Integrand size = 29, antiderivative size = 159 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2}{7} a^6 A x^{7/2}+\frac {2}{9} a^5 (6 A b+a B) x^{9/2}+\frac {6}{11} a^4 b (5 A b+2 a B) x^{11/2}+\frac {10}{13} a^3 b^2 (4 A b+3 a B) x^{13/2}+\frac {2}{3} a^2 b^3 (3 A b+4 a B) x^{15/2}+\frac {6}{17} a b^4 (2 A b+5 a B) x^{17/2}+\frac {2}{19} b^5 (A b+6 a B) x^{19/2}+\frac {2}{21} b^6 B x^{21/2} \]
2/7*a^6*A*x^(7/2)+2/9*a^5*(6*A*b+B*a)*x^(9/2)+6/11*a^4*b*(5*A*b+2*B*a)*x^( 11/2)+10/13*a^3*b^2*(4*A*b+3*B*a)*x^(13/2)+2/3*a^2*b^3*(3*A*b+4*B*a)*x^(15 /2)+6/17*a*b^4*(2*A*b+5*B*a)*x^(17/2)+2/19*b^5*(A*b+6*B*a)*x^(19/2)+2/21*b ^6*B*x^(21/2)
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.81 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 x^{7/2} \left (46189 a^6 (9 A+7 B x)+176358 a^5 b x (11 A+9 B x)+305235 a^4 b^2 x^2 (13 A+11 B x)+298452 a^3 b^3 x^3 (15 A+13 B x)+171171 a^2 b^4 x^4 (17 A+15 B x)+54054 a b^5 x^5 (19 A+17 B x)+7293 b^6 x^6 (21 A+19 B x)\right )}{2909907} \]
(2*x^(7/2)*(46189*a^6*(9*A + 7*B*x) + 176358*a^5*b*x*(11*A + 9*B*x) + 3052 35*a^4*b^2*x^2*(13*A + 11*B*x) + 298452*a^3*b^3*x^3*(15*A + 13*B*x) + 1711 71*a^2*b^4*x^4*(17*A + 15*B*x) + 54054*a*b^5*x^5*(19*A + 17*B*x) + 7293*b^ 6*x^6*(21*A + 19*B*x)))/2909907
Time = 0.31 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1184, 27, 85, 2009}
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.
| \(\displaystyle \int x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x) \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^6 x^{5/2} (a+b x)^6 (A+B x)dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int x^{5/2} (a+b x)^6 (A+B x)dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (a^6 A x^{5/2}+a^5 x^{7/2} (a B+6 A b)+3 a^4 b x^{9/2} (2 a B+5 A b)+5 a^3 b^2 x^{11/2} (3 a B+4 A b)+5 a^2 b^3 x^{13/2} (4 a B+3 A b)+b^5 x^{17/2} (6 a B+A b)+3 a b^4 x^{15/2} (5 a B+2 A b)+b^6 B x^{19/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{7} a^6 A x^{7/2}+\frac {2}{9} a^5 x^{9/2} (a B+6 A b)+\frac {6}{11} a^4 b x^{11/2} (2 a B+5 A b)+\frac {10}{13} a^3 b^2 x^{13/2} (3 a B+4 A b)+\frac {2}{3} a^2 b^3 x^{15/2} (4 a B+3 A b)+\frac {2}{19} b^5 x^{19/2} (6 a B+A b)+\frac {6}{17} a b^4 x^{17/2} (5 a B+2 A b)+\frac {2}{21} b^6 B x^{21/2}\) |
(2*a^6*A*x^(7/2))/7 + (2*a^5*(6*A*b + a*B)*x^(9/2))/9 + (6*a^4*b*(5*A*b + 2*a*B)*x^(11/2))/11 + (10*a^3*b^2*(4*A*b + 3*a*B)*x^(13/2))/13 + (2*a^2*b^ 3*(3*A*b + 4*a*B)*x^(15/2))/3 + (6*a*b^4*(2*A*b + 5*a*B)*x^(17/2))/17 + (2 *b^5*(A*b + 6*a*B)*x^(19/2))/19 + (2*b^6*B*x^(21/2))/21
3.8.48.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.93
| method | result | size |
| gosper | \(\frac {2 x^{\frac {7}{2}} \left (138567 b^{6} B \,x^{7}+153153 A \,b^{6} x^{6}+918918 x^{6} B a \,b^{5}+1027026 a A \,b^{5} x^{5}+2567565 x^{5} B \,b^{4} a^{2}+2909907 a^{2} A \,b^{4} x^{4}+3879876 x^{4} B \,a^{3} b^{3}+4476780 a^{3} A \,b^{3} x^{3}+3357585 x^{3} B \,a^{4} b^{2}+3968055 a^{4} A \,b^{2} x^{2}+1587222 x^{2} B \,a^{5} b +1939938 a^{5} A b x +323323 x B \,a^{6}+415701 A \,a^{6}\right )}{2909907}\) | \(148\) |
| derivativedivides | \(\frac {2 b^{6} B \,x^{\frac {21}{2}}}{21}+\frac {2 \left (A \,b^{6}+6 B a \,b^{5}\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (6 A a \,b^{5}+15 B \,b^{4} a^{2}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (15 A \,b^{4} a^{2}+20 B \,a^{3} b^{3}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (6 A \,a^{5} b +B \,a^{6}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{6} A \,x^{\frac {7}{2}}}{7}\) | \(148\) |
| default | \(\frac {2 b^{6} B \,x^{\frac {21}{2}}}{21}+\frac {2 \left (A \,b^{6}+6 B a \,b^{5}\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (6 A a \,b^{5}+15 B \,b^{4} a^{2}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (15 A \,b^{4} a^{2}+20 B \,a^{3} b^{3}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (6 A \,a^{5} b +B \,a^{6}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{6} A \,x^{\frac {7}{2}}}{7}\) | \(148\) |
| trager | \(\frac {2 x^{\frac {7}{2}} \left (138567 b^{6} B \,x^{7}+153153 A \,b^{6} x^{6}+918918 x^{6} B a \,b^{5}+1027026 a A \,b^{5} x^{5}+2567565 x^{5} B \,b^{4} a^{2}+2909907 a^{2} A \,b^{4} x^{4}+3879876 x^{4} B \,a^{3} b^{3}+4476780 a^{3} A \,b^{3} x^{3}+3357585 x^{3} B \,a^{4} b^{2}+3968055 a^{4} A \,b^{2} x^{2}+1587222 x^{2} B \,a^{5} b +1939938 a^{5} A b x +323323 x B \,a^{6}+415701 A \,a^{6}\right )}{2909907}\) | \(148\) |
| risch | \(\frac {2 x^{\frac {7}{2}} \left (138567 b^{6} B \,x^{7}+153153 A \,b^{6} x^{6}+918918 x^{6} B a \,b^{5}+1027026 a A \,b^{5} x^{5}+2567565 x^{5} B \,b^{4} a^{2}+2909907 a^{2} A \,b^{4} x^{4}+3879876 x^{4} B \,a^{3} b^{3}+4476780 a^{3} A \,b^{3} x^{3}+3357585 x^{3} B \,a^{4} b^{2}+3968055 a^{4} A \,b^{2} x^{2}+1587222 x^{2} B \,a^{5} b +1939938 a^{5} A b x +323323 x B \,a^{6}+415701 A \,a^{6}\right )}{2909907}\) | \(148\) |
2/2909907*x^(7/2)*(138567*B*b^6*x^7+153153*A*b^6*x^6+918918*B*a*b^5*x^6+10 27026*A*a*b^5*x^5+2567565*B*a^2*b^4*x^5+2909907*A*a^2*b^4*x^4+3879876*B*a^ 3*b^3*x^4+4476780*A*a^3*b^3*x^3+3357585*B*a^4*b^2*x^3+3968055*A*a^4*b^2*x^ 2+1587222*B*a^5*b*x^2+1939938*A*a^5*b*x+323323*B*a^6*x+415701*A*a^6)
Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.96 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2}{2909907} \, {\left (138567 \, B b^{6} x^{10} + 415701 \, A a^{6} x^{3} + 153153 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{9} + 513513 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{8} + 969969 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{7} + 1119195 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{6} + 793611 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{5} + 323323 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{4}\right )} \sqrt {x} \]
2/2909907*(138567*B*b^6*x^10 + 415701*A*a^6*x^3 + 153153*(6*B*a*b^5 + A*b^ 6)*x^9 + 513513*(5*B*a^2*b^4 + 2*A*a*b^5)*x^8 + 969969*(4*B*a^3*b^3 + 3*A* a^2*b^4)*x^7 + 1119195*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^6 + 793611*(2*B*a^5*b + 5*A*a^4*b^2)*x^5 + 323323*(B*a^6 + 6*A*a^5*b)*x^4)*sqrt(x)
Time = 0.73 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.35 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 A a^{6} x^{\frac {7}{2}}}{7} + \frac {4 A a^{5} b x^{\frac {9}{2}}}{3} + \frac {30 A a^{4} b^{2} x^{\frac {11}{2}}}{11} + \frac {40 A a^{3} b^{3} x^{\frac {13}{2}}}{13} + 2 A a^{2} b^{4} x^{\frac {15}{2}} + \frac {12 A a b^{5} x^{\frac {17}{2}}}{17} + \frac {2 A b^{6} x^{\frac {19}{2}}}{19} + \frac {2 B a^{6} x^{\frac {9}{2}}}{9} + \frac {12 B a^{5} b x^{\frac {11}{2}}}{11} + \frac {30 B a^{4} b^{2} x^{\frac {13}{2}}}{13} + \frac {8 B a^{3} b^{3} x^{\frac {15}{2}}}{3} + \frac {30 B a^{2} b^{4} x^{\frac {17}{2}}}{17} + \frac {12 B a b^{5} x^{\frac {19}{2}}}{19} + \frac {2 B b^{6} x^{\frac {21}{2}}}{21} \]
2*A*a**6*x**(7/2)/7 + 4*A*a**5*b*x**(9/2)/3 + 30*A*a**4*b**2*x**(11/2)/11 + 40*A*a**3*b**3*x**(13/2)/13 + 2*A*a**2*b**4*x**(15/2) + 12*A*a*b**5*x**( 17/2)/17 + 2*A*b**6*x**(19/2)/19 + 2*B*a**6*x**(9/2)/9 + 12*B*a**5*b*x**(1 1/2)/11 + 30*B*a**4*b**2*x**(13/2)/13 + 8*B*a**3*b**3*x**(15/2)/3 + 30*B*a **2*b**4*x**(17/2)/17 + 12*B*a*b**5*x**(19/2)/19 + 2*B*b**6*x**(21/2)/21
Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.92 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2}{21} \, B b^{6} x^{\frac {21}{2}} + \frac {2}{7} \, A a^{6} x^{\frac {7}{2}} + \frac {2}{19} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {19}{2}} + \frac {6}{17} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {17}{2}} + \frac {2}{3} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac {15}{2}} + \frac {10}{13} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac {13}{2}} + \frac {6}{11} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{\frac {9}{2}} \]
2/21*B*b^6*x^(21/2) + 2/7*A*a^6*x^(7/2) + 2/19*(6*B*a*b^5 + A*b^6)*x^(19/2 ) + 6/17*(5*B*a^2*b^4 + 2*A*a*b^5)*x^(17/2) + 2/3*(4*B*a^3*b^3 + 3*A*a^2*b ^4)*x^(15/2) + 10/13*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^(13/2) + 6/11*(2*B*a^5* b + 5*A*a^4*b^2)*x^(11/2) + 2/9*(B*a^6 + 6*A*a^5*b)*x^(9/2)
Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.94 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2}{21} \, B b^{6} x^{\frac {21}{2}} + \frac {12}{19} \, B a b^{5} x^{\frac {19}{2}} + \frac {2}{19} \, A b^{6} x^{\frac {19}{2}} + \frac {30}{17} \, B a^{2} b^{4} x^{\frac {17}{2}} + \frac {12}{17} \, A a b^{5} x^{\frac {17}{2}} + \frac {8}{3} \, B a^{3} b^{3} x^{\frac {15}{2}} + 2 \, A a^{2} b^{4} x^{\frac {15}{2}} + \frac {30}{13} \, B a^{4} b^{2} x^{\frac {13}{2}} + \frac {40}{13} \, A a^{3} b^{3} x^{\frac {13}{2}} + \frac {12}{11} \, B a^{5} b x^{\frac {11}{2}} + \frac {30}{11} \, A a^{4} b^{2} x^{\frac {11}{2}} + \frac {2}{9} \, B a^{6} x^{\frac {9}{2}} + \frac {4}{3} \, A a^{5} b x^{\frac {9}{2}} + \frac {2}{7} \, A a^{6} x^{\frac {7}{2}} \]
2/21*B*b^6*x^(21/2) + 12/19*B*a*b^5*x^(19/2) + 2/19*A*b^6*x^(19/2) + 30/17 *B*a^2*b^4*x^(17/2) + 12/17*A*a*b^5*x^(17/2) + 8/3*B*a^3*b^3*x^(15/2) + 2* A*a^2*b^4*x^(15/2) + 30/13*B*a^4*b^2*x^(13/2) + 40/13*A*a^3*b^3*x^(13/2) + 12/11*B*a^5*b*x^(11/2) + 30/11*A*a^4*b^2*x^(11/2) + 2/9*B*a^6*x^(9/2) + 4 /3*A*a^5*b*x^(9/2) + 2/7*A*a^6*x^(7/2)
Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.82 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^{9/2}\,\left (\frac {2\,B\,a^6}{9}+\frac {4\,A\,b\,a^5}{3}\right )+x^{19/2}\,\left (\frac {2\,A\,b^6}{19}+\frac {12\,B\,a\,b^5}{19}\right )+\frac {2\,A\,a^6\,x^{7/2}}{7}+\frac {2\,B\,b^6\,x^{21/2}}{21}+\frac {10\,a^3\,b^2\,x^{13/2}\,\left (4\,A\,b+3\,B\,a\right )}{13}+\frac {2\,a^2\,b^3\,x^{15/2}\,\left (3\,A\,b+4\,B\,a\right )}{3}+\frac {6\,a^4\,b\,x^{11/2}\,\left (5\,A\,b+2\,B\,a\right )}{11}+\frac {6\,a\,b^4\,x^{17/2}\,\left (2\,A\,b+5\,B\,a\right )}{17} \]
x^(9/2)*((2*B*a^6)/9 + (4*A*a^5*b)/3) + x^(19/2)*((2*A*b^6)/19 + (12*B*a*b ^5)/19) + (2*A*a^6*x^(7/2))/7 + (2*B*b^6*x^(21/2))/21 + (10*a^3*b^2*x^(13/ 2)*(4*A*b + 3*B*a))/13 + (2*a^2*b^3*x^(15/2)*(3*A*b + 4*B*a))/3 + (6*a^4*b *x^(11/2)*(5*A*b + 2*B*a))/11 + (6*a*b^4*x^(17/2)*(2*A*b + 5*B*a))/17