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

Optimal. Leaf size=79 \[ -\frac{2 a^3 b^2 c^4}{3 x^3}-\frac{a^2 b^3 c^4}{x^2}+\frac{3 a^4 b c^4}{4 x^4}-\frac{a^5 c^4}{5 x^5}+\frac{3 a b^4 c^4}{x}+b^5 c^4 \log (x) \]

[Out]

-(a^5*c^4)/(5*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]

________________________________________________________________________________________

Rubi [A]  time = 0.0354897, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {75} \[ -\frac{2 a^3 b^2 c^4}{3 x^3}-\frac{a^2 b^3 c^4}{x^2}+\frac{3 a^4 b c^4}{4 x^4}-\frac{a^5 c^4}{5 x^5}+\frac{3 a b^4 c^4}{x}+b^5 c^4 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^5*c^4)/(5*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 75

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*} \int \frac{(a+b x) (a c-b c x)^4}{x^6} \, dx &=\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]  time = 0.0078035, size = 79, normalized size = 1. \[ -\frac{2 a^3 b^2 c^4}{3 x^3}-\frac{a^2 b^3 c^4}{x^2}+\frac{3 a^4 b c^4}{4 x^4}-\frac{a^5 c^4}{5 x^5}+\frac{3 a b^4 c^4}{x}+b^5 c^4 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^5*c^4)/(5*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]  time = 0.005, size = 74, normalized size = 0.9 \begin{align*} -{\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}}}+3\,{\frac{a{b}^{4}{c}^{4}}{x}}+{b}^{5}{c}^{4}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Maxima [A]  time = 1.08663, size = 100, normalized size = 1.27 \begin{align*} 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]  time = 1.88507, size = 170, normalized size = 2.15 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0.556663, size = 78, normalized size = 0.99 \begin{align*} 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]  time = 1.21829, size = 101, normalized size = 1.28 \begin{align*} 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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