3.9 \(\int \frac{(a+b x) (a c-b c x)^3}{x^6} \, dx\)

Optimal. Leaf size=50 \[ -\frac{a^4 c^3}{5 x^5}+\frac{a^3 b c^3}{2 x^4}-\frac{a b^3 c^3}{x^2}+\frac{b^4 c^3}{x} \]

[Out]

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

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Rubi [A]  time = 0.0658317, antiderivative size = 50, 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 \[ -\frac{a^4 c^3}{5 x^5}+\frac{a^3 b c^3}{2 x^4}-\frac{a b^3 c^3}{x^2}+\frac{b^4 c^3}{x} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^3)/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 19.3438, size = 44, normalized size = 0.88 \[ - \frac{a^{4} c^{3}}{5 x^{5}} + \frac{a^{3} b c^{3}}{2 x^{4}} - \frac{a b^{3} c^{3}}{x^{2}} + \frac{b^{4} c^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**3/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.00945838, size = 42, normalized size = 0.84 \[ c^3 \left (-\frac{a^4}{5 x^5}+\frac{a^3 b}{2 x^4}-\frac{a b^3}{x^2}+\frac{b^4}{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^6,x]

[Out]

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

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Maple [A]  time = 0.009, size = 39, normalized size = 0.8 \[{c}^{3} \left ( -{\frac{a{b}^{3}}{{x}^{2}}}-{\frac{{a}^{4}}{5\,{x}^{5}}}+{\frac{{b}^{4}}{x}}+{\frac{{a}^{3}b}{2\,{x}^{4}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^3/x^6,x)

[Out]

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

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Maxima [A]  time = 1.33919, size = 63, normalized size = 1.26 \[ \frac{10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)/x^6,x, algorithm="maxima")

[Out]

1/10*(10*b^4*c^3*x^4 - 10*a*b^3*c^3*x^3 + 5*a^3*b*c^3*x - 2*a^4*c^3)/x^5

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Fricas [A]  time = 0.200824, size = 63, normalized size = 1.26 \[ \frac{10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)/x^6,x, algorithm="fricas")

[Out]

1/10*(10*b^4*c^3*x^4 - 10*a*b^3*c^3*x^3 + 5*a^3*b*c^3*x - 2*a^4*c^3)/x^5

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Sympy [A]  time = 0.967409, size = 49, normalized size = 0.98 \[ \frac{- 2 a^{4} c^{3} + 5 a^{3} b c^{3} x - 10 a b^{3} c^{3} x^{3} + 10 b^{4} c^{3} x^{4}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**3/x**6,x)

[Out]

(-2*a**4*c**3 + 5*a**3*b*c**3*x - 10*a*b**3*c**3*x**3 + 10*b**4*c**3*x**4)/(10*x
**5)

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GIAC/XCAS [A]  time = 0.257344, size = 63, normalized size = 1.26 \[ \frac{10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)/x^6,x, algorithm="giac")

[Out]

1/10*(10*b^4*c^3*x^4 - 10*a*b^3*c^3*x^3 + 5*a^3*b*c^3*x - 2*a^4*c^3)/x^5