Integrand size = 29, antiderivative size = 157 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=2 a^6 A \sqrt {x}+\frac {2}{3} a^5 (6 A b+a B) x^{3/2}+\frac {6}{5} a^4 b (5 A b+2 a B) x^{5/2}+\frac {10}{7} a^3 b^2 (4 A b+3 a B) x^{7/2}+\frac {10}{9} a^2 b^3 (3 A b+4 a B) x^{9/2}+\frac {6}{11} a b^4 (2 A b+5 a B) x^{11/2}+\frac {2}{13} b^5 (A b+6 a B) x^{13/2}+\frac {2}{15} b^6 B x^{15/2} \]
2/3*a^5*(6*A*b+B*a)*x^(3/2)+6/5*a^4*b*(5*A*b+2*B*a)*x^(5/2)+10/7*a^3*b^2*( 4*A*b+3*B*a)*x^(7/2)+10/9*a^2*b^3*(3*A*b+4*B*a)*x^(9/2)+6/11*a*b^4*(2*A*b+ 5*B*a)*x^(11/2)+2/13*b^5*(A*b+6*B*a)*x^(13/2)+2/15*b^6*B*x^(15/2)+2*a^6*A* x^(1/2)
Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (15015 a^6 (3 A+B x)+18018 a^5 b x (5 A+3 B x)+19305 a^4 b^2 x^2 (7 A+5 B x)+14300 a^3 b^3 x^3 (9 A+7 B x)+6825 a^2 b^4 x^4 (11 A+9 B x)+1890 a b^5 x^5 (13 A+11 B x)+231 b^6 x^6 (15 A+13 B x)\right )}{45045} \]
(2*Sqrt[x]*(15015*a^6*(3*A + B*x) + 18018*a^5*b*x*(5*A + 3*B*x) + 19305*a^ 4*b^2*x^2*(7*A + 5*B*x) + 14300*a^3*b^3*x^3*(9*A + 7*B*x) + 6825*a^2*b^4*x ^4*(11*A + 9*B*x) + 1890*a*b^5*x^5*(13*A + 11*B*x) + 231*b^6*x^6*(15*A + 1 3*B*x)))/45045
Time = 0.30 (sec) , antiderivative size = 157, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x)}{\sqrt {x}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6 (A+B x)}{\sqrt {x}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{\sqrt {x}}dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {a^6 A}{\sqrt {x}}+a^5 \sqrt {x} (a B+6 A b)+3 a^4 b x^{3/2} (2 a B+5 A b)+5 a^3 b^2 x^{5/2} (3 a B+4 A b)+5 a^2 b^3 x^{7/2} (4 a B+3 A b)+b^5 x^{11/2} (6 a B+A b)+3 a b^4 x^{9/2} (5 a B+2 A b)+b^6 B x^{13/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 a^6 A \sqrt {x}+\frac {2}{3} a^5 x^{3/2} (a B+6 A b)+\frac {6}{5} a^4 b x^{5/2} (2 a B+5 A b)+\frac {10}{7} a^3 b^2 x^{7/2} (3 a B+4 A b)+\frac {10}{9} a^2 b^3 x^{9/2} (4 a B+3 A b)+\frac {2}{13} b^5 x^{13/2} (6 a B+A b)+\frac {6}{11} a b^4 x^{11/2} (5 a B+2 A b)+\frac {2}{15} b^6 B x^{15/2}\) |
2*a^6*A*Sqrt[x] + (2*a^5*(6*A*b + a*B)*x^(3/2))/3 + (6*a^4*b*(5*A*b + 2*a* B)*x^(5/2))/5 + (10*a^3*b^2*(4*A*b + 3*a*B)*x^(7/2))/7 + (10*a^2*b^3*(3*A* b + 4*a*B)*x^(9/2))/9 + (6*a*b^4*(2*A*b + 5*a*B)*x^(11/2))/11 + (2*b^5*(A* b + 6*a*B)*x^(13/2))/13 + (2*b^6*B*x^(15/2))/15
3.8.51.3.1 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 = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94
| method | result | size |
| trager | \(\left (\frac {2}{15} b^{6} B \,x^{7}+\frac {2}{13} A \,b^{6} x^{6}+\frac {12}{13} x^{6} B a \,b^{5}+\frac {12}{11} a A \,b^{5} x^{5}+\frac {30}{11} x^{5} B \,b^{4} a^{2}+\frac {10}{3} a^{2} A \,b^{4} x^{4}+\frac {40}{9} x^{4} B \,a^{3} b^{3}+\frac {40}{7} a^{3} A \,b^{3} x^{3}+\frac {30}{7} x^{3} B \,a^{4} b^{2}+6 a^{4} A \,b^{2} x^{2}+\frac {12}{5} x^{2} B \,a^{5} b +4 a^{5} A b x +\frac {2}{3} x B \,a^{6}+2 A \,a^{6}\right ) \sqrt {x}\) | \(147\) |
| gosper | \(\frac {2 \sqrt {x}\, \left (3003 b^{6} B \,x^{7}+3465 A \,b^{6} x^{6}+20790 x^{6} B a \,b^{5}+24570 a A \,b^{5} x^{5}+61425 x^{5} B \,b^{4} a^{2}+75075 a^{2} A \,b^{4} x^{4}+100100 x^{4} B \,a^{3} b^{3}+128700 a^{3} A \,b^{3} x^{3}+96525 x^{3} B \,a^{4} b^{2}+135135 a^{4} A \,b^{2} x^{2}+54054 x^{2} B \,a^{5} b +90090 a^{5} A b x +15015 x B \,a^{6}+45045 A \,a^{6}\right )}{45045}\) | \(148\) |
| derivativedivides | \(\frac {2 b^{6} B \,x^{\frac {15}{2}}}{15}+\frac {2 \left (A \,b^{6}+6 B a \,b^{5}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (6 A a \,b^{5}+15 B \,b^{4} a^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (15 A \,b^{4} a^{2}+20 B \,a^{3} b^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (6 A \,a^{5} b +B \,a^{6}\right ) x^{\frac {3}{2}}}{3}+2 a^{6} A \sqrt {x}\) | \(148\) |
| default | \(\frac {2 b^{6} B \,x^{\frac {15}{2}}}{15}+\frac {2 \left (A \,b^{6}+6 B a \,b^{5}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (6 A a \,b^{5}+15 B \,b^{4} a^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (15 A \,b^{4} a^{2}+20 B \,a^{3} b^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (6 A \,a^{5} b +B \,a^{6}\right ) x^{\frac {3}{2}}}{3}+2 a^{6} A \sqrt {x}\) | \(148\) |
| risch | \(\frac {2 \sqrt {x}\, \left (3003 b^{6} B \,x^{7}+3465 A \,b^{6} x^{6}+20790 x^{6} B a \,b^{5}+24570 a A \,b^{5} x^{5}+61425 x^{5} B \,b^{4} a^{2}+75075 a^{2} A \,b^{4} x^{4}+100100 x^{4} B \,a^{3} b^{3}+128700 a^{3} A \,b^{3} x^{3}+96525 x^{3} B \,a^{4} b^{2}+135135 a^{4} A \,b^{2} x^{2}+54054 x^{2} B \,a^{5} b +90090 a^{5} A b x +15015 x B \,a^{6}+45045 A \,a^{6}\right )}{45045}\) | \(148\) |
(2/15*b^6*B*x^7+2/13*A*b^6*x^6+12/13*x^6*B*a*b^5+12/11*a*A*b^5*x^5+30/11*x ^5*B*b^4*a^2+10/3*a^2*A*b^4*x^4+40/9*x^4*B*a^3*b^3+40/7*a^3*A*b^3*x^3+30/7 *x^3*B*a^4*b^2+6*a^4*A*b^2*x^2+12/5*x^2*B*a^5*b+4*a^5*A*b*x+2/3*x*B*a^6+2* A*a^6)*x^(1/2)
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=\frac {2}{45045} \, {\left (3003 \, B b^{6} x^{7} + 45045 \, A a^{6} + 3465 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 12285 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 25025 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 32175 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 27027 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 15015 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} \sqrt {x} \]
2/45045*(3003*B*b^6*x^7 + 45045*A*a^6 + 3465*(6*B*a*b^5 + A*b^6)*x^6 + 122 85*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 25025*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 32175*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 27027*(2*B*a^5*b + 5*A*a^4*b^2)*x ^2 + 15015*(B*a^6 + 6*A*a^5*b)*x)*sqrt(x)
Time = 0.33 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=2 A a^{6} \sqrt {x} + 4 A a^{5} b x^{\frac {3}{2}} + 6 A a^{4} b^{2} x^{\frac {5}{2}} + \frac {40 A a^{3} b^{3} x^{\frac {7}{2}}}{7} + \frac {10 A a^{2} b^{4} x^{\frac {9}{2}}}{3} + \frac {12 A a b^{5} x^{\frac {11}{2}}}{11} + \frac {2 A b^{6} x^{\frac {13}{2}}}{13} + \frac {2 B a^{6} x^{\frac {3}{2}}}{3} + \frac {12 B a^{5} b x^{\frac {5}{2}}}{5} + \frac {30 B a^{4} b^{2} x^{\frac {7}{2}}}{7} + \frac {40 B a^{3} b^{3} x^{\frac {9}{2}}}{9} + \frac {30 B a^{2} b^{4} x^{\frac {11}{2}}}{11} + \frac {12 B a b^{5} x^{\frac {13}{2}}}{13} + \frac {2 B b^{6} x^{\frac {15}{2}}}{15} \]
2*A*a**6*sqrt(x) + 4*A*a**5*b*x**(3/2) + 6*A*a**4*b**2*x**(5/2) + 40*A*a** 3*b**3*x**(7/2)/7 + 10*A*a**2*b**4*x**(9/2)/3 + 12*A*a*b**5*x**(11/2)/11 + 2*A*b**6*x**(13/2)/13 + 2*B*a**6*x**(3/2)/3 + 12*B*a**5*b*x**(5/2)/5 + 30 *B*a**4*b**2*x**(7/2)/7 + 40*B*a**3*b**3*x**(9/2)/9 + 30*B*a**2*b**4*x**(1 1/2)/11 + 12*B*a*b**5*x**(13/2)/13 + 2*B*b**6*x**(15/2)/15
Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=\frac {2}{15} \, B b^{6} x^{\frac {15}{2}} + 2 \, A a^{6} \sqrt {x} + \frac {2}{13} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {13}{2}} + \frac {6}{11} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {11}{2}} + \frac {10}{9} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac {9}{2}} + \frac {10}{7} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac {7}{2}} + \frac {6}{5} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{\frac {3}{2}} \]
2/15*B*b^6*x^(15/2) + 2*A*a^6*sqrt(x) + 2/13*(6*B*a*b^5 + A*b^6)*x^(13/2) + 6/11*(5*B*a^2*b^4 + 2*A*a*b^5)*x^(11/2) + 10/9*(4*B*a^3*b^3 + 3*A*a^2*b^ 4)*x^(9/2) + 10/7*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^(7/2) + 6/5*(2*B*a^5*b + 5 *A*a^4*b^2)*x^(5/2) + 2/3*(B*a^6 + 6*A*a^5*b)*x^(3/2)
Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=\frac {2}{15} \, B b^{6} x^{\frac {15}{2}} + \frac {12}{13} \, B a b^{5} x^{\frac {13}{2}} + \frac {2}{13} \, A b^{6} x^{\frac {13}{2}} + \frac {30}{11} \, B a^{2} b^{4} x^{\frac {11}{2}} + \frac {12}{11} \, A a b^{5} x^{\frac {11}{2}} + \frac {40}{9} \, B a^{3} b^{3} x^{\frac {9}{2}} + \frac {10}{3} \, A a^{2} b^{4} x^{\frac {9}{2}} + \frac {30}{7} \, B a^{4} b^{2} x^{\frac {7}{2}} + \frac {40}{7} \, A a^{3} b^{3} x^{\frac {7}{2}} + \frac {12}{5} \, B a^{5} b x^{\frac {5}{2}} + 6 \, A a^{4} b^{2} x^{\frac {5}{2}} + \frac {2}{3} \, B a^{6} x^{\frac {3}{2}} + 4 \, A a^{5} b x^{\frac {3}{2}} + 2 \, A a^{6} \sqrt {x} \]
2/15*B*b^6*x^(15/2) + 12/13*B*a*b^5*x^(13/2) + 2/13*A*b^6*x^(13/2) + 30/11 *B*a^2*b^4*x^(11/2) + 12/11*A*a*b^5*x^(11/2) + 40/9*B*a^3*b^3*x^(9/2) + 10 /3*A*a^2*b^4*x^(9/2) + 30/7*B*a^4*b^2*x^(7/2) + 40/7*A*a^3*b^3*x^(7/2) + 1 2/5*B*a^5*b*x^(5/2) + 6*A*a^4*b^2*x^(5/2) + 2/3*B*a^6*x^(3/2) + 4*A*a^5*b* x^(3/2) + 2*A*a^6*sqrt(x)
Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.83 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {x}} \, dx=x^{3/2}\,\left (\frac {2\,B\,a^6}{3}+4\,A\,b\,a^5\right )+x^{13/2}\,\left (\frac {2\,A\,b^6}{13}+\frac {12\,B\,a\,b^5}{13}\right )+2\,A\,a^6\,\sqrt {x}+\frac {2\,B\,b^6\,x^{15/2}}{15}+\frac {10\,a^3\,b^2\,x^{7/2}\,\left (4\,A\,b+3\,B\,a\right )}{7}+\frac {10\,a^2\,b^3\,x^{9/2}\,\left (3\,A\,b+4\,B\,a\right )}{9}+\frac {6\,a^4\,b\,x^{5/2}\,\left (5\,A\,b+2\,B\,a\right )}{5}+\frac {6\,a\,b^4\,x^{11/2}\,\left (2\,A\,b+5\,B\,a\right )}{11} \]