Integrand size = 29, antiderivative size = 109 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {x}} \, dx=2 a^4 A \sqrt {x}+\frac {2}{3} a^3 (4 A b+a B) x^{3/2}+\frac {4}{5} a^2 b (3 A b+2 a B) x^{5/2}+\frac {4}{7} a b^2 (2 A b+3 a B) x^{7/2}+\frac {2}{9} b^3 (A b+4 a B) x^{9/2}+\frac {2}{11} b^4 B x^{11/2} \]
2/3*a^3*(4*A*b+B*a)*x^(3/2)+4/5*a^2*b*(3*A*b+2*B*a)*x^(5/2)+4/7*a*b^2*(2*A *b+3*B*a)*x^(7/2)+2/9*b^3*(A*b+4*B*a)*x^(9/2)+2/11*b^4*B*x^(11/2)+2*a^4*A* x^(1/2)
Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (1155 a^4 (3 A+B x)+924 a^3 b x (5 A+3 B x)+594 a^2 b^2 x^2 (7 A+5 B x)+220 a b^3 x^3 (9 A+7 B x)+35 b^4 x^4 (11 A+9 B x)\right )}{3465} \]
(2*Sqrt[x]*(1155*a^4*(3*A + B*x) + 924*a^3*b*x*(5*A + 3*B*x) + 594*a^2*b^2 *x^2*(7*A + 5*B*x) + 220*a*b^3*x^3*(9*A + 7*B*x) + 35*b^4*x^4*(11*A + 9*B* x)))/3465
Time = 0.24 (sec) , antiderivative size = 109, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x)}{\sqrt {x}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int \frac {b^4 (a+b x)^4 (A+B x)}{\sqrt {x}}dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^4 (A+B x)}{\sqrt {x}}dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (\frac {a^4 A}{\sqrt {x}}+a^3 \sqrt {x} (a B+4 A b)+2 a^2 b x^{3/2} (2 a B+3 A b)+b^3 x^{7/2} (4 a B+A b)+2 a b^2 x^{5/2} (3 a B+2 A b)+b^4 B x^{9/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 a^4 A \sqrt {x}+\frac {2}{3} a^3 x^{3/2} (a B+4 A b)+\frac {4}{5} a^2 b x^{5/2} (2 a B+3 A b)+\frac {2}{9} b^3 x^{9/2} (4 a B+A b)+\frac {4}{7} a b^2 x^{7/2} (3 a B+2 A b)+\frac {2}{11} b^4 B x^{11/2}\) |
2*a^4*A*Sqrt[x] + (2*a^3*(4*A*b + a*B)*x^(3/2))/3 + (4*a^2*b*(3*A*b + 2*a* B)*x^(5/2))/5 + (4*a*b^2*(2*A*b + 3*a*B)*x^(7/2))/7 + (2*b^3*(A*b + 4*a*B) *x^(9/2))/9 + (2*b^4*B*x^(11/2))/11
3.8.42.3.1 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 = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91
method | result | size |
trager | \(\left (\frac {2}{11} b^{4} B \,x^{5}+\frac {2}{9} A \,b^{4} x^{4}+\frac {8}{9} x^{4} B \,b^{3} a +\frac {8}{7} a A \,b^{3} x^{3}+\frac {12}{7} x^{3} B \,a^{2} b^{2}+\frac {12}{5} a^{2} A \,b^{2} x^{2}+\frac {8}{5} x^{2} B \,a^{3} b +\frac {8}{3} a^{3} A b x +\frac {2}{3} a^{4} B x +2 A \,a^{4}\right ) \sqrt {x}\) | \(99\) |
gosper | \(\frac {2 \sqrt {x}\, \left (315 b^{4} B \,x^{5}+385 A \,b^{4} x^{4}+1540 x^{4} B \,b^{3} a +1980 a A \,b^{3} x^{3}+2970 x^{3} B \,a^{2} b^{2}+4158 a^{2} A \,b^{2} x^{2}+2772 x^{2} B \,a^{3} b +4620 a^{3} A b x +1155 a^{4} B x +3465 A \,a^{4}\right )}{3465}\) | \(100\) |
derivativedivides | \(\frac {2 b^{4} B \,x^{\frac {11}{2}}}{11}+\frac {2 \left (A \,b^{4}+4 B \,b^{3} a \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (4 A \,b^{3} a +6 B \,a^{2} b^{2}\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (4 A \,a^{3} b +B \,a^{4}\right ) x^{\frac {3}{2}}}{3}+2 a^{4} A \sqrt {x}\) | \(100\) |
default | \(\frac {2 b^{4} B \,x^{\frac {11}{2}}}{11}+\frac {2 \left (A \,b^{4}+4 B \,b^{3} a \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (4 A \,b^{3} a +6 B \,a^{2} b^{2}\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (4 A \,a^{3} b +B \,a^{4}\right ) x^{\frac {3}{2}}}{3}+2 a^{4} A \sqrt {x}\) | \(100\) |
risch | \(\frac {2 \sqrt {x}\, \left (315 b^{4} B \,x^{5}+385 A \,b^{4} x^{4}+1540 x^{4} B \,b^{3} a +1980 a A \,b^{3} x^{3}+2970 x^{3} B \,a^{2} b^{2}+4158 a^{2} A \,b^{2} x^{2}+2772 x^{2} B \,a^{3} b +4620 a^{3} A b x +1155 a^{4} B x +3465 A \,a^{4}\right )}{3465}\) | \(100\) |
(2/11*b^4*B*x^5+2/9*A*b^4*x^4+8/9*x^4*B*b^3*a+8/7*a*A*b^3*x^3+12/7*x^3*B*a ^2*b^2+12/5*a^2*A*b^2*x^2+8/5*x^2*B*a^3*b+8/3*a^3*A*b*x+2/3*a^4*B*x+2*A*a^ 4)*x^(1/2)
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{3465} \, {\left (315 \, B b^{4} x^{5} + 3465 \, A a^{4} + 385 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 990 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 1386 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 1155 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )} \sqrt {x} \]
2/3465*(315*B*b^4*x^5 + 3465*A*a^4 + 385*(4*B*a*b^3 + A*b^4)*x^4 + 990*(3* B*a^2*b^2 + 2*A*a*b^3)*x^3 + 1386*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 1155*(B* a^4 + 4*A*a^3*b)*x)*sqrt(x)
Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {x}} \, dx=2 A a^{4} \sqrt {x} + \frac {8 A a^{3} b x^{\frac {3}{2}}}{3} + \frac {12 A a^{2} b^{2} x^{\frac {5}{2}}}{5} + \frac {8 A a b^{3} x^{\frac {7}{2}}}{7} + \frac {2 A b^{4} x^{\frac {9}{2}}}{9} + \frac {2 B a^{4} x^{\frac {3}{2}}}{3} + \frac {8 B a^{3} b x^{\frac {5}{2}}}{5} + \frac {12 B a^{2} b^{2} x^{\frac {7}{2}}}{7} + \frac {8 B a b^{3} x^{\frac {9}{2}}}{9} + \frac {2 B b^{4} x^{\frac {11}{2}}}{11} \]
2*A*a**4*sqrt(x) + 8*A*a**3*b*x**(3/2)/3 + 12*A*a**2*b**2*x**(5/2)/5 + 8*A *a*b**3*x**(7/2)/7 + 2*A*b**4*x**(9/2)/9 + 2*B*a**4*x**(3/2)/3 + 8*B*a**3* b*x**(5/2)/5 + 12*B*a**2*b**2*x**(7/2)/7 + 8*B*a*b**3*x**(9/2)/9 + 2*B*b** 4*x**(11/2)/11
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{11} \, B b^{4} x^{\frac {11}{2}} + 2 \, A a^{4} \sqrt {x} + \frac {2}{9} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac {9}{2}} + \frac {4}{7} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac {7}{2}} + \frac {4}{5} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{\frac {3}{2}} \]
2/11*B*b^4*x^(11/2) + 2*A*a^4*sqrt(x) + 2/9*(4*B*a*b^3 + A*b^4)*x^(9/2) + 4/7*(3*B*a^2*b^2 + 2*A*a*b^3)*x^(7/2) + 4/5*(2*B*a^3*b + 3*A*a^2*b^2)*x^(5 /2) + 2/3*(B*a^4 + 4*A*a^3*b)*x^(3/2)
Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{11} \, B b^{4} x^{\frac {11}{2}} + \frac {8}{9} \, B a b^{3} x^{\frac {9}{2}} + \frac {2}{9} \, A b^{4} x^{\frac {9}{2}} + \frac {12}{7} \, B a^{2} b^{2} x^{\frac {7}{2}} + \frac {8}{7} \, A a b^{3} x^{\frac {7}{2}} + \frac {8}{5} \, B a^{3} b x^{\frac {5}{2}} + \frac {12}{5} \, A a^{2} b^{2} x^{\frac {5}{2}} + \frac {2}{3} \, B a^{4} x^{\frac {3}{2}} + \frac {8}{3} \, A a^{3} b x^{\frac {3}{2}} + 2 \, A a^{4} \sqrt {x} \]
2/11*B*b^4*x^(11/2) + 8/9*B*a*b^3*x^(9/2) + 2/9*A*b^4*x^(9/2) + 12/7*B*a^2 *b^2*x^(7/2) + 8/7*A*a*b^3*x^(7/2) + 8/5*B*a^3*b*x^(5/2) + 12/5*A*a^2*b^2* x^(5/2) + 2/3*B*a^4*x^(3/2) + 8/3*A*a^3*b*x^(3/2) + 2*A*a^4*sqrt(x)
Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {x}} \, dx=x^{3/2}\,\left (\frac {2\,B\,a^4}{3}+\frac {8\,A\,b\,a^3}{3}\right )+x^{9/2}\,\left (\frac {2\,A\,b^4}{9}+\frac {8\,B\,a\,b^3}{9}\right )+2\,A\,a^4\,\sqrt {x}+\frac {2\,B\,b^4\,x^{11/2}}{11}+\frac {4\,a^2\,b\,x^{5/2}\,\left (3\,A\,b+2\,B\,a\right )}{5}+\frac {4\,a\,b^2\,x^{7/2}\,\left (2\,A\,b+3\,B\,a\right )}{7} \]