Integrand size = 26, antiderivative size = 112 \[ \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}-\frac {4 b^2 \sqrt {a+b x} \sqrt {a c-b c x}}{15 a^4 c x^3}-\frac {8 b^4 \sqrt {a+b x} \sqrt {a c-b c x}}{15 a^6 c x} \] Output:
-1/5*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a^2/c/x^5-4/15*b^2*(b*x+a)^(1/2)*(-b *c*x+a*c)^(1/2)/a^4/c/x^3-8/15*b^4*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a^6/c/ x
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {c (a-b x)} \sqrt {a+b x} \left (3 a^4+4 a^2 b^2 x^2+8 b^4 x^4\right )}{15 a^6 c x^5} \] Input:
Integrate[1/(x^6*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
Output:
-1/15*(Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*(3*a^4 + 4*a^2*b^2*x^2 + 8*b^4*x^4) )/(a^6*c*x^5)
Time = 0.19 (sec) , antiderivative size = 120, 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 = {114, 27, 114, 27, 106}
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.
\(\displaystyle \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\int -\frac {4 b^2 c}{x^4 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{5 a^2 c}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 b^2 \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {4 b^2 \left (-\frac {\int -\frac {2 b^2 c}{x^2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2 c}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 b^2 \left (\frac {2 b^2 \int \frac {1}{x^2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\) |
\(\Big \downarrow \) 106 |
\(\displaystyle \frac {4 b^2 \left (-\frac {2 b^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^4 c x}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\) |
Input:
Int[1/(x^6*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
Output:
-1/5*(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(a^2*c*x^5) + (4*b^2*(-1/3*(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(a^2*c*x^3) - (2*b^2*Sqrt[a + b*x]*Sqrt[a*c - b* c*x])/(3*a^4*c*x)))/(5*a^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_))^(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 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
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 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.48
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (8 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+3 a^{4}\right )}{15 c \,a^{6} x^{5}}\) | \(54\) |
gosper | \(-\frac {\sqrt {b x +a}\, \left (-b x +a \right ) \left (8 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+3 a^{4}\right )}{15 x^{5} a^{6} \sqrt {-b c x +a c}}\) | \(58\) |
orering | \(-\frac {\sqrt {b x +a}\, \left (-b x +a \right ) \left (8 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+3 a^{4}\right )}{15 x^{5} a^{6} \sqrt {-b c x +a c}}\) | \(58\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (-b x +a \right ) \left (8 b^{4} x^{4}+4 a^{2} b^{2} x^{2}+3 a^{4}\right )}{15 \sqrt {-c \left (b x -a \right )}\, a^{6} x^{5}}\) | \(59\) |
Input:
int(1/x^6/(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/a^6*(8*b^4*x^4+4*a^2*b^2*x^2+3*a^ 4)/x^5
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {{\left (8 \, b^{4} x^{4} + 4 \, a^{2} b^{2} x^{2} + 3 \, a^{4}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{15 \, a^{6} c x^{5}} \] Input:
integrate(1/x^6/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="fricas")
Output:
-1/15*(8*b^4*x^4 + 4*a^2*b^2*x^2 + 3*a^4)*sqrt(-b*c*x + a*c)*sqrt(b*x + a) /(a^6*c*x^5)
Result contains complex when optimal does not.
Time = 4.78 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {i b^{5} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {13}{4}, \frac {15}{4}, 1 & \frac {7}{2}, \frac {7}{2}, 4 \\3, \frac {13}{4}, \frac {7}{2}, \frac {15}{4}, 4 & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} a^{6} \sqrt {c}} + \frac {b^{5} {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {5}{2}, \frac {11}{4}, 3, \frac {13}{4}, \frac {7}{2}, 1 & \\\frac {11}{4}, \frac {13}{4} & \frac {5}{2}, 3, 3, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} a^{6} \sqrt {c}} \] Input:
integrate(1/x**6/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)
Output:
I*b**5*meijerg(((13/4, 15/4, 1), (7/2, 7/2, 4)), ((3, 13/4, 7/2, 15/4, 4), (0,)), a**2/(b**2*x**2))/(4*pi**(3/2)*a**6*sqrt(c)) + b**5*meijerg(((5/2, 11/4, 3, 13/4, 7/2, 1), ()), ((11/4, 13/4), (5/2, 3, 3, 0)), a**2*exp_pol ar(-2*I*pi)/(b**2*x**2))/(4*pi**(3/2)*a**6*sqrt(c))
Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {8 \, \sqrt {-b^{2} c x^{2} + a^{2} c} b^{4}}{15 \, a^{6} c x} - \frac {4 \, \sqrt {-b^{2} c x^{2} + a^{2} c} b^{2}}{15 \, a^{4} c x^{3}} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c}}{5 \, a^{2} c x^{5}} \] Input:
integrate(1/x^6/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="maxima")
Output:
-8/15*sqrt(-b^2*c*x^2 + a^2*c)*b^4/(a^6*c*x) - 4/15*sqrt(-b^2*c*x^2 + a^2* c)*b^2/(a^4*c*x^3) - 1/5*sqrt(-b^2*c*x^2 + a^2*c)/(a^2*c*x^5)
Time = 0.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {2048 \, {\left (5 \, {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{8} + 10 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} c^{2} + 8 \, a^{4} c^{4}\right )} b^{5} \sqrt {-c} c^{5}}{15 \, {\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{5}} \] Input:
integrate(1/x^6/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")
Output:
-2048/15*(5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^8 + 10*a ^2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*c^2 + 8*a^4*c^4 )*b^5*sqrt(-c)*c^5/((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^ 4 + 4*a^2*c^2)^5
Time = 0.67 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {a\,c-b\,c\,x}\,\left (\frac {1}{5\,a\,c}+\frac {4\,b^2\,x^2}{15\,a^3\,c}+\frac {4\,b^3\,x^3}{15\,a^4\,c}+\frac {8\,b^4\,x^4}{15\,a^5\,c}+\frac {8\,b^5\,x^5}{15\,a^6\,c}+\frac {b\,x}{5\,a^2\,c}\right )}{x^5\,\sqrt {a+b\,x}} \] Input:
int(1/(x^6*(a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)
Output:
-((a*c - b*c*x)^(1/2)*(1/(5*a*c) + (4*b^2*x^2)/(15*a^3*c) + (4*b^3*x^3)/(1 5*a^4*c) + (8*b^4*x^4)/(15*a^5*c) + (8*b^5*x^5)/(15*a^6*c) + (b*x)/(5*a^2* c)))/(x^5*(a + b*x)^(1/2))
Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.46 \[ \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {\sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, \left (-8 b^{4} x^{4}-4 a^{2} b^{2} x^{2}-3 a^{4}\right )}{15 a^{6} c \,x^{5}} \] Input:
int(1/x^6/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x)
Output:
(sqrt(c)*sqrt(a + b*x)*sqrt(a - b*x)*( - 3*a**4 - 4*a**2*b**2*x**2 - 8*b** 4*x**4))/(15*a**6*c*x**5)