3.24 \(\int \frac{(a+b x) (a c-b c x)^4}{x^8} \, dx\)

Optimal. Leaf size=84 \[ -\frac{a^5 c^4}{7 x^7}+\frac{a^4 b c^4}{2 x^6}-\frac{2 a^3 b^2 c^4}{5 x^5}-\frac{a^2 b^3 c^4}{2 x^4}+\frac{a b^4 c^4}{x^3}-\frac{b^5 c^4}{2 x^2} \]

[Out]

-(a^5*c^4)/(7*x^7) + (a^4*b*c^4)/(2*x^6) - (2*a^3*b^2*c^4)/(5*x^5) - (a^2*b^3*c^
4)/(2*x^4) + (a*b^4*c^4)/x^3 - (b^5*c^4)/(2*x^2)

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Rubi [A]  time = 0.0997115, antiderivative size = 84, 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 \[ -\frac{a^5 c^4}{7 x^7}+\frac{a^4 b c^4}{2 x^6}-\frac{2 a^3 b^2 c^4}{5 x^5}-\frac{a^2 b^3 c^4}{2 x^4}+\frac{a b^4 c^4}{x^3}-\frac{b^5 c^4}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^4)/x^8,x]

[Out]

-(a^5*c^4)/(7*x^7) + (a^4*b*c^4)/(2*x^6) - (2*a^3*b^2*c^4)/(5*x^5) - (a^2*b^3*c^
4)/(2*x^4) + (a*b^4*c^4)/x^3 - (b^5*c^4)/(2*x^2)

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Rubi in Sympy [A]  time = 30.4361, size = 80, normalized size = 0.95 \[ - \frac{a^{5} c^{4}}{7 x^{7}} + \frac{a^{4} b c^{4}}{2 x^{6}} - \frac{2 a^{3} b^{2} c^{4}}{5 x^{5}} - \frac{a^{2} b^{3} c^{4}}{2 x^{4}} + \frac{a b^{4} c^{4}}{x^{3}} - \frac{b^{5} c^{4}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**4/x**8,x)

[Out]

-a**5*c**4/(7*x**7) + a**4*b*c**4/(2*x**6) - 2*a**3*b**2*c**4/(5*x**5) - a**2*b*
*3*c**4/(2*x**4) + a*b**4*c**4/x**3 - b**5*c**4/(2*x**2)

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Mathematica [A]  time = 0.0123171, size = 84, normalized size = 1. \[ -\frac{a^5 c^4}{7 x^7}+\frac{a^4 b c^4}{2 x^6}-\frac{2 a^3 b^2 c^4}{5 x^5}-\frac{a^2 b^3 c^4}{2 x^4}+\frac{a b^4 c^4}{x^3}-\frac{b^5 c^4}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^8,x]

[Out]

-(a^5*c^4)/(7*x^7) + (a^4*b*c^4)/(2*x^6) - (2*a^3*b^2*c^4)/(5*x^5) - (a^2*b^3*c^
4)/(2*x^4) + (a*b^4*c^4)/x^3 - (b^5*c^4)/(2*x^2)

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Maple [A]  time = 0.007, size = 61, normalized size = 0.7 \[{c}^{4} \left ( -{\frac{{a}^{5}}{7\,{x}^{7}}}-{\frac{{b}^{5}}{2\,{x}^{2}}}-{\frac{2\,{a}^{3}{b}^{2}}{5\,{x}^{5}}}+{\frac{a{b}^{4}}{{x}^{3}}}-{\frac{{a}^{2}{b}^{3}}{2\,{x}^{4}}}+{\frac{{a}^{4}b}{2\,{x}^{6}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^4/x^8,x)

[Out]

c^4*(-1/7*a^5/x^7-1/2*b^5/x^2-2/5*a^3*b^2/x^5+a*b^4/x^3-1/2*a^2*b^3/x^4+1/2*a^4*
b/x^6)

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Maxima [A]  time = 1.3392, size = 101, normalized size = 1.2 \[ -\frac{35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^8,x, algorithm="maxima")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x
^2 - 35*a^4*b*c^4*x + 10*a^5*c^4)/x^7

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Fricas [A]  time = 0.20239, size = 101, normalized size = 1.2 \[ -\frac{35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^8,x, algorithm="fricas")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x
^2 - 35*a^4*b*c^4*x + 10*a^5*c^4)/x^7

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Sympy [A]  time = 2.35962, size = 82, normalized size = 0.98 \[ - \frac{10 a^{5} c^{4} - 35 a^{4} b c^{4} x + 28 a^{3} b^{2} c^{4} x^{2} + 35 a^{2} b^{3} c^{4} x^{3} - 70 a b^{4} c^{4} x^{4} + 35 b^{5} c^{4} x^{5}}{70 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**4/x**8,x)

[Out]

-(10*a**5*c**4 - 35*a**4*b*c**4*x + 28*a**3*b**2*c**4*x**2 + 35*a**2*b**3*c**4*x
**3 - 70*a*b**4*c**4*x**4 + 35*b**5*c**4*x**5)/(70*x**7)

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GIAC/XCAS [A]  time = 0.257668, size = 101, normalized size = 1.2 \[ -\frac{35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^8,x, algorithm="giac")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x
^2 - 35*a^4*b*c^4*x + 10*a^5*c^4)/x^7