\(\int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx\) [22]

Optimal result

Integrand size = 20, antiderivative size = 72 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx=-\frac {a^5 c^4}{4 x^4}+\frac {a^4 b c^4}{x^3}-\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}+b^5 c^4 x-3 a b^4 c^4 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 72, 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} \[ \int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx=-\frac {a^5 c^4}{4 x^4}+\frac {a^4 b c^4}{x^3}-\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}-3 a b^4 c^4 \log (x)+b^5 c^4 x \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^5,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx=-\frac {a^5 c^4}{4 x^4}+\frac {a^4 b c^4}{x^3}-\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}+b^5 c^4 x-3 a b^4 c^4 \log (x) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^5,x]

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79

method	result	size
default	\(c^{4} \left (b^{5} x -3 a \,b^{4} \ln \left (x \right )+\frac {a^{4} b}{x^{3}}-\frac {2 a^{2} b^{3}}{x}-\frac {a^{3} b^{2}}{x^{2}}-\frac {a^{5}}{4 x^{4}}\right )\)	\(57\)
risch	\(b^{5} c^{4} x +\frac {-2 a^{2} b^{3} c^{4} x^{3}-a^{3} b^{2} c^{4} x^{2}+a^{4} b \,c^{4} x -\frac {1}{4} a^{5} c^{4}}{x^{4}}-3 a \,b^{4} c^{4} \ln \left (x \right )\)	\(71\)
norman	\(\frac {b^{5} c^{4} x^{5}+a^{4} b \,c^{4} x -\frac {1}{4} a^{5} c^{4}-2 a^{2} b^{3} c^{4} x^{3}-a^{3} b^{2} c^{4} x^{2}}{x^{4}}-3 a \,b^{4} c^{4} \ln \left (x \right )\)	\(73\)
parallelrisch	\(-\frac {12 a \,b^{4} c^{4} \ln \left (x \right ) x^{4}-4 b^{5} c^{4} x^{5}+8 a^{2} b^{3} c^{4} x^{3}+4 a^{3} b^{2} c^{4} x^{2}-4 a^{4} b \,c^{4} x +a^{5} c^{4}}{4 x^{4}}\)	\(77\)

[In]

int((b*x+a)*(-b*c*x+a*c)^4/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx=\frac {4 \, b^{5} c^{4} x^{5} - 12 \, a b^{4} c^{4} x^{4} \log \left (x\right ) - 8 \, a^{2} b^{3} c^{4} x^{3} - 4 \, a^{3} b^{2} c^{4} x^{2} + 4 \, a^{4} b c^{4} x - a^{5} c^{4}}{4 \, x^{4}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^5,x, algorithm="fricas")

[Out]

1/4*(4*b^5*c^4*x^5 - 12*a*b^4*c^4*x^4*log(x) - 8*a^2*b^3*c^4*x^3 - 4*a^3*b^2*c^4*x^2 + 4*a^4*b*c^4*x - a^5*c^4
)/x^4

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx=- 3 a b^{4} c^{4} \log {\left (x \right )} + b^{5} c^{4} x + \frac {- a^{5} c^{4} + 4 a^{4} b c^{4} x - 4 a^{3} b^{2} c^{4} x^{2} - 8 a^{2} b^{3} c^{4} x^{3}}{4 x^{4}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**5,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx=b^{5} c^{4} x - 3 \, a b^{4} c^{4} \log \left (x\right ) - \frac {8 \, a^{2} b^{3} c^{4} x^{3} + 4 \, a^{3} b^{2} c^{4} x^{2} - 4 \, a^{4} b c^{4} x + a^{5} c^{4}}{4 \, x^{4}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^5,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx=b^{5} c^{4} x - 3 \, a b^{4} c^{4} \log \left ({\left | x \right |}\right ) - \frac {8 \, a^{2} b^{3} c^{4} x^{3} + 4 \, a^{3} b^{2} c^{4} x^{2} - 4 \, a^{4} b c^{4} x + a^{5} c^{4}}{4 \, x^{4}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^5,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx=-\frac {c^4\,\left (a^5-4\,b^5\,x^5+4\,a^3\,b^2\,x^2+8\,a^2\,b^3\,x^3-4\,a^4\,b\,x+12\,a\,b^4\,x^4\,\ln \left (x\right )\right )}{4\,x^4} \]

[In]

int(((a*c - b*c*x)^4*(a + b*x))/x^5,x)

[Out]

-(c^4*(a^5 - 4*b^5*x^5 + 4*a^3*b^2*x^2 + 8*a^2*b^3*x^3 - 4*a^4*b*x + 12*a*b^4*x^4*log(x)))/(4*x^4)