3.37 \(\int \frac{(a+b x) (a c-b c x)^5}{x^6} \, dx\)

Optimal. Leaf size=75 \[ -\frac{5 a^4 b^2 c^5}{3 x^3}+\frac{5 a^2 b^4 c^5}{x}+\frac{a^5 b c^5}{x^4}-\frac{a^6 c^5}{5 x^5}+4 a b^5 c^5 \log (x)-b^6 c^5 x \]

[Out]

-(a^6*c^5)/(5*x^5) + (a^5*b*c^5)/x^4 - (5*a^4*b^2*c^5)/(3*x^3) + (5*a^2*b^4*c^5)/x - b^6*c^5*x + 4*a*b^5*c^5*L
og[x]

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Rubi [A]  time = 0.0340368, antiderivative size = 75, 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} \[ -\frac{5 a^4 b^2 c^5}{3 x^3}+\frac{5 a^2 b^4 c^5}{x}+\frac{a^5 b c^5}{x^4}-\frac{a^6 c^5}{5 x^5}+4 a b^5 c^5 \log (x)-b^6 c^5 x \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^6,x]

[Out]

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

Mathematica [A]  time = 0.0067582, size = 61, normalized size = 0.81 \[ c^5 \left (-\frac{5 a^4 b^2}{3 x^3}+\frac{5 a^2 b^4}{x}+\frac{a^5 b}{x^4}-\frac{a^6}{5 x^5}+4 a b^5 \log (x)-b^6 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^6,x]

[Out]

c^5*(-a^6/(5*x^5) + (a^5*b)/x^4 - (5*a^4*b^2)/(3*x^3) + (5*a^2*b^4)/x - b^6*x + 4*a*b^5*Log[x])

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Maple [A]  time = 0.006, size = 72, normalized size = 1. \begin{align*} -{\frac{{a}^{6}{c}^{5}}{5\,{x}^{5}}}+{\frac{{a}^{5}b{c}^{5}}{{x}^{4}}}-{\frac{5\,{a}^{4}{b}^{2}{c}^{5}}{3\,{x}^{3}}}+5\,{\frac{{a}^{2}{b}^{4}{c}^{5}}{x}}-{b}^{6}{c}^{5}x+4\,a{b}^{5}{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^6,x)

[Out]

-1/5*a^6*c^5/x^5+a^5*b*c^5/x^4-5/3*a^4*b^2*c^5/x^3+5*a^2*b^4*c^5/x-b^6*c^5*x+4*a*b^5*c^5*ln(x)

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Maxima [A]  time = 1.09836, size = 99, normalized size = 1.32 \begin{align*} -b^{6} c^{5} x + 4 \, a b^{5} c^{5} \log \left (x\right ) + \frac{75 \, a^{2} b^{4} c^{5} x^{4} - 25 \, a^{4} b^{2} c^{5} x^{2} + 15 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^6,x, algorithm="maxima")

[Out]

-b^6*c^5*x + 4*a*b^5*c^5*log(x) + 1/15*(75*a^2*b^4*c^5*x^4 - 25*a^4*b^2*c^5*x^2 + 15*a^5*b*c^5*x - 3*a^6*c^5)/
x^5

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Fricas [A]  time = 1.73198, size = 169, normalized size = 2.25 \begin{align*} -\frac{15 \, b^{6} c^{5} x^{6} - 60 \, a b^{5} c^{5} x^{5} \log \left (x\right ) - 75 \, a^{2} b^{4} c^{5} x^{4} + 25 \, a^{4} b^{2} c^{5} x^{2} - 15 \, a^{5} b c^{5} x + 3 \, a^{6} c^{5}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^6,x, algorithm="fricas")

[Out]

-1/15*(15*b^6*c^5*x^6 - 60*a*b^5*c^5*x^5*log(x) - 75*a^2*b^4*c^5*x^4 + 25*a^4*b^2*c^5*x^2 - 15*a^5*b*c^5*x + 3
*a^6*c^5)/x^5

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Sympy [A]  time = 0.547812, size = 76, normalized size = 1.01 \begin{align*} 4 a b^{5} c^{5} \log{\left (x \right )} - b^{6} c^{5} x + \frac{- 3 a^{6} c^{5} + 15 a^{5} b c^{5} x - 25 a^{4} b^{2} c^{5} x^{2} + 75 a^{2} b^{4} c^{5} x^{4}}{15 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**6,x)

[Out]

4*a*b**5*c**5*log(x) - b**6*c**5*x + (-3*a**6*c**5 + 15*a**5*b*c**5*x - 25*a**4*b**2*c**5*x**2 + 75*a**2*b**4*
c**5*x**4)/(15*x**5)

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Giac [A]  time = 1.2897, size = 100, normalized size = 1.33 \begin{align*} -b^{6} c^{5} x + 4 \, a b^{5} c^{5} \log \left ({\left | x \right |}\right ) + \frac{75 \, a^{2} b^{4} c^{5} x^{4} - 25 \, a^{4} b^{2} c^{5} x^{2} + 15 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^6,x, algorithm="giac")

[Out]

-b^6*c^5*x + 4*a*b^5*c^5*log(abs(x)) + 1/15*(75*a^2*b^4*c^5*x^4 - 25*a^4*b^2*c^5*x^2 + 15*a^5*b*c^5*x - 3*a^6*
c^5)/x^5