Integrand size = 29, antiderivative size = 155 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx=-\frac {2 a^6 A}{9 x^{9/2}}-\frac {2 a^5 (6 A b+a B)}{7 x^{7/2}}-\frac {6 a^4 b (5 A b+2 a B)}{5 x^{5/2}}-\frac {10 a^3 b^2 (4 A b+3 a B)}{3 x^{3/2}}-\frac {10 a^2 b^3 (3 A b+4 a B)}{\sqrt {x}}+6 a b^4 (2 A b+5 a B) \sqrt {x}+\frac {2}{3} b^5 (A b+6 a B) x^{3/2}+\frac {2}{5} b^6 B x^{5/2} \] Output:
-2/9*a^6*A/x^(9/2)-2/7*a^5*(6*A*b+B*a)/x^(7/2)-6/5*a^4*b*(5*A*b+2*B*a)/x^( 5/2)-10/3*a^3*b^2*(4*A*b+3*B*a)/x^(3/2)-10*a^2*b^3*(3*A*b+4*B*a)/x^(1/2)+6 *a*b^4*(2*A*b+5*B*a)*x^(1/2)+2/3*b^5*(A*b+6*B*a)*x^(3/2)+2/5*b^6*B*x^(5/2)
Time = 0.14 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.79 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx=-\frac {2 \left (4725 a^2 b^4 x^4 (A-B x)-630 a b^5 x^5 (3 A+B x)+2100 a^3 b^3 x^3 (A+3 B x)-21 b^6 x^6 (5 A+3 B x)+315 a^4 b^2 x^2 (3 A+5 B x)+54 a^5 b x (5 A+7 B x)+5 a^6 (7 A+9 B x)\right )}{315 x^{9/2}} \] Input:
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(11/2),x]
Output:
(-2*(4725*a^2*b^4*x^4*(A - B*x) - 630*a*b^5*x^5*(3*A + B*x) + 2100*a^3*b^3 *x^3*(A + 3*B*x) - 21*b^6*x^6*(5*A + 3*B*x) + 315*a^4*b^2*x^2*(3*A + 5*B*x ) + 54*a^5*b*x*(5*A + 7*B*x) + 5*a^6*(7*A + 9*B*x)))/(315*x^(9/2))
Time = 0.49 (sec) , antiderivative size = 155, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x)}{x^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6 (A+B x)}{x^{11/2}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{x^{11/2}}dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {a^6 A}{x^{11/2}}+\frac {a^5 (a B+6 A b)}{x^{9/2}}+\frac {3 a^4 b (2 a B+5 A b)}{x^{7/2}}+\frac {5 a^3 b^2 (3 a B+4 A b)}{x^{5/2}}+\frac {5 a^2 b^3 (4 a B+3 A b)}{x^{3/2}}+b^5 \sqrt {x} (6 a B+A b)+\frac {3 a b^4 (5 a B+2 A b)}{\sqrt {x}}+b^6 B x^{3/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a^6 A}{9 x^{9/2}}-\frac {2 a^5 (a B+6 A b)}{7 x^{7/2}}-\frac {6 a^4 b (2 a B+5 A b)}{5 x^{5/2}}-\frac {10 a^3 b^2 (3 a B+4 A b)}{3 x^{3/2}}-\frac {10 a^2 b^3 (4 a B+3 A b)}{\sqrt {x}}+\frac {2}{3} b^5 x^{3/2} (6 a B+A b)+6 a b^4 \sqrt {x} (5 a B+2 A b)+\frac {2}{5} b^6 B x^{5/2}\) |
Input:
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(11/2),x]
Output:
(-2*a^6*A)/(9*x^(9/2)) - (2*a^5*(6*A*b + a*B))/(7*x^(7/2)) - (6*a^4*b*(5*A *b + 2*a*B))/(5*x^(5/2)) - (10*a^3*b^2*(4*A*b + 3*a*B))/(3*x^(3/2)) - (10* a^2*b^3*(3*A*b + 4*a*B))/Sqrt[x] + 6*a*b^4*(2*A*b + 5*a*B)*Sqrt[x] + (2*b^ 5*(A*b + 6*a*B)*x^(3/2))/3 + (2*b^6*B*x^(5/2))/5
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 = 0.88 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(\frac {2 b^{6} B \,x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{6} x^{\frac {3}{2}}}{3}+4 B a \,b^{5} x^{\frac {3}{2}}+12 A a \,b^{5} \sqrt {x}+30 B \,a^{2} b^{4} \sqrt {x}-\frac {6 a^{4} b \left (5 A b +2 B a \right )}{5 x^{\frac {5}{2}}}-\frac {2 a^{5} \left (6 A b +B a \right )}{7 x^{\frac {7}{2}}}-\frac {10 a^{3} b^{2} \left (4 A b +3 B a \right )}{3 x^{\frac {3}{2}}}-\frac {10 a^{2} b^{3} \left (3 A b +4 B a \right )}{\sqrt {x}}-\frac {2 a^{6} A}{9 x^{\frac {9}{2}}}\) | \(135\) |
| default | \(\frac {2 b^{6} B \,x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{6} x^{\frac {3}{2}}}{3}+4 B a \,b^{5} x^{\frac {3}{2}}+12 A a \,b^{5} \sqrt {x}+30 B \,a^{2} b^{4} \sqrt {x}-\frac {6 a^{4} b \left (5 A b +2 B a \right )}{5 x^{\frac {5}{2}}}-\frac {2 a^{5} \left (6 A b +B a \right )}{7 x^{\frac {7}{2}}}-\frac {10 a^{3} b^{2} \left (4 A b +3 B a \right )}{3 x^{\frac {3}{2}}}-\frac {10 a^{2} b^{3} \left (3 A b +4 B a \right )}{\sqrt {x}}-\frac {2 a^{6} A}{9 x^{\frac {9}{2}}}\) | \(135\) |
| gosper | \(-\frac {2 \left (-63 b^{6} B \,x^{7}-105 A \,b^{6} x^{6}-630 B a \,b^{5} x^{6}-1890 A a \,b^{5} x^{5}-4725 B \,a^{2} b^{4} x^{5}+4725 A \,a^{2} b^{4} x^{4}+6300 B \,a^{3} b^{3} x^{4}+2100 A \,a^{3} b^{3} x^{3}+1575 B \,a^{4} b^{2} x^{3}+945 A \,a^{4} b^{2} x^{2}+378 B \,a^{5} b \,x^{2}+270 A \,a^{5} b x +45 B \,a^{6} x +35 A \,a^{6}\right )}{315 x^{\frac {9}{2}}}\) | \(148\) |
| trager | \(-\frac {2 \left (-63 b^{6} B \,x^{7}-105 A \,b^{6} x^{6}-630 B a \,b^{5} x^{6}-1890 A a \,b^{5} x^{5}-4725 B \,a^{2} b^{4} x^{5}+4725 A \,a^{2} b^{4} x^{4}+6300 B \,a^{3} b^{3} x^{4}+2100 A \,a^{3} b^{3} x^{3}+1575 B \,a^{4} b^{2} x^{3}+945 A \,a^{4} b^{2} x^{2}+378 B \,a^{5} b \,x^{2}+270 A \,a^{5} b x +45 B \,a^{6} x +35 A \,a^{6}\right )}{315 x^{\frac {9}{2}}}\) | \(148\) |
| risch | \(-\frac {2 \left (-63 b^{6} B \,x^{7}-105 A \,b^{6} x^{6}-630 B a \,b^{5} x^{6}-1890 A a \,b^{5} x^{5}-4725 B \,a^{2} b^{4} x^{5}+4725 A \,a^{2} b^{4} x^{4}+6300 B \,a^{3} b^{3} x^{4}+2100 A \,a^{3} b^{3} x^{3}+1575 B \,a^{4} b^{2} x^{3}+945 A \,a^{4} b^{2} x^{2}+378 B \,a^{5} b \,x^{2}+270 A \,a^{5} b x +45 B \,a^{6} x +35 A \,a^{6}\right )}{315 x^{\frac {9}{2}}}\) | \(148\) |
| orering | \(-\frac {2 \left (-63 b^{6} B \,x^{7}-105 A \,b^{6} x^{6}-630 B a \,b^{5} x^{6}-1890 A a \,b^{5} x^{5}-4725 B \,a^{2} b^{4} x^{5}+4725 A \,a^{2} b^{4} x^{4}+6300 B \,a^{3} b^{3} x^{4}+2100 A \,a^{3} b^{3} x^{3}+1575 B \,a^{4} b^{2} x^{3}+945 A \,a^{4} b^{2} x^{2}+378 B \,a^{5} b \,x^{2}+270 A \,a^{5} b x +45 B \,a^{6} x +35 A \,a^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{315 x^{\frac {9}{2}} \left (b x +a \right )^{6}}\) | \(173\) |
Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(11/2),x,method=_RETURNVERBOSE)
Output:
2/5*b^6*B*x^(5/2)+2/3*A*b^6*x^(3/2)+4*B*a*b^5*x^(3/2)+12*A*a*b^5*x^(1/2)+3 0*B*a^2*b^4*x^(1/2)-6/5*a^4*b*(5*A*b+2*B*a)/x^(5/2)-2/7*a^5*(6*A*b+B*a)/x^ (7/2)-10/3*a^3*b^2*(4*A*b+3*B*a)/x^(3/2)-10*a^2*b^3*(3*A*b+4*B*a)/x^(1/2)- 2/9*a^6*A/x^(9/2)
Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx=\frac {2 \, {\left (63 \, B b^{6} x^{7} - 35 \, A a^{6} + 105 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 945 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 1575 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 189 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 45 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac {9}{2}}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(11/2),x, algorithm="fricas")
Output:
2/315*(63*B*b^6*x^7 - 35*A*a^6 + 105*(6*B*a*b^5 + A*b^6)*x^6 + 945*(5*B*a^ 2*b^4 + 2*A*a*b^5)*x^5 - 1575*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 - 525*(3*B*a ^4*b^2 + 4*A*a^3*b^3)*x^3 - 189*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 45*(B*a^6 + 6*A*a^5*b)*x)/x^(9/2)
Time = 0.80 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.32 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx=- \frac {2 A a^{6}}{9 x^{\frac {9}{2}}} - \frac {12 A a^{5} b}{7 x^{\frac {7}{2}}} - \frac {6 A a^{4} b^{2}}{x^{\frac {5}{2}}} - \frac {40 A a^{3} b^{3}}{3 x^{\frac {3}{2}}} - \frac {30 A a^{2} b^{4}}{\sqrt {x}} + 12 A a b^{5} \sqrt {x} + \frac {2 A b^{6} x^{\frac {3}{2}}}{3} - \frac {2 B a^{6}}{7 x^{\frac {7}{2}}} - \frac {12 B a^{5} b}{5 x^{\frac {5}{2}}} - \frac {10 B a^{4} b^{2}}{x^{\frac {3}{2}}} - \frac {40 B a^{3} b^{3}}{\sqrt {x}} + 30 B a^{2} b^{4} \sqrt {x} + 4 B a b^{5} x^{\frac {3}{2}} + \frac {2 B b^{6} x^{\frac {5}{2}}}{5} \] Input:
integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**(11/2),x)
Output:
-2*A*a**6/(9*x**(9/2)) - 12*A*a**5*b/(7*x**(7/2)) - 6*A*a**4*b**2/x**(5/2) - 40*A*a**3*b**3/(3*x**(3/2)) - 30*A*a**2*b**4/sqrt(x) + 12*A*a*b**5*sqrt (x) + 2*A*b**6*x**(3/2)/3 - 2*B*a**6/(7*x**(7/2)) - 12*B*a**5*b/(5*x**(5/2 )) - 10*B*a**4*b**2/x**(3/2) - 40*B*a**3*b**3/sqrt(x) + 30*B*a**2*b**4*sqr t(x) + 4*B*a*b**5*x**(3/2) + 2*B*b**6*x**(5/2)/5
Time = 0.05 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx=\frac {2}{5} \, B b^{6} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {3}{2}} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} \sqrt {x} - \frac {2 \, {\left (35 \, A a^{6} + 1575 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 189 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 45 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac {9}{2}}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(11/2),x, algorithm="maxima")
Output:
2/5*B*b^6*x^(5/2) + 2/3*(6*B*a*b^5 + A*b^6)*x^(3/2) + 6*(5*B*a^2*b^4 + 2*A *a*b^5)*sqrt(x) - 2/315*(35*A*a^6 + 1575*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 525*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 189*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 45*(B*a^6 + 6*A*a^5*b)*x)/x^(9/2)
Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx=\frac {2}{5} \, B b^{6} x^{\frac {5}{2}} + 4 \, B a b^{5} x^{\frac {3}{2}} + \frac {2}{3} \, A b^{6} x^{\frac {3}{2}} + 30 \, B a^{2} b^{4} \sqrt {x} + 12 \, A a b^{5} \sqrt {x} - \frac {2 \, {\left (6300 \, B a^{3} b^{3} x^{4} + 4725 \, A a^{2} b^{4} x^{4} + 1575 \, B a^{4} b^{2} x^{3} + 2100 \, A a^{3} b^{3} x^{3} + 378 \, B a^{5} b x^{2} + 945 \, A a^{4} b^{2} x^{2} + 45 \, B a^{6} x + 270 \, A a^{5} b x + 35 \, A a^{6}\right )}}{315 \, x^{\frac {9}{2}}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(11/2),x, algorithm="giac")
Output:
2/5*B*b^6*x^(5/2) + 4*B*a*b^5*x^(3/2) + 2/3*A*b^6*x^(3/2) + 30*B*a^2*b^4*s qrt(x) + 12*A*a*b^5*sqrt(x) - 2/315*(6300*B*a^3*b^3*x^4 + 4725*A*a^2*b^4*x ^4 + 1575*B*a^4*b^2*x^3 + 2100*A*a^3*b^3*x^3 + 378*B*a^5*b*x^2 + 945*A*a^4 *b^2*x^2 + 45*B*a^6*x + 270*A*a^5*b*x + 35*A*a^6)/x^(9/2)
Time = 10.68 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx=x^{3/2}\,\left (\frac {2\,A\,b^6}{3}+4\,B\,a\,b^5\right )-\frac {x\,\left (\frac {2\,B\,a^6}{7}+\frac {12\,A\,b\,a^5}{7}\right )+\frac {2\,A\,a^6}{9}+x^2\,\left (\frac {12\,B\,a^5\,b}{5}+6\,A\,a^4\,b^2\right )+x^3\,\left (10\,B\,a^4\,b^2+\frac {40\,A\,a^3\,b^3}{3}\right )+x^4\,\left (40\,B\,a^3\,b^3+30\,A\,a^2\,b^4\right )}{x^{9/2}}+\frac {2\,B\,b^6\,x^{5/2}}{5}+6\,a\,b^4\,\sqrt {x}\,\left (2\,A\,b+5\,B\,a\right ) \] Input:
int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/x^(11/2),x)
Output:
x^(3/2)*((2*A*b^6)/3 + 4*B*a*b^5) - (x*((2*B*a^6)/7 + (12*A*a^5*b)/7) + (2 *A*a^6)/9 + x^2*(6*A*a^4*b^2 + (12*B*a^5*b)/5) + x^3*((40*A*a^3*b^3)/3 + 1 0*B*a^4*b^2) + x^4*(30*A*a^2*b^4 + 40*B*a^3*b^3))/x^(9/2) + (2*B*b^6*x^(5/ 2))/5 + 6*a*b^4*x^(1/2)*(2*A*b + 5*B*a)
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.54 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx=\frac {\frac {2}{5} b^{7} x^{7}+\frac {14}{3} a \,b^{6} x^{6}+42 a^{2} b^{5} x^{5}-70 a^{3} b^{4} x^{4}-\frac {70}{3} a^{4} b^{3} x^{3}-\frac {42}{5} a^{5} b^{2} x^{2}-2 a^{6} b x -\frac {2}{9} a^{7}}{\sqrt {x}\, x^{4}} \] Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(11/2),x)
Output:
(2*( - 5*a**7 - 45*a**6*b*x - 189*a**5*b**2*x**2 - 525*a**4*b**3*x**3 - 15 75*a**3*b**4*x**4 + 945*a**2*b**5*x**5 + 105*a*b**6*x**6 + 9*b**7*x**7))/( 45*sqrt(x)*x**4)