Integrand size = 26, antiderivative size = 150 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{12}} \, dx=-\frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{11 a^2 c x^{11}}-\frac {2 b^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}{33 a^4 c x^9}-\frac {8 b^4 (a+b x)^{5/2} (a c-b c x)^{5/2}}{231 a^6 c x^7}-\frac {16 b^6 (a+b x)^{5/2} (a c-b c x)^{5/2}}{1155 a^8 c x^5} \] Output:
-1/11*(b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/a^2/c/x^11-2/33*b^2*(b*x+a)^(5/2)*( -b*c*x+a*c)^(5/2)/a^4/c/x^9-8/231*b^4*(b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/a^6 /c/x^7-16/1155*b^6*(b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/a^8/c/x^5
Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.47 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{12}} \, dx=-\frac {(c (a-b x))^{5/2} (a+b x)^{5/2} \left (105 a^6+70 a^4 b^2 x^2+40 a^2 b^4 x^4+16 b^6 x^6\right )}{1155 a^8 c x^{11}} \] Input:
Integrate[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^12,x]
Output:
-1/1155*((c*(a - b*x))^(5/2)*(a + b*x)^(5/2)*(105*a^6 + 70*a^4*b^2*x^2 + 4 0*a^2*b^4*x^4 + 16*b^6*x^6))/(a^8*c*x^11)
Time = 0.29 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.61, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {108, 27, 108, 25, 27, 114, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{12}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{11} \int -\frac {3 b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}{x^{10}}dx-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{11} b^2 c \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{10}}dx-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle -\frac {3}{11} b^2 c \left (\frac {1}{9} \int -\frac {b^2 c}{x^8 \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{9 x^9}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3}{11} b^2 c \left (-\frac {1}{9} \int \frac {b^2 c}{x^8 \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{9 x^9}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{11} b^2 c \left (-\frac {1}{9} b^2 c \int \frac {1}{x^8 \sqrt {a+b x} \sqrt {a c-b c x}}dx-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{9 x^9}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {3}{11} b^2 c \left (-\frac {1}{9} b^2 c \left (-\frac {\int -\frac {6 b^2 c}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{7 a^2 c}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 a^2 c x^7}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{9 x^9}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{11} b^2 c \left (-\frac {1}{9} b^2 c \left (\frac {6 b^2 \int \frac {1}{x^6 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{7 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 a^2 c x^7}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{9 x^9}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {3}{11} b^2 c \left (-\frac {1}{9} b^2 c \left (\frac {6 b^2 \left (-\frac {\int -\frac {4 b^2 c}{x^4 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{5 a^2 c}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\right )}{7 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 a^2 c x^7}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{9 x^9}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{11} b^2 c \left (-\frac {1}{9} b^2 c \left (\frac {6 b^2 \left (\frac {4 b^2 \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\right )}{7 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 a^2 c x^7}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{9 x^9}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {3}{11} b^2 c \left (-\frac {1}{9} b^2 c \left (\frac {6 b^2 \left (\frac {4 b^2 \left (-\frac {\int -\frac {2 b^2 c}{x^2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2 c}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\right )}{7 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 a^2 c x^7}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{9 x^9}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{11} b^2 c \left (-\frac {1}{9} b^2 c \left (\frac {6 b^2 \left (\frac {4 b^2 \left (\frac {2 b^2 \int \frac {1}{x^2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\right )}{7 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 a^2 c x^7}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{9 x^9}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
| \(\Big \downarrow \) 106 |
| \(\displaystyle -\frac {3}{11} b^2 c \left (-\frac {1}{9} b^2 c \left (\frac {6 b^2 \left (\frac {4 b^2 \left (-\frac {2 b^2 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^4 c x}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{3 a^2 c x^3}\right )}{5 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{5 a^2 c x^5}\right )}{7 a^2}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{7 a^2 c x^7}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{9 x^9}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{11 x^{11}}\) |
Input:
Int[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^12,x]
Output:
-1/11*((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^11 - (3*b^2*c*(-1/9*(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^9 - (b^2*c*(-1/7*(Sqrt[a + b*x]*Sqrt[a*c - b*c *x])/(a^2*c*x^7) + (6*b^2*(-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)))/(7*a^2)))/9))/11
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 = 0.36 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.46
| method | result | size |
| gosper | \(-\frac {\left (b x +a \right )^{\frac {5}{2}} \left (-b x +a \right ) \left (16 b^{6} x^{6}+40 a^{2} x^{4} b^{4}+70 a^{4} x^{2} b^{2}+105 a^{6}\right ) \left (-b c x +a c \right )^{\frac {3}{2}}}{1155 x^{11} a^{8}}\) | \(69\) |
| orering | \(-\frac {\left (b x +a \right )^{\frac {5}{2}} \left (-b x +a \right ) \left (16 b^{6} x^{6}+40 a^{2} x^{4} b^{4}+70 a^{4} x^{2} b^{2}+105 a^{6}\right ) \left (-b c x +a c \right )^{\frac {3}{2}}}{1155 x^{11} a^{8}}\) | \(69\) |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, c \left (-b^{2} x^{2}+a^{2}\right ) \left (-16 b^{8} x^{8}-24 a^{2} x^{6} b^{6}-30 a^{4} x^{4} b^{4}-35 a^{6} x^{2} b^{2}+105 a^{8}\right )}{1155 x^{11} a^{8}}\) | \(86\) |
| risch | \(-\frac {\sqrt {b x +a}\, c^{2} \left (-b x +a \right ) \left (16 b^{10} x^{10}+8 a^{2} b^{8} x^{8}+6 a^{4} b^{6} x^{6}+5 a^{6} b^{4} x^{4}-140 a^{8} b^{2} x^{2}+105 a^{10}\right )}{1155 \sqrt {-c \left (b x -a \right )}\, x^{11} a^{8}}\) | \(95\) |
Input:
int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^12,x,method=_RETURNVERBOSE)
Output:
-1/1155*(b*x+a)^(5/2)*(-b*x+a)*(16*b^6*x^6+40*a^2*b^4*x^4+70*a^4*b^2*x^2+1 05*a^6)*(-b*c*x+a*c)^(3/2)/x^11/a^8
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.60 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{12}} \, dx=-\frac {{\left (16 \, b^{10} c x^{10} + 8 \, a^{2} b^{8} c x^{8} + 6 \, a^{4} b^{6} c x^{6} + 5 \, a^{6} b^{4} c x^{4} - 140 \, a^{8} b^{2} c x^{2} + 105 \, a^{10} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{1155 \, a^{8} x^{11}} \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^12,x, algorithm="fricas")
Output:
-1/1155*(16*b^10*c*x^10 + 8*a^2*b^8*c*x^8 + 6*a^4*b^6*c*x^6 + 5*a^6*b^4*c* x^4 - 140*a^8*b^2*c*x^2 + 105*a^10*c)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(a^ 8*x^11)
\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{12}} \, dx=\int \frac {\left (- c \left (- a + b x\right )\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}}{x^{12}}\, dx \] Input:
integrate((b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2)/x**12,x)
Output:
Integral((-c*(-a + b*x))**(3/2)*(a + b*x)**(3/2)/x**12, x)
Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{12}} \, dx=-\frac {16 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} b^{6}}{1155 \, a^{8} c x^{5}} - \frac {8 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} b^{4}}{231 \, a^{6} c x^{7}} - \frac {2 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} b^{2}}{33 \, a^{4} c x^{9}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}}}{11 \, a^{2} c x^{11}} \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^12,x, algorithm="maxima")
Output:
-16/1155*(-b^2*c*x^2 + a^2*c)^(5/2)*b^6/(a^8*c*x^5) - 8/231*(-b^2*c*x^2 + a^2*c)^(5/2)*b^4/(a^6*c*x^7) - 2/33*(-b^2*c*x^2 + a^2*c)^(5/2)*b^2/(a^4*c* x^9) - 1/11*(-b^2*c*x^2 + a^2*c)^(5/2)/(a^2*c*x^11)
Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (126) = 252\).
Time = 0.27 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.71 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{12}} \, dx=-\frac {8192 \, {\left (1155 \, b^{12} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{28} \sqrt {-c} c^{9} - 8316 \, a^{2} b^{12} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{24} \sqrt {-c} c^{11} + 40656 \, a^{4} b^{12} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{20} \sqrt {-c} c^{13} - 52800 \, a^{6} b^{12} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{16} \sqrt {-c} c^{15} + 42240 \, a^{8} b^{12} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{12} \sqrt {-c} c^{17} + 56320 \, a^{10} b^{12} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{8} \sqrt {-c} c^{19} + 45056 \, a^{12} b^{12} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} \sqrt {-c} c^{21} + 16384 \, a^{14} b^{12} \sqrt {-c} c^{23}\right )}}{1155 \, {\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{11} b} \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^12,x, algorithm="giac")
Output:
-8192/1155*(1155*b^12*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c) )^28*sqrt(-c)*c^9 - 8316*a^2*b^12*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a )*c + 2*a*c))^24*sqrt(-c)*c^11 + 40656*a^4*b^12*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^20*sqrt(-c)*c^13 - 52800*a^6*b^12*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^16*sqrt(-c)*c^15 + 42240*a^8*b^1 2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^12*sqrt(-c)*c^17 + 56320*a^10*b^12*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^8*s qrt(-c)*c^19 + 45056*a^12*b^12*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*sqrt(-c)*c^21 + 16384*a^14*b^12*sqrt(-c)*c^23)/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4 + 4*a^2*c^2)^11*b)
Time = 0.55 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{12}} \, dx=-\frac {\sqrt {a\,c-b\,c\,x}\,\left (\frac {a^2\,c\,\sqrt {a+b\,x}}{11}-\frac {4\,b^2\,c\,x^2\,\sqrt {a+b\,x}}{33}+\frac {b^4\,c\,x^4\,\sqrt {a+b\,x}}{231\,a^2}+\frac {2\,b^6\,c\,x^6\,\sqrt {a+b\,x}}{385\,a^4}+\frac {8\,b^8\,c\,x^8\,\sqrt {a+b\,x}}{1155\,a^6}+\frac {16\,b^{10}\,c\,x^{10}\,\sqrt {a+b\,x}}{1155\,a^8}\right )}{x^{11}} \] Input:
int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x^12,x)
Output:
-((a*c - b*c*x)^(1/2)*((a^2*c*(a + b*x)^(1/2))/11 - (4*b^2*c*x^2*(a + b*x) ^(1/2))/33 + (b^4*c*x^4*(a + b*x)^(1/2))/(231*a^2) + (2*b^6*c*x^6*(a + b*x )^(1/2))/(385*a^4) + (8*b^8*c*x^8*(a + b*x)^(1/2))/(1155*a^6) + (16*b^10*c *x^10*(a + b*x)^(1/2))/(1155*a^8)))/x^11
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.55 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^{12}} \, dx=\frac {\sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, c \left (-16 b^{10} x^{10}-8 a^{2} b^{8} x^{8}-6 a^{4} b^{6} x^{6}-5 a^{6} b^{4} x^{4}+140 a^{8} b^{2} x^{2}-105 a^{10}\right )}{1155 a^{8} x^{11}} \] Input:
int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^12,x)
Output:
(sqrt(c)*sqrt(a + b*x)*sqrt(a - b*x)*c*( - 105*a**10 + 140*a**8*b**2*x**2 - 5*a**6*b**4*x**4 - 6*a**4*b**6*x**6 - 8*a**2*b**8*x**8 - 16*b**10*x**10) )/(1155*a**8*x**11)