\(\int \frac {x^5 (A+B x)}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [273]

Optimal result

Integrand size = 27, antiderivative size = 146 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {B x}{b^6}+\frac {a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac {a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac {5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac {5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac {5 a (A b-3 a B)}{b^7 (a+b x)}+\frac {(A b-6 a B) \log (a+b x)}{b^7} \] Output:

B*x/b^6+1/5*a^5*(A*b-B*a)/b^7/(b*x+a)^5-1/4*a^4*(5*A*b-6*B*a)/b^7/(b*x+a)^ 
4+5/3*a^3*(2*A*b-3*B*a)/b^7/(b*x+a)^3-5*a^2*(A*b-2*B*a)/b^7/(b*x+a)^2+5*a* 
(A*b-3*B*a)/b^7/(b*x+a)+(A*b-6*B*a)*ln(b*x+a)/b^7
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.89 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {60 b B x+\frac {12 a^5 (A b-a B)}{(a+b x)^5}+\frac {15 a^4 (-5 A b+6 a B)}{(a+b x)^4}+\frac {100 a^3 (2 A b-3 a B)}{(a+b x)^3}+\frac {300 a^2 (-A b+2 a B)}{(a+b x)^2}+\frac {300 a (A b-3 a B)}{a+b x}+60 (A b-6 a B) \log (a+b x)}{60 b^7} \] Input:

Integrate[(x^5*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
 

Output:

(60*b*B*x + (12*a^5*(A*b - a*B))/(a + b*x)^5 + (15*a^4*(-5*A*b + 6*a*B))/( 
a + b*x)^4 + (100*a^3*(2*A*b - 3*a*B))/(a + b*x)^3 + (300*a^2*(-(A*b) + 2* 
a*B))/(a + b*x)^2 + (300*a*(A*b - 3*a*B))/(a + b*x) + 60*(A*b - 6*a*B)*Log 
[a + b*x])/(60*b^7)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 146, 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.

\(\displaystyle \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1184

\(\displaystyle b^6 \int \frac {x^5 (A+B x)}{b^6 (a+b x)^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {x^5 (A+B x)}{(a+b x)^6}dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {a^5 (a B-A b)}{b^6 (a+b x)^6}-\frac {a^4 (6 a B-5 A b)}{b^6 (a+b x)^5}+\frac {5 a^3 (3 a B-2 A b)}{b^6 (a+b x)^4}-\frac {10 a^2 (2 a B-A b)}{b^6 (a+b x)^3}+\frac {5 a (3 a B-A b)}{b^6 (a+b x)^2}+\frac {A b-6 a B}{b^6 (a+b x)}+\frac {B}{b^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac {a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac {5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac {5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac {5 a (A b-3 a B)}{b^7 (a+b x)}+\frac {(A b-6 a B) \log (a+b x)}{b^7}+\frac {B x}{b^6}\)

Input:

Int[(x^5*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
 

Output:

(B*x)/b^6 + (a^5*(A*b - a*B))/(5*b^7*(a + b*x)^5) - (a^4*(5*A*b - 6*a*B))/ 
(4*b^7*(a + b*x)^4) + (5*a^3*(2*A*b - 3*a*B))/(3*b^7*(a + b*x)^3) - (5*a^2 
*(A*b - 2*a*B))/(b^7*(a + b*x)^2) + (5*a*(A*b - 3*a*B))/(b^7*(a + b*x)) + 
((A*b - 6*a*B)*Log[a + b*x])/b^7
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 86

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]))))
 

rule 1184

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.88

method	result	size
norman	\(\frac {\frac {B \,x^{6}}{b}+\frac {a^{5} \left (137 A b -822 B a \right )}{60 b^{7}}+\frac {5 a \left (A b -6 B a \right ) x^{4}}{b^{3}}+\frac {5 a^{2} \left (3 A b -18 B a \right ) x^{3}}{b^{4}}+\frac {5 a^{3} \left (11 A b -66 B a \right ) x^{2}}{3 b^{5}}+\frac {5 a^{4} \left (25 A b -150 B a \right ) x}{12 b^{6}}}{\left (b x +a \right )^{5}}+\frac {\left (A b -6 B a \right ) \ln \left (b x +a \right )}{b^{7}}\)	\(129\)
default	\(\frac {B x}{b^{6}}+\frac {a^{5} \left (A b -B a \right )}{5 b^{7} \left (b x +a \right )^{5}}-\frac {a^{4} \left (5 A b -6 B a \right )}{4 b^{7} \left (b x +a \right )^{4}}+\frac {5 a^{3} \left (2 A b -3 B a \right )}{3 b^{7} \left (b x +a \right )^{3}}-\frac {5 a^{2} \left (A b -2 B a \right )}{b^{7} \left (b x +a \right )^{2}}+\frac {5 a \left (A b -3 B a \right )}{b^{7} \left (b x +a \right )}+\frac {\left (A b -6 B a \right ) \ln \left (b x +a \right )}{b^{7}}\)	\(141\)
risch	\(\frac {B x}{b^{6}}+\frac {\left (5 A a \,b^{4}-15 B \,a^{2} b^{3}\right ) x^{4}+5 a^{2} b^{2} \left (3 A b -10 B a \right ) x^{3}+\left (\frac {55}{3} a^{3} A \,b^{2}-65 B \,a^{4} b \right ) x^{2}+\left (\frac {125}{12} A \,a^{4} b -\frac {77}{2} B \,a^{5}\right ) x +\frac {a^{5} \left (137 A b -522 B a \right )}{60 b}}{b^{6} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}+\frac {\ln \left (b x +a \right ) A}{b^{6}}-\frac {6 \ln \left (b x +a \right ) B a}{b^{7}}\)	\(157\)
parallelrisch	\(\frac {600 A \ln \left (b x +a \right ) x^{2} a^{3} b^{3}+600 A \ln \left (b x +a \right ) x^{3} a^{2} b^{4}-3600 B \ln \left (b x +a \right ) x^{3} a^{3} b^{3}-3750 B \,a^{5} b x +300 A a \,b^{5} x^{4}-6600 B \,a^{4} b^{2} x^{2}+300 A \ln \left (b x +a \right ) x \,a^{4} b^{2}-1800 B \ln \left (b x +a \right ) x \,a^{5} b -3600 B \ln \left (b x +a \right ) x^{2} a^{4} b^{2}-822 B \,a^{6}+60 A \ln \left (b x +a \right ) x^{5} b^{6}+1100 A \,a^{3} b^{3} x^{2}-1800 B \,a^{2} b^{4} x^{4}+625 A \,a^{4} b^{2} x -360 B \ln \left (b x +a \right ) x^{5} a \,b^{5}+300 A \ln \left (b x +a \right ) x^{4} a \,b^{5}-1800 B \ln \left (b x +a \right ) x^{4} a^{2} b^{4}+900 A \,a^{2} b^{4} x^{3}-5400 B \,a^{3} b^{3} x^{3}+60 A \ln \left (b x +a \right ) a^{5} b +60 b^{6} B \,x^{6}+137 A \,a^{5} b -360 B \ln \left (b x +a \right ) a^{6}}{60 b^{7} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}\)	\(334\)

Input:

int(x^5*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

(B/b*x^6+1/60*a^5*(137*A*b-822*B*a)/b^7+5*a*(A*b-6*B*a)/b^3*x^4+5*a^2*(3*A 
*b-18*B*a)/b^4*x^3+5/3*a^3*(11*A*b-66*B*a)/b^5*x^2+5/12*a^4*(25*A*b-150*B* 
a)/b^6*x)/(b*x+a)^5+(A*b-6*B*a)*ln(b*x+a)/b^7
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (144) = 288\).

Time = 0.08 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.13 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {60 \, B b^{6} x^{6} + 300 \, B a b^{5} x^{5} - 522 \, B a^{6} + 137 \, A a^{5} b - 300 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} x^{4} - 300 \, {\left (8 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} - 100 \, {\left (36 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} - 125 \, {\left (18 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x - 60 \, {\left (6 \, B a^{6} - A a^{5} b + {\left (6 \, B a b^{5} - A b^{6}\right )} x^{5} + 5 \, {\left (6 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 10 \, {\left (6 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (6 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (6 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \log \left (b x + a\right )}{60 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} \] Input:

integrate(x^5*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
 

Output:

1/60*(60*B*b^6*x^6 + 300*B*a*b^5*x^5 - 522*B*a^6 + 137*A*a^5*b - 300*(B*a^ 
2*b^4 - A*a*b^5)*x^4 - 300*(8*B*a^3*b^3 - 3*A*a^2*b^4)*x^3 - 100*(36*B*a^4 
*b^2 - 11*A*a^3*b^3)*x^2 - 125*(18*B*a^5*b - 5*A*a^4*b^2)*x - 60*(6*B*a^6 
- A*a^5*b + (6*B*a*b^5 - A*b^6)*x^5 + 5*(6*B*a^2*b^4 - A*a*b^5)*x^4 + 10*( 
6*B*a^3*b^3 - A*a^2*b^4)*x^3 + 10*(6*B*a^4*b^2 - A*a^3*b^3)*x^2 + 5*(6*B*a 
^5*b - A*a^4*b^2)*x)*log(b*x + a))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10* 
x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7)
 

Sympy [A] (verification not implemented)

Time = 1.41 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.30 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {B x}{b^{6}} + \frac {137 A a^{5} b - 522 B a^{6} + x^{4} \cdot \left (300 A a b^{5} - 900 B a^{2} b^{4}\right ) + x^{3} \cdot \left (900 A a^{2} b^{4} - 3000 B a^{3} b^{3}\right ) + x^{2} \cdot \left (1100 A a^{3} b^{3} - 3900 B a^{4} b^{2}\right ) + x \left (625 A a^{4} b^{2} - 2310 B a^{5} b\right )}{60 a^{5} b^{7} + 300 a^{4} b^{8} x + 600 a^{3} b^{9} x^{2} + 600 a^{2} b^{10} x^{3} + 300 a b^{11} x^{4} + 60 b^{12} x^{5}} - \frac {\left (- A b + 6 B a\right ) \log {\left (a + b x \right )}}{b^{7}} \] Input:

integrate(x**5*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3,x)
 

Output:

B*x/b**6 + (137*A*a**5*b - 522*B*a**6 + x**4*(300*A*a*b**5 - 900*B*a**2*b* 
*4) + x**3*(900*A*a**2*b**4 - 3000*B*a**3*b**3) + x**2*(1100*A*a**3*b**3 - 
 3900*B*a**4*b**2) + x*(625*A*a**4*b**2 - 2310*B*a**5*b))/(60*a**5*b**7 + 
300*a**4*b**8*x + 600*a**3*b**9*x**2 + 600*a**2*b**10*x**3 + 300*a*b**11*x 
**4 + 60*b**12*x**5) - (-A*b + 6*B*a)*log(a + b*x)/b**7
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.30 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {522 \, B a^{6} - 137 \, A a^{5} b + 300 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 300 \, {\left (10 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 100 \, {\left (39 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (462 \, B a^{5} b - 125 \, A a^{4} b^{2}\right )} x}{60 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} + \frac {B x}{b^{6}} - \frac {{\left (6 \, B a - A b\right )} \log \left (b x + a\right )}{b^{7}} \] Input:

integrate(x^5*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
 

Output:

-1/60*(522*B*a^6 - 137*A*a^5*b + 300*(3*B*a^2*b^4 - A*a*b^5)*x^4 + 300*(10 
*B*a^3*b^3 - 3*A*a^2*b^4)*x^3 + 100*(39*B*a^4*b^2 - 11*A*a^3*b^3)*x^2 + 5* 
(462*B*a^5*b - 125*A*a^4*b^2)*x)/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^ 
3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7) + B*x/b^6 - (6*B*a - A*b)*log( 
b*x + a)/b^7
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {B x}{b^{6}} - \frac {{\left (6 \, B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {522 \, B a^{6} - 137 \, A a^{5} b + 300 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 300 \, {\left (10 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 100 \, {\left (39 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (462 \, B a^{5} b - 125 \, A a^{4} b^{2}\right )} x}{60 \, {\left (b x + a\right )}^{5} b^{7}} \] Input:

integrate(x^5*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
 

Output:

B*x/b^6 - (6*B*a - A*b)*log(abs(b*x + a))/b^7 - 1/60*(522*B*a^6 - 137*A*a^ 
5*b + 300*(3*B*a^2*b^4 - A*a*b^5)*x^4 + 300*(10*B*a^3*b^3 - 3*A*a^2*b^4)*x 
^3 + 100*(39*B*a^4*b^2 - 11*A*a^3*b^3)*x^2 + 5*(462*B*a^5*b - 125*A*a^4*b^ 
2)*x)/((b*x + a)^5*b^7)
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.28 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {B\,x}{b^6}-\frac {x\,\left (\frac {77\,B\,a^5}{2}-\frac {125\,A\,a^4\,b}{12}\right )+x^4\,\left (15\,B\,a^2\,b^3-5\,A\,a\,b^4\right )-x^2\,\left (\frac {55\,A\,a^3\,b^2}{3}-65\,B\,a^4\,b\right )+\frac {522\,B\,a^6-137\,A\,a^5\,b}{60\,b}-x^3\,\left (15\,A\,a^2\,b^3-50\,B\,a^3\,b^2\right )}{a^5\,b^6+5\,a^4\,b^7\,x+10\,a^3\,b^8\,x^2+10\,a^2\,b^9\,x^3+5\,a\,b^{10}\,x^4+b^{11}\,x^5}+\frac {\ln \left (a+b\,x\right )\,\left (A\,b-6\,B\,a\right )}{b^7} \] Input:

int((x^5*(A + B*x))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
 

Output:

(B*x)/b^6 - (x*((77*B*a^5)/2 - (125*A*a^4*b)/12) + x^4*(15*B*a^2*b^3 - 5*A 
*a*b^4) - x^2*((55*A*a^3*b^2)/3 - 65*B*a^4*b) + (522*B*a^6 - 137*A*a^5*b)/ 
(60*b) - x^3*(15*A*a^2*b^3 - 50*B*a^3*b^2))/(a^5*b^6 + b^11*x^5 + 5*a^4*b^ 
7*x + 5*a*b^10*x^4 + 10*a^3*b^8*x^2 + 10*a^2*b^9*x^3) + (log(a + b*x)*(A*b 
 - 6*B*a))/b^7
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.09 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-60 \,\mathrm {log}\left (b x +a \right ) a^{5}-240 \,\mathrm {log}\left (b x +a \right ) a^{4} b x -360 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} x^{2}-240 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} x^{3}-60 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} x^{4}-65 a^{5}-200 a^{4} b x -180 a^{3} b^{2} x^{2}+60 a \,b^{4} x^{4}+12 b^{5} x^{5}}{12 b^{6} \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )} \] Input:

int(x^5*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x)
 

Output:

( - 60*log(a + b*x)*a**5 - 240*log(a + b*x)*a**4*b*x - 360*log(a + b*x)*a* 
*3*b**2*x**2 - 240*log(a + b*x)*a**2*b**3*x**3 - 60*log(a + b*x)*a*b**4*x* 
*4 - 65*a**5 - 200*a**4*b*x - 180*a**3*b**2*x**2 + 60*a*b**4*x**4 + 12*b** 
5*x**5)/(12*b**6*(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b 
**4*x**4))