\(\int \frac {x^5 (A+B x)}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [262]

Optimal result

Integrand size = 27, antiderivative size = 143 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {2 a (2 A b-5 a B) x}{b^6}+\frac {(A b-4 a B) x^2}{2 b^5}+\frac {B x^3}{3 b^4}+\frac {a^5 (A b-a B)}{3 b^7 (a+b x)^3}-\frac {a^4 (5 A b-6 a B)}{2 b^7 (a+b x)^2}+\frac {5 a^3 (2 A b-3 a B)}{b^7 (a+b x)}+\frac {10 a^2 (A b-2 a B) \log (a+b x)}{b^7} \] Output:

-2*a*(2*A*b-5*B*a)*x/b^6+1/2*(A*b-4*B*a)*x^2/b^5+1/3*B*x^3/b^4+1/3*a^5*(A* 
b-B*a)/b^7/(b*x+a)^3-1/2*a^4*(5*A*b-6*B*a)/b^7/(b*x+a)^2+5*a^3*(2*A*b-3*B* 
a)/b^7/(b*x+a)+10*a^2*(A*b-2*B*a)*ln(b*x+a)/b^7
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {-74 a^6 B+a^5 b (47 A-102 B x)+b^6 x^5 (3 A+2 B x)-3 a b^5 x^4 (5 A+2 B x)+3 a^2 b^4 x^3 (-21 A+10 B x)+3 a^4 b^2 x (27 A+26 B x)+a^3 b^3 x^2 (-9 A+146 B x)-60 a^2 (-A b+2 a B) (a+b x)^3 \log (a+b x)}{6 b^7 (a+b x)^3} \] Input:

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

Output:

(-74*a^6*B + a^5*b*(47*A - 102*B*x) + b^6*x^5*(3*A + 2*B*x) - 3*a*b^5*x^4* 
(5*A + 2*B*x) + 3*a^2*b^4*x^3*(-21*A + 10*B*x) + 3*a^4*b^2*x*(27*A + 26*B* 
x) + a^3*b^3*x^2*(-9*A + 146*B*x) - 60*a^2*(-(A*b) + 2*a*B)*(a + b*x)^3*Lo 
g[a + b*x])/(6*b^7*(a + b*x)^3)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 143, 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.

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

\(\Big \downarrow \) 1184

\(\displaystyle b^4 \int \frac {x^5 (A+B x)}{b^4 (a+b x)^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {x^5 (A+B x)}{(a+b x)^4}dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^5 (A b-a B)}{3 b^7 (a+b x)^3}-\frac {a^4 (5 A b-6 a B)}{2 b^7 (a+b x)^2}+\frac {5 a^3 (2 A b-3 a B)}{b^7 (a+b x)}+\frac {10 a^2 (A b-2 a B) \log (a+b x)}{b^7}-\frac {2 a x (2 A b-5 a B)}{b^6}+\frac {x^2 (A b-4 a B)}{2 b^5}+\frac {B x^3}{3 b^4}\)

Input:

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

Output:

(-2*a*(2*A*b - 5*a*B)*x)/b^6 + ((A*b - 4*a*B)*x^2)/(2*b^5) + (B*x^3)/(3*b^ 
4) + (a^5*(A*b - a*B))/(3*b^7*(a + b*x)^3) - (a^4*(5*A*b - 6*a*B))/(2*b^7* 
(a + b*x)^2) + (5*a^3*(2*A*b - 3*a*B))/(b^7*(a + b*x)) + (10*a^2*(A*b - 2* 
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 = 1.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {-\frac {1}{3} x^{3} B \,b^{2}-\frac {1}{2} x^{2} b^{2} A +2 B a \,x^{2} b +4 a b A x -10 a^{2} B x}{b^{6}}+\frac {a^{5} \left (A b -B a \right )}{3 b^{7} \left (b x +a \right )^{3}}+\frac {5 a^{3} \left (2 A b -3 B a \right )}{b^{7} \left (b x +a \right )}+\frac {10 a^{2} \left (A b -2 B a \right ) \ln \left (b x +a \right )}{b^{7}}-\frac {a^{4} \left (5 A b -6 B a \right )}{2 b^{7} \left (b x +a \right )^{2}}\)	\(140\)
norman	\(\frac {\frac {B \,x^{6}}{3 b}+\frac {a^{3} \left (55 A \,a^{2} b -110 B \,a^{3}\right )}{3 b^{7}}+\frac {\left (A b -2 B a \right ) x^{5}}{2 b^{2}}-\frac {5 a \left (A b -2 B a \right ) x^{4}}{2 b^{3}}+\frac {3 a \left (10 A \,a^{2} b -20 B \,a^{3}\right ) x^{2}}{b^{5}}+\frac {3 a^{2} \left (15 A \,a^{2} b -30 B \,a^{3}\right ) x}{b^{6}}}{\left (b x +a \right )^{3}}+\frac {10 a^{2} \left (A b -2 B a \right ) \ln \left (b x +a \right )}{b^{7}}\)	\(143\)
risch	\(\frac {B \,x^{3}}{3 b^{4}}+\frac {x^{2} A}{2 b^{4}}-\frac {2 B a \,x^{2}}{b^{5}}-\frac {4 a A x}{b^{5}}+\frac {10 a^{2} B x}{b^{6}}+\frac {\left (10 a^{3} A \,b^{2}-15 B \,a^{4} b \right ) x^{2}+\frac {a^{4} \left (35 A b -54 B a \right ) x}{2}+\frac {a^{5} \left (47 A b -74 B a \right )}{6 b}}{b^{6} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )}+\frac {10 a^{2} \ln \left (b x +a \right ) A}{b^{6}}-\frac {20 a^{3} \ln \left (b x +a \right ) B}{b^{7}}\)	\(161\)
parallelrisch	\(\frac {2 b^{6} B \,x^{6}+3 A \,b^{6} x^{5}-6 B a \,b^{5} x^{5}+60 A \ln \left (b x +a \right ) x^{3} a^{2} b^{4}-15 A a \,b^{5} x^{4}-120 B \ln \left (b x +a \right ) x^{3} a^{3} b^{3}+30 B \,a^{2} b^{4} x^{4}+180 A \ln \left (b x +a \right ) x^{2} a^{3} b^{3}-360 B \ln \left (b x +a \right ) x^{2} a^{4} b^{2}+180 A \ln \left (b x +a \right ) x \,a^{4} b^{2}+180 A \,a^{3} b^{3} x^{2}-360 B \ln \left (b x +a \right ) x \,a^{5} b -360 B \,a^{4} b^{2} x^{2}+60 A \ln \left (b x +a \right ) a^{5} b +270 A \,a^{4} b^{2} x -120 B \ln \left (b x +a \right ) a^{6}-540 B \,a^{5} b x +110 A \,a^{5} b -220 B \,a^{6}}{6 b^{7} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}\)	\(264\)

Input:

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

Output:

-1/b^6*(-1/3*x^3*B*b^2-1/2*x^2*b^2*A+2*B*a*x^2*b+4*a*b*A*x-10*a^2*B*x)+1/3 
*a^5*(A*b-B*a)/b^7/(b*x+a)^3+5*a^3*(2*A*b-3*B*a)/b^7/(b*x+a)+10*a^2*(A*b-2 
*B*a)*ln(b*x+a)/b^7-1/2*a^4*(5*A*b-6*B*a)/b^7/(b*x+a)^2
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.80 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {2 \, B b^{6} x^{6} - 74 \, B a^{6} + 47 \, A a^{5} b - 3 \, {\left (2 \, B a b^{5} - A b^{6}\right )} x^{5} + 15 \, {\left (2 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + {\left (146 \, B a^{3} b^{3} - 63 \, A a^{2} b^{4}\right )} x^{3} + 3 \, {\left (26 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} - 3 \, {\left (34 \, B a^{5} b - 27 \, A a^{4} b^{2}\right )} x - 60 \, {\left (2 \, B a^{6} - A a^{5} b + {\left (2 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 3 \, {\left (2 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 3 \, {\left (2 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} \] Input:

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

Output:

1/6*(2*B*b^6*x^6 - 74*B*a^6 + 47*A*a^5*b - 3*(2*B*a*b^5 - A*b^6)*x^5 + 15* 
(2*B*a^2*b^4 - A*a*b^5)*x^4 + (146*B*a^3*b^3 - 63*A*a^2*b^4)*x^3 + 3*(26*B 
*a^4*b^2 - 3*A*a^3*b^3)*x^2 - 3*(34*B*a^5*b - 27*A*a^4*b^2)*x - 60*(2*B*a^ 
6 - A*a^5*b + (2*B*a^3*b^3 - A*a^2*b^4)*x^3 + 3*(2*B*a^4*b^2 - A*a^3*b^3)* 
x^2 + 3*(2*B*a^5*b - A*a^4*b^2)*x)*log(b*x + a))/(b^10*x^3 + 3*a*b^9*x^2 + 
 3*a^2*b^8*x + a^3*b^7)
 

Sympy [A] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {B x^{3}}{3 b^{4}} - \frac {10 a^{2} \left (- A b + 2 B a\right ) \log {\left (a + b x \right )}}{b^{7}} + x^{2} \left (\frac {A}{2 b^{4}} - \frac {2 B a}{b^{5}}\right ) + x \left (- \frac {4 A a}{b^{5}} + \frac {10 B a^{2}}{b^{6}}\right ) + \frac {47 A a^{5} b - 74 B a^{6} + x^{2} \cdot \left (60 A a^{3} b^{3} - 90 B a^{4} b^{2}\right ) + x \left (105 A a^{4} b^{2} - 162 B a^{5} b\right )}{6 a^{3} b^{7} + 18 a^{2} b^{8} x + 18 a b^{9} x^{2} + 6 b^{10} x^{3}} \] Input:

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

Output:

B*x**3/(3*b**4) - 10*a**2*(-A*b + 2*B*a)*log(a + b*x)/b**7 + x**2*(A/(2*b* 
*4) - 2*B*a/b**5) + x*(-4*A*a/b**5 + 10*B*a**2/b**6) + (47*A*a**5*b - 74*B 
*a**6 + x**2*(60*A*a**3*b**3 - 90*B*a**4*b**2) + x*(105*A*a**4*b**2 - 162* 
B*a**5*b))/(6*a**3*b**7 + 18*a**2*b**8*x + 18*a*b**9*x**2 + 6*b**10*x**3)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {74 \, B a^{6} - 47 \, A a^{5} b + 30 \, {\left (3 \, B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{2} + 3 \, {\left (54 \, B a^{5} b - 35 \, A a^{4} b^{2}\right )} x}{6 \, {\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} + \frac {2 \, B b^{2} x^{3} - 3 \, {\left (4 \, B a b - A b^{2}\right )} x^{2} + 12 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} x}{6 \, b^{6}} - \frac {10 \, {\left (2 \, B a^{3} - A a^{2} b\right )} \log \left (b x + a\right )}{b^{7}} \] Input:

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

Output:

-1/6*(74*B*a^6 - 47*A*a^5*b + 30*(3*B*a^4*b^2 - 2*A*a^3*b^3)*x^2 + 3*(54*B 
*a^5*b - 35*A*a^4*b^2)*x)/(b^10*x^3 + 3*a*b^9*x^2 + 3*a^2*b^8*x + a^3*b^7) 
 + 1/6*(2*B*b^2*x^3 - 3*(4*B*a*b - A*b^2)*x^2 + 12*(5*B*a^2 - 2*A*a*b)*x)/ 
b^6 - 10*(2*B*a^3 - A*a^2*b)*log(b*x + a)/b^7
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.04 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {10 \, {\left (2 \, B a^{3} - A a^{2} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {74 \, B a^{6} - 47 \, A a^{5} b + 30 \, {\left (3 \, B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{2} + 3 \, {\left (54 \, B a^{5} b - 35 \, A a^{4} b^{2}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{7}} + \frac {2 \, B b^{8} x^{3} - 12 \, B a b^{7} x^{2} + 3 \, A b^{8} x^{2} + 60 \, B a^{2} b^{6} x - 24 \, A a b^{7} x}{6 \, b^{12}} \] Input:

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

Output:

-10*(2*B*a^3 - A*a^2*b)*log(abs(b*x + a))/b^7 - 1/6*(74*B*a^6 - 47*A*a^5*b 
 + 30*(3*B*a^4*b^2 - 2*A*a^3*b^3)*x^2 + 3*(54*B*a^5*b - 35*A*a^4*b^2)*x)/( 
(b*x + a)^3*b^7) + 1/6*(2*B*b^8*x^3 - 12*B*a*b^7*x^2 + 3*A*b^8*x^2 + 60*B* 
a^2*b^6*x - 24*A*a*b^7*x)/b^12
 

Mupad [B] (verification not implemented)

Time = 11.01 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.26 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=x^2\,\left (\frac {A}{2\,b^4}-\frac {2\,B\,a}{b^5}\right )-\frac {x\,\left (27\,B\,a^5-\frac {35\,A\,a^4\,b}{2}\right )-x^2\,\left (10\,A\,a^3\,b^2-15\,B\,a^4\,b\right )+\frac {74\,B\,a^6-47\,A\,a^5\,b}{6\,b}}{a^3\,b^6+3\,a^2\,b^7\,x+3\,a\,b^8\,x^2+b^9\,x^3}-x\,\left (\frac {4\,a\,\left (\frac {A}{b^4}-\frac {4\,B\,a}{b^5}\right )}{b}+\frac {6\,B\,a^2}{b^6}\right )-\frac {\ln \left (a+b\,x\right )\,\left (20\,B\,a^3-10\,A\,a^2\,b\right )}{b^7}+\frac {B\,x^3}{3\,b^4} \] Input:

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

Output:

x^2*(A/(2*b^4) - (2*B*a)/b^5) - (x*(27*B*a^5 - (35*A*a^4*b)/2) - x^2*(10*A 
*a^3*b^2 - 15*B*a^4*b) + (74*B*a^6 - 47*A*a^5*b)/(6*b))/(a^3*b^6 + b^9*x^3 
 + 3*a^2*b^7*x + 3*a*b^8*x^2) - x*((4*a*(A/b^4 - (4*B*a)/b^5))/b + (6*B*a^ 
2)/b^6) - (log(a + b*x)*(20*B*a^3 - 10*A*a^2*b))/b^7 + (B*x^3)/(3*b^4)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.76 \[ \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {-60 \,\mathrm {log}\left (b x +a \right ) a^{5}-120 \,\mathrm {log}\left (b x +a \right ) a^{4} b x -60 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} x^{2}-30 a^{5}+60 a^{3} b^{2} x^{2}+20 a^{2} b^{3} x^{3}-5 a \,b^{4} x^{4}+2 b^{5} x^{5}}{6 b^{6} \left (b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:

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

Output:

( - 60*log(a + b*x)*a**5 - 120*log(a + b*x)*a**4*b*x - 60*log(a + b*x)*a** 
3*b**2*x**2 - 30*a**5 + 60*a**3*b**2*x**2 + 20*a**2*b**3*x**3 - 5*a*b**4*x 
**4 + 2*b**5*x**5)/(6*b**6*(a**2 + 2*a*b*x + b**2*x**2))