Integrand size = 26, antiderivative size = 109 \[ \int \frac {x^5}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {8 a^4 \sqrt {a+b x} \sqrt {a c-b c x}}{15 b^6 c}-\frac {4 a^2 x^2 \sqrt {a+b x} \sqrt {a c-b c x}}{15 b^4 c}-\frac {x^4 \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c} \] Output:
-8/15*a^4*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^6/c-4/15*a^2*x^2*(b*x+a)^(1/2 )*(-b*c*x+a*c)^(1/2)/b^4/c-1/5*x^4*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^2/c
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.51 \[ \int \frac {x^5}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {c (a-b x)} \sqrt {a+b x} \left (8 a^4+4 a^2 b^2 x^2+3 b^4 x^4\right )}{15 b^6 c} \] Input:
Integrate[x^5/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
Output:
-1/15*(Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*(8*a^4 + 4*a^2*b^2*x^2 + 3*b^4*x^4) )/(b^6*c)
Time = 0.19 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {111, 27, 111, 27, 83}
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.
\(\displaystyle \int \frac {x^5}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\) |
\(\Big \downarrow \) 111 |
\(\displaystyle -\frac {\int -\frac {4 a^2 c x^3}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{5 b^2 c}-\frac {x^4 \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 a^2 \int \frac {x^3}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{5 b^2}-\frac {x^4 \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {4 a^2 \left (-\frac {\int -\frac {2 a^2 c x}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b^2 c}-\frac {x^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{5 b^2}-\frac {x^4 \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 a^2 \left (\frac {2 a^2 \int \frac {x}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b^2}-\frac {x^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{5 b^2}-\frac {x^4 \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {4 a^2 \left (-\frac {2 a^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^4 c}-\frac {x^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 b^2 c}\right )}{5 b^2}-\frac {x^4 \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\) |
Input:
Int[x^5/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
Output:
-1/5*(x^4*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(b^2*c) + (4*a^2*((-2*a^2*Sqrt[ a + b*x]*Sqrt[a*c - b*c*x])/(3*b^4*c) - (x^2*Sqrt[a + b*x]*Sqrt[a*c - b*c* x])/(3*b^2*c)))/(5*b^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.47
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (3 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+8 a^{4}\right )}{15 c \,b^{6}}\) | \(51\) |
gosper | \(-\frac {\left (-b x +a \right ) \sqrt {b x +a}\, \left (3 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+8 a^{4}\right )}{15 b^{6} \sqrt {-b c x +a c}}\) | \(55\) |
orering | \(-\frac {\left (-b x +a \right ) \sqrt {b x +a}\, \left (3 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+8 a^{4}\right )}{15 b^{6} \sqrt {-b c x +a c}}\) | \(55\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (3 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+8 a^{4}\right ) \left (-b x +a \right )}{15 \sqrt {-c \left (b x -a \right )}\, b^{6}}\) | \(56\) |
Input:
int(x^5/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/15*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)/c*(3*b^4*x^4+4*a^2*b^2*x^2+8*a^4)/b ^6
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.47 \[ \int \frac {x^5}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {{\left (3 \, b^{4} x^{4} + 4 \, a^{2} b^{2} x^{2} + 8 \, a^{4}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{15 \, b^{6} c} \] Input:
integrate(x^5/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="fricas")
Output:
-1/15*(3*b^4*x^4 + 4*a^2*b^2*x^2 + 8*a^4)*sqrt(-b*c*x + a*c)*sqrt(b*x + a) /(b^6*c)
Result contains complex when optimal does not.
Time = 3.63 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18 \[ \int \frac {x^5}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=- \frac {i a^{5} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {9}{4}, - \frac {7}{4} & -2, -2, - \frac {3}{2}, 1 \\- \frac {5}{2}, - \frac {9}{4}, -2, - \frac {7}{4}, - \frac {3}{2}, 0 & \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b^{6} \sqrt {c}} - \frac {a^{5} {G_{6, 6}^{2, 6}\left (\begin {matrix} -3, - \frac {11}{4}, - \frac {5}{2}, - \frac {9}{4}, -2, 1 & \\- \frac {11}{4}, - \frac {9}{4} & -3, - \frac {5}{2}, - \frac {5}{2}, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b^{6} \sqrt {c}} \] Input:
integrate(x**5/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)
Output:
-I*a**5*meijerg(((-9/4, -7/4), (-2, -2, -3/2, 1)), ((-5/2, -9/4, -2, -7/4, -3/2, 0), ()), a**2/(b**2*x**2))/(4*pi**(3/2)*b**6*sqrt(c)) - a**5*meijer g(((-3, -11/4, -5/2, -9/4, -2, 1), ()), ((-11/4, -9/4), (-3, -5/2, -5/2, 0 )), a**2*exp_polar(-2*I*pi)/(b**2*x**2))/(4*pi**(3/2)*b**6*sqrt(c))
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {-b^{2} c x^{2} + a^{2} c} x^{4}}{5 \, b^{2} c} - \frac {4 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{2} x^{2}}{15 \, b^{4} c} - \frac {8 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{4}}{15 \, b^{6} c} \] Input:
integrate(x^5/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="maxima")
Output:
-1/5*sqrt(-b^2*c*x^2 + a^2*c)*x^4/(b^2*c) - 4/15*sqrt(-b^2*c*x^2 + a^2*c)* a^2*x^2/(b^4*c) - 8/15*sqrt(-b^2*c*x^2 + a^2*c)*a^4/(b^6*c)
Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82 \[ \int \frac {x^5}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {{\left (\frac {15 \, a^{4}}{c} + {\left ({\left (b x + a\right )} {\left (3 \, {\left (b x + a\right )} {\left (\frac {b x + a}{c} - \frac {4 \, a}{c}\right )} + \frac {22 \, a^{2}}{c}\right )} - \frac {20 \, a^{3}}{c}\right )} {\left (b x + a\right )}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}}{15 \, b^{6}} \] Input:
integrate(x^5/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")
Output:
-1/15*(15*a^4/c + ((b*x + a)*(3*(b*x + a)*((b*x + a)/c - 4*a/c) + 22*a^2/c ) - 20*a^3/c)*(b*x + a))*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)/b^6
Time = 0.58 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.87 \[ \int \frac {x^5}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {a\,c-b\,c\,x}\,\left (\frac {8\,a^5}{15\,b^6\,c}+\frac {x^5}{5\,b\,c}+\frac {a\,x^4}{5\,b^2\,c}+\frac {8\,a^4\,x}{15\,b^5\,c}+\frac {4\,a^2\,x^3}{15\,b^3\,c}+\frac {4\,a^3\,x^2}{15\,b^4\,c}\right )}{\sqrt {a+b\,x}} \] Input:
int(x^5/((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)
Output:
-((a*c - b*c*x)^(1/2)*((8*a^5)/(15*b^6*c) + x^5/(5*b*c) + (a*x^4)/(5*b^2*c ) + (8*a^4*x)/(15*b^5*c) + (4*a^2*x^3)/(15*b^3*c) + (4*a^3*x^2)/(15*b^4*c) ))/(a + b*x)^(1/2)
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.44 \[ \int \frac {x^5}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {\sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, \left (-3 b^{4} x^{4}-4 a^{2} b^{2} x^{2}-8 a^{4}\right )}{15 b^{6} c} \] Input:
int(x^5/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x)
Output:
(sqrt(c)*sqrt(a + b*x)*sqrt(a - b*x)*( - 8*a**4 - 4*a**2*b**2*x**2 - 3*b** 4*x**4))/(15*b**6*c)