3.44 \(\int \frac{(a+b x) (a c-b c x)^6}{x^8} \, dx\)

Optimal. Leaf size=113 \[ -\frac{9 a^5 b^2 c^6}{5 x^5}+\frac{5 a^4 b^3 c^6}{4 x^4}+\frac{5 a^3 b^4 c^6}{3 x^3}-\frac{9 a^2 b^5 c^6}{2 x^2}+\frac{5 a^6 b c^6}{6 x^6}-\frac{a^7 c^6}{7 x^7}+\frac{5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]

[Out]

-(a^7*c^6)/(7*x^7) + (5*a^6*b*c^6)/(6*x^6) - (9*a^5*b^2*c^6)/(5*x^5) + (5*a^4*b^3*c^6)/(4*x^4) + (5*a^3*b^4*c^
6)/(3*x^3) - (9*a^2*b^5*c^6)/(2*x^2) + (5*a*b^6*c^6)/x + b^7*c^6*Log[x]

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Rubi [A]  time = 0.0616419, antiderivative size = 113, 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} \[ -\frac{9 a^5 b^2 c^6}{5 x^5}+\frac{5 a^4 b^3 c^6}{4 x^4}+\frac{5 a^3 b^4 c^6}{3 x^3}-\frac{9 a^2 b^5 c^6}{2 x^2}+\frac{5 a^6 b c^6}{6 x^6}-\frac{a^7 c^6}{7 x^7}+\frac{5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^6)/x^8,x]

[Out]

-(a^7*c^6)/(7*x^7) + (5*a^6*b*c^6)/(6*x^6) - (9*a^5*b^2*c^6)/(5*x^5) + (5*a^4*b^3*c^6)/(4*x^4) + (5*a^3*b^4*c^
6)/(3*x^3) - (9*a^2*b^5*c^6)/(2*x^2) + (5*a*b^6*c^6)/x + b^7*c^6*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)^6}{x^8} \, dx &=\int \left (\frac{a^7 c^6}{x^8}-\frac{5 a^6 b c^6}{x^7}+\frac{9 a^5 b^2 c^6}{x^6}-\frac{5 a^4 b^3 c^6}{x^5}-\frac{5 a^3 b^4 c^6}{x^4}+\frac{9 a^2 b^5 c^6}{x^3}-\frac{5 a b^6 c^6}{x^2}+\frac{b^7 c^6}{x}\right ) \, dx\\ &=-\frac{a^7 c^6}{7 x^7}+\frac{5 a^6 b c^6}{6 x^6}-\frac{9 a^5 b^2 c^6}{5 x^5}+\frac{5 a^4 b^3 c^6}{4 x^4}+\frac{5 a^3 b^4 c^6}{3 x^3}-\frac{9 a^2 b^5 c^6}{2 x^2}+\frac{5 a b^6 c^6}{x}+b^7 c^6 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0113898, size = 113, normalized size = 1. \[ -\frac{9 a^5 b^2 c^6}{5 x^5}+\frac{5 a^4 b^3 c^6}{4 x^4}+\frac{5 a^3 b^4 c^6}{3 x^3}-\frac{9 a^2 b^5 c^6}{2 x^2}+\frac{5 a^6 b c^6}{6 x^6}-\frac{a^7 c^6}{7 x^7}+\frac{5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^8,x]

[Out]

-(a^7*c^6)/(7*x^7) + (5*a^6*b*c^6)/(6*x^6) - (9*a^5*b^2*c^6)/(5*x^5) + (5*a^4*b^3*c^6)/(4*x^4) + (5*a^3*b^4*c^
6)/(3*x^3) - (9*a^2*b^5*c^6)/(2*x^2) + (5*a*b^6*c^6)/x + b^7*c^6*Log[x]

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Maple [A]  time = 0.008, size = 102, normalized size = 0.9 \begin{align*} -{\frac{{a}^{7}{c}^{6}}{7\,{x}^{7}}}+{\frac{5\,{a}^{6}b{c}^{6}}{6\,{x}^{6}}}-{\frac{9\,{a}^{5}{b}^{2}{c}^{6}}{5\,{x}^{5}}}+{\frac{5\,{a}^{4}{b}^{3}{c}^{6}}{4\,{x}^{4}}}+{\frac{5\,{a}^{3}{b}^{4}{c}^{6}}{3\,{x}^{3}}}-{\frac{9\,{a}^{2}{b}^{5}{c}^{6}}{2\,{x}^{2}}}+5\,{\frac{a{b}^{6}{c}^{6}}{x}}+{b}^{7}{c}^{6}\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)^6/x^8,x)

[Out]

-1/7*a^7*c^6/x^7+5/6*a^6*b*c^6/x^6-9/5*a^5*b^2*c^6/x^5+5/4*a^4*b^3*c^6/x^4+5/3*a^3*b^4*c^6/x^3-9/2*a^2*b^5*c^6
/x^2+5*a*b^6*c^6/x+b^7*c^6*ln(x)

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Maxima [A]  time = 1.13953, size = 138, normalized size = 1.22 \begin{align*} b^{7} c^{6} \log \left (x\right ) + \frac{2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^8,x, algorithm="maxima")

[Out]

b^7*c^6*log(x) + 1/420*(2100*a*b^6*c^6*x^6 - 1890*a^2*b^5*c^6*x^5 + 700*a^3*b^4*c^6*x^4 + 525*a^4*b^3*c^6*x^3
- 756*a^5*b^2*c^6*x^2 + 350*a^6*b*c^6*x - 60*a^7*c^6)/x^7

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Fricas [A]  time = 1.77647, size = 239, normalized size = 2.12 \begin{align*} \frac{420 \, b^{7} c^{6} x^{7} \log \left (x\right ) + 2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^8,x, algorithm="fricas")

[Out]

1/420*(420*b^7*c^6*x^7*log(x) + 2100*a*b^6*c^6*x^6 - 1890*a^2*b^5*c^6*x^5 + 700*a^3*b^4*c^6*x^4 + 525*a^4*b^3*
c^6*x^3 - 756*a^5*b^2*c^6*x^2 + 350*a^6*b*c^6*x - 60*a^7*c^6)/x^7

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Sympy [A]  time = 0.777313, size = 109, normalized size = 0.96 \begin{align*} b^{7} c^{6} \log{\left (x \right )} + \frac{- 60 a^{7} c^{6} + 350 a^{6} b c^{6} x - 756 a^{5} b^{2} c^{6} x^{2} + 525 a^{4} b^{3} c^{6} x^{3} + 700 a^{3} b^{4} c^{6} x^{4} - 1890 a^{2} b^{5} c^{6} x^{5} + 2100 a b^{6} c^{6} x^{6}}{420 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**6/x**8,x)

[Out]

b**7*c**6*log(x) + (-60*a**7*c**6 + 350*a**6*b*c**6*x - 756*a**5*b**2*c**6*x**2 + 525*a**4*b**3*c**6*x**3 + 70
0*a**3*b**4*c**6*x**4 - 1890*a**2*b**5*c**6*x**5 + 2100*a*b**6*c**6*x**6)/(420*x**7)

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Giac [A]  time = 1.16445, size = 139, normalized size = 1.23 \begin{align*} b^{7} c^{6} \log \left ({\left | x \right |}\right ) + \frac{2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^8,x, algorithm="giac")

[Out]

b^7*c^6*log(abs(x)) + 1/420*(2100*a*b^6*c^6*x^6 - 1890*a^2*b^5*c^6*x^5 + 700*a^3*b^4*c^6*x^4 + 525*a^4*b^3*c^6
*x^3 - 756*a^5*b^2*c^6*x^2 + 350*a^6*b*c^6*x - 60*a^7*c^6)/x^7