\(\int \frac {(a+b x) (a c-b c x)^5}{x^3} \, dx\) [34]

Optimal result

Integrand size = 20, antiderivative size = 82 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^3} \, dx=-\frac {a^6 c^5}{2 x^2}+\frac {4 a^5 b c^5}{x}-\frac {5}{2} a^2 b^4 c^5 x^2+\frac {4}{3} a b^5 c^5 x^3-\frac {1}{4} b^6 c^5 x^4+5 a^4 b^2 c^5 \log (x) \]

[Out]

-1/2*a^6*c^5/x^2+4*a^5*b*c^5/x-5/2*a^2*b^4*c^5*x^2+4/3*a*b^5*c^5*x^3-1/4*b^6*c^5*x^4+5*a^4*b^2*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} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^3} \, dx=-\frac {a^6 c^5}{2 x^2}+\frac {4 a^5 b c^5}{x}+5 a^4 b^2 c^5 \log (x)-\frac {5}{2} a^2 b^4 c^5 x^2+\frac {4}{3} a b^5 c^5 x^3-\frac {1}{4} b^6 c^5 x^4 \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^3} \, dx=c^5 \left (-\frac {a^6}{2 x^2}+\frac {4 a^5 b}{x}-\frac {5}{2} a^2 b^4 x^2+\frac {4}{3} a b^5 x^3-\frac {b^6 x^4}{4}+5 a^4 b^2 \log (x)\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74

method	result	size
default	\(c^{5} \left (-\frac {b^{6} x^{4}}{4}+\frac {4 a \,b^{5} x^{3}}{3}-\frac {5 a^{2} b^{4} x^{2}}{2}+5 a^{4} b^{2} \ln \left (x \right )+\frac {4 a^{5} b}{x}-\frac {a^{6}}{2 x^{2}}\right )\)	\(61\)
norman	\(\frac {-\frac {1}{2} a^{6} c^{5}-\frac {1}{4} b^{6} c^{5} x^{6}+\frac {4}{3} a \,b^{5} c^{5} x^{5}-\frac {5}{2} a^{2} b^{4} c^{5} x^{4}+4 a^{5} b \,c^{5} x}{x^{2}}+5 a^{4} b^{2} c^{5} \ln \left (x \right )\)	\(75\)
risch	\(-\frac {b^{6} c^{5} x^{4}}{4}+\frac {4 a \,b^{5} c^{5} x^{3}}{3}-\frac {5 a^{2} b^{4} c^{5} x^{2}}{2}+\frac {4 a^{5} b \,c^{5} x -\frac {1}{2} a^{6} c^{5}}{x^{2}}+5 a^{4} b^{2} c^{5} \ln \left (x \right )\)	\(75\)
parallelrisch	\(\frac {-3 b^{6} c^{5} x^{6}+16 a \,b^{5} c^{5} x^{5}-30 a^{2} b^{4} c^{5} x^{4}+60 a^{4} c^{5} b^{2} \ln \left (x \right ) x^{2}+48 a^{5} b \,c^{5} x -6 a^{6} c^{5}}{12 x^{2}}\)	\(78\)

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^3} \, dx=-\frac {3 \, b^{6} c^{5} x^{6} - 16 \, a b^{5} c^{5} x^{5} + 30 \, a^{2} b^{4} c^{5} x^{4} - 60 \, a^{4} b^{2} c^{5} x^{2} \log \left (x\right ) - 48 \, a^{5} b c^{5} x + 6 \, a^{6} c^{5}}{12 \, x^{2}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^3} \, dx=5 a^{4} b^{2} c^{5} \log {\left (x \right )} - \frac {5 a^{2} b^{4} c^{5} x^{2}}{2} + \frac {4 a b^{5} c^{5} x^{3}}{3} - \frac {b^{6} c^{5} x^{4}}{4} - \frac {a^{6} c^{5} - 8 a^{5} b c^{5} x}{2 x^{2}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**3,x)

[Out]

5*a**4*b**2*c**5*log(x) - 5*a**2*b**4*c**5*x**2/2 + 4*a*b**5*c**5*x**3/3 - b**6*c**5*x**4/4 - (a**6*c**5 - 8*a
**5*b*c**5*x)/(2*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^3} \, dx=-\frac {1}{4} \, b^{6} c^{5} x^{4} + \frac {4}{3} \, a b^{5} c^{5} x^{3} - \frac {5}{2} \, a^{2} b^{4} c^{5} x^{2} + 5 \, a^{4} b^{2} c^{5} \log \left (x\right ) + \frac {8 \, a^{5} b c^{5} x - a^{6} c^{5}}{2 \, x^{2}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^3,x, algorithm="maxima")

[Out]

-1/4*b^6*c^5*x^4 + 4/3*a*b^5*c^5*x^3 - 5/2*a^2*b^4*c^5*x^2 + 5*a^4*b^2*c^5*log(x) + 1/2*(8*a^5*b*c^5*x - a^6*c
^5)/x^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^3} \, dx=-\frac {1}{4} \, b^{6} c^{5} x^{4} + \frac {4}{3} \, a b^{5} c^{5} x^{3} - \frac {5}{2} \, a^{2} b^{4} c^{5} x^{2} + 5 \, a^{4} b^{2} c^{5} \log \left ({\left | x \right |}\right ) + \frac {8 \, a^{5} b c^{5} x - a^{6} c^{5}}{2 \, x^{2}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^3,x, algorithm="giac")

[Out]

-1/4*b^6*c^5*x^4 + 4/3*a*b^5*c^5*x^3 - 5/2*a^2*b^4*c^5*x^2 + 5*a^4*b^2*c^5*log(abs(x)) + 1/2*(8*a^5*b*c^5*x -
a^6*c^5)/x^2

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^3} \, dx=\frac {4\,a\,b^5\,c^5\,x^3}{3}-\frac {b^6\,c^5\,x^4}{4}-\frac {\frac {a^6\,c^5}{2}-4\,a^5\,b\,c^5\,x}{x^2}-\frac {5\,a^2\,b^4\,c^5\,x^2}{2}+5\,a^4\,b^2\,c^5\,\ln \left (x\right ) \]

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^3,x)

[Out]

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