Integrand size = 23, antiderivative size = 178 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11/2}} \, dx=-\frac {2 a^3 A}{9 x^{9/2}}-\frac {2 a^2 (3 A b+a B)}{7 x^{7/2}}-\frac {6 a \left (a b B+A \left (b^2+a c\right )\right )}{5 x^{5/2}}-\frac {2 \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right )}{3 x^{3/2}}-\frac {2 \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right )}{\sqrt {x}}+6 c \left (b^2 B+A b c+a B c\right ) \sqrt {x}+\frac {2}{3} c^2 (3 b B+A c) x^{3/2}+\frac {2}{5} B c^3 x^{5/2} \]
-2/9*a^3*A/x^(9/2)-2/7*a^2*(3*A*b+B*a)/x^(7/2)-6/5*a*(a*b*B+A*(a*c+b^2))/x ^(5/2)-2/3*(3*a*B*(a*c+b^2)+A*(6*a*b*c+b^3))/x^(3/2)+2/3*c^2*(A*c+3*B*b)*x ^(3/2)+2/5*B*c^3*x^(5/2)-2*(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+B*b^3)/x^(1/2)+6 *c*(A*b*c+B*a*c+B*b^2)*x^(1/2)
Time = 0.16 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.97 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11/2}} \, dx=-\frac {2 \left (5 a^3 (7 A+9 B x)+9 a^2 x (7 B x (3 b+5 c x)+3 A (5 b+7 c x))+63 a x^2 \left (5 B x \left (b^2+6 b c x-3 c^2 x^2\right )+A \left (3 b^2+10 b c x+15 c^2 x^2\right )\right )+21 x^3 \left (5 A \left (b^3+9 b^2 c x-9 b c^2 x^2-c^3 x^3\right )-3 B x \left (-5 b^3+15 b^2 c x+5 b c^2 x^2+c^3 x^3\right )\right )\right )}{315 x^{9/2}} \]
(-2*(5*a^3*(7*A + 9*B*x) + 9*a^2*x*(7*B*x*(3*b + 5*c*x) + 3*A*(5*b + 7*c*x )) + 63*a*x^2*(5*B*x*(b^2 + 6*b*c*x - 3*c^2*x^2) + A*(3*b^2 + 10*b*c*x + 1 5*c^2*x^2)) + 21*x^3*(5*A*(b^3 + 9*b^2*c*x - 9*b*c^2*x^2 - c^3*x^3) - 3*B* x*(-5*b^3 + 15*b^2*c*x + 5*b*c^2*x^2 + c^3*x^3))))/(315*x^(9/2))
Time = 0.31 (sec) , antiderivative size = 178, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {a^3 A}{x^{11/2}}+\frac {a^2 (a B+3 A b)}{x^{9/2}}+\frac {3 a \left (A \left (a c+b^2\right )+a b B\right )}{x^{7/2}}+\frac {3 c \left (a B c+A b c+b^2 B\right )}{\sqrt {x}}+\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{x^{3/2}}+\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{x^{5/2}}+c^2 \sqrt {x} (A c+3 b B)+B c^3 x^{3/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a^3 A}{9 x^{9/2}}-\frac {2 a^2 (a B+3 A b)}{7 x^{7/2}}-\frac {6 a \left (A \left (a c+b^2\right )+a b B\right )}{5 x^{5/2}}+6 c \sqrt {x} \left (a B c+A b c+b^2 B\right )-\frac {2 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )}{\sqrt {x}}-\frac {2 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )}{3 x^{3/2}}+\frac {2}{3} c^2 x^{3/2} (A c+3 b B)+\frac {2}{5} B c^3 x^{5/2}\) |
(-2*a^3*A)/(9*x^(9/2)) - (2*a^2*(3*A*b + a*B))/(7*x^(7/2)) - (6*a*(a*b*B + A*(b^2 + a*c)))/(5*x^(5/2)) - (2*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))) /(3*x^(3/2)) - (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2))/Sqrt[x] + 6 *c*(b^2*B + A*b*c + a*B*c)*Sqrt[x] + (2*c^2*(3*b*B + A*c)*x^(3/2))/3 + (2* B*c^3*x^(5/2))/5
3.11.8.3.1 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 = 0.26 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(\frac {2 B \,c^{3} x^{\frac {5}{2}}}{5}+\frac {2 A \,c^{3} x^{\frac {3}{2}}}{3}+2 B b \,c^{2} x^{\frac {3}{2}}+6 A b \,c^{2} \sqrt {x}+6 a B \,c^{2} \sqrt {x}+6 B \,b^{2} c \sqrt {x}-\frac {6 a \left (A a c +A \,b^{2}+a b B \right )}{5 x^{\frac {5}{2}}}-\frac {2 a^{3} A}{9 x^{\frac {9}{2}}}-\frac {2 \left (6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}\right )}{3 x^{\frac {3}{2}}}-\frac {2 a^{2} \left (3 A b +B a \right )}{7 x^{\frac {7}{2}}}-\frac {2 \left (3 A a \,c^{2}+3 A \,b^{2} c +6 B a b c +B \,b^{3}\right )}{\sqrt {x}}\) | \(167\) |
| default | \(\frac {2 B \,c^{3} x^{\frac {5}{2}}}{5}+\frac {2 A \,c^{3} x^{\frac {3}{2}}}{3}+2 B b \,c^{2} x^{\frac {3}{2}}+6 A b \,c^{2} \sqrt {x}+6 a B \,c^{2} \sqrt {x}+6 B \,b^{2} c \sqrt {x}-\frac {6 a \left (A a c +A \,b^{2}+a b B \right )}{5 x^{\frac {5}{2}}}-\frac {2 a^{3} A}{9 x^{\frac {9}{2}}}-\frac {2 \left (6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}\right )}{3 x^{\frac {3}{2}}}-\frac {2 a^{2} \left (3 A b +B a \right )}{7 x^{\frac {7}{2}}}-\frac {2 \left (3 A a \,c^{2}+3 A \,b^{2} c +6 B a b c +B \,b^{3}\right )}{\sqrt {x}}\) | \(167\) |
| gosper | \(-\frac {2 \left (-63 B \,c^{3} x^{7}-105 A \,c^{3} x^{6}-315 B b \,c^{2} x^{6}-945 A b \,c^{2} x^{5}-945 a B \,c^{2} x^{5}-945 B \,b^{2} c \,x^{5}+945 a A \,c^{2} x^{4}+945 A \,b^{2} c \,x^{4}+1890 B a b c \,x^{4}+315 x^{4} B \,b^{3}+630 A a b c \,x^{3}+105 A \,b^{3} x^{3}+315 a^{2} B c \,x^{3}+315 B a \,b^{2} x^{3}+189 a^{2} A c \,x^{2}+189 A a \,b^{2} x^{2}+189 B \,a^{2} b \,x^{2}+135 A \,a^{2} b x +45 a^{3} B x +35 A \,a^{3}\right )}{315 x^{\frac {9}{2}}}\) | \(192\) |
| trager | \(-\frac {2 \left (-63 B \,c^{3} x^{7}-105 A \,c^{3} x^{6}-315 B b \,c^{2} x^{6}-945 A b \,c^{2} x^{5}-945 a B \,c^{2} x^{5}-945 B \,b^{2} c \,x^{5}+945 a A \,c^{2} x^{4}+945 A \,b^{2} c \,x^{4}+1890 B a b c \,x^{4}+315 x^{4} B \,b^{3}+630 A a b c \,x^{3}+105 A \,b^{3} x^{3}+315 a^{2} B c \,x^{3}+315 B a \,b^{2} x^{3}+189 a^{2} A c \,x^{2}+189 A a \,b^{2} x^{2}+189 B \,a^{2} b \,x^{2}+135 A \,a^{2} b x +45 a^{3} B x +35 A \,a^{3}\right )}{315 x^{\frac {9}{2}}}\) | \(192\) |
| risch | \(-\frac {2 \left (-63 B \,c^{3} x^{7}-105 A \,c^{3} x^{6}-315 B b \,c^{2} x^{6}-945 A b \,c^{2} x^{5}-945 a B \,c^{2} x^{5}-945 B \,b^{2} c \,x^{5}+945 a A \,c^{2} x^{4}+945 A \,b^{2} c \,x^{4}+1890 B a b c \,x^{4}+315 x^{4} B \,b^{3}+630 A a b c \,x^{3}+105 A \,b^{3} x^{3}+315 a^{2} B c \,x^{3}+315 B a \,b^{2} x^{3}+189 a^{2} A c \,x^{2}+189 A a \,b^{2} x^{2}+189 B \,a^{2} b \,x^{2}+135 A \,a^{2} b x +45 a^{3} B x +35 A \,a^{3}\right )}{315 x^{\frac {9}{2}}}\) | \(192\) |
2/5*B*c^3*x^(5/2)+2/3*A*c^3*x^(3/2)+2*B*b*c^2*x^(3/2)+6*A*b*c^2*x^(1/2)+6* a*B*c^2*x^(1/2)+6*B*b^2*c*x^(1/2)-6/5*a*(A*a*c+A*b^2+B*a*b)/x^(5/2)-2/9*a^ 3*A/x^(9/2)-2/3*(6*A*a*b*c+A*b^3+3*B*a^2*c+3*B*a*b^2)/x^(3/2)-2/7*a^2*(3*A *b+B*a)/x^(7/2)-2*(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+B*b^3)/x^(1/2)
Time = 0.31 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11/2}} \, dx=\frac {2 \, {\left (63 \, B c^{3} x^{7} + 105 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 945 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} - 315 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 35 \, A a^{3} - 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 189 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 45 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{315 \, x^{\frac {9}{2}}} \]
2/315*(63*B*c^3*x^7 + 105*(3*B*b*c^2 + A*c^3)*x^6 + 945*(B*b^2*c + (B*a + A*b)*c^2)*x^5 - 315*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 35*A *a^3 - 105*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 - 189*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 - 45*(B*a^3 + 3*A*a^2*b)*x)/x^(9/2)
Time = 0.78 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11/2}} \, dx=- \frac {2 A a^{3}}{9 x^{\frac {9}{2}}} - \frac {6 A a^{2} b}{7 x^{\frac {7}{2}}} - \frac {6 A a^{2} c}{5 x^{\frac {5}{2}}} - \frac {6 A a b^{2}}{5 x^{\frac {5}{2}}} - \frac {4 A a b c}{x^{\frac {3}{2}}} - \frac {6 A a c^{2}}{\sqrt {x}} - \frac {2 A b^{3}}{3 x^{\frac {3}{2}}} - \frac {6 A b^{2} c}{\sqrt {x}} + 6 A b c^{2} \sqrt {x} + \frac {2 A c^{3} x^{\frac {3}{2}}}{3} - \frac {2 B a^{3}}{7 x^{\frac {7}{2}}} - \frac {6 B a^{2} b}{5 x^{\frac {5}{2}}} - \frac {2 B a^{2} c}{x^{\frac {3}{2}}} - \frac {2 B a b^{2}}{x^{\frac {3}{2}}} - \frac {12 B a b c}{\sqrt {x}} + 6 B a c^{2} \sqrt {x} - \frac {2 B b^{3}}{\sqrt {x}} + 6 B b^{2} c \sqrt {x} + 2 B b c^{2} x^{\frac {3}{2}} + \frac {2 B c^{3} x^{\frac {5}{2}}}{5} \]
-2*A*a**3/(9*x**(9/2)) - 6*A*a**2*b/(7*x**(7/2)) - 6*A*a**2*c/(5*x**(5/2)) - 6*A*a*b**2/(5*x**(5/2)) - 4*A*a*b*c/x**(3/2) - 6*A*a*c**2/sqrt(x) - 2*A *b**3/(3*x**(3/2)) - 6*A*b**2*c/sqrt(x) + 6*A*b*c**2*sqrt(x) + 2*A*c**3*x* *(3/2)/3 - 2*B*a**3/(7*x**(7/2)) - 6*B*a**2*b/(5*x**(5/2)) - 2*B*a**2*c/x* *(3/2) - 2*B*a*b**2/x**(3/2) - 12*B*a*b*c/sqrt(x) + 6*B*a*c**2*sqrt(x) - 2 *B*b**3/sqrt(x) + 6*B*b**2*c*sqrt(x) + 2*B*b*c**2*x**(3/2) + 2*B*c**3*x**( 5/2)/5
Time = 0.20 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11/2}} \, dx=\frac {2}{5} \, B c^{3} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {3}{2}} + 6 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} \sqrt {x} - \frac {2 \, {\left (315 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 35 \, A a^{3} + 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 189 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 45 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{315 \, x^{\frac {9}{2}}} \]
2/5*B*c^3*x^(5/2) + 2/3*(3*B*b*c^2 + A*c^3)*x^(3/2) + 6*(B*b^2*c + (B*a + A*b)*c^2)*sqrt(x) - 2/315*(315*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c) *x^4 + 35*A*a^3 + 105*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 18 9*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 45*(B*a^3 + 3*A*a^2*b)*x)/x^(9/2)
Time = 0.30 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11/2}} \, dx=\frac {2}{5} \, B c^{3} x^{\frac {5}{2}} + 2 \, B b c^{2} x^{\frac {3}{2}} + \frac {2}{3} \, A c^{3} x^{\frac {3}{2}} + 6 \, B b^{2} c \sqrt {x} + 6 \, B a c^{2} \sqrt {x} + 6 \, A b c^{2} \sqrt {x} - \frac {2 \, {\left (315 \, B b^{3} x^{4} + 1890 \, B a b c x^{4} + 945 \, A b^{2} c x^{4} + 945 \, A a c^{2} x^{4} + 315 \, B a b^{2} x^{3} + 105 \, A b^{3} x^{3} + 315 \, B a^{2} c x^{3} + 630 \, A a b c x^{3} + 189 \, B a^{2} b x^{2} + 189 \, A a b^{2} x^{2} + 189 \, A a^{2} c x^{2} + 45 \, B a^{3} x + 135 \, A a^{2} b x + 35 \, A a^{3}\right )}}{315 \, x^{\frac {9}{2}}} \]
2/5*B*c^3*x^(5/2) + 2*B*b*c^2*x^(3/2) + 2/3*A*c^3*x^(3/2) + 6*B*b^2*c*sqrt (x) + 6*B*a*c^2*sqrt(x) + 6*A*b*c^2*sqrt(x) - 2/315*(315*B*b^3*x^4 + 1890* B*a*b*c*x^4 + 945*A*b^2*c*x^4 + 945*A*a*c^2*x^4 + 315*B*a*b^2*x^3 + 105*A* b^3*x^3 + 315*B*a^2*c*x^3 + 630*A*a*b*c*x^3 + 189*B*a^2*b*x^2 + 189*A*a*b^ 2*x^2 + 189*A*a^2*c*x^2 + 45*B*a^3*x + 135*A*a^2*b*x + 35*A*a^3)/x^(9/2)
Time = 0.08 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{11/2}} \, dx=x^{3/2}\,\left (\frac {2\,A\,c^3}{3}+2\,B\,b\,c^2\right )-\frac {x^3\,\left (2\,B\,c\,a^2+2\,B\,a\,b^2+4\,A\,c\,a\,b+\frac {2\,A\,b^3}{3}\right )+x^4\,\left (2\,B\,b^3+6\,A\,b^2\,c+12\,B\,a\,b\,c+6\,A\,a\,c^2\right )+x\,\left (\frac {2\,B\,a^3}{7}+\frac {6\,A\,b\,a^2}{7}\right )+\frac {2\,A\,a^3}{9}+x^2\,\left (\frac {6\,B\,a^2\,b}{5}+\frac {6\,A\,c\,a^2}{5}+\frac {6\,A\,a\,b^2}{5}\right )}{x^{9/2}}+\sqrt {x}\,\left (6\,B\,b^2\,c+6\,A\,b\,c^2+6\,B\,a\,c^2\right )+\frac {2\,B\,c^3\,x^{5/2}}{5} \]
x^(3/2)*((2*A*c^3)/3 + 2*B*b*c^2) - (x^3*((2*A*b^3)/3 + 2*B*a*b^2 + 2*B*a^ 2*c + 4*A*a*b*c) + x^4*(2*B*b^3 + 6*A*a*c^2 + 6*A*b^2*c + 12*B*a*b*c) + x* ((2*B*a^3)/7 + (6*A*a^2*b)/7) + (2*A*a^3)/9 + x^2*((6*A*a*b^2)/5 + (6*A*a^ 2*c)/5 + (6*B*a^2*b)/5))/x^(9/2) + x^(1/2)*(6*A*b*c^2 + 6*B*a*c^2 + 6*B*b^ 2*c) + (2*B*c^3*x^(5/2))/5