Integrand size = 23, antiderivative size = 176 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{3/2}} \, dx=-\frac {2 a^3 A}{\sqrt {x}}+2 a^2 (3 A b+a B) \sqrt {x}+2 a \left (a b B+A \left (b^2+a c\right )\right ) x^{3/2}+\frac {2}{5} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{5/2}+\frac {2}{7} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{7/2}+\frac {2}{3} c \left (b^2 B+A b c+a B c\right ) x^{9/2}+\frac {2}{11} c^2 (3 b B+A c) x^{11/2}+\frac {2}{13} B c^3 x^{13/2} \]
2*a*(a*b*B+A*(a*c+b^2))*x^(3/2)+2/5*(3*a*B*(a*c+b^2)+A*(6*a*b*c+b^3))*x^(5 /2)+2/7*(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+B*b^3)*x^(7/2)+2/3*c*(A*b*c+B*a*c+B *b^2)*x^(9/2)+2/11*c^2*(A*c+3*B*b)*x^(11/2)+2/13*B*c^3*x^(13/2)-2*a^3*A/x^ (1/2)+2*a^2*(3*A*b+B*a)*x^(1/2)
Time = 0.15 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{3/2}} \, dx=\frac {-30030 a^3 (A-B x)+6006 a^2 x (5 A (3 b+c x)+B x (5 b+3 c x))+286 a x^2 \left (3 A \left (35 b^2+42 b c x+15 c^2 x^2\right )+B x \left (63 b^2+90 b c x+35 c^2 x^2\right )\right )+2 x^3 \left (13 A \left (231 b^3+495 b^2 c x+385 b c^2 x^2+105 c^3 x^3\right )+5 B x \left (429 b^3+1001 b^2 c x+819 b c^2 x^2+231 c^3 x^3\right )\right )}{15015 \sqrt {x}} \]
(-30030*a^3*(A - B*x) + 6006*a^2*x*(5*A*(3*b + c*x) + B*x*(5*b + 3*c*x)) + 286*a*x^2*(3*A*(35*b^2 + 42*b*c*x + 15*c^2*x^2) + B*x*(63*b^2 + 90*b*c*x + 35*c^2*x^2)) + 2*x^3*(13*A*(231*b^3 + 495*b^2*c*x + 385*b*c^2*x^2 + 105* c^3*x^3) + 5*B*x*(429*b^3 + 1001*b^2*c*x + 819*b*c^2*x^2 + 231*c^3*x^3)))/ (15015*Sqrt[x])
Time = 0.32 (sec) , antiderivative size = 176, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {a^3 A}{x^{3/2}}+\frac {a^2 (a B+3 A b)}{\sqrt {x}}+3 c x^{7/2} \left (a B c+A b c+b^2 B\right )+3 a \sqrt {x} \left (A \left (a c+b^2\right )+a b B\right )+x^{5/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+x^{3/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+c^2 x^{9/2} (A c+3 b B)+B c^3 x^{11/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a^3 A}{\sqrt {x}}+2 a^2 \sqrt {x} (a B+3 A b)+\frac {2}{3} c x^{9/2} \left (a B c+A b c+b^2 B\right )+2 a x^{3/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac {2}{7} x^{7/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {2}{5} x^{5/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {2}{11} c^2 x^{11/2} (A c+3 b B)+\frac {2}{13} B c^3 x^{13/2}\) |
(-2*a^3*A)/Sqrt[x] + 2*a^2*(3*A*b + a*B)*Sqrt[x] + 2*a*(a*b*B + A*(b^2 + a *c))*x^(3/2) + (2*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(5/2))/5 + (2* (b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(7/2))/7 + (2*c*(b^2*B + A*b *c + a*B*c)*x^(9/2))/3 + (2*c^2*(3*b*B + A*c)*x^(11/2))/11 + (2*B*c^3*x^(1 3/2))/13
3.11.4.3.1 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 = 0.27 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.09
| method | result | size |
| gosper | \(-\frac {2 \left (-1155 B \,c^{3} x^{7}-1365 A \,c^{3} x^{6}-4095 B b \,c^{2} x^{6}-5005 A b \,c^{2} x^{5}-5005 a B \,c^{2} x^{5}-5005 B \,b^{2} c \,x^{5}-6435 a A \,c^{2} x^{4}-6435 A \,b^{2} c \,x^{4}-12870 B a b c \,x^{4}-2145 x^{4} B \,b^{3}-18018 A a b c \,x^{3}-3003 A \,b^{3} x^{3}-9009 a^{2} B c \,x^{3}-9009 B a \,b^{2} x^{3}-15015 a^{2} A c \,x^{2}-15015 A a \,b^{2} x^{2}-15015 B \,a^{2} b \,x^{2}-45045 A \,a^{2} b x -15015 a^{3} B x +15015 A \,a^{3}\right )}{15015 \sqrt {x}}\) | \(192\) |
| trager | \(-\frac {2 \left (-1155 B \,c^{3} x^{7}-1365 A \,c^{3} x^{6}-4095 B b \,c^{2} x^{6}-5005 A b \,c^{2} x^{5}-5005 a B \,c^{2} x^{5}-5005 B \,b^{2} c \,x^{5}-6435 a A \,c^{2} x^{4}-6435 A \,b^{2} c \,x^{4}-12870 B a b c \,x^{4}-2145 x^{4} B \,b^{3}-18018 A a b c \,x^{3}-3003 A \,b^{3} x^{3}-9009 a^{2} B c \,x^{3}-9009 B a \,b^{2} x^{3}-15015 a^{2} A c \,x^{2}-15015 A a \,b^{2} x^{2}-15015 B \,a^{2} b \,x^{2}-45045 A \,a^{2} b x -15015 a^{3} B x +15015 A \,a^{3}\right )}{15015 \sqrt {x}}\) | \(192\) |
| risch | \(-\frac {2 \left (-1155 B \,c^{3} x^{7}-1365 A \,c^{3} x^{6}-4095 B b \,c^{2} x^{6}-5005 A b \,c^{2} x^{5}-5005 a B \,c^{2} x^{5}-5005 B \,b^{2} c \,x^{5}-6435 a A \,c^{2} x^{4}-6435 A \,b^{2} c \,x^{4}-12870 B a b c \,x^{4}-2145 x^{4} B \,b^{3}-18018 A a b c \,x^{3}-3003 A \,b^{3} x^{3}-9009 a^{2} B c \,x^{3}-9009 B a \,b^{2} x^{3}-15015 a^{2} A c \,x^{2}-15015 A a \,b^{2} x^{2}-15015 B \,a^{2} b \,x^{2}-45045 A \,a^{2} b x -15015 a^{3} B x +15015 A \,a^{3}\right )}{15015 \sqrt {x}}\) | \(192\) |
| derivativedivides | \(\frac {2 B \,c^{3} x^{\frac {13}{2}}}{13}+\frac {2 A \,c^{3} x^{\frac {11}{2}}}{11}+\frac {6 B b \,c^{2} x^{\frac {11}{2}}}{11}+\frac {2 A b \,c^{2} x^{\frac {9}{2}}}{3}+\frac {2 a B \,c^{2} x^{\frac {9}{2}}}{3}+\frac {2 B \,b^{2} c \,x^{\frac {9}{2}}}{3}+\frac {6 a A \,c^{2} x^{\frac {7}{2}}}{7}+\frac {6 A \,b^{2} c \,x^{\frac {7}{2}}}{7}+\frac {12 B a b c \,x^{\frac {7}{2}}}{7}+\frac {2 B \,b^{3} x^{\frac {7}{2}}}{7}+\frac {12 A a b c \,x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}+\frac {6 a^{2} B c \,x^{\frac {5}{2}}}{5}+\frac {6 B a \,b^{2} x^{\frac {5}{2}}}{5}+2 a^{2} A c \,x^{\frac {3}{2}}+2 A a \,b^{2} x^{\frac {3}{2}}+2 B \,a^{2} b \,x^{\frac {3}{2}}+6 A \,a^{2} b \sqrt {x}+2 a^{3} B \sqrt {x}-\frac {2 a^{3} A}{\sqrt {x}}\) | \(194\) |
| default | \(\frac {2 B \,c^{3} x^{\frac {13}{2}}}{13}+\frac {2 A \,c^{3} x^{\frac {11}{2}}}{11}+\frac {6 B b \,c^{2} x^{\frac {11}{2}}}{11}+\frac {2 A b \,c^{2} x^{\frac {9}{2}}}{3}+\frac {2 a B \,c^{2} x^{\frac {9}{2}}}{3}+\frac {2 B \,b^{2} c \,x^{\frac {9}{2}}}{3}+\frac {6 a A \,c^{2} x^{\frac {7}{2}}}{7}+\frac {6 A \,b^{2} c \,x^{\frac {7}{2}}}{7}+\frac {12 B a b c \,x^{\frac {7}{2}}}{7}+\frac {2 B \,b^{3} x^{\frac {7}{2}}}{7}+\frac {12 A a b c \,x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}+\frac {6 a^{2} B c \,x^{\frac {5}{2}}}{5}+\frac {6 B a \,b^{2} x^{\frac {5}{2}}}{5}+2 a^{2} A c \,x^{\frac {3}{2}}+2 A a \,b^{2} x^{\frac {3}{2}}+2 B \,a^{2} b \,x^{\frac {3}{2}}+6 A \,a^{2} b \sqrt {x}+2 a^{3} B \sqrt {x}-\frac {2 a^{3} A}{\sqrt {x}}\) | \(194\) |
-2/15015*(-1155*B*c^3*x^7-1365*A*c^3*x^6-4095*B*b*c^2*x^6-5005*A*b*c^2*x^5 -5005*B*a*c^2*x^5-5005*B*b^2*c*x^5-6435*A*a*c^2*x^4-6435*A*b^2*c*x^4-12870 *B*a*b*c*x^4-2145*B*b^3*x^4-18018*A*a*b*c*x^3-3003*A*b^3*x^3-9009*B*a^2*c* x^3-9009*B*a*b^2*x^3-15015*A*a^2*c*x^2-15015*A*a*b^2*x^2-15015*B*a^2*b*x^2 -45045*A*a^2*b*x-15015*B*a^3*x+15015*A*a^3)/x^(1/2)
Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{3/2}} \, dx=\frac {2 \, {\left (1155 \, B c^{3} x^{7} + 1365 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 5005 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 2145 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 15015 \, A a^{3} + 3003 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 15015 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 15015 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15015 \, \sqrt {x}} \]
2/15015*(1155*B*c^3*x^7 + 1365*(3*B*b*c^2 + A*c^3)*x^6 + 5005*(B*b^2*c + ( B*a + A*b)*c^2)*x^5 + 2145*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 15015*A*a^3 + 3003*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 15 015*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 15015*(B*a^3 + 3*A*a^2*b)*x)/sqrt( x)
Time = 0.41 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.61 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{3/2}} \, dx=- \frac {2 A a^{3}}{\sqrt {x}} + 6 A a^{2} b \sqrt {x} + 2 A a^{2} c x^{\frac {3}{2}} + 2 A a b^{2} x^{\frac {3}{2}} + \frac {12 A a b c x^{\frac {5}{2}}}{5} + \frac {6 A a c^{2} x^{\frac {7}{2}}}{7} + \frac {2 A b^{3} x^{\frac {5}{2}}}{5} + \frac {6 A b^{2} c x^{\frac {7}{2}}}{7} + \frac {2 A b c^{2} x^{\frac {9}{2}}}{3} + \frac {2 A c^{3} x^{\frac {11}{2}}}{11} + 2 B a^{3} \sqrt {x} + 2 B a^{2} b x^{\frac {3}{2}} + \frac {6 B a^{2} c x^{\frac {5}{2}}}{5} + \frac {6 B a b^{2} x^{\frac {5}{2}}}{5} + \frac {12 B a b c x^{\frac {7}{2}}}{7} + \frac {2 B a c^{2} x^{\frac {9}{2}}}{3} + \frac {2 B b^{3} x^{\frac {7}{2}}}{7} + \frac {2 B b^{2} c x^{\frac {9}{2}}}{3} + \frac {6 B b c^{2} x^{\frac {11}{2}}}{11} + \frac {2 B c^{3} x^{\frac {13}{2}}}{13} \]
-2*A*a**3/sqrt(x) + 6*A*a**2*b*sqrt(x) + 2*A*a**2*c*x**(3/2) + 2*A*a*b**2* x**(3/2) + 12*A*a*b*c*x**(5/2)/5 + 6*A*a*c**2*x**(7/2)/7 + 2*A*b**3*x**(5/ 2)/5 + 6*A*b**2*c*x**(7/2)/7 + 2*A*b*c**2*x**(9/2)/3 + 2*A*c**3*x**(11/2)/ 11 + 2*B*a**3*sqrt(x) + 2*B*a**2*b*x**(3/2) + 6*B*a**2*c*x**(5/2)/5 + 6*B* a*b**2*x**(5/2)/5 + 12*B*a*b*c*x**(7/2)/7 + 2*B*a*c**2*x**(9/2)/3 + 2*B*b* *3*x**(7/2)/7 + 2*B*b**2*c*x**(9/2)/3 + 6*B*b*c**2*x**(11/2)/11 + 2*B*c**3 *x**(13/2)/13
Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{3/2}} \, dx=\frac {2}{13} \, B c^{3} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac {7}{2}} - \frac {2 \, A a^{3}}{\sqrt {x}} + \frac {2}{5} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac {5}{2}} + 2 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac {3}{2}} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt {x} \]
2/13*B*c^3*x^(13/2) + 2/11*(3*B*b*c^2 + A*c^3)*x^(11/2) + 2/3*(B*b^2*c + ( B*a + A*b)*c^2)*x^(9/2) + 2/7*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)* x^(7/2) - 2*A*a^3/sqrt(x) + 2/5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c )*x^(5/2) + 2*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^(3/2) + 2*(B*a^3 + 3*A*a^2*b )*sqrt(x)
Time = 0.40 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{3/2}} \, dx=\frac {2}{13} \, B c^{3} x^{\frac {13}{2}} + \frac {6}{11} \, B b c^{2} x^{\frac {11}{2}} + \frac {2}{11} \, A c^{3} x^{\frac {11}{2}} + \frac {2}{3} \, B b^{2} c x^{\frac {9}{2}} + \frac {2}{3} \, B a c^{2} x^{\frac {9}{2}} + \frac {2}{3} \, A b c^{2} x^{\frac {9}{2}} + \frac {2}{7} \, B b^{3} x^{\frac {7}{2}} + \frac {12}{7} \, B a b c x^{\frac {7}{2}} + \frac {6}{7} \, A b^{2} c x^{\frac {7}{2}} + \frac {6}{7} \, A a c^{2} x^{\frac {7}{2}} + \frac {6}{5} \, B a b^{2} x^{\frac {5}{2}} + \frac {2}{5} \, A b^{3} x^{\frac {5}{2}} + \frac {6}{5} \, B a^{2} c x^{\frac {5}{2}} + \frac {12}{5} \, A a b c x^{\frac {5}{2}} + 2 \, B a^{2} b x^{\frac {3}{2}} + 2 \, A a b^{2} x^{\frac {3}{2}} + 2 \, A a^{2} c x^{\frac {3}{2}} + 2 \, B a^{3} \sqrt {x} + 6 \, A a^{2} b \sqrt {x} - \frac {2 \, A a^{3}}{\sqrt {x}} \]
2/13*B*c^3*x^(13/2) + 6/11*B*b*c^2*x^(11/2) + 2/11*A*c^3*x^(11/2) + 2/3*B* b^2*c*x^(9/2) + 2/3*B*a*c^2*x^(9/2) + 2/3*A*b*c^2*x^(9/2) + 2/7*B*b^3*x^(7 /2) + 12/7*B*a*b*c*x^(7/2) + 6/7*A*b^2*c*x^(7/2) + 6/7*A*a*c^2*x^(7/2) + 6 /5*B*a*b^2*x^(5/2) + 2/5*A*b^3*x^(5/2) + 6/5*B*a^2*c*x^(5/2) + 12/5*A*a*b* c*x^(5/2) + 2*B*a^2*b*x^(3/2) + 2*A*a*b^2*x^(3/2) + 2*A*a^2*c*x^(3/2) + 2* B*a^3*sqrt(x) + 6*A*a^2*b*sqrt(x) - 2*A*a^3/sqrt(x)
Time = 0.05 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^{3/2}} \, dx=x^{5/2}\,\left (\frac {6\,B\,c\,a^2}{5}+\frac {6\,B\,a\,b^2}{5}+\frac {12\,A\,c\,a\,b}{5}+\frac {2\,A\,b^3}{5}\right )+x^{7/2}\,\left (\frac {2\,B\,b^3}{7}+\frac {6\,A\,b^2\,c}{7}+\frac {12\,B\,a\,b\,c}{7}+\frac {6\,A\,a\,c^2}{7}\right )+\sqrt {x}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+x^{11/2}\,\left (\frac {2\,A\,c^3}{11}+\frac {6\,B\,b\,c^2}{11}\right )+x^{3/2}\,\left (2\,B\,a^2\,b+2\,A\,c\,a^2+2\,A\,a\,b^2\right )+x^{9/2}\,\left (\frac {2\,B\,b^2\,c}{3}+\frac {2\,A\,b\,c^2}{3}+\frac {2\,B\,a\,c^2}{3}\right )-\frac {2\,A\,a^3}{\sqrt {x}}+\frac {2\,B\,c^3\,x^{13/2}}{13} \]
x^(5/2)*((2*A*b^3)/5 + (6*B*a*b^2)/5 + (6*B*a^2*c)/5 + (12*A*a*b*c)/5) + x ^(7/2)*((2*B*b^3)/7 + (6*A*a*c^2)/7 + (6*A*b^2*c)/7 + (12*B*a*b*c)/7) + x^ (1/2)*(2*B*a^3 + 6*A*a^2*b) + x^(11/2)*((2*A*c^3)/11 + (6*B*b*c^2)/11) + x ^(3/2)*(2*A*a*b^2 + 2*A*a^2*c + 2*B*a^2*b) + x^(9/2)*((2*A*b*c^2)/3 + (2*B *a*c^2)/3 + (2*B*b^2*c)/3) - (2*A*a^3)/x^(1/2) + (2*B*c^3*x^(13/2))/13