\(\int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx\) [23]

Optimal result

Integrand size = 20, antiderivative size = 79 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx=-\frac {a^5 c^4}{5 x^5}+\frac {3 a^4 b c^4}{4 x^4}-\frac {2 a^3 b^2 c^4}{3 x^3}-\frac {a^2 b^3 c^4}{x^2}+\frac {3 a b^4 c^4}{x}+b^5 c^4 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 79, 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} \[ \int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx=-\frac {a^5 c^4}{5 x^5}+\frac {3 a^4 b c^4}{4 x^4}-\frac {2 a^3 b^2 c^4}{3 x^3}-\frac {a^2 b^3 c^4}{x^2}+\frac {3 a b^4 c^4}{x}+b^5 c^4 \log (x) \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^6,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx=-\frac {a^5 c^4}{5 x^5}+\frac {3 a^4 b c^4}{4 x^4}-\frac {2 a^3 b^2 c^4}{3 x^3}-\frac {a^2 b^3 c^4}{x^2}+\frac {3 a b^4 c^4}{x}+b^5 c^4 \log (x) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^6,x]

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76

method	result	size
default	\(c^{4} \left (b^{5} \ln \left (x \right )-\frac {2 a^{3} b^{2}}{3 x^{3}}+\frac {3 a \,b^{4}}{x}-\frac {a^{2} b^{3}}{x^{2}}+\frac {3 a^{4} b}{4 x^{4}}-\frac {a^{5}}{5 x^{5}}\right )\)	\(60\)
norman	\(\frac {-\frac {1}{5} a^{5} c^{4}+3 a \,b^{4} c^{4} x^{4}-a^{2} b^{3} c^{4} x^{3}-\frac {2}{3} a^{3} b^{2} c^{4} x^{2}+\frac {3}{4} a^{4} b \,c^{4} x}{x^{5}}+b^{5} c^{4} \ln \left (x \right )\)	\(74\)
risch	\(\frac {-\frac {1}{5} a^{5} c^{4}+3 a \,b^{4} c^{4} x^{4}-a^{2} b^{3} c^{4} x^{3}-\frac {2}{3} a^{3} b^{2} c^{4} x^{2}+\frac {3}{4} a^{4} b \,c^{4} x}{x^{5}}+b^{5} c^{4} \ln \left (x \right )\)	\(74\)
parallelrisch	\(\frac {60 b^{5} c^{4} \ln \left (x \right ) x^{5}+180 a \,b^{4} c^{4} x^{4}-60 a^{2} b^{3} c^{4} x^{3}-40 a^{3} b^{2} c^{4} x^{2}+45 a^{4} b \,c^{4} x -12 a^{5} c^{4}}{60 x^{5}}\)	\(78\)

[In]

int((b*x+a)*(-b*c*x+a*c)^4/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx=\frac {60 \, b^{5} c^{4} x^{5} \log \left (x\right ) + 180 \, a b^{4} c^{4} x^{4} - 60 \, a^{2} b^{3} c^{4} x^{3} - 40 \, a^{3} b^{2} c^{4} x^{2} + 45 \, a^{4} b c^{4} x - 12 \, a^{5} c^{4}}{60 \, x^{5}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^6,x, algorithm="fricas")

[Out]

1/60*(60*b^5*c^4*x^5*log(x) + 180*a*b^4*c^4*x^4 - 60*a^2*b^3*c^4*x^3 - 40*a^3*b^2*c^4*x^2 + 45*a^4*b*c^4*x - 1
2*a^5*c^4)/x^5

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx=b^{5} c^{4} \log {\left (x \right )} + \frac {- 12 a^{5} c^{4} + 45 a^{4} b c^{4} x - 40 a^{3} b^{2} c^{4} x^{2} - 60 a^{2} b^{3} c^{4} x^{3} + 180 a b^{4} c^{4} x^{4}}{60 x^{5}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**6,x)

[Out]

b**5*c**4*log(x) + (-12*a**5*c**4 + 45*a**4*b*c**4*x - 40*a**3*b**2*c**4*x**2 - 60*a**2*b**3*c**4*x**3 + 180*a
*b**4*c**4*x**4)/(60*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx=b^{5} c^{4} \log \left (x\right ) + \frac {180 \, a b^{4} c^{4} x^{4} - 60 \, a^{2} b^{3} c^{4} x^{3} - 40 \, a^{3} b^{2} c^{4} x^{2} + 45 \, a^{4} b c^{4} x - 12 \, a^{5} c^{4}}{60 \, x^{5}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^6,x, algorithm="maxima")

[Out]

b^5*c^4*log(x) + 1/60*(180*a*b^4*c^4*x^4 - 60*a^2*b^3*c^4*x^3 - 40*a^3*b^2*c^4*x^2 + 45*a^4*b*c^4*x - 12*a^5*c
^4)/x^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx=b^{5} c^{4} \log \left ({\left | x \right |}\right ) + \frac {180 \, a b^{4} c^{4} x^{4} - 60 \, a^{2} b^{3} c^{4} x^{3} - 40 \, a^{3} b^{2} c^{4} x^{2} + 45 \, a^{4} b c^{4} x - 12 \, a^{5} c^{4}}{60 \, x^{5}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^6,x, algorithm="giac")

[Out]

b^5*c^4*log(abs(x)) + 1/60*(180*a*b^4*c^4*x^4 - 60*a^2*b^3*c^4*x^3 - 40*a^3*b^2*c^4*x^2 + 45*a^4*b*c^4*x - 12*
a^5*c^4)/x^5

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx=-\frac {c^4\,\left (\frac {a^5}{5}-3\,a\,b^4\,x^4+\frac {2\,a^3\,b^2\,x^2}{3}+a^2\,b^3\,x^3-b^5\,x^5\,\ln \left (x\right )-\frac {3\,a^4\,b\,x}{4}\right )}{x^5} \]

[In]

int(((a*c - b*c*x)^4*(a + b*x))/x^6,x)

[Out]

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