\(\int \frac {(a+b x) (a c-b c x)^5}{x^{11}} \, dx\) [42]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{11}} \, dx=-\frac {a^6 c^5}{10 x^{10}}+\frac {4 a^5 b c^5}{9 x^9}-\frac {5 a^4 b^2 c^5}{8 x^8}+\frac {5 a^2 b^4 c^5}{6 x^6}-\frac {4 a b^5 c^5}{5 x^5}+\frac {b^6 c^5}{4 x^4} \]

[Out]

-1/10*a^6*c^5/x^10+4/9*a^5*b*c^5/x^9-5/8*a^4*b^2*c^5/x^8+5/6*a^2*b^4*c^5/x^6-4/5*a*b^5*c^5/x^5+1/4*b^6*c^5/x^4

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{11}} \, dx=-\frac {a^6 c^5}{10 x^{10}}+\frac {4 a^5 b c^5}{9 x^9}-\frac {5 a^4 b^2 c^5}{8 x^8}+\frac {5 a^2 b^4 c^5}{6 x^6}-\frac {4 a b^5 c^5}{5 x^5}+\frac {b^6 c^5}{4 x^4} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^11,x]

[Out]

-1/10*(a^6*c^5)/x^10 + (4*a^5*b*c^5)/(9*x^9) - (5*a^4*b^2*c^5)/(8*x^8) + (5*a^2*b^4*c^5)/(6*x^6) - (4*a*b^5*c^
5)/(5*x^5) + (b^6*c^5)/(4*x^4)

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^6 c^5}{x^{11}}-\frac {4 a^5 b c^5}{x^{10}}+\frac {5 a^4 b^2 c^5}{x^9}-\frac {5 a^2 b^4 c^5}{x^7}+\frac {4 a b^5 c^5}{x^6}-\frac {b^6 c^5}{x^5}\right ) \, dx \\ & = -\frac {a^6 c^5}{10 x^{10}}+\frac {4 a^5 b c^5}{9 x^9}-\frac {5 a^4 b^2 c^5}{8 x^8}+\frac {5 a^2 b^4 c^5}{6 x^6}-\frac {4 a b^5 c^5}{5 x^5}+\frac {b^6 c^5}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{11}} \, dx=c^5 \left (-\frac {a^6}{10 x^{10}}+\frac {4 a^5 b}{9 x^9}-\frac {5 a^4 b^2}{8 x^8}+\frac {5 a^2 b^4}{6 x^6}-\frac {4 a b^5}{5 x^5}+\frac {b^6}{4 x^4}\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^11,x]

[Out]

c^5*(-1/10*a^6/x^10 + (4*a^5*b)/(9*x^9) - (5*a^4*b^2)/(8*x^8) + (5*a^2*b^4)/(6*x^6) - (4*a*b^5)/(5*x^5) + b^6/
(4*x^4))

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.70

method	result	size
gosper	\(-\frac {c^{5} \left (-90 b^{6} x^{6}+288 a \,x^{5} b^{5}-300 a^{2} x^{4} b^{4}+225 a^{4} x^{2} b^{2}-160 a^{5} x b +36 a^{6}\right )}{360 x^{10}}\)	\(61\)
default	\(c^{5} \left (\frac {5 a^{2} b^{4}}{6 x^{6}}-\frac {5 a^{4} b^{2}}{8 x^{8}}-\frac {a^{6}}{10 x^{10}}+\frac {b^{6}}{4 x^{4}}-\frac {4 a \,b^{5}}{5 x^{5}}+\frac {4 a^{5} b}{9 x^{9}}\right )\)	\(62\)
norman	\(\frac {-\frac {1}{10} a^{6} c^{5}+\frac {1}{4} b^{6} c^{5} x^{6}-\frac {4}{5} a \,b^{5} c^{5} x^{5}+\frac {5}{6} a^{2} b^{4} c^{5} x^{4}-\frac {5}{8} a^{4} b^{2} c^{5} x^{2}+\frac {4}{9} a^{5} b \,c^{5} x}{x^{10}}\)	\(75\)
risch	\(\frac {-\frac {1}{10} a^{6} c^{5}+\frac {1}{4} b^{6} c^{5} x^{6}-\frac {4}{5} a \,b^{5} c^{5} x^{5}+\frac {5}{6} a^{2} b^{4} c^{5} x^{4}-\frac {5}{8} a^{4} b^{2} c^{5} x^{2}+\frac {4}{9} a^{5} b \,c^{5} x}{x^{10}}\)	\(75\)
parallelrisch	\(\frac {90 b^{6} c^{5} x^{6}-288 a \,b^{5} c^{5} x^{5}+300 a^{2} b^{4} c^{5} x^{4}-225 a^{4} b^{2} c^{5} x^{2}+160 a^{5} b \,c^{5} x -36 a^{6} c^{5}}{360 x^{10}}\)	\(76\)

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^11,x,method=_RETURNVERBOSE)

[Out]

-1/360*c^5*(-90*b^6*x^6+288*a*b^5*x^5-300*a^2*b^4*x^4+225*a^4*b^2*x^2-160*a^5*b*x+36*a^6)/x^10

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{11}} \, dx=\frac {90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^11,x, algorithm="fricas")

[Out]

1/360*(90*b^6*c^5*x^6 - 288*a*b^5*c^5*x^5 + 300*a^2*b^4*c^5*x^4 - 225*a^4*b^2*c^5*x^2 + 160*a^5*b*c^5*x - 36*a
^6*c^5)/x^10

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{11}} \, dx=- \frac {36 a^{6} c^{5} - 160 a^{5} b c^{5} x + 225 a^{4} b^{2} c^{5} x^{2} - 300 a^{2} b^{4} c^{5} x^{4} + 288 a b^{5} c^{5} x^{5} - 90 b^{6} c^{5} x^{6}}{360 x^{10}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**11,x)

[Out]

-(36*a**6*c**5 - 160*a**5*b*c**5*x + 225*a**4*b**2*c**5*x**2 - 300*a**2*b**4*c**5*x**4 + 288*a*b**5*c**5*x**5
- 90*b**6*c**5*x**6)/(360*x**10)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{11}} \, dx=\frac {90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^11,x, algorithm="maxima")

[Out]

1/360*(90*b^6*c^5*x^6 - 288*a*b^5*c^5*x^5 + 300*a^2*b^4*c^5*x^4 - 225*a^4*b^2*c^5*x^2 + 160*a^5*b*c^5*x - 36*a
^6*c^5)/x^10

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{11}} \, dx=\frac {90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^11,x, algorithm="giac")

[Out]

1/360*(90*b^6*c^5*x^6 - 288*a*b^5*c^5*x^5 + 300*a^2*b^4*c^5*x^4 - 225*a^4*b^2*c^5*x^2 + 160*a^5*b*c^5*x - 36*a
^6*c^5)/x^10

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{11}} \, dx=-\frac {\frac {a^6\,c^5}{10}-\frac {4\,a^5\,b\,c^5\,x}{9}+\frac {5\,a^4\,b^2\,c^5\,x^2}{8}-\frac {5\,a^2\,b^4\,c^5\,x^4}{6}+\frac {4\,a\,b^5\,c^5\,x^5}{5}-\frac {b^6\,c^5\,x^6}{4}}{x^{10}} \]

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^11,x)

[Out]

-((a^6*c^5)/10 - (b^6*c^5*x^6)/4 + (4*a*b^5*c^5*x^5)/5 + (5*a^4*b^2*c^5*x^2)/8 - (5*a^2*b^4*c^5*x^4)/6 - (4*a^
5*b*c^5*x)/9)/x^10