Integrand size = 26, antiderivative size = 74 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^8} \, dx=-\frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{7 a^2 c x^7}-\frac {2 b^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}{35 a^4 c x^5} \] Output:
-1/7*(b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/a^2/c/x^7-2/35*b^2*(b*x+a)^(5/2)*(-b *c*x+a*c)^(5/2)/a^4/c/x^5
Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.65 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^8} \, dx=-\frac {(c (a-b x))^{5/2} (a+b x)^{5/2} \left (5 a^2+2 b^2 x^2\right )}{35 a^4 c x^7} \] Input:
Integrate[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^8,x]
Output:
-1/35*((c*(a - b*x))^(5/2)*(a + b*x)^(5/2)*(5*a^2 + 2*b^2*x^2))/(a^4*c*x^7 )
Leaf count is larger than twice the leaf count of optimal. \(150\) vs. \(2(74)=148\).
Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {108, 27, 108, 25, 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.
\(\displaystyle \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^8} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {1}{7} \int -\frac {3 b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}{x^6}dx-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {3}{7} b^2 c \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^6}dx-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^7}\) |
\(\Big \downarrow \) 108 |
\(\displaystyle -\frac {3}{7} b^2 c \left (\frac {1}{5} \int -\frac {b^2 c}{x^4 \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 x^5}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^7}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {3}{7} b^2 c \left (-\frac {1}{5} \int \frac {b^2 c}{x^4 \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 x^5}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {3}{7} b^2 c \left (-\frac {1}{5} b^2 c \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 x^5}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^7}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {3}{7} b^2 c \left (-\frac {1}{5} b^2 c \left (-\frac {\int -\frac {2 b^2 c}{x^2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2 c}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 x^5}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {3}{7} b^2 c \left (-\frac {1}{5} b^2 c \left (\frac {2 b^2 \int \frac {1}{x^2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 x^5}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^7}\) |
\(\Big \downarrow \) 106 |
\(\displaystyle -\frac {3}{7} b^2 c \left (-\frac {1}{5} b^2 c \left (-\frac {2 b^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^4 c x}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 x^5}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{7 x^7}\) |
Input:
Int[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^8,x]
Output:
-1/7*((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^7 - (3*b^2*c*(-1/5*(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^5 - (b^2*c*(-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)) /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 = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(-\frac {\left (b x +a \right )^{\frac {5}{2}} \left (-b x +a \right ) \left (2 b^{2} x^{2}+5 a^{2}\right ) \left (-b c x +a c \right )^{\frac {3}{2}}}{35 x^{7} a^{4}}\) | \(47\) |
orering | \(-\frac {\left (b x +a \right )^{\frac {5}{2}} \left (-b x +a \right ) \left (2 b^{2} x^{2}+5 a^{2}\right ) \left (-b c x +a c \right )^{\frac {3}{2}}}{35 x^{7} a^{4}}\) | \(47\) |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, c \left (-b^{2} x^{2}+a^{2}\right ) \left (-2 b^{4} x^{4}-3 a^{2} b^{2} x^{2}+5 a^{4}\right )}{35 x^{7} a^{4}}\) | \(64\) |
risch | \(-\frac {\sqrt {b x +a}\, c^{2} \left (-b x +a \right ) \left (2 b^{6} x^{6}+a^{2} x^{4} b^{4}-8 a^{4} x^{2} b^{2}+5 a^{6}\right )}{35 \sqrt {-c \left (b x -a \right )}\, x^{7} a^{4}}\) | \(72\) |
Input:
int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^8,x,method=_RETURNVERBOSE)
Output:
-1/35*(b*x+a)^(5/2)*(-b*x+a)*(2*b^2*x^2+5*a^2)*(-b*c*x+a*c)^(3/2)/x^7/a^4
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^8} \, dx=-\frac {{\left (2 \, b^{6} c x^{6} + a^{2} b^{4} c x^{4} - 8 \, a^{4} b^{2} c x^{2} + 5 \, a^{6} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{35 \, a^{4} x^{7}} \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^8,x, algorithm="fricas")
Output:
-1/35*(2*b^6*c*x^6 + a^2*b^4*c*x^4 - 8*a^4*b^2*c*x^2 + 5*a^6*c)*sqrt(-b*c* x + a*c)*sqrt(b*x + a)/(a^4*x^7)
\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^8} \, dx=\int \frac {\left (- c \left (- a + b x\right )\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}}{x^{8}}\, dx \] Input:
integrate((b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2)/x**8,x)
Output:
Integral((-c*(-a + b*x))**(3/2)*(a + b*x)**(3/2)/x**8, x)
Time = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^8} \, dx=-\frac {2 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} b^{2}}{35 \, a^{4} c x^{5}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}}}{7 \, a^{2} c x^{7}} \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^8,x, algorithm="maxima")
Output:
-2/35*(-b^2*c*x^2 + a^2*c)^(5/2)*b^2/(a^4*c*x^5) - 1/7*(-b^2*c*x^2 + a^2*c )^(5/2)/(a^2*c*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (62) = 124\).
Time = 0.24 (sec) , antiderivative size = 308, normalized size of antiderivative = 4.16 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^8} \, dx=-\frac {64 \, {\left (35 \, b^{8} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{20} \sqrt {-c} c^{5} - 140 \, a^{2} b^{8} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{16} \sqrt {-c} c^{7} + 1120 \, a^{4} b^{8} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{12} \sqrt {-c} c^{9} - 896 \, a^{6} b^{8} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{8} \sqrt {-c} c^{11} + 1792 \, a^{8} b^{8} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} \sqrt {-c} c^{13} + 1024 \, a^{10} b^{8} \sqrt {-c} c^{15}\right )}}{35 \, {\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{7} b} \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^8,x, algorithm="giac")
Output:
-64/35*(35*b^8*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^20*sq rt(-c)*c^5 - 140*a^2*b^8*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a *c))^16*sqrt(-c)*c^7 + 1120*a^4*b^8*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^12*sqrt(-c)*c^9 - 896*a^6*b^8*(sqrt(b*x + a)*sqrt(-c) - sq rt(-(b*x + a)*c + 2*a*c))^8*sqrt(-c)*c^11 + 1792*a^8*b^8*(sqrt(b*x + a)*sq rt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*sqrt(-c)*c^13 + 1024*a^10*b^8*sqrt( -c)*c^15)/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4 + 4*a^ 2*c^2)^7*b)
Time = 0.53 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^8} \, dx=-\frac {\sqrt {a\,c-b\,c\,x}\,\left (\frac {a^2\,c\,\sqrt {a+b\,x}}{7}-\frac {8\,b^2\,c\,x^2\,\sqrt {a+b\,x}}{35}+\frac {b^4\,c\,x^4\,\sqrt {a+b\,x}}{35\,a^2}+\frac {2\,b^6\,c\,x^6\,\sqrt {a+b\,x}}{35\,a^4}\right )}{x^7} \] Input:
int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x^8,x)
Output:
-((a*c - b*c*x)^(1/2)*((a^2*c*(a + b*x)^(1/2))/7 - (8*b^2*c*x^2*(a + b*x)^ (1/2))/35 + (b^4*c*x^4*(a + b*x)^(1/2))/(35*a^2) + (2*b^6*c*x^6*(a + b*x)^ (1/2))/(35*a^4)))/x^7
Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^8} \, dx=\frac {\sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, c \left (-2 b^{6} x^{6}-a^{2} b^{4} x^{4}+8 a^{4} b^{2} x^{2}-5 a^{6}\right )}{35 a^{4} x^{7}} \] Input:
int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^8,x)
Output:
(sqrt(c)*sqrt(a + b*x)*sqrt(a - b*x)*c*( - 5*a**6 + 8*a**4*b**2*x**2 - a** 2*b**4*x**4 - 2*b**6*x**6))/(35*a**4*x**7)