Integrand size = 24, antiderivative size = 207 \[ \int x^{5/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=-\frac {256 b^4 (2 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{45045 c^6 x^{5/2}}+\frac {128 b^3 (2 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac {32 b^2 (2 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt {x}}+\frac {16 b (2 b B-3 A c) \sqrt {x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac {2 (2 b B-3 A c) x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac {2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]
-256/45045*b^4*(-3*A*c+2*B*b)*(c*x^2+b*x)^(5/2)/c^6/x^(5/2)+128/9009*b^3*( -3*A*c+2*B*b)*(c*x^2+b*x)^(5/2)/c^5/x^(3/2)-2/39*(-3*A*c+2*B*b)*x^(3/2)*(c *x^2+b*x)^(5/2)/c^2+2/15*B*x^(5/2)*(c*x^2+b*x)^(5/2)/c-32/1287*b^2*(-3*A*c +2*B*b)*(c*x^2+b*x)^(5/2)/c^4/x^(1/2)+16/429*b*(-3*A*c+2*B*b)*(c*x^2+b*x)^ (5/2)*x^(1/2)/c^3
Time = 0.07 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.53 \[ \int x^{5/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 (x (b+c x))^{5/2} \left (-256 b^5 B+1680 b^2 c^3 x^2 (A+B x)+128 b^4 c (3 A+5 B x)-160 b^3 c^2 x (6 A+7 B x)-210 b c^4 x^3 (12 A+11 B x)+231 c^5 x^4 (15 A+13 B x)\right )}{45045 c^6 x^{5/2}} \]
(2*(x*(b + c*x))^(5/2)*(-256*b^5*B + 1680*b^2*c^3*x^2*(A + B*x) + 128*b^4* c*(3*A + 5*B*x) - 160*b^3*c^2*x*(6*A + 7*B*x) - 210*b*c^4*x^3*(12*A + 11*B *x) + 231*c^5*x^4*(15*A + 13*B*x)))/(45045*c^6*x^(5/2))
Time = 0.33 (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.
| \(\displaystyle \int x^{5/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac {(2 b B-3 A c) \int x^{5/2} \left (c x^2+b x\right )^{3/2}dx}{3 c}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac {(2 b B-3 A c) \left (\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}-\frac {8 b \int x^{3/2} \left (c x^2+b x\right )^{3/2}dx}{13 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac {(2 b B-3 A c) \left (\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}-\frac {8 b \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {6 b \int \sqrt {x} \left (c x^2+b x\right )^{3/2}dx}{11 c}\right )}{13 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac {(2 b B-3 A c) \left (\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}-\frac {8 b \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {6 b \left (\frac {2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt {x}}-\frac {4 b \int \frac {\left (c x^2+b x\right )^{3/2}}{\sqrt {x}}dx}{9 c}\right )}{11 c}\right )}{13 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac {(2 b B-3 A c) \left (\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}-\frac {8 b \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {6 b \left (\frac {2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt {x}}-\frac {4 b \left (\frac {2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac {2 b \int \frac {\left (c x^2+b x\right )^{3/2}}{x^{3/2}}dx}{7 c}\right )}{9 c}\right )}{11 c}\right )}{13 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac {\left (\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}-\frac {8 b \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {6 b \left (\frac {2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt {x}}-\frac {4 b \left (\frac {2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac {4 b \left (b x+c x^2\right )^{5/2}}{35 c^2 x^{5/2}}\right )}{9 c}\right )}{11 c}\right )}{13 c}\right ) (2 b B-3 A c)}{3 c}\) |
(2*B*x^(5/2)*(b*x + c*x^2)^(5/2))/(15*c) - ((2*b*B - 3*A*c)*((2*x^(3/2)*(b *x + c*x^2)^(5/2))/(13*c) - (8*b*((2*Sqrt[x]*(b*x + c*x^2)^(5/2))/(11*c) - (6*b*((2*(b*x + c*x^2)^(5/2))/(9*c*Sqrt[x]) - (4*b*((-4*b*(b*x + c*x^2)^( 5/2))/(35*c^2*x^(5/2)) + (2*(b*x + c*x^2)^(5/2))/(7*c*x^(3/2))))/(9*c)))/( 11*c)))/(13*c)))/(3*c)
3.3.3.3.1 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 | \(\frac {2 \left (c x +b \right ) \left (3003 B \,c^{5} x^{5}+3465 A \,c^{5} x^{4}-2310 B b \,c^{4} x^{4}-2520 A b \,c^{4} x^{3}+1680 B \,b^{2} c^{3} x^{3}+1680 A \,b^{2} c^{3} x^{2}-1120 B \,b^{3} c^{2} x^{2}-960 A \,b^{3} c^{2} x +640 B \,b^{4} c x +384 A \,b^{4} c -256 B \,b^{5}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{45045 c^{6} x^{\frac {3}{2}}}\) | \(131\) |
| default | \(\frac {2 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right )^{2} \left (3003 B \,c^{5} x^{5}+3465 A \,c^{5} x^{4}-2310 B b \,c^{4} x^{4}-2520 A b \,c^{4} x^{3}+1680 B \,b^{2} c^{3} x^{3}+1680 A \,b^{2} c^{3} x^{2}-1120 B \,b^{3} c^{2} x^{2}-960 A \,b^{3} c^{2} x +640 B \,b^{4} c x +384 A \,b^{4} c -256 B \,b^{5}\right )}{45045 \sqrt {x}\, c^{6}}\) | \(131\) |
| risch | \(\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (3003 B \,c^{7} x^{7}+3465 A \,c^{7} x^{6}+3696 B b \,c^{6} x^{6}+4410 A b \,c^{6} x^{5}+63 B \,b^{2} c^{5} x^{5}+105 A \,b^{2} c^{5} x^{4}-70 B \,b^{3} c^{4} x^{4}-120 A \,b^{3} c^{4} x^{3}+80 B \,b^{4} c^{3} x^{3}+144 A \,b^{4} c^{3} x^{2}-96 B \,b^{5} c^{2} x^{2}-192 A \,b^{5} c^{2} x +128 B \,b^{6} c x +384 A \,b^{6} c -256 B \,b^{7}\right )}{45045 \sqrt {x \left (c x +b \right )}\, c^{6}}\) | \(177\) |
2/45045*(c*x+b)*(3003*B*c^5*x^5+3465*A*c^5*x^4-2310*B*b*c^4*x^4-2520*A*b*c ^4*x^3+1680*B*b^2*c^3*x^3+1680*A*b^2*c^3*x^2-1120*B*b^3*c^2*x^2-960*A*b^3* c^2*x+640*B*b^4*c*x+384*A*b^4*c-256*B*b^5)*(c*x^2+b*x)^(3/2)/c^6/x^(3/2)
Time = 0.30 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.84 \[ \int x^{5/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left (3003 \, B c^{7} x^{7} - 256 \, B b^{7} + 384 \, A b^{6} c + 231 \, {\left (16 \, B b c^{6} + 15 \, A c^{7}\right )} x^{6} + 63 \, {\left (B b^{2} c^{5} + 70 \, A b c^{6}\right )} x^{5} - 35 \, {\left (2 \, B b^{3} c^{4} - 3 \, A b^{2} c^{5}\right )} x^{4} + 40 \, {\left (2 \, B b^{4} c^{3} - 3 \, A b^{3} c^{4}\right )} x^{3} - 48 \, {\left (2 \, B b^{5} c^{2} - 3 \, A b^{4} c^{3}\right )} x^{2} + 64 \, {\left (2 \, B b^{6} c - 3 \, A b^{5} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{45045 \, c^{6} \sqrt {x}} \]
2/45045*(3003*B*c^7*x^7 - 256*B*b^7 + 384*A*b^6*c + 231*(16*B*b*c^6 + 15*A *c^7)*x^6 + 63*(B*b^2*c^5 + 70*A*b*c^6)*x^5 - 35*(2*B*b^3*c^4 - 3*A*b^2*c^ 5)*x^4 + 40*(2*B*b^4*c^3 - 3*A*b^3*c^4)*x^3 - 48*(2*B*b^5*c^2 - 3*A*b^4*c^ 3)*x^2 + 64*(2*B*b^6*c - 3*A*b^5*c^2)*x)*sqrt(c*x^2 + b*x)/(c^6*sqrt(x))
\[ \int x^{5/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\int x^{\frac {5}{2}} \left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )\, dx \]
Time = 0.21 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.54 \[ \int x^{5/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left (5 \, {\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} x^{5} + 13 \, {\left (315 \, b c^{5} x^{6} + 35 \, b^{2} c^{4} x^{5} - 40 \, b^{3} c^{3} x^{4} + 48 \, b^{4} c^{2} x^{3} - 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )} x^{4}\right )} \sqrt {c x + b} A}{45045 \, c^{5} x^{5}} + \frac {2 \, {\left ({\left (3003 \, c^{7} x^{7} + 231 \, b c^{6} x^{6} - 252 \, b^{2} c^{5} x^{5} + 280 \, b^{3} c^{4} x^{4} - 320 \, b^{4} c^{3} x^{3} + 384 \, b^{5} c^{2} x^{2} - 512 \, b^{6} c x + 1024 \, b^{7}\right )} x^{6} + 5 \, {\left (693 \, b c^{6} x^{7} + 63 \, b^{2} c^{5} x^{6} - 70 \, b^{3} c^{4} x^{5} + 80 \, b^{4} c^{3} x^{4} - 96 \, b^{5} c^{2} x^{3} + 128 \, b^{6} c x^{2} - 256 \, b^{7} x\right )} x^{5}\right )} \sqrt {c x + b} B}{45045 \, c^{6} x^{6}} \]
2/45045*(5*(693*c^6*x^6 + 63*b*c^5*x^5 - 70*b^2*c^4*x^4 + 80*b^3*c^3*x^3 - 96*b^4*c^2*x^2 + 128*b^5*c*x - 256*b^6)*x^5 + 13*(315*b*c^5*x^6 + 35*b^2* c^4*x^5 - 40*b^3*c^3*x^4 + 48*b^4*c^2*x^3 - 64*b^5*c*x^2 + 128*b^6*x)*x^4) *sqrt(c*x + b)*A/(c^5*x^5) + 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 + 5*(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)*x^5)*sqrt( c*x + b)*B/(c^6*x^6)
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (171) = 342\).
Time = 0.29 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.66 \[ \int x^{5/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=-\frac {2}{45045} \, B c {\left (\frac {1024 \, b^{\frac {15}{2}}}{c^{7}} - \frac {3003 \, {\left (c x + b\right )}^{\frac {15}{2}} - 20790 \, {\left (c x + b\right )}^{\frac {13}{2}} b + 61425 \, {\left (c x + b\right )}^{\frac {11}{2}} b^{2} - 100100 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{3} + 96525 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{4} - 54054 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{6}}{c^{7}}\right )} + \frac {2}{9009} \, B b {\left (\frac {256 \, b^{\frac {13}{2}}}{c^{6}} + \frac {693 \, {\left (c x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (c x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{5}}{c^{6}}\right )} + \frac {2}{9009} \, A c {\left (\frac {256 \, b^{\frac {13}{2}}}{c^{6}} + \frac {693 \, {\left (c x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (c x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{5}}{c^{6}}\right )} - \frac {2}{3465} \, A b {\left (\frac {128 \, b^{\frac {11}{2}}}{c^{5}} - \frac {315 \, {\left (c x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (c x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{4}}{c^{5}}\right )} \]
-2/45045*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 + 9652 5*(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*(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) + 2/9009*A*c*(2 56*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*(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. \[ \int x^{5/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\int x^{5/2}\,{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \]