\(\int \frac {(a+b x) (a c-b c x)^5}{x^7} \, dx\) [38]

Optimal result

Integrand size = 20, antiderivative size = 82 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^7} \, dx=-\frac {a^6 c^5}{6 x^6}+\frac {4 a^5 b c^5}{5 x^5}-\frac {5 a^4 b^2 c^5}{4 x^4}+\frac {5 a^2 b^4 c^5}{2 x^2}-\frac {4 a b^5 c^5}{x}-b^6 c^5 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 82, 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} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^7} \, dx=-\frac {a^6 c^5}{6 x^6}+\frac {4 a^5 b c^5}{5 x^5}-\frac {5 a^4 b^2 c^5}{4 x^4}+\frac {5 a^2 b^4 c^5}{2 x^2}-\frac {4 a b^5 c^5}{x}-b^6 c^5 \log (x) \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^7,x]

[Out]

-1/6*(a^6*c^5)/x^6 + (4*a^5*b*c^5)/(5*x^5) - (5*a^4*b^2*c^5)/(4*x^4) + (5*a^2*b^4*c^5)/(2*x^2) - (4*a*b^5*c^5)
/x - b^6*c^5*Log[x]

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^7}-\frac {4 a^5 b c^5}{x^6}+\frac {5 a^4 b^2 c^5}{x^5}-\frac {5 a^2 b^4 c^5}{x^3}+\frac {4 a b^5 c^5}{x^2}-\frac {b^6 c^5}{x}\right ) \, dx \\ & = -\frac {a^6 c^5}{6 x^6}+\frac {4 a^5 b c^5}{5 x^5}-\frac {5 a^4 b^2 c^5}{4 x^4}+\frac {5 a^2 b^4 c^5}{2 x^2}-\frac {4 a b^5 c^5}{x}-b^6 c^5 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^7} \, dx=c^5 \left (-\frac {a^6}{6 x^6}+\frac {4 a^5 b}{5 x^5}-\frac {5 a^4 b^2}{4 x^4}+\frac {5 a^2 b^4}{2 x^2}-\frac {4 a b^5}{x}-b^6 \log (x)\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^7,x]

[Out]

c^5*(-1/6*a^6/x^6 + (4*a^5*b)/(5*x^5) - (5*a^4*b^2)/(4*x^4) + (5*a^2*b^4)/(2*x^2) - (4*a*b^5)/x - b^6*Log[x])

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74

method	result	size
default	\(c^{5} \left (-b^{6} \ln \left (x \right )-\frac {a^{6}}{6 x^{6}}-\frac {4 a \,b^{5}}{x}+\frac {5 a^{2} b^{4}}{2 x^{2}}-\frac {5 a^{4} b^{2}}{4 x^{4}}+\frac {4 a^{5} b}{5 x^{5}}\right )\)	\(61\)
norman	\(\frac {-\frac {1}{6} a^{6} c^{5}-4 a \,b^{5} c^{5} x^{5}+\frac {5}{2} a^{2} b^{4} c^{5} x^{4}-\frac {5}{4} a^{4} b^{2} c^{5} x^{2}+\frac {4}{5} a^{5} b \,c^{5} x}{x^{6}}-b^{6} c^{5} \ln \left (x \right )\)	\(75\)
risch	\(\frac {-\frac {1}{6} a^{6} c^{5}-4 a \,b^{5} c^{5} x^{5}+\frac {5}{2} a^{2} b^{4} c^{5} x^{4}-\frac {5}{4} a^{4} b^{2} c^{5} x^{2}+\frac {4}{5} a^{5} b \,c^{5} x}{x^{6}}-b^{6} c^{5} \ln \left (x \right )\)	\(75\)
parallelrisch	\(-\frac {60 b^{6} c^{5} \ln \left (x \right ) x^{6}+240 a \,b^{5} c^{5} x^{5}-150 a^{2} b^{4} c^{5} x^{4}+75 a^{4} b^{2} c^{5} x^{2}-48 a^{5} b \,c^{5} x +10 a^{6} c^{5}}{60 x^{6}}\)	\(78\)

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^7} \, dx=-\frac {60 \, b^{6} c^{5} x^{6} \log \left (x\right ) + 240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^7,x, algorithm="fricas")

[Out]

-1/60*(60*b^6*c^5*x^6*log(x) + 240*a*b^5*c^5*x^5 - 150*a^2*b^4*c^5*x^4 + 75*a^4*b^2*c^5*x^2 - 48*a^5*b*c^5*x +
10*a^6*c^5)/x^6

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^7} \, dx=- b^{6} c^{5} \log {\left (x \right )} - \frac {10 a^{6} c^{5} - 48 a^{5} b c^{5} x + 75 a^{4} b^{2} c^{5} x^{2} - 150 a^{2} b^{4} c^{5} x^{4} + 240 a b^{5} c^{5} x^{5}}{60 x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**7,x)

[Out]

-b**6*c**5*log(x) - (10*a**6*c**5 - 48*a**5*b*c**5*x + 75*a**4*b**2*c**5*x**2 - 150*a**2*b**4*c**5*x**4 + 240*
a*b**5*c**5*x**5)/(60*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^7} \, dx=-b^{6} c^{5} \log \left (x\right ) - \frac {240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^7,x, algorithm="maxima")

[Out]

-b^6*c^5*log(x) - 1/60*(240*a*b^5*c^5*x^5 - 150*a^2*b^4*c^5*x^4 + 75*a^4*b^2*c^5*x^2 - 48*a^5*b*c^5*x + 10*a^6
*c^5)/x^6

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^7} \, dx=-b^{6} c^{5} \log \left ({\left | x \right |}\right ) - \frac {240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^7,x, algorithm="giac")

[Out]

-b^6*c^5*log(abs(x)) - 1/60*(240*a*b^5*c^5*x^5 - 150*a^2*b^4*c^5*x^4 + 75*a^4*b^2*c^5*x^2 - 48*a^5*b*c^5*x + 1
0*a^6*c^5)/x^6

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^7} \, dx=-\frac {c^5\,\left (10\,a^6+240\,a\,b^5\,x^5+75\,a^4\,b^2\,x^2-150\,a^2\,b^4\,x^4+60\,b^6\,x^6\,\ln \left (x\right )-48\,a^5\,b\,x\right )}{60\,x^6} \]

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^7,x)

[Out]

-(c^5*(10*a^6 + 240*a*b^5*x^5 + 75*a^4*b^2*x^2 - 150*a^2*b^4*x^4 + 60*b^6*x^6*log(x) - 48*a^5*b*x))/(60*x^6)