3.11 \(\int \frac{(a+b x) (a c-b c x)^3}{x^8} \, dx\)

Optimal. Leaf size=55 \[ -\frac{a^4 c^3}{7 x^7}+\frac{a^3 b c^3}{3 x^6}-\frac{a b^3 c^3}{2 x^4}+\frac{b^4 c^3}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0664601, antiderivative size = 55, 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 \[ -\frac{a^4 c^3}{7 x^7}+\frac{a^3 b c^3}{3 x^6}-\frac{a b^3 c^3}{2 x^4}+\frac{b^4 c^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^3)/x^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 19.5576, size = 49, normalized size = 0.89 \[ - \frac{a^{4} c^{3}}{7 x^{7}} + \frac{a^{3} b c^{3}}{3 x^{6}} - \frac{a b^{3} c^{3}}{2 x^{4}} + \frac{b^{4} c^{3}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**3/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.00962125, size = 47, normalized size = 0.85 \[ c^3 \left (-\frac{a^4}{7 x^7}+\frac{a^3 b}{3 x^6}-\frac{a b^3}{2 x^4}+\frac{b^4}{3 x^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^8,x]

[Out]

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

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Maple [A]  time = 0.007, size = 40, normalized size = 0.7 \[{c}^{3} \left ( -{\frac{{a}^{4}}{7\,{x}^{7}}}+{\frac{{b}^{4}}{3\,{x}^{3}}}-{\frac{a{b}^{3}}{2\,{x}^{4}}}+{\frac{{a}^{3}b}{3\,{x}^{6}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^3/x^8,x)

[Out]

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

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Maxima [A]  time = 1.34668, size = 63, normalized size = 1.15 \[ \frac{14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)/x^8,x, algorithm="maxima")

[Out]

1/42*(14*b^4*c^3*x^4 - 21*a*b^3*c^3*x^3 + 14*a^3*b*c^3*x - 6*a^4*c^3)/x^7

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Fricas [A]  time = 0.196409, size = 63, normalized size = 1.15 \[ \frac{14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)/x^8,x, algorithm="fricas")

[Out]

1/42*(14*b^4*c^3*x^4 - 21*a*b^3*c^3*x^3 + 14*a^3*b*c^3*x - 6*a^4*c^3)/x^7

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Sympy [A]  time = 0.968414, size = 49, normalized size = 0.89 \[ \frac{- 6 a^{4} c^{3} + 14 a^{3} b c^{3} x - 21 a b^{3} c^{3} x^{3} + 14 b^{4} c^{3} x^{4}}{42 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**3/x**8,x)

[Out]

(-6*a**4*c**3 + 14*a**3*b*c**3*x - 21*a*b**3*c**3*x**3 + 14*b**4*c**3*x**4)/(42*
x**7)

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GIAC/XCAS [A]  time = 0.270836, size = 63, normalized size = 1.15 \[ \frac{14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)/x^8,x, algorithm="giac")

[Out]

1/42*(14*b^4*c^3*x^4 - 21*a*b^3*c^3*x^3 + 14*a^3*b*c^3*x - 6*a^4*c^3)/x^7