3.33 \(\int \frac{(a+b x) (a c-b c x)^5}{x^3} \, dx\)

Optimal. Leaf size=82 \[ -\frac{a^6 c^5}{2 x^2}+\frac{4 a^5 b c^5}{x}+5 a^4 b^2 c^5 \log (x)-\frac{5}{2} a^2 b^4 c^5 x^2+\frac{4}{3} a b^5 c^5 x^3-\frac{1}{4} b^6 c^5 x^4 \]

[Out]

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

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Rubi [A]  time = 0.0996539, 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 \[ -\frac{a^6 c^5}{2 x^2}+\frac{4 a^5 b c^5}{x}+5 a^4 b^2 c^5 \log (x)-\frac{5}{2} a^2 b^4 c^5 x^2+\frac{4}{3} a b^5 c^5 x^3-\frac{1}{4} b^6 c^5 x^4 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^5)/x^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{6} c^{5}}{2 x^{2}} + \frac{4 a^{5} b c^{5}}{x} + 5 a^{4} b^{2} c^{5} \log{\left (x \right )} - 5 a^{2} b^{4} c^{5} \int x\, dx + \frac{4 a b^{5} c^{5} x^{3}}{3} - \frac{b^{6} c^{5} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**5/x**3,x)

[Out]

-a**6*c**5/(2*x**2) + 4*a**5*b*c**5/x + 5*a**4*b**2*c**5*log(x) - 5*a**2*b**4*c*
*5*Integral(x, x) + 4*a*b**5*c**5*x**3/3 - b**6*c**5*x**4/4

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Mathematica [A]  time = 0.0116138, size = 68, normalized size = 0.83 \[ c^5 \left (-\frac{a^6}{2 x^2}+\frac{4 a^5 b}{x}+5 a^4 b^2 \log (x)-\frac{5}{2} a^2 b^4 x^2+\frac{4}{3} a b^5 x^3-\frac{b^6 x^4}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^3,x]

[Out]

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

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Maple [A]  time = 0.012, size = 75, normalized size = 0.9 \[ -{\frac{{a}^{6}{c}^{5}}{2\,{x}^{2}}}+4\,{\frac{{a}^{5}b{c}^{5}}{x}}-{\frac{5\,{a}^{2}{b}^{4}{c}^{5}{x}^{2}}{2}}+{\frac{4\,a{b}^{5}{c}^{5}{x}^{3}}{3}}-{\frac{{b}^{6}{c}^{5}{x}^{4}}{4}}+5\,{a}^{4}{b}^{2}{c}^{5}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^5/x^3,x)

[Out]

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

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Maxima [A]  time = 1.43133, size = 101, normalized size = 1.23 \[ -\frac{1}{4} \, b^{6} c^{5} x^{4} + \frac{4}{3} \, a b^{5} c^{5} x^{3} - \frac{5}{2} \, a^{2} b^{4} c^{5} x^{2} + 5 \, a^{4} b^{2} c^{5} \log \left (x\right ) + \frac{8 \, a^{5} b c^{5} x - a^{6} c^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^3,x, algorithm="maxima")

[Out]

-1/4*b^6*c^5*x^4 + 4/3*a*b^5*c^5*x^3 - 5/2*a^2*b^4*c^5*x^2 + 5*a^4*b^2*c^5*log(x
) + 1/2*(8*a^5*b*c^5*x - a^6*c^5)/x^2

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Fricas [A]  time = 0.205664, size = 104, normalized size = 1.27 \[ -\frac{3 \, b^{6} c^{5} x^{6} - 16 \, a b^{5} c^{5} x^{5} + 30 \, a^{2} b^{4} c^{5} x^{4} - 60 \, a^{4} b^{2} c^{5} x^{2} \log \left (x\right ) - 48 \, a^{5} b c^{5} x + 6 \, a^{6} c^{5}}{12 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^3,x, algorithm="fricas")

[Out]

-1/12*(3*b^6*c^5*x^6 - 16*a*b^5*c^5*x^5 + 30*a^2*b^4*c^5*x^4 - 60*a^4*b^2*c^5*x^
2*log(x) - 48*a^5*b*c^5*x + 6*a^6*c^5)/x^2

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Sympy [A]  time = 1.67691, size = 82, normalized size = 1. \[ 5 a^{4} b^{2} c^{5} \log{\left (x \right )} - \frac{5 a^{2} b^{4} c^{5} x^{2}}{2} + \frac{4 a b^{5} c^{5} x^{3}}{3} - \frac{b^{6} c^{5} x^{4}}{4} + \frac{- a^{6} c^{5} + 8 a^{5} b c^{5} x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**5/x**3,x)

[Out]

5*a**4*b**2*c**5*log(x) - 5*a**2*b**4*c**5*x**2/2 + 4*a*b**5*c**5*x**3/3 - b**6*
c**5*x**4/4 + (-a**6*c**5 + 8*a**5*b*c**5*x)/(2*x**2)

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GIAC/XCAS [A]  time = 0.239884, size = 103, normalized size = 1.26 \[ -\frac{1}{4} \, b^{6} c^{5} x^{4} + \frac{4}{3} \, a b^{5} c^{5} x^{3} - \frac{5}{2} \, a^{2} b^{4} c^{5} x^{2} + 5 \, a^{4} b^{2} c^{5}{\rm ln}\left ({\left | x \right |}\right ) + \frac{8 \, a^{5} b c^{5} x - a^{6} c^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^3,x, algorithm="giac")

[Out]

-1/4*b^6*c^5*x^4 + 4/3*a*b^5*c^5*x^3 - 5/2*a^2*b^4*c^5*x^2 + 5*a^4*b^2*c^5*ln(ab
s(x)) + 1/2*(8*a^5*b*c^5*x - a^6*c^5)/x^2