3.21 \(\int \frac{(a+b x) (a c-b c x)^4}{x^5} \, dx\)

Optimal. Leaf size=72 \[ -\frac{a^5 c^4}{4 x^4}+\frac{a^4 b c^4}{x^3}-\frac{a^3 b^2 c^4}{x^2}-\frac{2 a^2 b^3 c^4}{x}-3 a b^4 c^4 \log (x)+b^5 c^4 x \]

[Out]

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

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Rubi [A]  time = 0.0953821, antiderivative size = 72, 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 \[ -\frac{a^5 c^4}{4 x^4}+\frac{a^4 b c^4}{x^3}-\frac{a^3 b^2 c^4}{x^2}-\frac{2 a^2 b^3 c^4}{x}-3 a b^4 c^4 \log (x)+b^5 c^4 x \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^4)/x^5,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5} c^{4}}{4 x^{4}} + \frac{a^{4} b c^{4}}{x^{3}} - \frac{a^{3} b^{2} c^{4}}{x^{2}} - \frac{2 a^{2} b^{3} c^{4}}{x} - 3 a b^{4} c^{4} \log{\left (x \right )} + c^{4} \int b^{5}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**4/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.0129897, size = 72, normalized size = 1. \[ -\frac{a^5 c^4}{4 x^4}+\frac{a^4 b c^4}{x^3}-\frac{a^3 b^2 c^4}{x^2}-\frac{2 a^2 b^3 c^4}{x}-3 a b^4 c^4 \log (x)+b^5 c^4 x \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^5,x]

[Out]

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

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Maple [A]  time = 0.01, size = 71, normalized size = 1. \[ -{\frac{{a}^{5}{c}^{4}}{4\,{x}^{4}}}+{\frac{{a}^{4}b{c}^{4}}{{x}^{3}}}-{\frac{{a}^{3}{b}^{2}{c}^{4}}{{x}^{2}}}-2\,{\frac{{a}^{2}{b}^{3}{c}^{4}}{x}}+{b}^{5}{c}^{4}x-3\,a{b}^{4}{c}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^4/x^5,x)

[Out]

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

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Maxima [A]  time = 1.36321, size = 96, normalized size = 1.33 \[ b^{5} c^{4} x - 3 \, a b^{4} c^{4} \log \left (x\right ) - \frac{8 \, a^{2} b^{3} c^{4} x^{3} + 4 \, a^{3} b^{2} c^{4} x^{2} - 4 \, a^{4} b c^{4} x + a^{5} c^{4}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^5,x, algorithm="maxima")

[Out]

b^5*c^4*x - 3*a*b^4*c^4*log(x) - 1/4*(8*a^2*b^3*c^4*x^3 + 4*a^3*b^2*c^4*x^2 - 4*
a^4*b*c^4*x + a^5*c^4)/x^4

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Fricas [A]  time = 0.211525, size = 104, normalized size = 1.44 \[ \frac{4 \, b^{5} c^{4} x^{5} - 12 \, a b^{4} c^{4} x^{4} \log \left (x\right ) - 8 \, a^{2} b^{3} c^{4} x^{3} - 4 \, a^{3} b^{2} c^{4} x^{2} + 4 \, a^{4} b c^{4} x - a^{5} c^{4}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^5,x, algorithm="fricas")

[Out]

1/4*(4*b^5*c^4*x^5 - 12*a*b^4*c^4*x^4*log(x) - 8*a^2*b^3*c^4*x^3 - 4*a^3*b^2*c^4
*x^2 + 4*a^4*b*c^4*x - a^5*c^4)/x^4

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Sympy [A]  time = 0.988506, size = 75, normalized size = 1.04 \[ - 3 a b^{4} c^{4} \log{\left (x \right )} + b^{5} c^{4} x - \frac{a^{5} c^{4} - 4 a^{4} b c^{4} x + 4 a^{3} b^{2} c^{4} x^{2} + 8 a^{2} b^{3} c^{4} x^{3}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**4/x**5,x)

[Out]

-3*a*b**4*c**4*log(x) + b**5*c**4*x - (a**5*c**4 - 4*a**4*b*c**4*x + 4*a**3*b**2
*c**4*x**2 + 8*a**2*b**3*c**4*x**3)/(4*x**4)

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GIAC/XCAS [A]  time = 0.274291, size = 97, normalized size = 1.35 \[ b^{5} c^{4} x - 3 \, a b^{4} c^{4}{\rm ln}\left ({\left | x \right |}\right ) - \frac{8 \, a^{2} b^{3} c^{4} x^{3} + 4 \, a^{3} b^{2} c^{4} x^{2} - 4 \, a^{4} b c^{4} x + a^{5} c^{4}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^5,x, algorithm="giac")

[Out]

b^5*c^4*x - 3*a*b^4*c^4*ln(abs(x)) - 1/4*(8*a^2*b^3*c^4*x^3 + 4*a^3*b^2*c^4*x^2
- 4*a^4*b*c^4*x + a^5*c^4)/x^4