\(\int \frac {(a+b x) (a c-b c x)^5}{x^5} \, dx\) [36]

Optimal result

Integrand size = 20, antiderivative size = 80 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^5} \, dx=-\frac {a^6 c^5}{4 x^4}+\frac {4 a^5 b c^5}{3 x^3}-\frac {5 a^4 b^2 c^5}{2 x^2}+4 a b^5 c^5 x-\frac {1}{2} b^6 c^5 x^2-5 a^2 b^4 c^5 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, 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} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^5} \, dx=-\frac {a^6 c^5}{4 x^4}+\frac {4 a^5 b c^5}{3 x^3}-\frac {5 a^4 b^2 c^5}{2 x^2}-5 a^2 b^4 c^5 \log (x)+4 a b^5 c^5 x-\frac {1}{2} b^6 c^5 x^2 \]

[In]

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

[Out]

-1/4*(a^6*c^5)/x^4 + (4*a^5*b*c^5)/(3*x^3) - (5*a^4*b^2*c^5)/(2*x^2) + 4*a*b^5*c^5*x - (b^6*c^5*x^2)/2 - 5*a^2
*b^4*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 (4 a b^5 c^5+\frac {a^6 c^5}{x^5}-\frac {4 a^5 b c^5}{x^4}+\frac {5 a^4 b^2 c^5}{x^3}-\frac {5 a^2 b^4 c^5}{x}-b^6 c^5 x\right ) \, dx \\ & = -\frac {a^6 c^5}{4 x^4}+\frac {4 a^5 b c^5}{3 x^3}-\frac {5 a^4 b^2 c^5}{2 x^2}+4 a b^5 c^5 x-\frac {1}{2} b^6 c^5 x^2-5 a^2 b^4 c^5 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^5} \, dx=c^5 \left (-\frac {a^6}{4 x^4}+\frac {4 a^5 b}{3 x^3}-\frac {5 a^4 b^2}{2 x^2}+4 a b^5 x-\frac {b^6 x^2}{2}-5 a^2 b^4 \log (x)\right ) \]

[In]

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

[Out]

c^5*(-1/4*a^6/x^4 + (4*a^5*b)/(3*x^3) - (5*a^4*b^2)/(2*x^2) + 4*a*b^5*x - (b^6*x^2)/2 - 5*a^2*b^4*Log[x])

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.74

method	result	size
default	\(c^{5} \left (-\frac {b^{6} x^{2}}{2}+4 a \,b^{5} x -5 a^{2} b^{4} \ln \left (x \right )+\frac {4 a^{5} b}{3 x^{3}}-\frac {5 a^{4} b^{2}}{2 x^{2}}-\frac {a^{6}}{4 x^{4}}\right )\)	\(59\)
risch	\(-\frac {b^{6} c^{5} x^{2}}{2}+4 a \,b^{5} c^{5} x +\frac {-\frac {5}{2} a^{4} b^{2} c^{5} x^{2}+\frac {4}{3} a^{5} b \,c^{5} x -\frac {1}{4} a^{6} c^{5}}{x^{4}}-5 a^{2} b^{4} c^{5} \ln \left (x \right )\)	\(73\)
norman	\(\frac {-\frac {1}{4} a^{6} c^{5}-\frac {1}{2} b^{6} c^{5} x^{6}+4 a \,b^{5} c^{5} x^{5}-\frac {5}{2} a^{4} b^{2} c^{5} x^{2}+\frac {4}{3} a^{5} b \,c^{5} x}{x^{4}}-5 a^{2} b^{4} c^{5} \ln \left (x \right )\)	\(75\)
parallelrisch	\(-\frac {6 b^{6} c^{5} x^{6}+60 a^{2} c^{5} b^{4} \ln \left (x \right ) x^{4}-48 a \,b^{5} c^{5} x^{5}+30 a^{4} b^{2} c^{5} x^{2}-16 a^{5} b \,c^{5} x +3 a^{6} c^{5}}{12 x^{4}}\)	\(78\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^5} \, dx=-\frac {6 \, b^{6} c^{5} x^{6} - 48 \, a b^{5} c^{5} x^{5} + 60 \, a^{2} b^{4} c^{5} x^{4} \log \left (x\right ) + 30 \, a^{4} b^{2} c^{5} x^{2} - 16 \, a^{5} b c^{5} x + 3 \, a^{6} c^{5}}{12 \, x^{4}} \]

[In]

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

[Out]

-1/12*(6*b^6*c^5*x^6 - 48*a*b^5*c^5*x^5 + 60*a^2*b^4*c^5*x^4*log(x) + 30*a^4*b^2*c^5*x^2 - 16*a^5*b*c^5*x + 3*
a^6*c^5)/x^4

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^5} \, dx=- 5 a^{2} b^{4} c^{5} \log {\left (x \right )} + 4 a b^{5} c^{5} x - \frac {b^{6} c^{5} x^{2}}{2} - \frac {3 a^{6} c^{5} - 16 a^{5} b c^{5} x + 30 a^{4} b^{2} c^{5} x^{2}}{12 x^{4}} \]

[In]

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

[Out]

-5*a**2*b**4*c**5*log(x) + 4*a*b**5*c**5*x - b**6*c**5*x**2/2 - (3*a**6*c**5 - 16*a**5*b*c**5*x + 30*a**4*b**2
*c**5*x**2)/(12*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^5} \, dx=-\frac {1}{2} \, b^{6} c^{5} x^{2} + 4 \, a b^{5} c^{5} x - 5 \, a^{2} b^{4} c^{5} \log \left (x\right ) - \frac {30 \, a^{4} b^{2} c^{5} x^{2} - 16 \, a^{5} b c^{5} x + 3 \, a^{6} c^{5}}{12 \, x^{4}} \]

[In]

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

[Out]

-1/2*b^6*c^5*x^2 + 4*a*b^5*c^5*x - 5*a^2*b^4*c^5*log(x) - 1/12*(30*a^4*b^2*c^5*x^2 - 16*a^5*b*c^5*x + 3*a^6*c^
5)/x^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^5} \, dx=-\frac {1}{2} \, b^{6} c^{5} x^{2} + 4 \, a b^{5} c^{5} x - 5 \, a^{2} b^{4} c^{5} \log \left ({\left | x \right |}\right ) - \frac {30 \, a^{4} b^{2} c^{5} x^{2} - 16 \, a^{5} b c^{5} x + 3 \, a^{6} c^{5}}{12 \, x^{4}} \]

[In]

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

[Out]

-1/2*b^6*c^5*x^2 + 4*a*b^5*c^5*x - 5*a^2*b^4*c^5*log(abs(x)) - 1/12*(30*a^4*b^2*c^5*x^2 - 16*a^5*b*c^5*x + 3*a
^6*c^5)/x^4

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^5} \, dx=4\,a\,b^5\,c^5\,x-\frac {b^6\,c^5\,x^2}{2}-5\,a^2\,b^4\,c^5\,\ln \left (x\right )-\frac {\frac {a^6\,c^5}{4}-\frac {4\,a^5\,b\,c^5\,x}{3}+\frac {5\,a^4\,b^2\,c^5\,x^2}{2}}{x^4} \]

[In]

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

[Out]

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