Integrand size = 22, antiderivative size = 195 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^8} \, dx=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}-\frac {2 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{143 b^2 x^7}+\frac {16 c (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{1287 b^3 x^6}-\frac {32 c^2 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{3003 b^4 x^5}+\frac {128 c^3 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{15015 b^5 x^4}-\frac {256 c^4 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{45045 b^6 x^3} \]
-2/13*A*(c*x^2+b*x)^(3/2)/b/x^8-2/143*(-10*A*c+13*B*b)*(c*x^2+b*x)^(3/2)/b ^2/x^7+16/1287*c*(-10*A*c+13*B*b)*(c*x^2+b*x)^(3/2)/b^3/x^6-32/3003*c^2*(- 10*A*c+13*B*b)*(c*x^2+b*x)^(3/2)/b^4/x^5+128/15015*c^3*(-10*A*c+13*B*b)*(c *x^2+b*x)^(3/2)/b^5/x^4-256/45045*c^4*(-10*A*c+13*B*b)*(c*x^2+b*x)^(3/2)/b ^6/x^3
Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.63 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^8} \, dx=-\frac {2 (x (b+c x))^{3/2} \left (13 b B x \left (315 b^4-280 b^3 c x+240 b^2 c^2 x^2-192 b c^3 x^3+128 c^4 x^4\right )+5 A \left (693 b^5-630 b^4 c x+560 b^3 c^2 x^2-480 b^2 c^3 x^3+384 b c^4 x^4-256 c^5 x^5\right )\right )}{45045 b^6 x^8} \]
(-2*(x*(b + c*x))^(3/2)*(13*b*B*x*(315*b^4 - 280*b^3*c*x + 240*b^2*c^2*x^2 - 192*b*c^3*x^3 + 128*c^4*x^4) + 5*A*(693*b^5 - 630*b^4*c*x + 560*b^3*c^2 *x^2 - 480*b^2*c^3*x^3 + 384*b*c^4*x^4 - 256*c^5*x^5)))/(45045*b^6*x^8)
Time = 0.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1220, 1129, 1129, 1129, 1129, 1123}
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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^8} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(13 b B-10 A c) \int \frac {\sqrt {c x^2+b x}}{x^7}dx}{13 b}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(13 b B-10 A c) \left (-\frac {8 c \int \frac {\sqrt {c x^2+b x}}{x^6}dx}{11 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}\right )}{13 b}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(13 b B-10 A c) \left (-\frac {8 c \left (-\frac {2 c \int \frac {\sqrt {c x^2+b x}}{x^5}dx}{3 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}\right )}{11 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}\right )}{13 b}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(13 b B-10 A c) \left (-\frac {8 c \left (-\frac {2 c \left (-\frac {4 c \int \frac {\sqrt {c x^2+b x}}{x^4}dx}{7 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}\right )}{3 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}\right )}{11 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}\right )}{13 b}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(13 b B-10 A c) \left (-\frac {8 c \left (-\frac {2 c \left (-\frac {4 c \left (-\frac {2 c \int \frac {\sqrt {c x^2+b x}}{x^3}dx}{5 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 b x^4}\right )}{7 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}\right )}{3 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}\right )}{11 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}\right )}{13 b}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {\left (-\frac {8 c \left (-\frac {2 c \left (-\frac {4 c \left (\frac {4 c \left (b x+c x^2\right )^{3/2}}{15 b^2 x^3}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 b x^4}\right )}{7 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}\right )}{3 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}\right )}{11 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}\right ) (13 b B-10 A c)}{13 b}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}\) |
(-2*A*(b*x + c*x^2)^(3/2))/(13*b*x^8) + ((13*b*B - 10*A*c)*((-2*(b*x + c*x ^2)^(3/2))/(11*b*x^7) - (8*c*((-2*(b*x + c*x^2)^(3/2))/(9*b*x^6) - (2*c*(( -2*(b*x + c*x^2)^(3/2))/(7*b*x^5) - (4*c*((-2*(b*x + c*x^2)^(3/2))/(5*b*x^ 4) + (4*c*(b*x + c*x^2)^(3/2))/(15*b^2*x^3)))/(7*b)))/(3*b)))/(11*b)))/(13 *b)
3.1.78.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.54
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (\frac {13 B x}{11}+A \right ) b^{5}-\frac {10 c x \left (\frac {52 B x}{45}+A \right ) b^{4}}{11}+\frac {80 c^{2} x^{2} \left (\frac {39 B x}{35}+A \right ) b^{3}}{99}-\frac {160 c^{3} \left (\frac {26 B x}{25}+A \right ) x^{3} b^{2}}{231}+\frac {128 c^{4} \left (\frac {13 B x}{15}+A \right ) x^{4} b}{231}-\frac {256 A \,c^{5} x^{5}}{693}\right ) \left (c x +b \right ) \sqrt {x \left (c x +b \right )}}{13 x^{7} b^{6}}\) | \(105\) |
| gosper | \(-\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{5} x^{5}+1664 B b \,c^{4} x^{5}+1920 A b \,c^{4} x^{4}-2496 B \,b^{2} c^{3} x^{4}-2400 A \,b^{2} c^{3} x^{3}+3120 B \,b^{3} c^{2} x^{3}+2800 A \,b^{3} c^{2} x^{2}-3640 B \,b^{4} c \,x^{2}-3150 A \,b^{4} c x +4095 B \,b^{5} x +3465 A \,b^{5}\right ) \sqrt {c \,x^{2}+b x}}{45045 x^{7} b^{6}}\) | \(134\) |
| trager | \(-\frac {2 \left (-1280 A \,c^{6} x^{6}+1664 B b \,c^{5} x^{6}+640 A b \,c^{5} x^{5}-832 B \,b^{2} c^{4} x^{5}-480 A \,b^{2} c^{4} x^{4}+624 B \,b^{3} c^{3} x^{4}+400 A \,b^{3} c^{3} x^{3}-520 B \,b^{4} c^{2} x^{3}-350 A \,b^{4} c^{2} x^{2}+455 B \,b^{5} c \,x^{2}+315 A \,b^{5} c x +4095 b^{6} B x +3465 A \,b^{6}\right ) \sqrt {c \,x^{2}+b x}}{45045 x^{7} b^{6}}\) | \(153\) |
| risch | \(-\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{6} x^{6}+1664 B b \,c^{5} x^{6}+640 A b \,c^{5} x^{5}-832 B \,b^{2} c^{4} x^{5}-480 A \,b^{2} c^{4} x^{4}+624 B \,b^{3} c^{3} x^{4}+400 A \,b^{3} c^{3} x^{3}-520 B \,b^{4} c^{2} x^{3}-350 A \,b^{4} c^{2} x^{2}+455 B \,b^{5} c \,x^{2}+315 A \,b^{5} c x +4095 b^{6} B x +3465 A \,b^{6}\right )}{45045 x^{6} \sqrt {x \left (c x +b \right )}\, b^{6}}\) | \(156\) |
| default | \(B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{11 b \,x^{7}}-\frac {8 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{9 b \,x^{6}}-\frac {2 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{7 b \,x^{5}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 b \,x^{4}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15 b^{2} x^{3}}\right )}{7 b}\right )}{3 b}\right )}{11 b}\right )+A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{13 b \,x^{8}}-\frac {10 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{11 b \,x^{7}}-\frac {8 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{9 b \,x^{6}}-\frac {2 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{7 b \,x^{5}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 b \,x^{4}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15 b^{2} x^{3}}\right )}{7 b}\right )}{3 b}\right )}{11 b}\right )}{13 b}\right )\) | \(268\) |
-2/13*((13/11*B*x+A)*b^5-10/11*c*x*(52/45*B*x+A)*b^4+80/99*c^2*x^2*(39/35* B*x+A)*b^3-160/231*c^3*(26/25*B*x+A)*x^3*b^2+128/231*c^4*(13/15*B*x+A)*x^4 *b-256/693*A*c^5*x^5)*(c*x+b)*(x*(c*x+b))^(1/2)/x^7/b^6
Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.78 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^8} \, dx=-\frac {2 \, {\left (3465 \, A b^{6} + 128 \, {\left (13 \, B b c^{5} - 10 \, A c^{6}\right )} x^{6} - 64 \, {\left (13 \, B b^{2} c^{4} - 10 \, A b c^{5}\right )} x^{5} + 48 \, {\left (13 \, B b^{3} c^{3} - 10 \, A b^{2} c^{4}\right )} x^{4} - 40 \, {\left (13 \, B b^{4} c^{2} - 10 \, A b^{3} c^{3}\right )} x^{3} + 35 \, {\left (13 \, B b^{5} c - 10 \, A b^{4} c^{2}\right )} x^{2} + 315 \, {\left (13 \, B b^{6} + A b^{5} c\right )} x\right )} \sqrt {c x^{2} + b x}}{45045 \, b^{6} x^{7}} \]
-2/45045*(3465*A*b^6 + 128*(13*B*b*c^5 - 10*A*c^6)*x^6 - 64*(13*B*b^2*c^4 - 10*A*b*c^5)*x^5 + 48*(13*B*b^3*c^3 - 10*A*b^2*c^4)*x^4 - 40*(13*B*b^4*c^ 2 - 10*A*b^3*c^3)*x^3 + 35*(13*B*b^5*c - 10*A*b^4*c^2)*x^2 + 315*(13*B*b^6 + A*b^5*c)*x)*sqrt(c*x^2 + b*x)/(b^6*x^7)
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^8} \, dx=\int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{x^{8}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^8} \, dx=-\frac {256 \, \sqrt {c x^{2} + b x} B c^{5}}{3465 \, b^{5} x} + \frac {512 \, \sqrt {c x^{2} + b x} A c^{6}}{9009 \, b^{6} x} + \frac {128 \, \sqrt {c x^{2} + b x} B c^{4}}{3465 \, b^{4} x^{2}} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{5}}{9009 \, b^{5} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} B c^{3}}{1155 \, b^{3} x^{3}} + \frac {64 \, \sqrt {c x^{2} + b x} A c^{4}}{3003 \, b^{4} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} B c^{2}}{693 \, b^{2} x^{4}} - \frac {160 \, \sqrt {c x^{2} + b x} A c^{3}}{9009 \, b^{3} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} B c}{99 \, b x^{5}} + \frac {20 \, \sqrt {c x^{2} + b x} A c^{2}}{1287 \, b^{2} x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{11 \, x^{6}} - \frac {2 \, \sqrt {c x^{2} + b x} A c}{143 \, b x^{6}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{13 \, x^{7}} \]
-256/3465*sqrt(c*x^2 + b*x)*B*c^5/(b^5*x) + 512/9009*sqrt(c*x^2 + b*x)*A*c ^6/(b^6*x) + 128/3465*sqrt(c*x^2 + b*x)*B*c^4/(b^4*x^2) - 256/9009*sqrt(c* x^2 + b*x)*A*c^5/(b^5*x^2) - 32/1155*sqrt(c*x^2 + b*x)*B*c^3/(b^3*x^3) + 6 4/3003*sqrt(c*x^2 + b*x)*A*c^4/(b^4*x^3) + 16/693*sqrt(c*x^2 + b*x)*B*c^2/ (b^2*x^4) - 160/9009*sqrt(c*x^2 + b*x)*A*c^3/(b^3*x^4) - 2/99*sqrt(c*x^2 + b*x)*B*c/(b*x^5) + 20/1287*sqrt(c*x^2 + b*x)*A*c^2/(b^2*x^5) - 2/11*sqrt( c*x^2 + b*x)*B/x^6 - 2/143*sqrt(c*x^2 + b*x)*A*c/(b*x^6) - 2/13*sqrt(c*x^2 + b*x)*A/x^7
Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (171) = 342\).
Time = 0.27 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.21 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^8} \, dx=\frac {2 \, {\left (144144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B c^{3} + 480480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b c^{\frac {5}{2}} + 240240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A c^{\frac {7}{2}} + 669240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{2} c^{2} + 926640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b c^{3} + 495495 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{3} c^{\frac {3}{2}} + 1531530 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{2} c^{\frac {5}{2}} + 205205 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{4} c + 1401400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{3} c^{2} + 45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{5} \sqrt {c} + 765765 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{4} c^{\frac {3}{2}} + 4095 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{6} + 249795 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{5} c + 45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{6} \sqrt {c} + 3465 \, A b^{7}\right )}}{45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{13}} \]
2/45045*(144144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*c^3 + 480480*(sqrt(c)* x - sqrt(c*x^2 + b*x))^7*B*b*c^(5/2) + 240240*(sqrt(c)*x - sqrt(c*x^2 + b* x))^7*A*c^(7/2) + 669240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^2*c^2 + 926 640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b*c^3 + 495495*(sqrt(c)*x - sqrt(c *x^2 + b*x))^5*B*b^3*c^(3/2) + 1531530*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A *b^2*c^(5/2) + 205205*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^4*c + 1401400* (sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^3*c^2 + 45045*(sqrt(c)*x - sqrt(c*x^ 2 + b*x))^3*B*b^5*sqrt(c) + 765765*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^4 *c^(3/2) + 4095*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^6 + 249795*(sqrt(c)* x - sqrt(c*x^2 + b*x))^2*A*b^5*c + 45045*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A *b^6*sqrt(c) + 3465*A*b^7)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^13
Time = 11.56 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^8} \, dx=\frac {20\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{1287\,b^2\,x^5}-\frac {2\,B\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {2\,A\,c\,\sqrt {c\,x^2+b\,x}}{143\,b\,x^6}-\frac {2\,B\,c\,\sqrt {c\,x^2+b\,x}}{99\,b\,x^5}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{13\,x^7}-\frac {160\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{9009\,b^3\,x^4}+\frac {64\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{3003\,b^4\,x^3}-\frac {256\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{9009\,b^5\,x^2}+\frac {512\,A\,c^6\,\sqrt {c\,x^2+b\,x}}{9009\,b^6\,x}+\frac {16\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{693\,b^2\,x^4}-\frac {32\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{1155\,b^3\,x^3}+\frac {128\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{3465\,b^4\,x^2}-\frac {256\,B\,c^5\,\sqrt {c\,x^2+b\,x}}{3465\,b^5\,x} \]
(20*A*c^2*(b*x + c*x^2)^(1/2))/(1287*b^2*x^5) - (2*B*(b*x + c*x^2)^(1/2))/ (11*x^6) - (2*A*c*(b*x + c*x^2)^(1/2))/(143*b*x^6) - (2*B*c*(b*x + c*x^2)^ (1/2))/(99*b*x^5) - (2*A*(b*x + c*x^2)^(1/2))/(13*x^7) - (160*A*c^3*(b*x + c*x^2)^(1/2))/(9009*b^3*x^4) + (64*A*c^4*(b*x + c*x^2)^(1/2))/(3003*b^4*x ^3) - (256*A*c^5*(b*x + c*x^2)^(1/2))/(9009*b^5*x^2) + (512*A*c^6*(b*x + c *x^2)^(1/2))/(9009*b^6*x) + (16*B*c^2*(b*x + c*x^2)^(1/2))/(693*b^2*x^4) - (32*B*c^3*(b*x + c*x^2)^(1/2))/(1155*b^3*x^3) + (128*B*c^4*(b*x + c*x^2)^ (1/2))/(3465*b^4*x^2) - (256*B*c^5*(b*x + c*x^2)^(1/2))/(3465*b^5*x)