Integrand size = 29, antiderivative size = 111 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2}{9} a^4 A x^{9/2}+\frac {2}{11} a^3 (4 A b+a B) x^{11/2}+\frac {4}{13} a^2 b (3 A b+2 a B) x^{13/2}+\frac {4}{15} a b^2 (2 A b+3 a B) x^{15/2}+\frac {2}{17} b^3 (A b+4 a B) x^{17/2}+\frac {2}{19} b^4 B x^{19/2} \] Output:
2/9*a^4*A*x^(9/2)+2/11*a^3*(4*A*b+B*a)*x^(11/2)+4/13*a^2*b*(3*A*b+2*B*a)*x ^(13/2)+4/15*a*b^2*(2*A*b+3*B*a)*x^(15/2)+2/17*b^3*(A*b+4*B*a)*x^(17/2)+2/ 19*b^4*B*x^(19/2)
Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.81 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 x^{9/2} \left (20995 a^4 (11 A+9 B x)+58140 a^3 b x (13 A+11 B x)+63954 a^2 b^2 x^2 (15 A+13 B x)+32604 a b^3 x^3 (17 A+15 B x)+6435 b^4 x^4 (19 A+17 B x)\right )}{2078505} \] Input:
Integrate[x^(7/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
(2*x^(9/2)*(20995*a^4*(11*A + 9*B*x) + 58140*a^3*b*x*(13*A + 11*B*x) + 639 54*a^2*b^2*x^2*(15*A + 13*B*x) + 32604*a*b^3*x^3*(17*A + 15*B*x) + 6435*b^ 4*x^4*(19*A + 17*B*x)))/2078505
Time = 0.43 (sec) , antiderivative size = 111, 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.
\(\displaystyle \int x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x) \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int b^4 x^{7/2} (a+b x)^4 (A+B x)dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int x^{7/2} (a+b x)^4 (A+B x)dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (a^4 A x^{7/2}+a^3 x^{9/2} (a B+4 A b)+2 a^2 b x^{11/2} (2 a B+3 A b)+b^3 x^{15/2} (4 a B+A b)+2 a b^2 x^{13/2} (3 a B+2 A b)+b^4 B x^{17/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{9} a^4 A x^{9/2}+\frac {2}{11} a^3 x^{11/2} (a B+4 A b)+\frac {4}{13} a^2 b x^{13/2} (2 a B+3 A b)+\frac {2}{17} b^3 x^{17/2} (4 a B+A b)+\frac {4}{15} a b^2 x^{15/2} (3 a B+2 A b)+\frac {2}{19} b^4 B x^{19/2}\) |
Input:
Int[x^(7/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
(2*a^4*A*x^(9/2))/9 + (2*a^3*(4*A*b + a*B)*x^(11/2))/11 + (4*a^2*b*(3*A*b + 2*a*B)*x^(13/2))/13 + (4*a*b^2*(2*A*b + 3*a*B)*x^(15/2))/15 + (2*b^3*(A* b + 4*a*B)*x^(17/2))/17 + (2*b^4*B*x^(19/2))/19
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 = 0.92 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(\frac {2 x^{\frac {9}{2}} \left (109395 b^{4} B \,x^{5}+122265 A \,b^{4} x^{4}+489060 B a \,b^{3} x^{4}+554268 A a \,b^{3} x^{3}+831402 B \,a^{2} b^{2} x^{3}+959310 A \,a^{2} b^{2} x^{2}+639540 B \,a^{3} b \,x^{2}+755820 A \,a^{3} b x +188955 a^{4} B x +230945 a^{4} A \right )}{2078505}\) | \(100\) |
derivativedivides | \(\frac {2 b^{4} B \,x^{\frac {19}{2}}}{19}+\frac {2 \left (A \,b^{4}+4 B a \,b^{3}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (6 a^{2} A \,b^{2}+4 B \,a^{3} b \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (4 A \,a^{3} b +a^{4} B \right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{4} A \,x^{\frac {9}{2}}}{9}\) | \(100\) |
default | \(\frac {2 b^{4} B \,x^{\frac {19}{2}}}{19}+\frac {2 \left (A \,b^{4}+4 B a \,b^{3}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (6 a^{2} A \,b^{2}+4 B \,a^{3} b \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (4 A \,a^{3} b +a^{4} B \right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{4} A \,x^{\frac {9}{2}}}{9}\) | \(100\) |
trager | \(\frac {2 x^{\frac {9}{2}} \left (109395 b^{4} B \,x^{5}+122265 A \,b^{4} x^{4}+489060 B a \,b^{3} x^{4}+554268 A a \,b^{3} x^{3}+831402 B \,a^{2} b^{2} x^{3}+959310 A \,a^{2} b^{2} x^{2}+639540 B \,a^{3} b \,x^{2}+755820 A \,a^{3} b x +188955 a^{4} B x +230945 a^{4} A \right )}{2078505}\) | \(100\) |
risch | \(\frac {2 x^{\frac {9}{2}} \left (109395 b^{4} B \,x^{5}+122265 A \,b^{4} x^{4}+489060 B a \,b^{3} x^{4}+554268 A a \,b^{3} x^{3}+831402 B \,a^{2} b^{2} x^{3}+959310 A \,a^{2} b^{2} x^{2}+639540 B \,a^{3} b \,x^{2}+755820 A \,a^{3} b x +188955 a^{4} B x +230945 a^{4} A \right )}{2078505}\) | \(100\) |
orering | \(\frac {2 \left (109395 b^{4} B \,x^{5}+122265 A \,b^{4} x^{4}+489060 B a \,b^{3} x^{4}+554268 A a \,b^{3} x^{3}+831402 B \,a^{2} b^{2} x^{3}+959310 A \,a^{2} b^{2} x^{2}+639540 B \,a^{3} b \,x^{2}+755820 A \,a^{3} b x +188955 a^{4} B x +230945 a^{4} A \right ) x^{\frac {9}{2}} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}{2078505 \left (b x +a \right )^{4}}\) | \(125\) |
Input:
int(x^(7/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
Output:
2/2078505*x^(9/2)*(109395*B*b^4*x^5+122265*A*b^4*x^4+489060*B*a*b^3*x^4+55 4268*A*a*b^3*x^3+831402*B*a^2*b^2*x^3+959310*A*a^2*b^2*x^2+639540*B*a^3*b* x^2+755820*A*a^3*b*x+188955*B*a^4*x+230945*A*a^4)
Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2}{2078505} \, {\left (109395 \, B b^{4} x^{9} + 230945 \, A a^{4} x^{4} + 122265 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{8} + 277134 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{7} + 319770 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{6} + 188955 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{5}\right )} \sqrt {x} \] Input:
integrate(x^(7/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
Output:
2/2078505*(109395*B*b^4*x^9 + 230945*A*a^4*x^4 + 122265*(4*B*a*b^3 + A*b^4 )*x^8 + 277134*(3*B*a^2*b^2 + 2*A*a*b^3)*x^7 + 319770*(2*B*a^3*b + 3*A*a^2 *b^2)*x^6 + 188955*(B*a^4 + 4*A*a^3*b)*x^5)*sqrt(x)
Time = 0.71 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.33 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 A a^{4} x^{\frac {9}{2}}}{9} + \frac {8 A a^{3} b x^{\frac {11}{2}}}{11} + \frac {12 A a^{2} b^{2} x^{\frac {13}{2}}}{13} + \frac {8 A a b^{3} x^{\frac {15}{2}}}{15} + \frac {2 A b^{4} x^{\frac {17}{2}}}{17} + \frac {2 B a^{4} x^{\frac {11}{2}}}{11} + \frac {8 B a^{3} b x^{\frac {13}{2}}}{13} + \frac {4 B a^{2} b^{2} x^{\frac {15}{2}}}{5} + \frac {8 B a b^{3} x^{\frac {17}{2}}}{17} + \frac {2 B b^{4} x^{\frac {19}{2}}}{19} \] Input:
integrate(x**(7/2)*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2,x)
Output:
2*A*a**4*x**(9/2)/9 + 8*A*a**3*b*x**(11/2)/11 + 12*A*a**2*b**2*x**(13/2)/1 3 + 8*A*a*b**3*x**(15/2)/15 + 2*A*b**4*x**(17/2)/17 + 2*B*a**4*x**(11/2)/1 1 + 8*B*a**3*b*x**(13/2)/13 + 4*B*a**2*b**2*x**(15/2)/5 + 8*B*a*b**3*x**(1 7/2)/17 + 2*B*b**4*x**(19/2)/19
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.89 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2}{19} \, B b^{4} x^{\frac {19}{2}} + \frac {2}{9} \, A a^{4} x^{\frac {9}{2}} + \frac {2}{17} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac {17}{2}} + \frac {4}{15} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac {15}{2}} + \frac {4}{13} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{\frac {11}{2}} \] Input:
integrate(x^(7/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
Output:
2/19*B*b^4*x^(19/2) + 2/9*A*a^4*x^(9/2) + 2/17*(4*B*a*b^3 + A*b^4)*x^(17/2 ) + 4/15*(3*B*a^2*b^2 + 2*A*a*b^3)*x^(15/2) + 4/13*(2*B*a^3*b + 3*A*a^2*b^ 2)*x^(13/2) + 2/11*(B*a^4 + 4*A*a^3*b)*x^(11/2)
Time = 0.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.91 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2}{19} \, B b^{4} x^{\frac {19}{2}} + \frac {8}{17} \, B a b^{3} x^{\frac {17}{2}} + \frac {2}{17} \, A b^{4} x^{\frac {17}{2}} + \frac {4}{5} \, B a^{2} b^{2} x^{\frac {15}{2}} + \frac {8}{15} \, A a b^{3} x^{\frac {15}{2}} + \frac {8}{13} \, B a^{3} b x^{\frac {13}{2}} + \frac {12}{13} \, A a^{2} b^{2} x^{\frac {13}{2}} + \frac {2}{11} \, B a^{4} x^{\frac {11}{2}} + \frac {8}{11} \, A a^{3} b x^{\frac {11}{2}} + \frac {2}{9} \, A a^{4} x^{\frac {9}{2}} \] Input:
integrate(x^(7/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
Output:
2/19*B*b^4*x^(19/2) + 8/17*B*a*b^3*x^(17/2) + 2/17*A*b^4*x^(17/2) + 4/5*B* a^2*b^2*x^(15/2) + 8/15*A*a*b^3*x^(15/2) + 8/13*B*a^3*b*x^(13/2) + 12/13*A *a^2*b^2*x^(13/2) + 2/11*B*a^4*x^(11/2) + 8/11*A*a^3*b*x^(11/2) + 2/9*A*a^ 4*x^(9/2)
Time = 10.76 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.82 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^{11/2}\,\left (\frac {2\,B\,a^4}{11}+\frac {8\,A\,b\,a^3}{11}\right )+x^{17/2}\,\left (\frac {2\,A\,b^4}{17}+\frac {8\,B\,a\,b^3}{17}\right )+\frac {2\,A\,a^4\,x^{9/2}}{9}+\frac {2\,B\,b^4\,x^{19/2}}{19}+\frac {4\,a^2\,b\,x^{13/2}\,\left (3\,A\,b+2\,B\,a\right )}{13}+\frac {4\,a\,b^2\,x^{15/2}\,\left (2\,A\,b+3\,B\,a\right )}{15} \] Input:
int(x^(7/2)*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
Output:
x^(11/2)*((2*B*a^4)/11 + (8*A*a^3*b)/11) + x^(17/2)*((2*A*b^4)/17 + (8*B*a *b^3)/17) + (2*A*a^4*x^(9/2))/9 + (2*B*b^4*x^(19/2))/19 + (4*a^2*b*x^(13/2 )*(3*A*b + 2*B*a))/13 + (4*a*b^2*x^(15/2)*(2*A*b + 3*B*a))/15
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.53 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \sqrt {x}\, x^{4} \left (21879 b^{5} x^{5}+122265 a \,b^{4} x^{4}+277134 a^{2} b^{3} x^{3}+319770 a^{3} b^{2} x^{2}+188955 a^{4} b x +46189 a^{5}\right )}{415701} \] Input:
int(x^(7/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
(2*sqrt(x)*x**4*(46189*a**5 + 188955*a**4*b*x + 319770*a**3*b**2*x**2 + 27 7134*a**2*b**3*x**3 + 122265*a*b**4*x**4 + 21879*b**5*x**5))/415701