Integrand size = 29, antiderivative size = 155 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{7/2}} \, dx=-\frac {2 a^6 A}{5 x^{5/2}}-\frac {2 a^5 (6 A b+a B)}{3 x^{3/2}}-\frac {6 a^4 b (5 A b+2 a B)}{\sqrt {x}}+10 a^3 b^2 (4 A b+3 a B) \sqrt {x}+\frac {10}{3} a^2 b^3 (3 A b+4 a B) x^{3/2}+\frac {6}{5} a b^4 (2 A b+5 a B) x^{5/2}+\frac {2}{7} b^5 (A b+6 a B) x^{7/2}+\frac {2}{9} b^6 B x^{9/2} \]
-2/5*a^6*A/x^(5/2)-2/3*a^5*(6*A*b+B*a)/x^(3/2)+10/3*a^2*b^3*(3*A*b+4*B*a)* x^(3/2)+6/5*a*b^4*(2*A*b+5*B*a)*x^(5/2)+2/7*b^5*(A*b+6*B*a)*x^(7/2)+2/9*b^ 6*B*x^(9/2)-6*a^4*b*(5*A*b+2*B*a)/x^(1/2)+10*a^3*b^2*(4*A*b+3*B*a)*x^(1/2)
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.80 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{7/2}} \, dx=\frac {2 \left (4725 a^4 b^2 x^2 (-A+B x)+2100 a^3 b^3 x^3 (3 A+B x)-630 a^5 b x (A+3 B x)+315 a^2 b^4 x^4 (5 A+3 B x)-21 a^6 (3 A+5 B x)+54 a b^5 x^5 (7 A+5 B x)+5 b^6 x^6 (9 A+7 B x)\right )}{315 x^{5/2}} \]
(2*(4725*a^4*b^2*x^2*(-A + B*x) + 2100*a^3*b^3*x^3*(3*A + B*x) - 630*a^5*b *x*(A + 3*B*x) + 315*a^2*b^4*x^4*(5*A + 3*B*x) - 21*a^6*(3*A + 5*B*x) + 54 *a*b^5*x^5*(7*A + 5*B*x) + 5*b^6*x^6*(9*A + 7*B*x)))/(315*x^(5/2))
Time = 0.30 (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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x)}{x^{7/2}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int \frac {b^6 (a+b x)^6 (A+B x)}{x^{7/2}}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{x^{7/2}}dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (\frac {a^6 A}{x^{7/2}}+\frac {a^5 (a B+6 A b)}{x^{5/2}}+\frac {3 a^4 b (2 a B+5 A b)}{x^{3/2}}+\frac {5 a^3 b^2 (3 a B+4 A b)}{\sqrt {x}}+5 a^2 b^3 \sqrt {x} (4 a B+3 A b)+b^5 x^{5/2} (6 a B+A b)+3 a b^4 x^{3/2} (5 a B+2 A b)+b^6 B x^{7/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^6 A}{5 x^{5/2}}-\frac {2 a^5 (a B+6 A b)}{3 x^{3/2}}-\frac {6 a^4 b (2 a B+5 A b)}{\sqrt {x}}+10 a^3 b^2 \sqrt {x} (3 a B+4 A b)+\frac {10}{3} a^2 b^3 x^{3/2} (4 a B+3 A b)+\frac {2}{7} b^5 x^{7/2} (6 a B+A b)+\frac {6}{5} a b^4 x^{5/2} (5 a B+2 A b)+\frac {2}{9} b^6 B x^{9/2}\) |
(-2*a^6*A)/(5*x^(5/2)) - (2*a^5*(6*A*b + a*B))/(3*x^(3/2)) - (6*a^4*b*(5*A *b + 2*a*B))/Sqrt[x] + 10*a^3*b^2*(4*A*b + 3*a*B)*Sqrt[x] + (10*a^2*b^3*(3 *A*b + 4*a*B)*x^(3/2))/3 + (6*a*b^4*(2*A*b + 5*a*B)*x^(5/2))/5 + (2*b^5*(A *b + 6*a*B)*x^(7/2))/7 + (2*b^6*B*x^(9/2))/9
3.8.54.3.1 Defintions of rubi rules used
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 = 0.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {2 b^{6} B \,x^{\frac {9}{2}}}{9}+\frac {2 A \,b^{6} x^{\frac {7}{2}}}{7}+\frac {12 B a \,b^{5} x^{\frac {7}{2}}}{7}+\frac {12 A a \,b^{5} x^{\frac {5}{2}}}{5}+6 B \,a^{2} b^{4} x^{\frac {5}{2}}+10 A \,a^{2} b^{4} x^{\frac {3}{2}}+\frac {40 B \,a^{3} b^{3} x^{\frac {3}{2}}}{3}+40 A \,a^{3} b^{3} \sqrt {x}+30 B \,a^{4} b^{2} \sqrt {x}-\frac {2 a^{5} \left (6 A b +B a \right )}{3 x^{\frac {3}{2}}}-\frac {2 a^{6} A}{5 x^{\frac {5}{2}}}-\frac {6 a^{4} b \left (5 A b +2 B a \right )}{\sqrt {x}}\) | \(143\) |
default | \(\frac {2 b^{6} B \,x^{\frac {9}{2}}}{9}+\frac {2 A \,b^{6} x^{\frac {7}{2}}}{7}+\frac {12 B a \,b^{5} x^{\frac {7}{2}}}{7}+\frac {12 A a \,b^{5} x^{\frac {5}{2}}}{5}+6 B \,a^{2} b^{4} x^{\frac {5}{2}}+10 A \,a^{2} b^{4} x^{\frac {3}{2}}+\frac {40 B \,a^{3} b^{3} x^{\frac {3}{2}}}{3}+40 A \,a^{3} b^{3} \sqrt {x}+30 B \,a^{4} b^{2} \sqrt {x}-\frac {2 a^{5} \left (6 A b +B a \right )}{3 x^{\frac {3}{2}}}-\frac {2 a^{6} A}{5 x^{\frac {5}{2}}}-\frac {6 a^{4} b \left (5 A b +2 B a \right )}{\sqrt {x}}\) | \(143\) |
gosper | \(-\frac {2 \left (-35 b^{6} B \,x^{7}-45 A \,b^{6} x^{6}-270 x^{6} B a \,b^{5}-378 a A \,b^{5} x^{5}-945 x^{5} B \,b^{4} a^{2}-1575 a^{2} A \,b^{4} x^{4}-2100 x^{4} B \,a^{3} b^{3}-6300 a^{3} A \,b^{3} x^{3}-4725 x^{3} B \,a^{4} b^{2}+4725 a^{4} A \,b^{2} x^{2}+1890 x^{2} B \,a^{5} b +630 a^{5} A b x +105 x B \,a^{6}+63 A \,a^{6}\right )}{315 x^{\frac {5}{2}}}\) | \(148\) |
trager | \(-\frac {2 \left (-35 b^{6} B \,x^{7}-45 A \,b^{6} x^{6}-270 x^{6} B a \,b^{5}-378 a A \,b^{5} x^{5}-945 x^{5} B \,b^{4} a^{2}-1575 a^{2} A \,b^{4} x^{4}-2100 x^{4} B \,a^{3} b^{3}-6300 a^{3} A \,b^{3} x^{3}-4725 x^{3} B \,a^{4} b^{2}+4725 a^{4} A \,b^{2} x^{2}+1890 x^{2} B \,a^{5} b +630 a^{5} A b x +105 x B \,a^{6}+63 A \,a^{6}\right )}{315 x^{\frac {5}{2}}}\) | \(148\) |
risch | \(-\frac {2 \left (-35 b^{6} B \,x^{7}-45 A \,b^{6} x^{6}-270 x^{6} B a \,b^{5}-378 a A \,b^{5} x^{5}-945 x^{5} B \,b^{4} a^{2}-1575 a^{2} A \,b^{4} x^{4}-2100 x^{4} B \,a^{3} b^{3}-6300 a^{3} A \,b^{3} x^{3}-4725 x^{3} B \,a^{4} b^{2}+4725 a^{4} A \,b^{2} x^{2}+1890 x^{2} B \,a^{5} b +630 a^{5} A b x +105 x B \,a^{6}+63 A \,a^{6}\right )}{315 x^{\frac {5}{2}}}\) | \(148\) |
2/9*b^6*B*x^(9/2)+2/7*A*b^6*x^(7/2)+12/7*B*a*b^5*x^(7/2)+12/5*A*a*b^5*x^(5 /2)+6*B*a^2*b^4*x^(5/2)+10*A*a^2*b^4*x^(3/2)+40/3*B*a^3*b^3*x^(3/2)+40*A*a ^3*b^3*x^(1/2)+30*B*a^4*b^2*x^(1/2)-2/3*a^5*(6*A*b+B*a)/x^(3/2)-2/5*a^6*A/ x^(5/2)-6*a^4*b*(5*A*b+2*B*a)/x^(1/2)
Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{7/2}} \, dx=\frac {2 \, {\left (35 \, B b^{6} x^{7} - 63 \, A a^{6} + 45 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 189 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 525 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 1575 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 945 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 105 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac {5}{2}}} \]
2/315*(35*B*b^6*x^7 - 63*A*a^6 + 45*(6*B*a*b^5 + A*b^6)*x^6 + 189*(5*B*a^2 *b^4 + 2*A*a*b^5)*x^5 + 525*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 1575*(3*B*a^ 4*b^2 + 4*A*a^3*b^3)*x^3 - 945*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 105*(B*a^6 + 6*A*a^5*b)*x)/x^(5/2)
Time = 0.61 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.32 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{7/2}} \, dx=- \frac {2 A a^{6}}{5 x^{\frac {5}{2}}} - \frac {4 A a^{5} b}{x^{\frac {3}{2}}} - \frac {30 A a^{4} b^{2}}{\sqrt {x}} + 40 A a^{3} b^{3} \sqrt {x} + 10 A a^{2} b^{4} x^{\frac {3}{2}} + \frac {12 A a b^{5} x^{\frac {5}{2}}}{5} + \frac {2 A b^{6} x^{\frac {7}{2}}}{7} - \frac {2 B a^{6}}{3 x^{\frac {3}{2}}} - \frac {12 B a^{5} b}{\sqrt {x}} + 30 B a^{4} b^{2} \sqrt {x} + \frac {40 B a^{3} b^{3} x^{\frac {3}{2}}}{3} + 6 B a^{2} b^{4} x^{\frac {5}{2}} + \frac {12 B a b^{5} x^{\frac {7}{2}}}{7} + \frac {2 B b^{6} x^{\frac {9}{2}}}{9} \]
-2*A*a**6/(5*x**(5/2)) - 4*A*a**5*b/x**(3/2) - 30*A*a**4*b**2/sqrt(x) + 40 *A*a**3*b**3*sqrt(x) + 10*A*a**2*b**4*x**(3/2) + 12*A*a*b**5*x**(5/2)/5 + 2*A*b**6*x**(7/2)/7 - 2*B*a**6/(3*x**(3/2)) - 12*B*a**5*b/sqrt(x) + 30*B*a **4*b**2*sqrt(x) + 40*B*a**3*b**3*x**(3/2)/3 + 6*B*a**2*b**4*x**(5/2) + 12 *B*a*b**5*x**(7/2)/7 + 2*B*b**6*x**(9/2)/9
Time = 0.19 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{7/2}} \, dx=\frac {2}{9} \, B b^{6} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {7}{2}} + \frac {6}{5} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {5}{2}} + \frac {10}{3} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac {3}{2}} + 10 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} \sqrt {x} - \frac {2 \, {\left (3 \, A a^{6} + 45 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \]
2/9*B*b^6*x^(9/2) + 2/7*(6*B*a*b^5 + A*b^6)*x^(7/2) + 6/5*(5*B*a^2*b^4 + 2 *A*a*b^5)*x^(5/2) + 10/3*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^(3/2) + 10*(3*B*a^4 *b^2 + 4*A*a^3*b^3)*sqrt(x) - 2/15*(3*A*a^6 + 45*(2*B*a^5*b + 5*A*a^4*b^2) *x^2 + 5*(B*a^6 + 6*A*a^5*b)*x)/x^(5/2)
Time = 0.30 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{7/2}} \, dx=\frac {2}{9} \, B b^{6} x^{\frac {9}{2}} + \frac {12}{7} \, B a b^{5} x^{\frac {7}{2}} + \frac {2}{7} \, A b^{6} x^{\frac {7}{2}} + 6 \, B a^{2} b^{4} x^{\frac {5}{2}} + \frac {12}{5} \, A a b^{5} x^{\frac {5}{2}} + \frac {40}{3} \, B a^{3} b^{3} x^{\frac {3}{2}} + 10 \, A a^{2} b^{4} x^{\frac {3}{2}} + 30 \, B a^{4} b^{2} \sqrt {x} + 40 \, A a^{3} b^{3} \sqrt {x} - \frac {2 \, {\left (90 \, B a^{5} b x^{2} + 225 \, A a^{4} b^{2} x^{2} + 5 \, B a^{6} x + 30 \, A a^{5} b x + 3 \, A a^{6}\right )}}{15 \, x^{\frac {5}{2}}} \]
2/9*B*b^6*x^(9/2) + 12/7*B*a*b^5*x^(7/2) + 2/7*A*b^6*x^(7/2) + 6*B*a^2*b^4 *x^(5/2) + 12/5*A*a*b^5*x^(5/2) + 40/3*B*a^3*b^3*x^(3/2) + 10*A*a^2*b^4*x^ (3/2) + 30*B*a^4*b^2*sqrt(x) + 40*A*a^3*b^3*sqrt(x) - 2/15*(90*B*a^5*b*x^2 + 225*A*a^4*b^2*x^2 + 5*B*a^6*x + 30*A*a^5*b*x + 3*A*a^6)/x^(5/2)
Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{7/2}} \, dx=x^{7/2}\,\left (\frac {2\,A\,b^6}{7}+\frac {12\,B\,a\,b^5}{7}\right )-\frac {x\,\left (\frac {2\,B\,a^6}{3}+4\,A\,b\,a^5\right )+\frac {2\,A\,a^6}{5}+x^2\,\left (12\,B\,a^5\,b+30\,A\,a^4\,b^2\right )}{x^{5/2}}+\frac {2\,B\,b^6\,x^{9/2}}{9}+10\,a^3\,b^2\,\sqrt {x}\,\left (4\,A\,b+3\,B\,a\right )+\frac {10\,a^2\,b^3\,x^{3/2}\,\left (3\,A\,b+4\,B\,a\right )}{3}+\frac {6\,a\,b^4\,x^{5/2}\,\left (2\,A\,b+5\,B\,a\right )}{5} \]