Integrand size = 26, antiderivative size = 112 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^8} \, dx=-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{7 a^2 c x^7}-\frac {4 b^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{35 a^4 c x^5}-\frac {8 b^4 (a+b x)^{3/2} (a c-b c x)^{3/2}}{105 a^6 c x^3} \] Output:
-1/7*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/a^2/c/x^7-4/35*b^2*(b*x+a)^(3/2)*(-b *c*x+a*c)^(3/2)/a^4/c/x^5-8/105*b^4*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/a^6/c /x^3
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^8} \, dx=-\frac {(c (a-b x))^{3/2} (a+b x)^{3/2} \left (15 a^4+12 a^2 b^2 x^2+8 b^4 x^4\right )}{105 a^6 c x^7} \] Input:
Integrate[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^8,x]
Output:
-1/105*((c*(a - b*x))^(3/2)*(a + b*x)^(3/2)*(15*a^4 + 12*a^2*b^2*x^2 + 8*b ^4*x^4))/(a^6*c*x^7)
Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.41, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {108, 25, 27, 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.
\(\displaystyle \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^8} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {1}{7} \int -\frac {b^2 c}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 x^7}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{7} \int \frac {b^2 c}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 x^7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{7} b^2 c \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 x^7}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {1}{7} b^2 c \left (-\frac {\int -\frac {4 b^2 c}{x^4 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{5 a^2 c}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 x^7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{7} b^2 c \left (\frac {4 b^2 \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 x^7}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {1}{7} b^2 c \left (\frac {4 b^2 \left (-\frac {\int -\frac {2 b^2 c}{x^2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2 c}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 x^7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{7} b^2 c \left (\frac {4 b^2 \left (\frac {2 b^2 \int \frac {1}{x^2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 x^7}\) |
\(\Big \downarrow \) 106 |
\(\displaystyle -\frac {1}{7} b^2 c \left (\frac {4 b^2 \left (-\frac {2 b^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^4 c x}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 x^7}\) |
Input:
Int[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^8,x]
Output:
-1/7*(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^7 - (b^2*c*(-1/5*(Sqrt[a + b*x]*S qrt[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)))/7
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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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 = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.52
method | result | size |
gosper | \(-\frac {\left (b x +a \right )^{\frac {3}{2}} \left (-b x +a \right ) \left (8 b^{4} x^{4}+12 a^{2} b^{2} x^{2}+15 a^{4}\right ) \sqrt {-b c x +a c}}{105 x^{7} a^{6}}\) | \(58\) |
orering | \(-\frac {\left (b x +a \right )^{\frac {3}{2}} \left (-b x +a \right ) \left (8 b^{4} x^{4}+12 a^{2} b^{2} x^{2}+15 a^{4}\right ) \sqrt {-b c x +a c}}{105 x^{7} a^{6}}\) | \(58\) |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (-b^{2} x^{2}+a^{2}\right ) \left (8 b^{4} x^{4}+12 a^{2} b^{2} x^{2}+15 a^{4}\right )}{105 a^{6} x^{7}}\) | \(63\) |
risch | \(-\frac {\sqrt {b x +a}\, c \left (-b x +a \right ) \left (-8 b^{6} x^{6}-4 a^{2} x^{4} b^{4}-3 a^{4} x^{2} b^{2}+15 a^{6}\right )}{105 \sqrt {-c \left (b x -a \right )}\, x^{7} a^{6}}\) | \(71\) |
Input:
int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^8,x,method=_RETURNVERBOSE)
Output:
-1/105*(b*x+a)^(3/2)*(-b*x+a)*(8*b^4*x^4+12*a^2*b^2*x^2+15*a^4)*(-b*c*x+a* c)^(1/2)/x^7/a^6
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^8} \, dx=\frac {{\left (8 \, b^{6} x^{6} + 4 \, a^{2} b^{4} x^{4} + 3 \, a^{4} b^{2} x^{2} - 15 \, a^{6}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{105 \, a^{6} x^{7}} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^8,x, algorithm="fricas")
Output:
1/105*(8*b^6*x^6 + 4*a^2*b^4*x^4 + 3*a^4*b^2*x^2 - 15*a^6)*sqrt(-b*c*x + a *c)*sqrt(b*x + a)/(a^6*x^7)
\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^8} \, dx=\int \frac {\sqrt {- c \left (- a + b x\right )} \sqrt {a + b x}}{x^{8}}\, dx \] Input:
integrate((b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2)/x**8,x)
Output:
Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x)/x**8, x)
Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^8} \, dx=-\frac {8 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} b^{4}}{105 \, a^{6} c x^{3}} - \frac {4 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} b^{2}}{35 \, a^{4} c x^{5}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}}}{7 \, a^{2} c x^{7}} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^8,x, algorithm="maxima")
Output:
-8/105*(-b^2*c*x^2 + a^2*c)^(3/2)*b^4/(a^6*c*x^3) - 4/35*(-b^2*c*x^2 + a^2 *c)^(3/2)*b^2/(a^4*c*x^5) - 1/7*(-b^2*c*x^2 + a^2*c)^(3/2)/(a^2*c*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (94) = 188\).
Time = 0.17 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^8} \, dx=\frac {2048 \, {\left (35 \, b^{8} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{16} \sqrt {-c} c^{6} - 70 \, a^{2} b^{8} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{12} \sqrt {-c} c^{8} + 168 \, a^{4} b^{8} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{8} \sqrt {-c} c^{10} + 224 \, a^{6} b^{8} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} \sqrt {-c} c^{12} + 128 \, a^{8} b^{8} \sqrt {-c} c^{14}\right )}}{105 \, {\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{7} b} \] Input:
integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^8,x, algorithm="giac")
Output:
2048/105*(35*b^8*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^16* sqrt(-c)*c^6 - 70*a^2*b^8*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2* a*c))^12*sqrt(-c)*c^8 + 168*a^4*b^8*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^8*sqrt(-c)*c^10 + 224*a^6*b^8*(sqrt(b*x + a)*sqrt(-c) - sq rt(-(b*x + a)*c + 2*a*c))^4*sqrt(-c)*c^12 + 128*a^8*b^8*sqrt(-c)*c^14)/((( sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4 + 4*a^2*c^2)^7*b)
Time = 0.48 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^8} \, dx=\frac {\sqrt {a\,c-b\,c\,x}\,\left (\frac {b^2\,x^2\,\sqrt {a+b\,x}}{35\,a^2}-\frac {\sqrt {a+b\,x}}{7}+\frac {4\,b^4\,x^4\,\sqrt {a+b\,x}}{105\,a^4}+\frac {8\,b^6\,x^6\,\sqrt {a+b\,x}}{105\,a^6}\right )}{x^7} \] Input:
int(((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2))/x^8,x)
Output:
((a*c - b*c*x)^(1/2)*((b^2*x^2*(a + b*x)^(1/2))/(35*a^2) - (a + b*x)^(1/2) /7 + (4*b^4*x^4*(a + b*x)^(1/2))/(105*a^4) + (8*b^6*x^6*(a + b*x)^(1/2))/( 105*a^6)))/x^7
Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^8} \, dx=\frac {\sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, \left (8 b^{6} x^{6}+4 a^{2} b^{4} x^{4}+3 a^{4} b^{2} x^{2}-15 a^{6}\right )}{105 a^{6} x^{7}} \] Input:
int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^8,x)
Output:
(sqrt(c)*sqrt(a + b*x)*sqrt(a - b*x)*( - 15*a**6 + 3*a**4*b**2*x**2 + 4*a* *2*b**4*x**4 + 8*b**6*x**6))/(105*a**6*x**7)