Integrand size = 23, antiderivative size = 111 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{\sqrt {x}} \, dx=2 a^2 A \sqrt {x}+\frac {2}{3} a (2 A b+a B) x^{3/2}+\frac {2}{5} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{5/2}+\frac {2}{7} \left (b^2 B+2 A b c+2 a B c\right ) x^{7/2}+\frac {2}{9} c (2 b B+A c) x^{9/2}+\frac {2}{11} B c^2 x^{11/2} \]
2/3*a*(2*A*b+B*a)*x^(3/2)+2/5*(2*a*b*B+A*(2*a*c+b^2))*x^(5/2)+2/7*(2*A*b*c +2*B*a*c+B*b^2)*x^(7/2)+2/9*c*(A*c+2*B*b)*x^(9/2)+2/11*B*c^2*x^(11/2)+2*a^ 2*A*x^(1/2)
Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.90 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (1155 a^2 (3 A+B x)+66 a x (7 A (5 b+3 c x)+3 B x (7 b+5 c x))+x^2 \left (11 A \left (63 b^2+90 b c x+35 c^2 x^2\right )+5 B x \left (99 b^2+154 b c x+63 c^2 x^2\right )\right )\right )}{3465} \]
(2*Sqrt[x]*(1155*a^2*(3*A + B*x) + 66*a*x*(7*A*(5*b + 3*c*x) + 3*B*x*(7*b + 5*c*x)) + x^2*(11*A*(63*b^2 + 90*b*c*x + 35*c^2*x^2) + 5*B*x*(99*b^2 + 1 54*b*c*x + 63*c^2*x^2))))/3465
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1195, 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.
\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{\sqrt {x}} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {a^2 A}{\sqrt {x}}+x^{5/2} \left (2 a B c+2 A b c+b^2 B\right )+x^{3/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+a \sqrt {x} (a B+2 A b)+c x^{7/2} (A c+2 b B)+B c^2 x^{9/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 a^2 A \sqrt {x}+\frac {2}{7} x^{7/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{5} x^{5/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{9} c x^{9/2} (A c+2 b B)+\frac {2}{11} B c^2 x^{11/2}\) |
2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*(2*a*b*B + A*(b^2 + 2 *a*c))*x^(5/2))/5 + (2*(b^2*B + 2*A*b*c + 2*a*B*c)*x^(7/2))/7 + (2*c*(2*b* B + A*c)*x^(9/2))/9 + (2*B*c^2*x^(11/2))/11
3.10.94.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.51 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {2 B \,c^{2} x^{\frac {11}{2}}}{11}+\frac {2 \left (A \,c^{2}+2 B b c \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (2 A b a +B \,a^{2}\right ) x^{\frac {3}{2}}}{3}+2 a^{2} A \sqrt {x}\) | \(94\) |
default | \(\frac {2 B \,c^{2} x^{\frac {11}{2}}}{11}+\frac {2 \left (A \,c^{2}+2 B b c \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (2 A b a +B \,a^{2}\right ) x^{\frac {3}{2}}}{3}+2 a^{2} A \sqrt {x}\) | \(94\) |
trager | \(\left (\frac {2}{11} B \,c^{2} x^{5}+\frac {2}{9} A \,c^{2} x^{4}+\frac {4}{9} x^{4} B b c +\frac {4}{7} x^{3} A b c +\frac {4}{7} a B c \,x^{3}+\frac {2}{7} B \,b^{2} x^{3}+\frac {4}{5} a A c \,x^{2}+\frac {2}{5} A \,b^{2} x^{2}+\frac {4}{5} B a b \,x^{2}+\frac {4}{3} a A b x +\frac {2}{3} a^{2} B x +2 A \,a^{2}\right ) \sqrt {x}\) | \(101\) |
gosper | \(\frac {2 \sqrt {x}\, \left (315 B \,c^{2} x^{5}+385 A \,c^{2} x^{4}+770 x^{4} B b c +990 x^{3} A b c +990 a B c \,x^{3}+495 B \,b^{2} x^{3}+1386 a A c \,x^{2}+693 A \,b^{2} x^{2}+1386 B a b \,x^{2}+2310 a A b x +1155 a^{2} B x +3465 A \,a^{2}\right )}{3465}\) | \(102\) |
risch | \(\frac {2 \sqrt {x}\, \left (315 B \,c^{2} x^{5}+385 A \,c^{2} x^{4}+770 x^{4} B b c +990 x^{3} A b c +990 a B c \,x^{3}+495 B \,b^{2} x^{3}+1386 a A c \,x^{2}+693 A \,b^{2} x^{2}+1386 B a b \,x^{2}+2310 a A b x +1155 a^{2} B x +3465 A \,a^{2}\right )}{3465}\) | \(102\) |
2/11*B*c^2*x^(11/2)+2/9*(A*c^2+2*B*b*c)*x^(9/2)+2/7*(2*A*b*c+B*(2*a*c+b^2) )*x^(7/2)+2/5*(2*a*b*B+A*(2*a*c+b^2))*x^(5/2)+2/3*(2*A*a*b+B*a^2)*x^(3/2)+ 2*a^2*A*x^(1/2)
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{3465} \, {\left (315 \, B c^{2} x^{5} + 385 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 495 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 3465 \, A a^{2} + 693 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 1155 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )} \sqrt {x} \]
2/3465*(315*B*c^2*x^5 + 385*(2*B*b*c + A*c^2)*x^4 + 495*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 3465*A*a^2 + 693*(2*B*a*b + A*b^2 + 2*A*a*c)*x^2 + 1155*(B*a ^2 + 2*A*a*b)*x)*sqrt(x)
Time = 0.19 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.44 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{\sqrt {x}} \, dx=2 A a^{2} \sqrt {x} + \frac {4 A a b x^{\frac {3}{2}}}{3} + \frac {4 A a c x^{\frac {5}{2}}}{5} + \frac {2 A b^{2} x^{\frac {5}{2}}}{5} + \frac {4 A b c x^{\frac {7}{2}}}{7} + \frac {2 A c^{2} x^{\frac {9}{2}}}{9} + \frac {2 B a^{2} x^{\frac {3}{2}}}{3} + \frac {4 B a b x^{\frac {5}{2}}}{5} + \frac {4 B a c x^{\frac {7}{2}}}{7} + \frac {2 B b^{2} x^{\frac {7}{2}}}{7} + \frac {4 B b c x^{\frac {9}{2}}}{9} + \frac {2 B c^{2} x^{\frac {11}{2}}}{11} \]
2*A*a**2*sqrt(x) + 4*A*a*b*x**(3/2)/3 + 4*A*a*c*x**(5/2)/5 + 2*A*b**2*x**( 5/2)/5 + 4*A*b*c*x**(7/2)/7 + 2*A*c**2*x**(9/2)/9 + 2*B*a**2*x**(3/2)/3 + 4*B*a*b*x**(5/2)/5 + 4*B*a*c*x**(7/2)/7 + 2*B*b**2*x**(7/2)/7 + 4*B*b*c*x* *(9/2)/9 + 2*B*c**2*x**(11/2)/11
Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{11} \, B c^{2} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{\frac {7}{2}} + 2 \, A a^{2} \sqrt {x} + \frac {2}{5} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {3}{2}} \]
2/11*B*c^2*x^(11/2) + 2/9*(2*B*b*c + A*c^2)*x^(9/2) + 2/7*(B*b^2 + 2*(B*a + A*b)*c)*x^(7/2) + 2*A*a^2*sqrt(x) + 2/5*(2*B*a*b + A*b^2 + 2*A*a*c)*x^(5 /2) + 2/3*(B*a^2 + 2*A*a*b)*x^(3/2)
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{11} \, B c^{2} x^{\frac {11}{2}} + \frac {4}{9} \, B b c x^{\frac {9}{2}} + \frac {2}{9} \, A c^{2} x^{\frac {9}{2}} + \frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + \frac {4}{7} \, B a c x^{\frac {7}{2}} + \frac {4}{7} \, A b c x^{\frac {7}{2}} + \frac {4}{5} \, B a b x^{\frac {5}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} + \frac {4}{5} \, A a c x^{\frac {5}{2}} + \frac {2}{3} \, B a^{2} x^{\frac {3}{2}} + \frac {4}{3} \, A a b x^{\frac {3}{2}} + 2 \, A a^{2} \sqrt {x} \]
2/11*B*c^2*x^(11/2) + 4/9*B*b*c*x^(9/2) + 2/9*A*c^2*x^(9/2) + 2/7*B*b^2*x^ (7/2) + 4/7*B*a*c*x^(7/2) + 4/7*A*b*c*x^(7/2) + 4/5*B*a*b*x^(5/2) + 2/5*A* b^2*x^(5/2) + 4/5*A*a*c*x^(5/2) + 2/3*B*a^2*x^(3/2) + 4/3*A*a*b*x^(3/2) + 2*A*a^2*sqrt(x)
Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{\sqrt {x}} \, dx=x^{3/2}\,\left (\frac {2\,B\,a^2}{3}+\frac {4\,A\,b\,a}{3}\right )+x^{9/2}\,\left (\frac {2\,A\,c^2}{9}+\frac {4\,B\,b\,c}{9}\right )+x^{5/2}\,\left (\frac {2\,A\,b^2}{5}+\frac {4\,B\,a\,b}{5}+\frac {4\,A\,a\,c}{5}\right )+x^{7/2}\,\left (\frac {2\,B\,b^2}{7}+\frac {4\,A\,c\,b}{7}+\frac {4\,B\,a\,c}{7}\right )+2\,A\,a^2\,\sqrt {x}+\frac {2\,B\,c^2\,x^{11/2}}{11} \]