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

Optimal. Leaf size=72 \[ -\frac{a^3 b^2 c^4}{x^2}-\frac{2 a^2 b^3 c^4}{x}+\frac{a^4 b c^4}{x^3}-\frac{a^5 c^4}{4 x^4}-3 a b^4 c^4 \log (x)+b^5 c^4 x \]

[Out]

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

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Rubi [A]  time = 0.0345822, antiderivative size = 72, 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} \[ -\frac{a^3 b^2 c^4}{x^2}-\frac{2 a^2 b^3 c^4}{x}+\frac{a^4 b c^4}{x^3}-\frac{a^5 c^4}{4 x^4}-3 a b^4 c^4 \log (x)+b^5 c^4 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^5*c^4)/(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 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^5} \, dx &=\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]  time = 0.00846, size = 72, normalized size = 1. \[ -\frac{a^3 b^2 c^4}{x^2}-\frac{2 a^2 b^3 c^4}{x}+\frac{a^4 b c^4}{x^3}-\frac{a^5 c^4}{4 x^4}-3 a b^4 c^4 \log (x)+b^5 c^4 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 71, normalized size = 1. \begin{align*} -{\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}}}-2\,{\frac{{a}^{2}{b}^{3}{c}^{4}}{x}}+{b}^{5}{c}^{4}x-3\,a{b}^{4}{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^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*ln(x)

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Maxima [A]  time = 1.06708, size = 96, normalized size = 1.33 \begin{align*} 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.95553, size = 158, normalized size = 2.19 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.503976, size = 75, normalized size = 1.04 \begin{align*} - 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.26259, size = 97, normalized size = 1.35 \begin{align*} 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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