Integrand size = 23, antiderivative size = 113 \[ \int x^{7/2} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{9} a^2 A x^{9/2}+\frac {2}{11} a (2 A b+a B) x^{11/2}+\frac {2}{13} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{13/2}+\frac {2}{15} \left (b^2 B+2 A b c+2 a B c\right ) x^{15/2}+\frac {2}{17} c (2 b B+A c) x^{17/2}+\frac {2}{19} B c^2 x^{19/2} \]
2/9*a^2*A*x^(9/2)+2/11*a*(2*A*b+B*a)*x^(11/2)+2/13*(2*a*b*B+A*(2*a*c+b^2)) *x^(13/2)+2/15*(2*A*b*c+2*B*a*c+B*b^2)*x^(15/2)+2/17*c*(A*c+2*B*b)*x^(17/2 )+2/19*B*c^2*x^(19/2)
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90 \[ \int x^{7/2} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2 x^{9/2} \left (20995 a^2 (11 A+9 B x)+1938 a x (15 A (13 b+11 c x)+11 B x (15 b+13 c x))+33 x^2 \left (19 A \left (255 b^2+442 b c x+195 c^2 x^2\right )+13 B x \left (323 b^2+570 b c x+255 c^2 x^2\right )\right )\right )}{2078505} \]
(2*x^(9/2)*(20995*a^2*(11*A + 9*B*x) + 1938*a*x*(15*A*(13*b + 11*c*x) + 11 *B*x*(15*b + 13*c*x)) + 33*x^2*(19*A*(255*b^2 + 442*b*c*x + 195*c^2*x^2) + 13*B*x*(323*b^2 + 570*b*c*x + 255*c^2*x^2))))/2078505
Time = 0.25 (sec) , antiderivative size = 113, 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.
\(\displaystyle \int x^{7/2} (A+B x) \left (a+b x+c x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (a^2 A x^{7/2}+x^{13/2} \left (2 a B c+2 A b c+b^2 B\right )+x^{11/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+a x^{9/2} (a B+2 A b)+c x^{15/2} (A c+2 b B)+B c^2 x^{17/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{9} a^2 A x^{9/2}+\frac {2}{15} x^{15/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{13} x^{13/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{11} a x^{11/2} (a B+2 A b)+\frac {2}{17} c x^{17/2} (A c+2 b B)+\frac {2}{19} B c^2 x^{19/2}\) |
(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(11/2))/11 + (2*(2*a*b*B + A*(b ^2 + 2*a*c))*x^(13/2))/13 + (2*(b^2*B + 2*A*b*c + 2*a*B*c)*x^(15/2))/15 + (2*c*(2*b*B + A*c)*x^(17/2))/17 + (2*B*c^2*x^(19/2))/19
3.10.90.3.1 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 = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {2 B \,c^{2} x^{\frac {19}{2}}}{19}+\frac {2 \left (A \,c^{2}+2 B b c \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (2 A b a +B \,a^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} A \,x^{\frac {9}{2}}}{9}\) | \(94\) |
default | \(\frac {2 B \,c^{2} x^{\frac {19}{2}}}{19}+\frac {2 \left (A \,c^{2}+2 B b c \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (2 A b a +B \,a^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} A \,x^{\frac {9}{2}}}{9}\) | \(94\) |
gosper | \(\frac {2 x^{\frac {9}{2}} \left (109395 B \,c^{2} x^{5}+122265 A \,c^{2} x^{4}+244530 x^{4} B b c +277134 x^{3} A b c +277134 a B c \,x^{3}+138567 B \,b^{2} x^{3}+319770 a A c \,x^{2}+159885 A \,b^{2} x^{2}+319770 B a b \,x^{2}+377910 a A b x +188955 a^{2} B x +230945 A \,a^{2}\right )}{2078505}\) | \(102\) |
trager | \(\frac {2 x^{\frac {9}{2}} \left (109395 B \,c^{2} x^{5}+122265 A \,c^{2} x^{4}+244530 x^{4} B b c +277134 x^{3} A b c +277134 a B c \,x^{3}+138567 B \,b^{2} x^{3}+319770 a A c \,x^{2}+159885 A \,b^{2} x^{2}+319770 B a b \,x^{2}+377910 a A b x +188955 a^{2} B x +230945 A \,a^{2}\right )}{2078505}\) | \(102\) |
risch | \(\frac {2 x^{\frac {9}{2}} \left (109395 B \,c^{2} x^{5}+122265 A \,c^{2} x^{4}+244530 x^{4} B b c +277134 x^{3} A b c +277134 a B c \,x^{3}+138567 B \,b^{2} x^{3}+319770 a A c \,x^{2}+159885 A \,b^{2} x^{2}+319770 B a b \,x^{2}+377910 a A b x +188955 a^{2} B x +230945 A \,a^{2}\right )}{2078505}\) | \(102\) |
2/19*B*c^2*x^(19/2)+2/17*(A*c^2+2*B*b*c)*x^(17/2)+2/15*(2*A*b*c+B*(2*a*c+b ^2))*x^(15/2)+2/13*(2*a*b*B+A*(2*a*c+b^2))*x^(13/2)+2/11*(2*A*a*b+B*a^2)*x ^(11/2)+2/9*a^2*A*x^(9/2)
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.87 \[ \int x^{7/2} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{2078505} \, {\left (109395 \, B c^{2} x^{9} + 122265 \, {\left (2 \, B b c + A c^{2}\right )} x^{8} + 138567 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{7} + 230945 \, A a^{2} x^{4} + 159885 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{6} + 188955 \, {\left (B a^{2} + 2 \, A a b\right )} x^{5}\right )} \sqrt {x} \]
2/2078505*(109395*B*c^2*x^9 + 122265*(2*B*b*c + A*c^2)*x^8 + 138567*(B*b^2 + 2*(B*a + A*b)*c)*x^7 + 230945*A*a^2*x^4 + 159885*(2*B*a*b + A*b^2 + 2*A *a*c)*x^6 + 188955*(B*a^2 + 2*A*a*b)*x^5)*sqrt(x)
Time = 0.67 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.43 \[ \int x^{7/2} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2 A a^{2} x^{\frac {9}{2}}}{9} + \frac {4 A a b x^{\frac {11}{2}}}{11} + \frac {4 A a c x^{\frac {13}{2}}}{13} + \frac {2 A b^{2} x^{\frac {13}{2}}}{13} + \frac {4 A b c x^{\frac {15}{2}}}{15} + \frac {2 A c^{2} x^{\frac {17}{2}}}{17} + \frac {2 B a^{2} x^{\frac {11}{2}}}{11} + \frac {4 B a b x^{\frac {13}{2}}}{13} + \frac {4 B a c x^{\frac {15}{2}}}{15} + \frac {2 B b^{2} x^{\frac {15}{2}}}{15} + \frac {4 B b c x^{\frac {17}{2}}}{17} + \frac {2 B c^{2} x^{\frac {19}{2}}}{19} \]
2*A*a**2*x**(9/2)/9 + 4*A*a*b*x**(11/2)/11 + 4*A*a*c*x**(13/2)/13 + 2*A*b* *2*x**(13/2)/13 + 4*A*b*c*x**(15/2)/15 + 2*A*c**2*x**(17/2)/17 + 2*B*a**2* x**(11/2)/11 + 4*B*a*b*x**(13/2)/13 + 4*B*a*c*x**(15/2)/15 + 2*B*b**2*x**( 15/2)/15 + 4*B*b*c*x**(17/2)/17 + 2*B*c**2*x**(19/2)/19
Time = 0.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int x^{7/2} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{19} \, B c^{2} x^{\frac {19}{2}} + \frac {2}{17} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {17}{2}} + \frac {2}{15} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{\frac {15}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{2}} + \frac {2}{13} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {11}{2}} \]
2/19*B*c^2*x^(19/2) + 2/17*(2*B*b*c + A*c^2)*x^(17/2) + 2/15*(B*b^2 + 2*(B *a + A*b)*c)*x^(15/2) + 2/9*A*a^2*x^(9/2) + 2/13*(2*B*a*b + A*b^2 + 2*A*a* c)*x^(13/2) + 2/11*(B*a^2 + 2*A*a*b)*x^(11/2)
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91 \[ \int x^{7/2} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{19} \, B c^{2} x^{\frac {19}{2}} + \frac {4}{17} \, B b c x^{\frac {17}{2}} + \frac {2}{17} \, A c^{2} x^{\frac {17}{2}} + \frac {2}{15} \, B b^{2} x^{\frac {15}{2}} + \frac {4}{15} \, B a c x^{\frac {15}{2}} + \frac {4}{15} \, A b c x^{\frac {15}{2}} + \frac {4}{13} \, B a b x^{\frac {13}{2}} + \frac {2}{13} \, A b^{2} x^{\frac {13}{2}} + \frac {4}{13} \, A a c x^{\frac {13}{2}} + \frac {2}{11} \, B a^{2} x^{\frac {11}{2}} + \frac {4}{11} \, A a b x^{\frac {11}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{2}} \]
2/19*B*c^2*x^(19/2) + 4/17*B*b*c*x^(17/2) + 2/17*A*c^2*x^(17/2) + 2/15*B*b ^2*x^(15/2) + 4/15*B*a*c*x^(15/2) + 4/15*A*b*c*x^(15/2) + 4/13*B*a*b*x^(13 /2) + 2/13*A*b^2*x^(13/2) + 4/13*A*a*c*x^(13/2) + 2/11*B*a^2*x^(11/2) + 4/ 11*A*a*b*x^(11/2) + 2/9*A*a^2*x^(9/2)
Time = 10.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int x^{7/2} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=x^{11/2}\,\left (\frac {2\,B\,a^2}{11}+\frac {4\,A\,b\,a}{11}\right )+x^{17/2}\,\left (\frac {2\,A\,c^2}{17}+\frac {4\,B\,b\,c}{17}\right )+x^{13/2}\,\left (\frac {2\,A\,b^2}{13}+\frac {4\,B\,a\,b}{13}+\frac {4\,A\,a\,c}{13}\right )+x^{15/2}\,\left (\frac {2\,B\,b^2}{15}+\frac {4\,A\,c\,b}{15}+\frac {4\,B\,a\,c}{15}\right )+\frac {2\,A\,a^2\,x^{9/2}}{9}+\frac {2\,B\,c^2\,x^{19/2}}{19} \]