\(\int \frac {(a+b x) (a c-b c x)^5}{x^{12}} \, dx\) [43]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{12}} \, dx=-\frac {a^6 c^5}{11 x^{11}}+\frac {2 a^5 b c^5}{5 x^{10}}-\frac {5 a^4 b^2 c^5}{9 x^9}+\frac {5 a^2 b^4 c^5}{7 x^7}-\frac {2 a b^5 c^5}{3 x^6}+\frac {b^6 c^5}{5 x^5} \]

[Out]

-1/11*a^6*c^5/x^11+2/5*a^5*b*c^5/x^10-5/9*a^4*b^2*c^5/x^9+5/7*a^2*b^4*c^5/x^7-2/3*a*b^5*c^5/x^6+1/5*b^6*c^5/x^
5

Rubi [A] (verified)

Time = 0.02 (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} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{12}} \, dx=-\frac {a^6 c^5}{11 x^{11}}+\frac {2 a^5 b c^5}{5 x^{10}}-\frac {5 a^4 b^2 c^5}{9 x^9}+\frac {5 a^2 b^4 c^5}{7 x^7}-\frac {2 a b^5 c^5}{3 x^6}+\frac {b^6 c^5}{5 x^5} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^12,x]

[Out]

-1/11*(a^6*c^5)/x^11 + (2*a^5*b*c^5)/(5*x^10) - (5*a^4*b^2*c^5)/(9*x^9) + (5*a^2*b^4*c^5)/(7*x^7) - (2*a*b^5*c
^5)/(3*x^6) + (b^6*c^5)/(5*x^5)

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^{12}}-\frac {4 a^5 b c^5}{x^{11}}+\frac {5 a^4 b^2 c^5}{x^{10}}-\frac {5 a^2 b^4 c^5}{x^8}+\frac {4 a b^5 c^5}{x^7}-\frac {b^6 c^5}{x^6}\right ) \, dx \\ & = -\frac {a^6 c^5}{11 x^{11}}+\frac {2 a^5 b c^5}{5 x^{10}}-\frac {5 a^4 b^2 c^5}{9 x^9}+\frac {5 a^2 b^4 c^5}{7 x^7}-\frac {2 a b^5 c^5}{3 x^6}+\frac {b^6 c^5}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{12}} \, dx=c^5 \left (-\frac {a^6}{11 x^{11}}+\frac {2 a^5 b}{5 x^{10}}-\frac {5 a^4 b^2}{9 x^9}+\frac {5 a^2 b^4}{7 x^7}-\frac {2 a b^5}{3 x^6}+\frac {b^6}{5 x^5}\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^12,x]

[Out]

c^5*(-1/11*a^6/x^11 + (2*a^5*b)/(5*x^10) - (5*a^4*b^2)/(9*x^9) + (5*a^2*b^4)/(7*x^7) - (2*a*b^5)/(3*x^6) + b^6
/(5*x^5))

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.70

method	result	size
gosper	\(-\frac {c^{5} \left (-693 b^{6} x^{6}+2310 a \,x^{5} b^{5}-2475 a^{2} x^{4} b^{4}+1925 a^{4} x^{2} b^{2}-1386 a^{5} x b +315 a^{6}\right )}{3465 x^{11}}\)	\(61\)
default	\(c^{5} \left (-\frac {2 a \,b^{5}}{3 x^{6}}+\frac {5 a^{2} b^{4}}{7 x^{7}}+\frac {2 a^{5} b}{5 x^{10}}+\frac {b^{6}}{5 x^{5}}-\frac {5 a^{4} b^{2}}{9 x^{9}}-\frac {a^{6}}{11 x^{11}}\right )\)	\(62\)
norman	\(\frac {-\frac {1}{11} a^{6} c^{5}+\frac {1}{5} b^{6} c^{5} x^{6}-\frac {2}{3} a \,b^{5} c^{5} x^{5}+\frac {5}{7} a^{2} b^{4} c^{5} x^{4}-\frac {5}{9} a^{4} b^{2} c^{5} x^{2}+\frac {2}{5} a^{5} b \,c^{5} x}{x^{11}}\)	\(75\)
risch	\(\frac {-\frac {1}{11} a^{6} c^{5}+\frac {1}{5} b^{6} c^{5} x^{6}-\frac {2}{3} a \,b^{5} c^{5} x^{5}+\frac {5}{7} a^{2} b^{4} c^{5} x^{4}-\frac {5}{9} a^{4} b^{2} c^{5} x^{2}+\frac {2}{5} a^{5} b \,c^{5} x}{x^{11}}\)	\(75\)
parallelrisch	\(\frac {693 b^{6} c^{5} x^{6}-2310 a \,b^{5} c^{5} x^{5}+2475 a^{2} b^{4} c^{5} x^{4}-1925 a^{4} b^{2} c^{5} x^{2}+1386 a^{5} b \,c^{5} x -315 a^{6} c^{5}}{3465 x^{11}}\)	\(76\)

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^12,x,method=_RETURNVERBOSE)

[Out]

-1/3465*c^5*(-693*b^6*x^6+2310*a*b^5*x^5-2475*a^2*b^4*x^4+1925*a^4*b^2*x^2-1386*a^5*b*x+315*a^6)/x^11

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{12}} \, dx=\frac {693 \, b^{6} c^{5} x^{6} - 2310 \, a b^{5} c^{5} x^{5} + 2475 \, a^{2} b^{4} c^{5} x^{4} - 1925 \, a^{4} b^{2} c^{5} x^{2} + 1386 \, a^{5} b c^{5} x - 315 \, a^{6} c^{5}}{3465 \, x^{11}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^12,x, algorithm="fricas")

[Out]

1/3465*(693*b^6*c^5*x^6 - 2310*a*b^5*c^5*x^5 + 2475*a^2*b^4*c^5*x^4 - 1925*a^4*b^2*c^5*x^2 + 1386*a^5*b*c^5*x
- 315*a^6*c^5)/x^11

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{12}} \, dx=- \frac {315 a^{6} c^{5} - 1386 a^{5} b c^{5} x + 1925 a^{4} b^{2} c^{5} x^{2} - 2475 a^{2} b^{4} c^{5} x^{4} + 2310 a b^{5} c^{5} x^{5} - 693 b^{6} c^{5} x^{6}}{3465 x^{11}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**12,x)

[Out]

-(315*a**6*c**5 - 1386*a**5*b*c**5*x + 1925*a**4*b**2*c**5*x**2 - 2475*a**2*b**4*c**5*x**4 + 2310*a*b**5*c**5*
x**5 - 693*b**6*c**5*x**6)/(3465*x**11)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{12}} \, dx=\frac {693 \, b^{6} c^{5} x^{6} - 2310 \, a b^{5} c^{5} x^{5} + 2475 \, a^{2} b^{4} c^{5} x^{4} - 1925 \, a^{4} b^{2} c^{5} x^{2} + 1386 \, a^{5} b c^{5} x - 315 \, a^{6} c^{5}}{3465 \, x^{11}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^12,x, algorithm="maxima")

[Out]

1/3465*(693*b^6*c^5*x^6 - 2310*a*b^5*c^5*x^5 + 2475*a^2*b^4*c^5*x^4 - 1925*a^4*b^2*c^5*x^2 + 1386*a^5*b*c^5*x
- 315*a^6*c^5)/x^11

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{12}} \, dx=\frac {693 \, b^{6} c^{5} x^{6} - 2310 \, a b^{5} c^{5} x^{5} + 2475 \, a^{2} b^{4} c^{5} x^{4} - 1925 \, a^{4} b^{2} c^{5} x^{2} + 1386 \, a^{5} b c^{5} x - 315 \, a^{6} c^{5}}{3465 \, x^{11}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^12,x, algorithm="giac")

[Out]

1/3465*(693*b^6*c^5*x^6 - 2310*a*b^5*c^5*x^5 + 2475*a^2*b^4*c^5*x^4 - 1925*a^4*b^2*c^5*x^2 + 1386*a^5*b*c^5*x
- 315*a^6*c^5)/x^11

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{12}} \, dx=-\frac {\frac {a^6\,c^5}{11}-\frac {2\,a^5\,b\,c^5\,x}{5}+\frac {5\,a^4\,b^2\,c^5\,x^2}{9}-\frac {5\,a^2\,b^4\,c^5\,x^4}{7}+\frac {2\,a\,b^5\,c^5\,x^5}{3}-\frac {b^6\,c^5\,x^6}{5}}{x^{11}} \]

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^12,x)

[Out]

-((a^6*c^5)/11 - (b^6*c^5*x^6)/5 + (2*a*b^5*c^5*x^5)/3 + (5*a^4*b^2*c^5*x^2)/9 - (5*a^2*b^4*c^5*x^4)/7 - (2*a^
5*b*c^5*x)/5)/x^11