Integrand size = 22, antiderivative size = 195 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}-\frac {2 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{255 b^2 x^{11}}+\frac {16 c (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{3315 b^3 x^{10}}-\frac {32 c^2 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{12155 b^4 x^9}+\frac {128 c^3 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{109395 b^5 x^8}-\frac {256 c^4 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{765765 b^6 x^7} \]
-2/17*A*(c*x^2+b*x)^(7/2)/b/x^12-2/255*(-10*A*c+17*B*b)*(c*x^2+b*x)^(7/2)/ b^2/x^11+16/3315*c*(-10*A*c+17*B*b)*(c*x^2+b*x)^(7/2)/b^3/x^10-32/12155*c^ 2*(-10*A*c+17*B*b)*(c*x^2+b*x)^(7/2)/b^4/x^9+128/109395*c^3*(-10*A*c+17*B* b)*(c*x^2+b*x)^(7/2)/b^5/x^8-256/765765*c^4*(-10*A*c+17*B*b)*(c*x^2+b*x)^( 7/2)/b^6/x^7
Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.67 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {2 (b+c x)^3 \sqrt {x (b+c x)} \left (17 b B x \left (3003 b^4-1848 b^3 c x+1008 b^2 c^2 x^2-448 b c^3 x^3+128 c^4 x^4\right )+5 A \left (9009 b^5-6006 b^4 c x+3696 b^3 c^2 x^2-2016 b^2 c^3 x^3+896 b c^4 x^4-256 c^5 x^5\right )\right )}{765765 b^6 x^9} \]
(-2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(17*b*B*x*(3003*b^4 - 1848*b^3*c*x + 100 8*b^2*c^2*x^2 - 448*b*c^3*x^3 + 128*c^4*x^4) + 5*A*(9009*b^5 - 6006*b^4*c* x + 3696*b^3*c^2*x^2 - 2016*b^2*c^3*x^3 + 896*b*c^4*x^4 - 256*c^5*x^5)))/( 765765*b^6*x^9)
Time = 0.35 (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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{12}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(17 b B-10 A c) \int \frac {\left (c x^2+b x\right )^{5/2}}{x^{11}}dx}{17 b}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(17 b B-10 A c) \left (-\frac {8 c \int \frac {\left (c x^2+b x\right )^{5/2}}{x^{10}}dx}{15 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}\right )}{17 b}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(17 b B-10 A c) \left (-\frac {8 c \left (-\frac {6 c \int \frac {\left (c x^2+b x\right )^{5/2}}{x^9}dx}{13 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}\right )}{15 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}\right )}{17 b}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(17 b B-10 A c) \left (-\frac {8 c \left (-\frac {6 c \left (-\frac {4 c \int \frac {\left (c x^2+b x\right )^{5/2}}{x^8}dx}{11 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{11 b x^9}\right )}{13 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}\right )}{15 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}\right )}{17 b}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(17 b B-10 A c) \left (-\frac {8 c \left (-\frac {6 c \left (-\frac {4 c \left (-\frac {2 c \int \frac {\left (c x^2+b x\right )^{5/2}}{x^7}dx}{9 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{9 b x^8}\right )}{11 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{11 b x^9}\right )}{13 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}\right )}{15 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}\right )}{17 b}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {\left (-\frac {8 c \left (-\frac {6 c \left (-\frac {4 c \left (\frac {4 c \left (b x+c x^2\right )^{7/2}}{63 b^2 x^7}-\frac {2 \left (b x+c x^2\right )^{7/2}}{9 b x^8}\right )}{11 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{11 b x^9}\right )}{13 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}\right )}{15 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}\right ) (17 b B-10 A c)}{17 b}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}\) |
(-2*A*(b*x + c*x^2)^(7/2))/(17*b*x^12) + ((17*b*B - 10*A*c)*((-2*(b*x + c* x^2)^(7/2))/(15*b*x^11) - (8*c*((-2*(b*x + c*x^2)^(7/2))/(13*b*x^10) - (6* c*((-2*(b*x + c*x^2)^(7/2))/(11*b*x^9) - (4*c*((-2*(b*x + c*x^2)^(7/2))/(9 *b*x^8) + (4*c*(b*x + c*x^2)^(7/2))/(63*b^2*x^7)))/(11*b)))/(13*b)))/(15*b )))/(17*b)
3.2.8.3.1 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 = 0.65 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.55
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (\frac {17 B x}{15}+A \right ) b^{5}-\frac {2 c \left (\frac {68 B x}{65}+A \right ) x \,b^{4}}{3}+\frac {16 c^{2} x^{2} \left (\frac {51 B x}{55}+A \right ) b^{3}}{39}-\frac {32 c^{3} \left (\frac {34 B x}{45}+A \right ) x^{3} b^{2}}{143}+\frac {128 c^{4} \left (\frac {17 B x}{35}+A \right ) x^{4} b}{1287}-\frac {256 A \,c^{5} x^{5}}{9009}\right ) \left (c x +b \right )^{3} \sqrt {x \left (c x +b \right )}}{17 x^{9} b^{6}}\) | \(107\) |
| gosper | \(-\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{5} x^{5}+2176 B b \,c^{4} x^{5}+4480 A b \,c^{4} x^{4}-7616 B \,b^{2} c^{3} x^{4}-10080 A \,b^{2} c^{3} x^{3}+17136 B \,b^{3} c^{2} x^{3}+18480 A \,b^{3} c^{2} x^{2}-31416 B \,b^{4} c \,x^{2}-30030 A \,b^{4} c x +51051 B \,b^{5} x +45045 A \,b^{5}\right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{765765 x^{11} b^{6}}\) | \(134\) |
| trager | \(-\frac {2 \left (-1280 A \,c^{8} x^{8}+2176 B b \,c^{7} x^{8}+640 A b \,c^{7} x^{7}-1088 B \,b^{2} c^{6} x^{7}-480 A \,b^{2} c^{6} x^{6}+816 B \,b^{3} c^{5} x^{6}+400 A \,b^{3} c^{5} x^{5}-680 B \,b^{4} c^{4} x^{5}-350 A \,b^{4} c^{4} x^{4}+595 B \,b^{5} c^{3} x^{4}+315 A \,b^{5} c^{3} x^{3}+76041 B \,b^{6} c^{2} x^{3}+63525 A \,b^{6} c^{2} x^{2}+121737 B \,b^{7} c \,x^{2}+105105 A \,b^{7} c x +51051 B \,b^{8} x +45045 A \,b^{8}\right ) \sqrt {c \,x^{2}+b x}}{765765 b^{6} x^{9}}\) | \(201\) |
| risch | \(-\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{8} x^{8}+2176 B b \,c^{7} x^{8}+640 A b \,c^{7} x^{7}-1088 B \,b^{2} c^{6} x^{7}-480 A \,b^{2} c^{6} x^{6}+816 B \,b^{3} c^{5} x^{6}+400 A \,b^{3} c^{5} x^{5}-680 B \,b^{4} c^{4} x^{5}-350 A \,b^{4} c^{4} x^{4}+595 B \,b^{5} c^{3} x^{4}+315 A \,b^{5} c^{3} x^{3}+76041 B \,b^{6} c^{2} x^{3}+63525 A \,b^{6} c^{2} x^{2}+121737 B \,b^{7} c \,x^{2}+105105 A \,b^{7} c x +51051 B \,b^{8} x +45045 A \,b^{8}\right )}{765765 x^{8} \sqrt {x \left (c x +b \right )}\, b^{6}}\) | \(204\) |
| default | \(B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{15 b \,x^{11}}-\frac {8 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{13 b \,x^{10}}-\frac {6 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{11 b \,x^{9}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 b \,x^{8}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{63 b^{2} x^{7}}\right )}{11 b}\right )}{13 b}\right )}{15 b}\right )+A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{17 b \,x^{12}}-\frac {10 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{15 b \,x^{11}}-\frac {8 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{13 b \,x^{10}}-\frac {6 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{11 b \,x^{9}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 b \,x^{8}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{63 b^{2} x^{7}}\right )}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}\right )\) | \(268\) |
-2/17*((17/15*B*x+A)*b^5-2/3*c*(68/65*B*x+A)*x*b^4+16/39*c^2*x^2*(51/55*B* x+A)*b^3-32/143*c^3*(34/45*B*x+A)*x^3*b^2+128/1287*c^4*(17/35*B*x+A)*x^4*b -256/9009*A*c^5*x^5)*(c*x+b)^3*(x*(c*x+b))^(1/2)/x^9/b^6
Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {2 \, {\left (45045 \, A b^{8} + 128 \, {\left (17 \, B b c^{7} - 10 \, A c^{8}\right )} x^{8} - 64 \, {\left (17 \, B b^{2} c^{6} - 10 \, A b c^{7}\right )} x^{7} + 48 \, {\left (17 \, B b^{3} c^{5} - 10 \, A b^{2} c^{6}\right )} x^{6} - 40 \, {\left (17 \, B b^{4} c^{4} - 10 \, A b^{3} c^{5}\right )} x^{5} + 35 \, {\left (17 \, B b^{5} c^{3} - 10 \, A b^{4} c^{4}\right )} x^{4} + 63 \, {\left (1207 \, B b^{6} c^{2} + 5 \, A b^{5} c^{3}\right )} x^{3} + 231 \, {\left (527 \, B b^{7} c + 275 \, A b^{6} c^{2}\right )} x^{2} + 3003 \, {\left (17 \, B b^{8} + 35 \, A b^{7} c\right )} x\right )} \sqrt {c x^{2} + b x}}{765765 \, b^{6} x^{9}} \]
-2/765765*(45045*A*b^8 + 128*(17*B*b*c^7 - 10*A*c^8)*x^8 - 64*(17*B*b^2*c^ 6 - 10*A*b*c^7)*x^7 + 48*(17*B*b^3*c^5 - 10*A*b^2*c^6)*x^6 - 40*(17*B*b^4* c^4 - 10*A*b^3*c^5)*x^5 + 35*(17*B*b^5*c^3 - 10*A*b^4*c^4)*x^4 + 63*(1207* B*b^6*c^2 + 5*A*b^5*c^3)*x^3 + 231*(527*B*b^7*c + 275*A*b^6*c^2)*x^2 + 300 3*(17*B*b^8 + 35*A*b^7*c)*x)*sqrt(c*x^2 + b*x)/(b^6*x^9)
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{12}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x^{12}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (171) = 342\).
Time = 0.20 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.27 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {256 \, \sqrt {c x^{2} + b x} B c^{7}}{45045 \, b^{5} x} + \frac {512 \, \sqrt {c x^{2} + b x} A c^{8}}{153153 \, b^{6} x} + \frac {128 \, \sqrt {c x^{2} + b x} B c^{6}}{45045 \, b^{4} x^{2}} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{7}}{153153 \, b^{5} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} B c^{5}}{15015 \, b^{3} x^{3}} + \frac {64 \, \sqrt {c x^{2} + b x} A c^{6}}{51051 \, b^{4} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} B c^{4}}{9009 \, b^{2} x^{4}} - \frac {160 \, \sqrt {c x^{2} + b x} A c^{5}}{153153 \, b^{3} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} B c^{3}}{1287 \, b x^{5}} + \frac {20 \, \sqrt {c x^{2} + b x} A c^{4}}{21879 \, b^{2} x^{5}} + \frac {\sqrt {c x^{2} + b x} B c^{2}}{715 \, x^{6}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{3}}{2431 \, b x^{6}} - \frac {\sqrt {c x^{2} + b x} B b c}{780 \, x^{7}} + \frac {\sqrt {c x^{2} + b x} A c^{2}}{1326 \, x^{7}} - \frac {\sqrt {c x^{2} + b x} B b^{2}}{60 \, x^{8}} - \frac {\sqrt {c x^{2} + b x} A b c}{1428 \, x^{8}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{12 \, x^{9}} - \frac {5 \, \sqrt {c x^{2} + b x} A b^{2}}{476 \, x^{9}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{5 \, x^{10}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{84 \, x^{10}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{6 \, x^{11}} \]
-256/45045*sqrt(c*x^2 + b*x)*B*c^7/(b^5*x) + 512/153153*sqrt(c*x^2 + b*x)* A*c^8/(b^6*x) + 128/45045*sqrt(c*x^2 + b*x)*B*c^6/(b^4*x^2) - 256/153153*s qrt(c*x^2 + b*x)*A*c^7/(b^5*x^2) - 32/15015*sqrt(c*x^2 + b*x)*B*c^5/(b^3*x ^3) + 64/51051*sqrt(c*x^2 + b*x)*A*c^6/(b^4*x^3) + 16/9009*sqrt(c*x^2 + b* x)*B*c^4/(b^2*x^4) - 160/153153*sqrt(c*x^2 + b*x)*A*c^5/(b^3*x^4) - 2/1287 *sqrt(c*x^2 + b*x)*B*c^3/(b*x^5) + 20/21879*sqrt(c*x^2 + b*x)*A*c^4/(b^2*x ^5) + 1/715*sqrt(c*x^2 + b*x)*B*c^2/x^6 - 2/2431*sqrt(c*x^2 + b*x)*A*c^3/( b*x^6) - 1/780*sqrt(c*x^2 + b*x)*B*b*c/x^7 + 1/1326*sqrt(c*x^2 + b*x)*A*c^ 2/x^7 - 1/60*sqrt(c*x^2 + b*x)*B*b^2/x^8 - 1/1428*sqrt(c*x^2 + b*x)*A*b*c/ x^8 + 1/12*(c*x^2 + b*x)^(3/2)*B*b/x^9 - 5/476*sqrt(c*x^2 + b*x)*A*b^2/x^9 - 1/5*(c*x^2 + b*x)^(5/2)*B/x^10 + 5/84*(c*x^2 + b*x)^(3/2)*A*b/x^10 - 1/ 6*(c*x^2 + b*x)^(5/2)*A/x^11
Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (171) = 342\).
Time = 0.29 (sec) , antiderivative size = 671, normalized size of antiderivative = 3.44 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{12}} \, dx=\frac {2 \, {\left (2450448 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{12} B c^{5} + 16336320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11} B b c^{\frac {9}{2}} + 4084080 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11} A c^{\frac {11}{2}} + 49884120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{10} B b^{2} c^{4} + 29755440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{10} A b c^{5} + 91126035 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} B b^{3} c^{\frac {7}{2}} + 99549450 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} A b^{2} c^{\frac {9}{2}} + 109674565 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B b^{4} c^{3} + 200800600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} A b^{3} c^{4} + 90513423 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b^{5} c^{\frac {5}{2}} + 270315045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A b^{4} c^{\frac {7}{2}} + 51723945 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{6} c^{2} + 254303595 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b^{5} c^{3} + 20165145 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{7} c^{\frac {3}{2}} + 170255085 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{6} c^{\frac {5}{2}} + 5124735 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{8} c + 80994375 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{7} c^{2} + 765765 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{9} \sqrt {c} + 26801775 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{8} c^{\frac {3}{2}} + 51051 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{10} + 5870865 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{9} c + 765765 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{10} \sqrt {c} + 45045 \, A b^{11}\right )}}{765765 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{17}} \]
2/765765*(2450448*(sqrt(c)*x - sqrt(c*x^2 + b*x))^12*B*c^5 + 16336320*(sqr t(c)*x - sqrt(c*x^2 + b*x))^11*B*b*c^(9/2) + 4084080*(sqrt(c)*x - sqrt(c*x ^2 + b*x))^11*A*c^(11/2) + 49884120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b ^2*c^4 + 29755440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b*c^5 + 91126035*(s qrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^3*c^(7/2) + 99549450*(sqrt(c)*x - sqrt (c*x^2 + b*x))^9*A*b^2*c^(9/2) + 109674565*(sqrt(c)*x - sqrt(c*x^2 + b*x)) ^8*B*b^4*c^3 + 200800600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^3*c^4 + 905 13423*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^5*c^(5/2) + 270315045*(sqrt(c) *x - sqrt(c*x^2 + b*x))^7*A*b^4*c^(7/2) + 51723945*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^6*c^2 + 254303595*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^5*c ^3 + 20165145*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^7*c^(3/2) + 170255085* (sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^6*c^(5/2) + 5124735*(sqrt(c)*x - sqr t(c*x^2 + b*x))^4*B*b^8*c + 80994375*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b ^7*c^2 + 765765*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^9*sqrt(c) + 26801775 *(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^8*c^(3/2) + 51051*(sqrt(c)*x - sqrt (c*x^2 + b*x))^2*B*b^10 + 5870865*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^9* c + 765765*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^10*sqrt(c) + 45045*A*b^11)/ (sqrt(c)*x - sqrt(c*x^2 + b*x))^17
Time = 14.68 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.91 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{12}} \, dx=\frac {20\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{21879\,b^2\,x^5}-\frac {110\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{663\,x^7}-\frac {2\,B\,b^2\,\sqrt {c\,x^2+b\,x}}{15\,x^8}-\frac {142\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{715\,x^6}-\frac {2\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{2431\,b\,x^6}-\frac {2\,A\,b^2\,\sqrt {c\,x^2+b\,x}}{17\,x^9}-\frac {160\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{153153\,b^3\,x^4}+\frac {64\,A\,c^6\,\sqrt {c\,x^2+b\,x}}{51051\,b^4\,x^3}-\frac {256\,A\,c^7\,\sqrt {c\,x^2+b\,x}}{153153\,b^5\,x^2}+\frac {512\,A\,c^8\,\sqrt {c\,x^2+b\,x}}{153153\,b^6\,x}-\frac {2\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{1287\,b\,x^5}+\frac {16\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{9009\,b^2\,x^4}-\frac {32\,B\,c^5\,\sqrt {c\,x^2+b\,x}}{15015\,b^3\,x^3}+\frac {128\,B\,c^6\,\sqrt {c\,x^2+b\,x}}{45045\,b^4\,x^2}-\frac {256\,B\,c^7\,\sqrt {c\,x^2+b\,x}}{45045\,b^5\,x}-\frac {14\,A\,b\,c\,\sqrt {c\,x^2+b\,x}}{51\,x^8}-\frac {62\,B\,b\,c\,\sqrt {c\,x^2+b\,x}}{195\,x^7} \]
(20*A*c^4*(b*x + c*x^2)^(1/2))/(21879*b^2*x^5) - (110*A*c^2*(b*x + c*x^2)^ (1/2))/(663*x^7) - (2*B*b^2*(b*x + c*x^2)^(1/2))/(15*x^8) - (142*B*c^2*(b* x + c*x^2)^(1/2))/(715*x^6) - (2*A*c^3*(b*x + c*x^2)^(1/2))/(2431*b*x^6) - (2*A*b^2*(b*x + c*x^2)^(1/2))/(17*x^9) - (160*A*c^5*(b*x + c*x^2)^(1/2))/ (153153*b^3*x^4) + (64*A*c^6*(b*x + c*x^2)^(1/2))/(51051*b^4*x^3) - (256*A *c^7*(b*x + c*x^2)^(1/2))/(153153*b^5*x^2) + (512*A*c^8*(b*x + c*x^2)^(1/2 ))/(153153*b^6*x) - (2*B*c^3*(b*x + c*x^2)^(1/2))/(1287*b*x^5) + (16*B*c^4 *(b*x + c*x^2)^(1/2))/(9009*b^2*x^4) - (32*B*c^5*(b*x + c*x^2)^(1/2))/(150 15*b^3*x^3) + (128*B*c^6*(b*x + c*x^2)^(1/2))/(45045*b^4*x^2) - (256*B*c^7 *(b*x + c*x^2)^(1/2))/(45045*b^5*x) - (14*A*b*c*(b*x + c*x^2)^(1/2))/(51*x ^8) - (62*B*b*c*(b*x + c*x^2)^(1/2))/(195*x^7)