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

Optimal. Leaf size=79 \[ \frac{5}{2} a^4 b^2 c^5 x^2-\frac{5}{4} a^2 b^4 c^5 x^4-4 a^5 b c^5 x+a^6 c^5 \log (x)+\frac{4}{5} a b^5 c^5 x^5-\frac{1}{6} b^6 c^5 x^6 \]

[Out]

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

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Rubi [A]  time = 0.030425, 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} \[ \frac{5}{2} a^4 b^2 c^5 x^2-\frac{5}{4} a^2 b^4 c^5 x^4-4 a^5 b c^5 x+a^6 c^5 \log (x)+\frac{4}{5} a b^5 c^5 x^5-\frac{1}{6} b^6 c^5 x^6 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x,x]

[Out]

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

Mathematica [A]  time = 0.0138068, size = 75, normalized size = 0.95 \[ c^5 \left (\frac{5}{2} a^4 b^2 x^2-\frac{5}{4} a^2 b^4 x^4-4 a^5 b x+a^6 \log (-b x)+\frac{127 a^6}{60}+\frac{4}{5} a b^5 x^5-\frac{b^6 x^6}{6}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^5*((127*a^6)/60 - 4*a^5*b*x + (5*a^4*b^2*x^2)/2 - (5*a^2*b^4*x^4)/4 + (4*a*b^5*x^5)/5 - (b^6*x^6)/6 + a^6*Lo
g[-(b*x)])

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Maple [A]  time = 0.002, size = 72, normalized size = 0.9 \begin{align*} -4\,{a}^{5}b{c}^{5}x+{\frac{5\,{a}^{4}{b}^{2}{c}^{5}{x}^{2}}{2}}-{\frac{5\,{a}^{2}{b}^{4}{c}^{5}{x}^{4}}{4}}+{\frac{4\,a{b}^{5}{c}^{5}{x}^{5}}{5}}-{\frac{{b}^{6}{c}^{5}{x}^{6}}{6}}+{a}^{6}{c}^{5}\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)^5/x,x)

[Out]

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

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Maxima [A]  time = 1.10545, size = 96, normalized size = 1.22 \begin{align*} -\frac{1}{6} \, b^{6} c^{5} x^{6} + \frac{4}{5} \, a b^{5} c^{5} x^{5} - \frac{5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac{5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.24573, size = 155, normalized size = 1.96 \begin{align*} -\frac{1}{6} \, b^{6} c^{5} x^{6} + \frac{4}{5} \, a b^{5} c^{5} x^{5} - \frac{5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac{5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.32173, size = 82, normalized size = 1.04 \begin{align*} a^{6} c^{5} \log{\left (x \right )} - 4 a^{5} b c^{5} x + \frac{5 a^{4} b^{2} c^{5} x^{2}}{2} - \frac{5 a^{2} b^{4} c^{5} x^{4}}{4} + \frac{4 a b^{5} c^{5} x^{5}}{5} - \frac{b^{6} c^{5} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26387, size = 97, normalized size = 1.23 \begin{align*} -\frac{1}{6} \, b^{6} c^{5} x^{6} + \frac{4}{5} \, a b^{5} c^{5} x^{5} - \frac{5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac{5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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