Integrand size = 27, antiderivative size = 171 \[ \int \frac {x^6 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {(A b-6 a B) x}{b^7}+\frac {B x^2}{2 b^6}-\frac {a^6 (A b-a B)}{5 b^8 (a+b x)^5}+\frac {a^5 (6 A b-7 a B)}{4 b^8 (a+b x)^4}-\frac {a^4 (5 A b-7 a B)}{b^8 (a+b x)^3}+\frac {5 a^3 (4 A b-7 a B)}{2 b^8 (a+b x)^2}-\frac {5 a^2 (3 A b-7 a B)}{b^8 (a+b x)}-\frac {3 a (2 A b-7 a B) \log (a+b x)}{b^8} \]
(A*b-6*B*a)*x/b^7+1/2*B*x^2/b^6-1/5*a^6*(A*b-B*a)/b^8/(b*x+a)^5+1/4*a^5*(6 *A*b-7*B*a)/b^8/(b*x+a)^4-a^4*(5*A*b-7*B*a)/b^8/(b*x+a)^3+5/2*a^3*(4*A*b-7 *B*a)/b^8/(b*x+a)^2-5*a^2*(3*A*b-7*B*a)/b^8/(b*x+a)-3*a*(2*A*b-7*B*a)*ln(b *x+a)/b^8
Time = 0.07 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.88 \[ \int \frac {x^6 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {20 b (A b-6 a B) x+10 b^2 B x^2+\frac {4 a^6 (-A b+a B)}{(a+b x)^5}+\frac {5 a^5 (6 A b-7 a B)}{(a+b x)^4}+\frac {20 a^4 (-5 A b+7 a B)}{(a+b x)^3}-\frac {50 a^3 (-4 A b+7 a B)}{(a+b x)^2}+\frac {100 a^2 (-3 A b+7 a B)}{a+b x}+60 a (-2 A b+7 a B) \log (a+b x)}{20 b^8} \]
(20*b*(A*b - 6*a*B)*x + 10*b^2*B*x^2 + (4*a^6*(-(A*b) + a*B))/(a + b*x)^5 + (5*a^5*(6*A*b - 7*a*B))/(a + b*x)^4 + (20*a^4*(-5*A*b + 7*a*B))/(a + b*x )^3 - (50*a^3*(-4*A*b + 7*a*B))/(a + b*x)^2 + (100*a^2*(-3*A*b + 7*a*B))/( a + b*x) + 60*a*(-2*A*b + 7*a*B)*Log[a + b*x])/(20*b^8)
Time = 0.43 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1184, 27, 86, 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.
| \(\displaystyle \int \frac {x^6 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {x^6 (A+B x)}{b^6 (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {x^6 (A+B x)}{(a+b x)^6}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (-\frac {a^6 (a B-A b)}{b^7 (a+b x)^6}+\frac {a^5 (7 a B-6 A b)}{b^7 (a+b x)^5}-\frac {3 a^4 (7 a B-5 A b)}{b^7 (a+b x)^4}+\frac {5 a^3 (7 a B-4 A b)}{b^7 (a+b x)^3}-\frac {5 a^2 (7 a B-3 A b)}{b^7 (a+b x)^2}+\frac {3 a (7 a B-2 A b)}{b^7 (a+b x)}+\frac {A b-6 a B}{b^7}+\frac {B x}{b^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^6 (A b-a B)}{5 b^8 (a+b x)^5}+\frac {a^5 (6 A b-7 a B)}{4 b^8 (a+b x)^4}-\frac {a^4 (5 A b-7 a B)}{b^8 (a+b x)^3}+\frac {5 a^3 (4 A b-7 a B)}{2 b^8 (a+b x)^2}-\frac {5 a^2 (3 A b-7 a B)}{b^8 (a+b x)}-\frac {3 a (2 A b-7 a B) \log (a+b x)}{b^8}+\frac {x (A b-6 a B)}{b^7}+\frac {B x^2}{2 b^6}\) |
((A*b - 6*a*B)*x)/b^7 + (B*x^2)/(2*b^6) - (a^6*(A*b - a*B))/(5*b^8*(a + b* x)^5) + (a^5*(6*A*b - 7*a*B))/(4*b^8*(a + b*x)^4) - (a^4*(5*A*b - 7*a*B))/ (b^8*(a + b*x)^3) + (5*a^3*(4*A*b - 7*a*B))/(2*b^8*(a + b*x)^2) - (5*a^2*( 3*A*b - 7*a*B))/(b^8*(a + b*x)) - (3*a*(2*A*b - 7*a*B)*Log[a + b*x])/b^8
3.7.43.3.1 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[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 163, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {\frac {1}{2} B b \,x^{2}+A b x -6 a B x}{b^{7}}+\frac {a^{5} \left (6 A b -7 B a \right )}{4 b^{8} \left (b x +a \right )^{4}}-\frac {a^{6} \left (A b -B a \right )}{5 b^{8} \left (b x +a \right )^{5}}-\frac {3 a \left (2 A b -7 B a \right ) \ln \left (b x +a \right )}{b^{8}}-\frac {5 a^{2} \left (3 A b -7 B a \right )}{b^{8} \left (b x +a \right )}+\frac {5 a^{3} \left (4 A b -7 B a \right )}{2 b^{8} \left (b x +a \right )^{2}}-\frac {a^{4} \left (5 A b -7 B a \right )}{b^{8} \left (b x +a \right )^{3}}\) | \(163\) |
| norman | \(\frac {\frac {B \,x^{7}}{2 b}-\frac {a^{5} \left (274 A b a -959 B \,a^{2}\right )}{20 b^{8}}+\frac {\left (2 A b -7 B a \right ) x^{6}}{2 b^{2}}-\frac {5 a \left (6 A b a -21 B \,a^{2}\right ) x^{4}}{b^{4}}-\frac {5 a^{2} \left (18 A b a -63 B \,a^{2}\right ) x^{3}}{b^{5}}-\frac {5 a^{3} \left (22 A b a -77 B \,a^{2}\right ) x^{2}}{b^{6}}-\frac {5 a^{4} \left (50 A b a -175 B \,a^{2}\right ) x}{4 b^{7}}}{\left (b x +a \right )^{5}}-\frac {3 a \left (2 A b -7 B a \right ) \ln \left (b x +a \right )}{b^{8}}\) | \(166\) |
| risch | \(\frac {B \,x^{2}}{2 b^{6}}+\frac {A x}{b^{6}}-\frac {6 a B x}{b^{7}}+\frac {\left (-15 A \,b^{4} a^{2}+35 B \,a^{3} b^{3}\right ) x^{4}-\frac {5 a^{3} b^{2} \left (20 A b -49 B a \right ) x^{3}}{2}+\left (-65 A \,a^{4} b^{2}+\frac {329}{2} B \,a^{5} b \right ) x^{2}+\left (-\frac {77}{2} A \,a^{5} b +\frac {399}{4} B \,a^{6}\right ) x -\frac {3 a^{6} \left (58 A b -153 B a \right )}{20 b}}{b^{7} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}-\frac {6 a \ln \left (b x +a \right ) A}{b^{7}}+\frac {21 a^{2} \ln \left (b x +a \right ) B}{b^{8}}\) | \(180\) |
| parallelrisch | \(-\frac {120 A \ln \left (b x +a \right ) x^{5} a \,b^{6}-420 B \ln \left (b x +a \right ) x^{5} a^{2} b^{5}+600 A \ln \left (b x +a \right ) x^{4} a^{2} b^{5}-2100 B \ln \left (b x +a \right ) x^{4} a^{3} b^{4}+1200 A \ln \left (b x +a \right ) x^{3} a^{3} b^{4}-4200 B \ln \left (b x +a \right ) x^{3} a^{4} b^{3}+1200 A \ln \left (b x +a \right ) x^{2} a^{4} b^{3}-4200 B \ln \left (b x +a \right ) x^{2} a^{5} b^{2}+600 A \ln \left (b x +a \right ) x \,a^{5} b^{2}-2100 B \ln \left (b x +a \right ) x \,a^{6} b +120 A \ln \left (b x +a \right ) a^{6} b +70 B \,x^{6} a \,b^{6}-959 B \,a^{7}-6300 B \,a^{4} b^{3} x^{3}+2200 A \,a^{4} b^{3} x^{2}-7700 B \,a^{5} b^{2} x^{2}+1250 A \,a^{5} b^{2} x -4375 B \,a^{6} b x +600 A \,a^{2} b^{5} x^{4}-2100 B \,a^{3} b^{4} x^{4}+1800 A \,a^{3} b^{4} x^{3}+274 A \,a^{6} b -10 B \,x^{7} b^{7}-20 A \,x^{6} b^{7}-420 B \ln \left (b x +a \right ) a^{7}}{20 b^{8} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}\) | \(360\) |
1/b^7*(1/2*B*b*x^2+A*b*x-6*a*B*x)+1/4*a^5*(6*A*b-7*B*a)/b^8/(b*x+a)^4-1/5* a^6*(A*b-B*a)/b^8/(b*x+a)^5-3*a*(2*A*b-7*B*a)*ln(b*x+a)/b^8-5*a^2*(3*A*b-7 *B*a)/b^8/(b*x+a)+5/2*a^3*(4*A*b-7*B*a)/b^8/(b*x+a)^2-a^4*(5*A*b-7*B*a)/b^ 8/(b*x+a)^3
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (164) = 328\).
Time = 0.28 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.04 \[ \int \frac {x^6 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {10 \, B b^{7} x^{7} + 459 \, B a^{7} - 174 \, A a^{6} b - 10 \, {\left (7 \, B a b^{6} - 2 \, A b^{7}\right )} x^{6} - 100 \, {\left (5 \, B a^{2} b^{5} - A a b^{6}\right )} x^{5} - 100 \, {\left (4 \, B a^{3} b^{4} + A a^{2} b^{5}\right )} x^{4} + 100 \, {\left (13 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{3} + 300 \, {\left (9 \, B a^{5} b^{2} - 4 \, A a^{4} b^{3}\right )} x^{2} + 375 \, {\left (5 \, B a^{6} b - 2 \, A a^{5} b^{2}\right )} x + 60 \, {\left (7 \, B a^{7} - 2 \, A a^{6} b + {\left (7 \, B a^{2} b^{5} - 2 \, A a b^{6}\right )} x^{5} + 5 \, {\left (7 \, B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )} x^{4} + 10 \, {\left (7 \, B a^{4} b^{3} - 2 \, A a^{3} b^{4}\right )} x^{3} + 10 \, {\left (7 \, B a^{5} b^{2} - 2 \, A a^{4} b^{3}\right )} x^{2} + 5 \, {\left (7 \, B a^{6} b - 2 \, A a^{5} b^{2}\right )} x\right )} \log \left (b x + a\right )}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} \]
1/20*(10*B*b^7*x^7 + 459*B*a^7 - 174*A*a^6*b - 10*(7*B*a*b^6 - 2*A*b^7)*x^ 6 - 100*(5*B*a^2*b^5 - A*a*b^6)*x^5 - 100*(4*B*a^3*b^4 + A*a^2*b^5)*x^4 + 100*(13*B*a^4*b^3 - 8*A*a^3*b^4)*x^3 + 300*(9*B*a^5*b^2 - 4*A*a^4*b^3)*x^2 + 375*(5*B*a^6*b - 2*A*a^5*b^2)*x + 60*(7*B*a^7 - 2*A*a^6*b + (7*B*a^2*b^ 5 - 2*A*a*b^6)*x^5 + 5*(7*B*a^3*b^4 - 2*A*a^2*b^5)*x^4 + 10*(7*B*a^4*b^3 - 2*A*a^3*b^4)*x^3 + 10*(7*B*a^5*b^2 - 2*A*a^4*b^3)*x^2 + 5*(7*B*a^6*b - 2* A*a^5*b^2)*x)*log(b*x + a))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 1 0*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b^8)
Time = 1.61 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.26 \[ \int \frac {x^6 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {B x^{2}}{2 b^{6}} + \frac {3 a \left (- 2 A b + 7 B a\right ) \log {\left (a + b x \right )}}{b^{8}} + x \left (\frac {A}{b^{6}} - \frac {6 B a}{b^{7}}\right ) + \frac {- 174 A a^{6} b + 459 B a^{7} + x^{4} \left (- 300 A a^{2} b^{5} + 700 B a^{3} b^{4}\right ) + x^{3} \left (- 1000 A a^{3} b^{4} + 2450 B a^{4} b^{3}\right ) + x^{2} \left (- 1300 A a^{4} b^{3} + 3290 B a^{5} b^{2}\right ) + x \left (- 770 A a^{5} b^{2} + 1995 B a^{6} b\right )}{20 a^{5} b^{8} + 100 a^{4} b^{9} x + 200 a^{3} b^{10} x^{2} + 200 a^{2} b^{11} x^{3} + 100 a b^{12} x^{4} + 20 b^{13} x^{5}} \]
B*x**2/(2*b**6) + 3*a*(-2*A*b + 7*B*a)*log(a + b*x)/b**8 + x*(A/b**6 - 6*B *a/b**7) + (-174*A*a**6*b + 459*B*a**7 + x**4*(-300*A*a**2*b**5 + 700*B*a* *3*b**4) + x**3*(-1000*A*a**3*b**4 + 2450*B*a**4*b**3) + x**2*(-1300*A*a** 4*b**3 + 3290*B*a**5*b**2) + x*(-770*A*a**5*b**2 + 1995*B*a**6*b))/(20*a** 5*b**8 + 100*a**4*b**9*x + 200*a**3*b**10*x**2 + 200*a**2*b**11*x**3 + 100 *a*b**12*x**4 + 20*b**13*x**5)
Time = 0.21 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.25 \[ \int \frac {x^6 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {459 \, B a^{7} - 174 \, A a^{6} b + 100 \, {\left (7 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} x^{4} + 50 \, {\left (49 \, B a^{4} b^{3} - 20 \, A a^{3} b^{4}\right )} x^{3} + 10 \, {\left (329 \, B a^{5} b^{2} - 130 \, A a^{4} b^{3}\right )} x^{2} + 35 \, {\left (57 \, B a^{6} b - 22 \, A a^{5} b^{2}\right )} x}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} + \frac {B b x^{2} - 2 \, {\left (6 \, B a - A b\right )} x}{2 \, b^{7}} + \frac {3 \, {\left (7 \, B a^{2} - 2 \, A a b\right )} \log \left (b x + a\right )}{b^{8}} \]
1/20*(459*B*a^7 - 174*A*a^6*b + 100*(7*B*a^3*b^4 - 3*A*a^2*b^5)*x^4 + 50*( 49*B*a^4*b^3 - 20*A*a^3*b^4)*x^3 + 10*(329*B*a^5*b^2 - 130*A*a^4*b^3)*x^2 + 35*(57*B*a^6*b - 22*A*a^5*b^2)*x)/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11 *x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b^8) + 1/2*(B*b*x^2 - 2*(6*B*a - A*b)*x)/b^7 + 3*(7*B*a^2 - 2*A*a*b)*log(b*x + a)/b^8
Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.01 \[ \int \frac {x^6 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3 \, {\left (7 \, B a^{2} - 2 \, A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} + \frac {B b^{6} x^{2} - 12 \, B a b^{5} x + 2 \, A b^{6} x}{2 \, b^{12}} + \frac {459 \, B a^{7} - 174 \, A a^{6} b + 100 \, {\left (7 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} x^{4} + 50 \, {\left (49 \, B a^{4} b^{3} - 20 \, A a^{3} b^{4}\right )} x^{3} + 10 \, {\left (329 \, B a^{5} b^{2} - 130 \, A a^{4} b^{3}\right )} x^{2} + 35 \, {\left (57 \, B a^{6} b - 22 \, A a^{5} b^{2}\right )} x}{20 \, {\left (b x + a\right )}^{5} b^{8}} \]
3*(7*B*a^2 - 2*A*a*b)*log(abs(b*x + a))/b^8 + 1/2*(B*b^6*x^2 - 12*B*a*b^5* x + 2*A*b^6*x)/b^12 + 1/20*(459*B*a^7 - 174*A*a^6*b + 100*(7*B*a^3*b^4 - 3 *A*a^2*b^5)*x^4 + 50*(49*B*a^4*b^3 - 20*A*a^3*b^4)*x^3 + 10*(329*B*a^5*b^2 - 130*A*a^4*b^3)*x^2 + 35*(57*B*a^6*b - 22*A*a^5*b^2)*x)/((b*x + a)^5*b^8 )
Time = 9.95 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.23 \[ \int \frac {x^6 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=x\,\left (\frac {A}{b^6}-\frac {6\,B\,a}{b^7}\right )-\frac {x^2\,\left (65\,A\,a^4\,b^2-\frac {329\,B\,a^5\,b}{2}\right )-x\,\left (\frac {399\,B\,a^6}{4}-\frac {77\,A\,a^5\,b}{2}\right )-\frac {3\,\left (153\,B\,a^7-58\,A\,a^6\,b\right )}{20\,b}+x^4\,\left (15\,A\,a^2\,b^4-35\,B\,a^3\,b^3\right )+x^3\,\left (50\,A\,a^3\,b^3-\frac {245\,B\,a^4\,b^2}{2}\right )}{a^5\,b^7+5\,a^4\,b^8\,x+10\,a^3\,b^9\,x^2+10\,a^2\,b^{10}\,x^3+5\,a\,b^{11}\,x^4+b^{12}\,x^5}+\frac {B\,x^2}{2\,b^6}+\frac {\ln \left (a+b\,x\right )\,\left (21\,B\,a^2-6\,A\,a\,b\right )}{b^8} \]
x*(A/b^6 - (6*B*a)/b^7) - (x^2*(65*A*a^4*b^2 - (329*B*a^5*b)/2) - x*((399* B*a^6)/4 - (77*A*a^5*b)/2) - (3*(153*B*a^7 - 58*A*a^6*b))/(20*b) + x^4*(15 *A*a^2*b^4 - 35*B*a^3*b^3) + x^3*(50*A*a^3*b^3 - (245*B*a^4*b^2)/2))/(a^5* b^7 + b^12*x^5 + 5*a^4*b^8*x + 5*a*b^11*x^4 + 10*a^3*b^9*x^2 + 10*a^2*b^10 *x^3) + (B*x^2)/(2*b^6) + (log(a + b*x)*(21*B*a^2 - 6*A*a*b))/b^8