b - c)^3)*(a + c*x)^(1/2))/(3*c) + (2*x^2*(b + 3*c)*(a + b*x)^( 1/2))/(5*(b - c)^3) - (2*x^2*(3*b + c)*(a + c*x)^(1/2))/(5*(b - c)^3)
Time = 0.15 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.33 \[ \int \frac {x^3}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {\frac {12 \sqrt {b x +a}\, a^{2} b \,c^{2}}{5}-\frac {4 \sqrt {b x +a}\, a^{2} c^{3}}{5}+\frac {14 \sqrt {b x +a}\, a \,b^{2} c^{2} x}{5}+\frac {2 \sqrt {b x +a}\, a b \,c^{3} x}{5}+\frac {2 \sqrt {b x +a}\, b^{3} c^{2} x^{2}}{5}+\frac {6 \sqrt {b x +a}\, b^{2} c^{3} x^{2}}{5}+\frac {4 \sqrt {c x +a}\, a^{2} b^{3}}{5}-\frac {12 \sqrt {c x +a}\, a^{2} b^{2} c}{5}-\frac {2 \sqrt {c x +a}\, a \,b^{3} c x}{5}-\frac {14 \sqrt {c x +a}\, a \,b^{2} c^{2} x}{5}-\frac {6 \sqrt {c x +a}\, b^{3} c^{2} x^{2}}{5}-\frac {2 \sqrt {c x +a}\, b^{2} c^{3} x^{2}}{5}}{b^{2} c^{2} \left (b^{3}-3 b^{2} c +3 b \,c^{2}-c^{3}\right )} \] Input:
int(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x)
Output:
(2*(6*sqrt(a + b*x)*a**2*b*c**2 - 2*sqrt(a + b*x)*a**2*c**3 + 7*sqrt(a + b *x)*a*b**2*c**2*x + sqrt(a + b*x)*a*b*c**3*x + sqrt(a + b*x)*b**3*c**2*x** 2 + 3*sqrt(a + b*x)*b**2*c**3*x**2 + 2*sqrt(a + c*x)*a**2*b**3 - 6*sqrt(a + c*x)*a**2*b**2*c - sqrt(a + c*x)*a*b**3*c*x - 7*sqrt(a + c*x)*a*b**2*c** 2*x - 3*sqrt(a + c*x)*b**3*c**2*x**2 - sqrt(a + c*x)*b**2*c**3*x**2))/(5*b **2*c**2*(b**3 - 3*b**2*c + 3*b*c**2 - c**3))