3.38 \(\int \frac{(a+b x) (a c-b c x)^5}{x^7} \, dx\)

Optimal. Leaf size=82 \[ -\frac{5 a^4 b^2 c^5}{4 x^4}+\frac{5 a^2 b^4 c^5}{2 x^2}+\frac{4 a^5 b c^5}{5 x^5}-\frac{a^6 c^5}{6 x^6}-\frac{4 a b^5 c^5}{x}-b^6 c^5 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0351833, antiderivative size = 82, 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} \[ -\frac{5 a^4 b^2 c^5}{4 x^4}+\frac{5 a^2 b^4 c^5}{2 x^2}+\frac{4 a^5 b c^5}{5 x^5}-\frac{a^6 c^5}{6 x^6}-\frac{4 a b^5 c^5}{x}-b^6 c^5 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^7,x]

[Out]

-(a^6*c^5)/(6*x^6) + (4*a^5*b*c^5)/(5*x^5) - (5*a^4*b^2*c^5)/(4*x^4) + (5*a^2*b^4*c^5)/(2*x^2) - (4*a*b^5*c^5)
/x - b^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^7} \, dx &=\int \left (\frac{a^6 c^5}{x^7}-\frac{4 a^5 b c^5}{x^6}+\frac{5 a^4 b^2 c^5}{x^5}-\frac{5 a^2 b^4 c^5}{x^3}+\frac{4 a b^5 c^5}{x^2}-\frac{b^6 c^5}{x}\right ) \, dx\\ &=-\frac{a^6 c^5}{6 x^6}+\frac{4 a^5 b c^5}{5 x^5}-\frac{5 a^4 b^2 c^5}{4 x^4}+\frac{5 a^2 b^4 c^5}{2 x^2}-\frac{4 a b^5 c^5}{x}-b^6 c^5 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0067883, size = 68, normalized size = 0.83 \[ c^5 \left (-\frac{5 a^4 b^2}{4 x^4}+\frac{5 a^2 b^4}{2 x^2}+\frac{4 a^5 b}{5 x^5}-\frac{a^6}{6 x^6}-\frac{4 a b^5}{x}-b^6 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^7,x]

[Out]

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

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Maple [A]  time = 0.007, size = 75, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}{c}^{5}}{6\,{x}^{6}}}+{\frac{4\,{a}^{5}b{c}^{5}}{5\,{x}^{5}}}-{\frac{5\,{a}^{4}{b}^{2}{c}^{5}}{4\,{x}^{4}}}+{\frac{5\,{a}^{2}{b}^{4}{c}^{5}}{2\,{x}^{2}}}-4\,{\frac{a{b}^{5}{c}^{5}}{x}}-{b}^{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^7,x)

[Out]

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

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Maxima [A]  time = 1.1054, size = 101, normalized size = 1.23 \begin{align*} -b^{6} c^{5} \log \left (x\right ) - \frac{240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{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^7,x, algorithm="maxima")

[Out]

-b^6*c^5*log(x) - 1/60*(240*a*b^5*c^5*x^5 - 150*a^2*b^4*c^5*x^4 + 75*a^4*b^2*c^5*x^2 - 48*a^5*b*c^5*x + 10*a^6
*c^5)/x^6

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Fricas [A]  time = 1.76026, size = 173, normalized size = 2.11 \begin{align*} -\frac{60 \, b^{6} c^{5} x^{6} \log \left (x\right ) + 240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{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^7,x, algorithm="fricas")

[Out]

-1/60*(60*b^6*c^5*x^6*log(x) + 240*a*b^5*c^5*x^5 - 150*a^2*b^4*c^5*x^4 + 75*a^4*b^2*c^5*x^2 - 48*a^5*b*c^5*x +
10*a^6*c^5)/x^6

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Sympy [A]  time = 0.649149, size = 80, normalized size = 0.98 \begin{align*} - b^{6} c^{5} \log{\left (x \right )} - \frac{10 a^{6} c^{5} - 48 a^{5} b c^{5} x + 75 a^{4} b^{2} c^{5} x^{2} - 150 a^{2} b^{4} c^{5} x^{4} + 240 a b^{5} c^{5} x^{5}}{60 x^{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**7,x)

[Out]

-b**6*c**5*log(x) - (10*a**6*c**5 - 48*a**5*b*c**5*x + 75*a**4*b**2*c**5*x**2 - 150*a**2*b**4*c**5*x**4 + 240*
a*b**5*c**5*x**5)/(60*x**6)

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Giac [A]  time = 1.31999, size = 103, normalized size = 1.26 \begin{align*} -b^{6} c^{5} \log \left ({\left | x \right |}\right ) - \frac{240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{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^7,x, algorithm="giac")

[Out]

-b^6*c^5*log(abs(x)) - 1/60*(240*a*b^5*c^5*x^5 - 150*a^2*b^4*c^5*x^4 + 75*a^4*b^2*c^5*x^2 - 48*a^5*b*c^5*x + 1
0*a^6*c^5)/x^6