Integrand size = 24, antiderivative size = 207 \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=-\frac {256 b^4 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{765765 c^6 x^{7/2}}+\frac {128 b^3 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{109395 c^5 x^{5/2}}-\frac {32 b^2 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{12155 c^4 x^{3/2}}+\frac {16 b (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{3315 c^3 \sqrt {x}}-\frac {2 (10 b B-17 A c) \sqrt {x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c} \]
-256/765765*b^4*(-17*A*c+10*B*b)*(c*x^2+b*x)^(7/2)/c^6/x^(7/2)+128/109395* b^3*(-17*A*c+10*B*b)*(c*x^2+b*x)^(7/2)/c^5/x^(5/2)-32/12155*b^2*(-17*A*c+1 0*B*b)*(c*x^2+b*x)^(7/2)/c^4/x^(3/2)+2/17*B*x^(3/2)*(c*x^2+b*x)^(7/2)/c+16 /3315*b*(-17*A*c+10*B*b)*(c*x^2+b*x)^(7/2)/c^3/x^(1/2)-2/255*(-17*A*c+10*B *b)*(c*x^2+b*x)^(7/2)*x^(1/2)/c^2
Time = 0.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.58 \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {2 (b+c x)^3 \sqrt {x (b+c x)} \left (-1280 b^5 B+3003 c^5 x^4 (17 A+15 B x)+128 b^4 c (17 A+35 B x)-224 b^3 c^2 x (34 A+45 B x)+336 b^2 c^3 x^2 (51 A+55 B x)-462 b c^4 x^3 (68 A+65 B x)\right )}{765765 c^6 \sqrt {x}} \]
(2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(-1280*b^5*B + 3003*c^5*x^4*(17*A + 15*B* x) + 128*b^4*c*(17*A + 35*B*x) - 224*b^3*c^2*x*(34*A + 45*B*x) + 336*b^2*c ^3*x^2*(51*A + 55*B*x) - 462*b*c^4*x^3*(68*A + 65*B*x)))/(765765*c^6*Sqrt[ x])
Time = 0.34 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1221, 1128, 1128, 1128, 1128, 1122}
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.
| \(\displaystyle \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}-\frac {(10 b B-17 A c) \int x^{3/2} \left (c x^2+b x\right )^{5/2}dx}{17 c}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}-\frac {(10 b B-17 A c) \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{7/2}}{15 c}-\frac {8 b \int \sqrt {x} \left (c x^2+b x\right )^{5/2}dx}{15 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}-\frac {(10 b B-17 A c) \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{7/2}}{15 c}-\frac {8 b \left (\frac {2 \left (b x+c x^2\right )^{7/2}}{13 c \sqrt {x}}-\frac {6 b \int \frac {\left (c x^2+b x\right )^{5/2}}{\sqrt {x}}dx}{13 c}\right )}{15 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}-\frac {(10 b B-17 A c) \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{7/2}}{15 c}-\frac {8 b \left (\frac {2 \left (b x+c x^2\right )^{7/2}}{13 c \sqrt {x}}-\frac {6 b \left (\frac {2 \left (b x+c x^2\right )^{7/2}}{11 c x^{3/2}}-\frac {4 b \int \frac {\left (c x^2+b x\right )^{5/2}}{x^{3/2}}dx}{11 c}\right )}{13 c}\right )}{15 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}-\frac {(10 b B-17 A c) \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{7/2}}{15 c}-\frac {8 b \left (\frac {2 \left (b x+c x^2\right )^{7/2}}{13 c \sqrt {x}}-\frac {6 b \left (\frac {2 \left (b x+c x^2\right )^{7/2}}{11 c x^{3/2}}-\frac {4 b \left (\frac {2 \left (b x+c x^2\right )^{7/2}}{9 c x^{5/2}}-\frac {2 b \int \frac {\left (c x^2+b x\right )^{5/2}}{x^{5/2}}dx}{9 c}\right )}{11 c}\right )}{13 c}\right )}{15 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}-\frac {\left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{7/2}}{15 c}-\frac {8 b \left (\frac {2 \left (b x+c x^2\right )^{7/2}}{13 c \sqrt {x}}-\frac {6 b \left (\frac {2 \left (b x+c x^2\right )^{7/2}}{11 c x^{3/2}}-\frac {4 b \left (\frac {2 \left (b x+c x^2\right )^{7/2}}{9 c x^{5/2}}-\frac {4 b \left (b x+c x^2\right )^{7/2}}{63 c^2 x^{7/2}}\right )}{11 c}\right )}{13 c}\right )}{15 c}\right ) (10 b B-17 A c)}{17 c}\) |
(2*B*x^(3/2)*(b*x + c*x^2)^(7/2))/(17*c) - ((10*b*B - 17*A*c)*((2*Sqrt[x]* (b*x + c*x^2)^(7/2))/(15*c) - (8*b*((2*(b*x + c*x^2)^(7/2))/(13*c*Sqrt[x]) - (6*b*((2*(b*x + c*x^2)^(7/2))/(11*c*x^(3/2)) - (4*b*((-4*b*(b*x + c*x^2 )^(7/2))/(63*c^2*x^(7/2)) + (2*(b*x + c*x^2)^(7/2))/(9*c*x^(5/2))))/(11*c) ))/(13*c)))/(15*c)))/(17*c)
3.3.14.3.1 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 - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))) 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] && IGtQ[Simplify[m + p], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) Int[(d + e*x)^m*(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] && NeQ[m + 2*p + 2, 0]
Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.63
| method | result | size |
| gosper | \(\frac {2 \left (c x +b \right ) \left (45045 B \,c^{5} x^{5}+51051 A \,c^{5} x^{4}-30030 B b \,c^{4} x^{4}-31416 A b \,c^{4} x^{3}+18480 B \,b^{2} c^{3} x^{3}+17136 A \,b^{2} c^{3} x^{2}-10080 B \,b^{3} c^{2} x^{2}-7616 A \,b^{3} c^{2} x +4480 B \,b^{4} c x +2176 A \,b^{4} c -1280 B \,b^{5}\right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{765765 c^{6} x^{\frac {5}{2}}}\) | \(131\) |
| default | \(\frac {2 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right )^{3} \left (45045 B \,c^{5} x^{5}+51051 A \,c^{5} x^{4}-30030 B b \,c^{4} x^{4}-31416 A b \,c^{4} x^{3}+18480 B \,b^{2} c^{3} x^{3}+17136 A \,b^{2} c^{3} x^{2}-10080 B \,b^{3} c^{2} x^{2}-7616 A \,b^{3} c^{2} x +4480 B \,b^{4} c x +2176 A \,b^{4} c -1280 B \,b^{5}\right )}{765765 \sqrt {x}\, c^{6}}\) | \(131\) |
| risch | \(\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (45045 B \,c^{8} x^{8}+51051 A \,c^{8} x^{7}+105105 B b \,c^{7} x^{7}+121737 A b \,c^{7} x^{6}+63525 B \,b^{2} c^{6} x^{6}+76041 A \,b^{2} c^{6} x^{5}+315 B \,b^{3} c^{5} x^{5}+595 A \,b^{3} c^{5} x^{4}-350 B \,b^{4} c^{4} x^{4}-680 A \,b^{4} c^{4} x^{3}+400 B \,b^{5} c^{3} x^{3}+816 A \,b^{5} c^{3} x^{2}-480 B \,b^{6} c^{2} x^{2}-1088 A \,b^{6} c^{2} x +640 B \,b^{7} c x +2176 A \,b^{7} c -1280 B \,b^{8}\right )}{765765 \sqrt {x \left (c x +b \right )}\, c^{6}}\) | \(201\) |
2/765765*(c*x+b)*(45045*B*c^5*x^5+51051*A*c^5*x^4-30030*B*b*c^4*x^4-31416* A*b*c^4*x^3+18480*B*b^2*c^3*x^3+17136*A*b^2*c^3*x^2-10080*B*b^3*c^2*x^2-76 16*A*b^3*c^2*x+4480*B*b^4*c*x+2176*A*b^4*c-1280*B*b^5)*(c*x^2+b*x)^(5/2)/c ^6/x^(5/2)
Time = 0.37 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.96 \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {2 \, {\left (45045 \, B c^{8} x^{8} - 1280 \, B b^{8} + 2176 \, A b^{7} c + 3003 \, {\left (35 \, B b c^{7} + 17 \, A c^{8}\right )} x^{7} + 231 \, {\left (275 \, B b^{2} c^{6} + 527 \, A b c^{7}\right )} x^{6} + 63 \, {\left (5 \, B b^{3} c^{5} + 1207 \, A b^{2} c^{6}\right )} x^{5} - 35 \, {\left (10 \, B b^{4} c^{4} - 17 \, A b^{3} c^{5}\right )} x^{4} + 40 \, {\left (10 \, B b^{5} c^{3} - 17 \, A b^{4} c^{4}\right )} x^{3} - 48 \, {\left (10 \, B b^{6} c^{2} - 17 \, A b^{5} c^{3}\right )} x^{2} + 64 \, {\left (10 \, B b^{7} c - 17 \, A b^{6} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{765765 \, c^{6} \sqrt {x}} \]
2/765765*(45045*B*c^8*x^8 - 1280*B*b^8 + 2176*A*b^7*c + 3003*(35*B*b*c^7 + 17*A*c^8)*x^7 + 231*(275*B*b^2*c^6 + 527*A*b*c^7)*x^6 + 63*(5*B*b^3*c^5 + 1207*A*b^2*c^6)*x^5 - 35*(10*B*b^4*c^4 - 17*A*b^3*c^5)*x^4 + 40*(10*B*b^5 *c^3 - 17*A*b^4*c^4)*x^3 - 48*(10*B*b^6*c^2 - 17*A*b^5*c^3)*x^2 + 64*(10*B *b^7*c - 17*A*b^6*c^2)*x)*sqrt(c*x^2 + b*x)/(c^6*sqrt(x))
Timed out. \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (171) = 342\).
Time = 0.22 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.45 \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {2 \, {\left ({\left (3003 \, c^{7} x^{7} + 231 \, b c^{6} x^{6} - 252 \, b^{2} c^{5} x^{5} + 280 \, b^{3} c^{4} x^{4} - 320 \, b^{4} c^{3} x^{3} + 384 \, b^{5} c^{2} x^{2} - 512 \, b^{6} c x + 1024 \, b^{7}\right )} x^{6} + 10 \, {\left (693 \, b c^{6} x^{7} + 63 \, b^{2} c^{5} x^{6} - 70 \, b^{3} c^{4} x^{5} + 80 \, b^{4} c^{3} x^{4} - 96 \, b^{5} c^{2} x^{3} + 128 \, b^{6} c x^{2} - 256 \, b^{7} x\right )} x^{5} + 13 \, {\left (315 \, b^{2} c^{5} x^{7} + 35 \, b^{3} c^{4} x^{6} - 40 \, b^{4} c^{3} x^{5} + 48 \, b^{5} c^{2} x^{4} - 64 \, b^{6} c x^{3} + 128 \, b^{7} x^{2}\right )} x^{4}\right )} \sqrt {c x + b} A}{45045 \, c^{5} x^{6}} + \frac {2 \, {\left (7 \, {\left (6435 \, c^{8} x^{8} + 429 \, b c^{7} x^{7} - 462 \, b^{2} c^{6} x^{6} + 504 \, b^{3} c^{5} x^{5} - 560 \, b^{4} c^{4} x^{4} + 640 \, b^{5} c^{3} x^{3} - 768 \, b^{6} c^{2} x^{2} + 1024 \, b^{7} c x - 2048 \, b^{8}\right )} x^{7} + 34 \, {\left (3003 \, b c^{7} x^{8} + 231 \, b^{2} c^{6} x^{7} - 252 \, b^{3} c^{5} x^{6} + 280 \, b^{4} c^{4} x^{5} - 320 \, b^{5} c^{3} x^{4} + 384 \, b^{6} c^{2} x^{3} - 512 \, b^{7} c x^{2} + 1024 \, b^{8} x\right )} x^{6} + 85 \, {\left (693 \, b^{2} c^{6} x^{8} + 63 \, b^{3} c^{5} x^{7} - 70 \, b^{4} c^{4} x^{6} + 80 \, b^{5} c^{3} x^{5} - 96 \, b^{6} c^{2} x^{4} + 128 \, b^{7} c x^{3} - 256 \, b^{8} x^{2}\right )} x^{5}\right )} \sqrt {c x + b} B}{765765 \, c^{6} x^{7}} \]
2/45045*((3003*c^7*x^7 + 231*b*c^6*x^6 - 252*b^2*c^5*x^5 + 280*b^3*c^4*x^4 - 320*b^4*c^3*x^3 + 384*b^5*c^2*x^2 - 512*b^6*c*x + 1024*b^7)*x^6 + 10*(6 93*b*c^6*x^7 + 63*b^2*c^5*x^6 - 70*b^3*c^4*x^5 + 80*b^4*c^3*x^4 - 96*b^5*c ^2*x^3 + 128*b^6*c*x^2 - 256*b^7*x)*x^5 + 13*(315*b^2*c^5*x^7 + 35*b^3*c^4 *x^6 - 40*b^4*c^3*x^5 + 48*b^5*c^2*x^4 - 64*b^6*c*x^3 + 128*b^7*x^2)*x^4)* sqrt(c*x + b)*A/(c^5*x^6) + 2/765765*(7*(6435*c^8*x^8 + 429*b*c^7*x^7 - 46 2*b^2*c^6*x^6 + 504*b^3*c^5*x^5 - 560*b^4*c^4*x^4 + 640*b^5*c^3*x^3 - 768* b^6*c^2*x^2 + 1024*b^7*c*x - 2048*b^8)*x^7 + 34*(3003*b*c^7*x^8 + 231*b^2* c^6*x^7 - 252*b^3*c^5*x^6 + 280*b^4*c^4*x^5 - 320*b^5*c^3*x^4 + 384*b^6*c^ 2*x^3 - 512*b^7*c*x^2 + 1024*b^8*x)*x^6 + 85*(693*b^2*c^6*x^8 + 63*b^3*c^5 *x^7 - 70*b^4*c^4*x^6 + 80*b^5*c^3*x^5 - 96*b^6*c^2*x^4 + 128*b^7*c*x^3 - 256*b^8*x^2)*x^5)*sqrt(c*x + b)*B/(c^6*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 560 vs. \(2 (171) = 342\).
Time = 0.31 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.71 \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {2}{109395} \, B c^{2} {\left (\frac {2048 \, b^{\frac {17}{2}}}{c^{8}} + \frac {6435 \, {\left (c x + b\right )}^{\frac {17}{2}} - 51051 \, {\left (c x + b\right )}^{\frac {15}{2}} b + 176715 \, {\left (c x + b\right )}^{\frac {13}{2}} b^{2} - 348075 \, {\left (c x + b\right )}^{\frac {11}{2}} b^{3} + 425425 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{4} - 328185 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{5} + 153153 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{6} - 36465 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{7}}{c^{8}}\right )} - \frac {4}{45045} \, B b c {\left (\frac {1024 \, b^{\frac {15}{2}}}{c^{7}} - \frac {3003 \, {\left (c x + b\right )}^{\frac {15}{2}} - 20790 \, {\left (c x + b\right )}^{\frac {13}{2}} b + 61425 \, {\left (c x + b\right )}^{\frac {11}{2}} b^{2} - 100100 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{3} + 96525 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{4} - 54054 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{6}}{c^{7}}\right )} - \frac {2}{45045} \, A c^{2} {\left (\frac {1024 \, b^{\frac {15}{2}}}{c^{7}} - \frac {3003 \, {\left (c x + b\right )}^{\frac {15}{2}} - 20790 \, {\left (c x + b\right )}^{\frac {13}{2}} b + 61425 \, {\left (c x + b\right )}^{\frac {11}{2}} b^{2} - 100100 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{3} + 96525 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{4} - 54054 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{6}}{c^{7}}\right )} + \frac {2}{9009} \, B b^{2} {\left (\frac {256 \, b^{\frac {13}{2}}}{c^{6}} + \frac {693 \, {\left (c x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (c x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{5}}{c^{6}}\right )} + \frac {4}{9009} \, A b c {\left (\frac {256 \, b^{\frac {13}{2}}}{c^{6}} + \frac {693 \, {\left (c x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (c x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{5}}{c^{6}}\right )} - \frac {2}{3465} \, A b^{2} {\left (\frac {128 \, b^{\frac {11}{2}}}{c^{5}} - \frac {315 \, {\left (c x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (c x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{4}}{c^{5}}\right )} \]
2/109395*B*c^2*(2048*b^(17/2)/c^8 + (6435*(c*x + b)^(17/2) - 51051*(c*x + b)^(15/2)*b + 176715*(c*x + b)^(13/2)*b^2 - 348075*(c*x + b)^(11/2)*b^3 + 425425*(c*x + b)^(9/2)*b^4 - 328185*(c*x + b)^(7/2)*b^5 + 153153*(c*x + b) ^(5/2)*b^6 - 36465*(c*x + b)^(3/2)*b^7)/c^8) - 4/45045*B*b*c*(1024*b^(15/2 )/c^7 - (3003*(c*x + b)^(15/2) - 20790*(c*x + b)^(13/2)*b + 61425*(c*x + b )^(11/2)*b^2 - 100100*(c*x + b)^(9/2)*b^3 + 96525*(c*x + b)^(7/2)*b^4 - 54 054*(c*x + b)^(5/2)*b^5 + 15015*(c*x + b)^(3/2)*b^6)/c^7) - 2/45045*A*c^2* (1024*b^(15/2)/c^7 - (3003*(c*x + b)^(15/2) - 20790*(c*x + b)^(13/2)*b + 6 1425*(c*x + b)^(11/2)*b^2 - 100100*(c*x + b)^(9/2)*b^3 + 96525*(c*x + b)^( 7/2)*b^4 - 54054*(c*x + b)^(5/2)*b^5 + 15015*(c*x + b)^(3/2)*b^6)/c^7) + 2 /9009*B*b^2*(256*b^(13/2)/c^6 + (693*(c*x + b)^(13/2) - 4095*(c*x + b)^(11 /2)*b + 10010*(c*x + b)^(9/2)*b^2 - 12870*(c*x + b)^(7/2)*b^3 + 9009*(c*x + b)^(5/2)*b^4 - 3003*(c*x + b)^(3/2)*b^5)/c^6) + 4/9009*A*b*c*(256*b^(13/ 2)/c^6 + (693*(c*x + b)^(13/2) - 4095*(c*x + b)^(11/2)*b + 10010*(c*x + b) ^(9/2)*b^2 - 12870*(c*x + b)^(7/2)*b^3 + 9009*(c*x + b)^(5/2)*b^4 - 3003*( c*x + b)^(3/2)*b^5)/c^6) - 2/3465*A*b^2*(128*b^(11/2)/c^5 - (315*(c*x + b) ^(11/2) - 1540*(c*x + b)^(9/2)*b + 2970*(c*x + b)^(7/2)*b^2 - 2772*(c*x + b)^(5/2)*b^3 + 1155*(c*x + b)^(3/2)*b^4)/c^5)
Timed out. \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\int x^{3/2}\,{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]