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

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

[Out]

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

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Rubi [A]  time = 0.0588129, 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 \[ -\frac{a^4 c^3}{4 x^4}+\frac{2 a^3 b c^3}{3 x^3}-\frac{2 a b^3 c^3}{x}-b^4 c^3 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.6312, size = 48, normalized size = 0.96 \[ - \frac{a^{4} c^{3}}{4 x^{4}} + \frac{2 a^{3} b c^{3}}{3 x^{3}} - \frac{2 a b^{3} c^{3}}{x} - b^{4} c^{3} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0102081, size = 42, normalized size = 0.84 \[ c^3 \left (-\frac{a^4}{4 x^4}+\frac{2 a^3 b}{3 x^3}-\frac{2 a b^3}{x}-b^4 \log (x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 47, normalized size = 0.9 \[ -{\frac{{a}^{4}{c}^{3}}{4\,{x}^{4}}}+{\frac{2\,{a}^{3}b{c}^{3}}{3\,{x}^{3}}}-2\,{\frac{a{b}^{3}{c}^{3}}{x}}-{b}^{4}{c}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34589, size = 63, normalized size = 1.26 \[ -b^{4} c^{3} \log \left (x\right ) - \frac{24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^4*c^3*log(x) - 1/12*(24*a*b^3*c^3*x^3 - 8*a^3*b*c^3*x + 3*a^4*c^3)/x^4

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Fricas [A]  time = 0.207032, size = 66, normalized size = 1.32 \[ -\frac{12 \, b^{4} c^{3} x^{4} \log \left (x\right ) + 24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(12*b^4*c^3*x^4*log(x) + 24*a*b^3*c^3*x^3 - 8*a^3*b*c^3*x + 3*a^4*c^3)/x^4

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Sympy [A]  time = 0.859972, size = 49, normalized size = 0.98 \[ - b^{4} c^{3} \log{\left (x \right )} - \frac{3 a^{4} c^{3} - 8 a^{3} b c^{3} x + 24 a b^{3} c^{3} x^{3}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**4*c**3*log(x) - (3*a**4*c**3 - 8*a**3*b*c**3*x + 24*a*b**3*c**3*x**3)/(12*x*
*4)

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GIAC/XCAS [A]  time = 0.30132, size = 65, normalized size = 1.3 \[ -b^{4} c^{3}{\rm ln}\left ({\left | x \right |}\right ) - \frac{24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^4*c^3*ln(abs(x)) - 1/12*(24*a*b^3*c^3*x^3 - 8*a^3*b*c^3*x + 3*a^4*c^3)/x^4