Integrand size = 22, antiderivative size = 195 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{10}} \, dx=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{15 b x^{10}}-\frac {2 (3 b B-2 A c) \left (b x+c x^2\right )^{5/2}}{39 b^2 x^9}+\frac {16 c (3 b B-2 A c) \left (b x+c x^2\right )^{5/2}}{429 b^3 x^8}-\frac {32 c^2 (3 b B-2 A c) \left (b x+c x^2\right )^{5/2}}{1287 b^4 x^7}+\frac {128 c^3 (3 b B-2 A c) \left (b x+c x^2\right )^{5/2}}{9009 b^5 x^6}-\frac {256 c^4 (3 b B-2 A c) \left (b x+c x^2\right )^{5/2}}{45045 b^6 x^5} \]
-2/15*A*(c*x^2+b*x)^(5/2)/b/x^10-2/39*(-2*A*c+3*B*b)*(c*x^2+b*x)^(5/2)/b^2 /x^9+16/429*c*(-2*A*c+3*B*b)*(c*x^2+b*x)^(5/2)/b^3/x^8-32/1287*c^2*(-2*A*c +3*B*b)*(c*x^2+b*x)^(5/2)/b^4/x^7+128/9009*c^3*(-2*A*c+3*B*b)*(c*x^2+b*x)^ (5/2)/b^5/x^6-256/45045*c^4*(-2*A*c+3*B*b)*(c*x^2+b*x)^(5/2)/b^6/x^5
Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.63 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{10}} \, dx=-\frac {2 (x (b+c x))^{5/2} \left (3 b B x \left (1155 b^4-840 b^3 c x+560 b^2 c^2 x^2-320 b c^3 x^3+128 c^4 x^4\right )+A \left (3003 b^5-2310 b^4 c x+1680 b^3 c^2 x^2-1120 b^2 c^3 x^3+640 b c^4 x^4-256 c^5 x^5\right )\right )}{45045 b^6 x^{10}} \]
(-2*(x*(b + c*x))^(5/2)*(3*b*B*x*(1155*b^4 - 840*b^3*c*x + 560*b^2*c^2*x^2 - 320*b*c^3*x^3 + 128*c^4*x^4) + A*(3003*b^5 - 2310*b^4*c*x + 1680*b^3*c^ 2*x^2 - 1120*b^2*c^3*x^3 + 640*b*c^4*x^4 - 256*c^5*x^5)))/(45045*b^6*x^10)
Time = 0.33 (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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(3 b B-2 A c) \int \frac {\left (c x^2+b x\right )^{3/2}}{x^9}dx}{3 b}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{15 b x^{10}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(3 b B-2 A c) \left (-\frac {8 c \int \frac {\left (c x^2+b x\right )^{3/2}}{x^8}dx}{13 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}\right )}{3 b}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{15 b x^{10}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(3 b B-2 A c) \left (-\frac {8 c \left (-\frac {6 c \int \frac {\left (c x^2+b x\right )^{3/2}}{x^7}dx}{11 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{11 b x^8}\right )}{13 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}\right )}{3 b}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{15 b x^{10}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(3 b B-2 A c) \left (-\frac {8 c \left (-\frac {6 c \left (-\frac {4 c \int \frac {\left (c x^2+b x\right )^{3/2}}{x^6}dx}{9 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 b x^7}\right )}{11 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{11 b x^8}\right )}{13 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}\right )}{3 b}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{15 b x^{10}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(3 b B-2 A c) \left (-\frac {8 c \left (-\frac {6 c \left (-\frac {4 c \left (-\frac {2 c \int \frac {\left (c x^2+b x\right )^{3/2}}{x^5}dx}{7 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 b x^6}\right )}{9 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 b x^7}\right )}{11 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{11 b x^8}\right )}{13 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}\right )}{3 b}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{15 b x^{10}}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {\left (-\frac {8 c \left (-\frac {6 c \left (-\frac {4 c \left (\frac {4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 b x^6}\right )}{9 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 b x^7}\right )}{11 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{11 b x^8}\right )}{13 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}\right ) (3 b B-2 A c)}{3 b}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{15 b x^{10}}\) |
(-2*A*(b*x + c*x^2)^(5/2))/(15*b*x^10) + ((3*b*B - 2*A*c)*((-2*(b*x + c*x^ 2)^(5/2))/(13*b*x^9) - (8*c*((-2*(b*x + c*x^2)^(5/2))/(11*b*x^8) - (6*c*(( -2*(b*x + c*x^2)^(5/2))/(9*b*x^7) - (4*c*((-2*(b*x + c*x^2)^(5/2))/(7*b*x^ 6) + (4*c*(b*x + c*x^2)^(5/2))/(35*b^2*x^5)))/(9*b)))/(11*b)))/(13*b)))/(3 *b)
3.1.92.3.1 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 = 0.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.54
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (\frac {15 B x}{13}+A \right ) b^{5}-\frac {10 c \left (\frac {12 B x}{11}+A \right ) x \,b^{4}}{13}+\frac {80 c^{2} x^{2} \left (B x +A \right ) b^{3}}{143}-\frac {160 c^{3} \left (\frac {6 B x}{7}+A \right ) x^{3} b^{2}}{429}+\frac {640 c^{4} \left (\frac {3 B x}{5}+A \right ) x^{4} b}{3003}-\frac {256 A \,c^{5} x^{5}}{3003}\right ) \left (c x +b \right )^{2} \sqrt {x \left (c x +b \right )}}{15 x^{8} b^{6}}\) | \(106\) |
| gosper | \(-\frac {2 \left (c x +b \right ) \left (-256 A \,c^{5} x^{5}+384 B b \,c^{4} x^{5}+640 A b \,c^{4} x^{4}-960 B \,b^{2} c^{3} x^{4}-1120 A \,b^{2} c^{3} x^{3}+1680 B \,b^{3} c^{2} x^{3}+1680 A \,b^{3} c^{2} x^{2}-2520 B \,b^{4} c \,x^{2}-2310 A \,b^{4} c x +3465 B \,b^{5} x +3003 A \,b^{5}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{45045 x^{9} b^{6}}\) | \(134\) |
| trager | \(-\frac {2 \left (-256 A \,c^{7} x^{7}+384 B b \,c^{6} x^{7}+128 A b \,c^{6} x^{6}-192 B \,b^{2} c^{5} x^{6}-96 A \,b^{2} c^{5} x^{5}+144 B \,b^{3} c^{4} x^{5}+80 A \,b^{3} c^{4} x^{4}-120 B \,b^{4} c^{3} x^{4}-70 A \,b^{4} c^{3} x^{3}+105 B \,b^{5} c^{2} x^{3}+63 A \,b^{5} c^{2} x^{2}+4410 B \,b^{6} c \,x^{2}+3696 A \,b^{6} c x +3465 B \,b^{7} x +3003 A \,b^{7}\right ) \sqrt {c \,x^{2}+b x}}{45045 b^{6} x^{8}}\) | \(177\) |
| risch | \(-\frac {2 \left (c x +b \right ) \left (-256 A \,c^{7} x^{7}+384 B b \,c^{6} x^{7}+128 A b \,c^{6} x^{6}-192 B \,b^{2} c^{5} x^{6}-96 A \,b^{2} c^{5} x^{5}+144 B \,b^{3} c^{4} x^{5}+80 A \,b^{3} c^{4} x^{4}-120 B \,b^{4} c^{3} x^{4}-70 A \,b^{4} c^{3} x^{3}+105 B \,b^{5} c^{2} x^{3}+63 A \,b^{5} c^{2} x^{2}+4410 B \,b^{6} c \,x^{2}+3696 A \,b^{6} c x +3465 B \,b^{7} x +3003 A \,b^{7}\right )}{45045 x^{7} \sqrt {x \left (c x +b \right )}\, b^{6}}\) | \(180\) |
| default | \(B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{13 b \,x^{9}}-\frac {8 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{11 b \,x^{8}}-\frac {6 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{9 b \,x^{7}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 b \,x^{6}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{35 b^{2} x^{5}}\right )}{9 b}\right )}{11 b}\right )}{13 b}\right )+A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{15 b \,x^{10}}-\frac {2 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{13 b \,x^{9}}-\frac {8 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{11 b \,x^{8}}-\frac {6 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{9 b \,x^{7}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 b \,x^{6}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{35 b^{2} x^{5}}\right )}{9 b}\right )}{11 b}\right )}{13 b}\right )}{3 b}\right )\) | \(268\) |
-2/15*((15/13*B*x+A)*b^5-10/13*c*(12/11*B*x+A)*x*b^4+80/143*c^2*x^2*(B*x+A )*b^3-160/429*c^3*(6/7*B*x+A)*x^3*b^2+640/3003*c^4*(3/5*B*x+A)*x^4*b-256/3 003*A*c^5*x^5)*(c*x+b)^2*(x*(c*x+b))^(1/2)/x^8/b^6
Time = 0.28 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{10}} \, dx=-\frac {2 \, {\left (3003 \, A b^{7} + 128 \, {\left (3 \, B b c^{6} - 2 \, A c^{7}\right )} x^{7} - 64 \, {\left (3 \, B b^{2} c^{5} - 2 \, A b c^{6}\right )} x^{6} + 48 \, {\left (3 \, B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} x^{5} - 40 \, {\left (3 \, B b^{4} c^{3} - 2 \, A b^{3} c^{4}\right )} x^{4} + 35 \, {\left (3 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} x^{3} + 63 \, {\left (70 \, B b^{6} c + A b^{5} c^{2}\right )} x^{2} + 231 \, {\left (15 \, B b^{7} + 16 \, A b^{6} c\right )} x\right )} \sqrt {c x^{2} + b x}}{45045 \, b^{6} x^{8}} \]
-2/45045*(3003*A*b^7 + 128*(3*B*b*c^6 - 2*A*c^7)*x^7 - 64*(3*B*b^2*c^5 - 2 *A*b*c^6)*x^6 + 48*(3*B*b^3*c^4 - 2*A*b^2*c^5)*x^5 - 40*(3*B*b^4*c^3 - 2*A *b^3*c^4)*x^4 + 35*(3*B*b^5*c^2 - 2*A*b^4*c^3)*x^3 + 63*(70*B*b^6*c + A*b^ 5*c^2)*x^2 + 231*(15*B*b^7 + 16*A*b^6*c)*x)*sqrt(c*x^2 + b*x)/(b^6*x^8)
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{10}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{x^{10}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (171) = 342\).
Time = 0.20 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.85 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{10}} \, dx=-\frac {256 \, \sqrt {c x^{2} + b x} B c^{6}}{15015 \, b^{5} x} + \frac {512 \, \sqrt {c x^{2} + b x} A c^{7}}{45045 \, b^{6} x} + \frac {128 \, \sqrt {c x^{2} + b x} B c^{5}}{15015 \, b^{4} x^{2}} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{6}}{45045 \, b^{5} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} B c^{4}}{5005 \, b^{3} x^{3}} + \frac {64 \, \sqrt {c x^{2} + b x} A c^{5}}{15015 \, b^{4} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} B c^{3}}{3003 \, b^{2} x^{4}} - \frac {32 \, \sqrt {c x^{2} + b x} A c^{4}}{9009 \, b^{3} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} B c^{2}}{429 \, b x^{5}} + \frac {4 \, \sqrt {c x^{2} + b x} A c^{3}}{1287 \, b^{2} x^{5}} + \frac {3 \, \sqrt {c x^{2} + b x} B c}{715 \, x^{6}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{2}}{715 \, b x^{6}} + \frac {3 \, \sqrt {c x^{2} + b x} B b}{65 \, x^{7}} + \frac {\sqrt {c x^{2} + b x} A c}{390 \, x^{7}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B}{5 \, x^{8}} + \frac {\sqrt {c x^{2} + b x} A b}{30 \, x^{8}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A}{6 \, x^{9}} \]
-256/15015*sqrt(c*x^2 + b*x)*B*c^6/(b^5*x) + 512/45045*sqrt(c*x^2 + b*x)*A *c^7/(b^6*x) + 128/15015*sqrt(c*x^2 + b*x)*B*c^5/(b^4*x^2) - 256/45045*sqr t(c*x^2 + b*x)*A*c^6/(b^5*x^2) - 32/5005*sqrt(c*x^2 + b*x)*B*c^4/(b^3*x^3) + 64/15015*sqrt(c*x^2 + b*x)*A*c^5/(b^4*x^3) + 16/3003*sqrt(c*x^2 + b*x)* B*c^3/(b^2*x^4) - 32/9009*sqrt(c*x^2 + b*x)*A*c^4/(b^3*x^4) - 2/429*sqrt(c *x^2 + b*x)*B*c^2/(b*x^5) + 4/1287*sqrt(c*x^2 + b*x)*A*c^3/(b^2*x^5) + 3/7 15*sqrt(c*x^2 + b*x)*B*c/x^6 - 2/715*sqrt(c*x^2 + b*x)*A*c^2/(b*x^6) + 3/6 5*sqrt(c*x^2 + b*x)*B*b/x^7 + 1/390*sqrt(c*x^2 + b*x)*A*c/x^7 - 1/5*(c*x^2 + b*x)^(3/2)*B/x^8 + 1/30*sqrt(c*x^2 + b*x)*A*b/x^8 - 1/6*(c*x^2 + b*x)^( 3/2)*A/x^9
Leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (171) = 342\).
Time = 0.30 (sec) , antiderivative size = 551, normalized size of antiderivative = 2.83 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{10}} \, dx=\frac {2 \, {\left (144144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{10} B c^{4} + 720720 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} B b c^{\frac {7}{2}} + 240240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} A c^{\frac {9}{2}} + 1595880 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B b^{2} c^{3} + 1338480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} A b c^{4} + 2027025 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b^{3} c^{\frac {5}{2}} + 3333330 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A b^{2} c^{\frac {7}{2}} + 1606605 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{4} c^{2} + 4844840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b^{3} c^{3} + 810810 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{5} c^{\frac {3}{2}} + 4513509 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{4} c^{\frac {5}{2}} + 253890 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{6} c + 2788695 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{5} c^{2} + 45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{7} \sqrt {c} + 1141140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{6} c^{\frac {3}{2}} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{8} + 297990 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{7} c + 45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{8} \sqrt {c} + 3003 \, A b^{9}\right )}}{45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{15}} \]
2/45045*(144144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*c^4 + 720720*(sqrt(c) *x - sqrt(c*x^2 + b*x))^9*B*b*c^(7/2) + 240240*(sqrt(c)*x - sqrt(c*x^2 + b *x))^9*A*c^(9/2) + 1595880*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^2*c^3 + 1 338480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b*c^4 + 2027025*(sqrt(c)*x - sq rt(c*x^2 + b*x))^7*B*b^3*c^(5/2) + 3333330*(sqrt(c)*x - sqrt(c*x^2 + b*x)) ^7*A*b^2*c^(7/2) + 1606605*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^4*c^2 + 4 844840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^3*c^3 + 810810*(sqrt(c)*x - s qrt(c*x^2 + b*x))^5*B*b^5*c^(3/2) + 4513509*(sqrt(c)*x - sqrt(c*x^2 + b*x) )^5*A*b^4*c^(5/2) + 253890*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^6*c + 278 8695*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^5*c^2 + 45045*(sqrt(c)*x - sqrt (c*x^2 + b*x))^3*B*b^7*sqrt(c) + 1141140*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3 *A*b^6*c^(3/2) + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^8 + 297990*(sq rt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^7*c + 45045*(sqrt(c)*x - sqrt(c*x^2 + b *x))*A*b^8*sqrt(c) + 3003*A*b^9)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^15
Time = 13.12 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.67 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{10}} \, dx=\frac {4\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{1287\,b^2\,x^5}-\frac {32\,A\,c\,\sqrt {c\,x^2+b\,x}}{195\,x^7}-\frac {2\,B\,b\,\sqrt {c\,x^2+b\,x}}{13\,x^7}-\frac {28\,B\,c\,\sqrt {c\,x^2+b\,x}}{143\,x^6}-\frac {2\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{715\,b\,x^6}-\frac {2\,A\,b\,\sqrt {c\,x^2+b\,x}}{15\,x^8}-\frac {32\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{9009\,b^3\,x^4}+\frac {64\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{15015\,b^4\,x^3}-\frac {256\,A\,c^6\,\sqrt {c\,x^2+b\,x}}{45045\,b^5\,x^2}+\frac {512\,A\,c^7\,\sqrt {c\,x^2+b\,x}}{45045\,b^6\,x}-\frac {2\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{429\,b\,x^5}+\frac {16\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{3003\,b^2\,x^4}-\frac {32\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{5005\,b^3\,x^3}+\frac {128\,B\,c^5\,\sqrt {c\,x^2+b\,x}}{15015\,b^4\,x^2}-\frac {256\,B\,c^6\,\sqrt {c\,x^2+b\,x}}{15015\,b^5\,x} \]
(4*A*c^3*(b*x + c*x^2)^(1/2))/(1287*b^2*x^5) - (32*A*c*(b*x + c*x^2)^(1/2) )/(195*x^7) - (2*B*b*(b*x + c*x^2)^(1/2))/(13*x^7) - (28*B*c*(b*x + c*x^2) ^(1/2))/(143*x^6) - (2*A*c^2*(b*x + c*x^2)^(1/2))/(715*b*x^6) - (2*A*b*(b* x + c*x^2)^(1/2))/(15*x^8) - (32*A*c^4*(b*x + c*x^2)^(1/2))/(9009*b^3*x^4) + (64*A*c^5*(b*x + c*x^2)^(1/2))/(15015*b^4*x^3) - (256*A*c^6*(b*x + c*x^ 2)^(1/2))/(45045*b^5*x^2) + (512*A*c^7*(b*x + c*x^2)^(1/2))/(45045*b^6*x) - (2*B*c^2*(b*x + c*x^2)^(1/2))/(429*b*x^5) + (16*B*c^3*(b*x + c*x^2)^(1/2 ))/(3003*b^2*x^4) - (32*B*c^4*(b*x + c*x^2)^(1/2))/(5005*b^3*x^3) + (128*B *c^5*(b*x + c*x^2)^(1/2))/(15015*b^4*x^2) - (256*B*c^6*(b*x + c*x^2)^(1/2) )/(15015*b^5*x)