3.22 \(\int \frac{(a+b x) (a c-b c x)^4}{x^6} \, dx\)

Optimal. Leaf size=79 \[ -\frac{a^5 c^4}{5 x^5}+\frac{3 a^4 b c^4}{4 x^4}-\frac{2 a^3 b^2 c^4}{3 x^3}-\frac{a^2 b^3 c^4}{x^2}+\frac{3 a b^4 c^4}{x}+b^5 c^4 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0883102, 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 \[ -\frac{a^5 c^4}{5 x^5}+\frac{3 a^4 b c^4}{4 x^4}-\frac{2 a^3 b^2 c^4}{3 x^3}-\frac{a^2 b^3 c^4}{x^2}+\frac{3 a b^4 c^4}{x}+b^5 c^4 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^4)/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 29.5247, size = 78, normalized size = 0.99 \[ - \frac{a^{5} c^{4}}{5 x^{5}} + \frac{3 a^{4} b c^{4}}{4 x^{4}} - \frac{2 a^{3} b^{2} c^{4}}{3 x^{3}} - \frac{a^{2} b^{3} c^{4}}{x^{2}} + \frac{3 a b^{4} c^{4}}{x} + b^{5} c^{4} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**4/x**6,x)

[Out]

-a**5*c**4/(5*x**5) + 3*a**4*b*c**4/(4*x**4) - 2*a**3*b**2*c**4/(3*x**3) - a**2*
b**3*c**4/x**2 + 3*a*b**4*c**4/x + b**5*c**4*log(x)

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Mathematica [A]  time = 0.0119251, size = 79, normalized size = 1. \[ -\frac{a^5 c^4}{5 x^5}+\frac{3 a^4 b c^4}{4 x^4}-\frac{2 a^3 b^2 c^4}{3 x^3}-\frac{a^2 b^3 c^4}{x^2}+\frac{3 a b^4 c^4}{x}+b^5 c^4 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^6,x]

[Out]

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

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Maple [A]  time = 0.01, size = 74, normalized size = 0.9 \[ -{\frac{{a}^{5}{c}^{4}}{5\,{x}^{5}}}+{\frac{3\,{a}^{4}b{c}^{4}}{4\,{x}^{4}}}-{\frac{2\,{a}^{3}{b}^{2}{c}^{4}}{3\,{x}^{3}}}-{\frac{{a}^{2}{b}^{3}{c}^{4}}{{x}^{2}}}+3\,{\frac{a{b}^{4}{c}^{4}}{x}}+{b}^{5}{c}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^4/x^6,x)

[Out]

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

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Maxima [A]  time = 1.35063, size = 100, normalized size = 1.27 \[ b^{5} c^{4} \log \left (x\right ) + \frac{180 \, a b^{4} c^{4} x^{4} - 60 \, a^{2} b^{3} c^{4} x^{3} - 40 \, a^{3} b^{2} c^{4} x^{2} + 45 \, a^{4} b c^{4} x - 12 \, a^{5} c^{4}}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^6,x, algorithm="maxima")

[Out]

b^5*c^4*log(x) + 1/60*(180*a*b^4*c^4*x^4 - 60*a^2*b^3*c^4*x^3 - 40*a^3*b^2*c^4*x
^2 + 45*a^4*b*c^4*x - 12*a^5*c^4)/x^5

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Fricas [A]  time = 0.210444, size = 104, normalized size = 1.32 \[ \frac{60 \, b^{5} c^{4} x^{5} \log \left (x\right ) + 180 \, a b^{4} c^{4} x^{4} - 60 \, a^{2} b^{3} c^{4} x^{3} - 40 \, a^{3} b^{2} c^{4} x^{2} + 45 \, a^{4} b c^{4} x - 12 \, a^{5} c^{4}}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^6,x, algorithm="fricas")

[Out]

1/60*(60*b^5*c^4*x^5*log(x) + 180*a*b^4*c^4*x^4 - 60*a^2*b^3*c^4*x^3 - 40*a^3*b^
2*c^4*x^2 + 45*a^4*b*c^4*x - 12*a^5*c^4)/x^5

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Sympy [A]  time = 1.22647, size = 78, normalized size = 0.99 \[ b^{5} c^{4} \log{\left (x \right )} + \frac{- 12 a^{5} c^{4} + 45 a^{4} b c^{4} x - 40 a^{3} b^{2} c^{4} x^{2} - 60 a^{2} b^{3} c^{4} x^{3} + 180 a b^{4} c^{4} x^{4}}{60 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**4/x**6,x)

[Out]

b**5*c**4*log(x) + (-12*a**5*c**4 + 45*a**4*b*c**4*x - 40*a**3*b**2*c**4*x**2 -
60*a**2*b**3*c**4*x**3 + 180*a*b**4*c**4*x**4)/(60*x**5)

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GIAC/XCAS [A]  time = 0.265893, size = 101, normalized size = 1.28 \[ b^{5} c^{4}{\rm ln}\left ({\left | x \right |}\right ) + \frac{180 \, a b^{4} c^{4} x^{4} - 60 \, a^{2} b^{3} c^{4} x^{3} - 40 \, a^{3} b^{2} c^{4} x^{2} + 45 \, a^{4} b c^{4} x - 12 \, a^{5} c^{4}}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^6,x, algorithm="giac")

[Out]

b^5*c^4*ln(abs(x)) + 1/60*(180*a*b^4*c^4*x^4 - 60*a^2*b^3*c^4*x^3 - 40*a^3*b^2*c
^4*x^2 + 45*a^4*b*c^4*x - 12*a^5*c^4)/x^5