3.39 \(\int \frac{(a+b x) (a c-b c x)^5}{x^9} \, dx\)

Optimal. Leaf size=65 \[ -\frac{5 b^2 c^5 (a-b x)^6}{168 a^2 x^6}-\frac{c^5 (a-b x)^6}{8 x^8}-\frac{5 b c^5 (a-b x)^6}{28 a x^7} \]

[Out]

-(c^5*(a - b*x)^6)/(8*x^8) - (5*b*c^5*(a - b*x)^6)/(28*a*x^7) - (5*b^2*c^5*(a -
b*x)^6)/(168*a^2*x^6)

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Rubi [A]  time = 0.072663, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{5 b^2 c^5 (a-b x)^6}{168 a^2 x^6}-\frac{c^5 (a-b x)^6}{8 x^8}-\frac{5 b c^5 (a-b x)^6}{28 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^5)/x^9,x]

[Out]

-(c^5*(a - b*x)^6)/(8*x^8) - (5*b*c^5*(a - b*x)^6)/(28*a*x^7) - (5*b^2*c^5*(a -
b*x)^6)/(168*a^2*x^6)

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Rubi in Sympy [A]  time = 17.0452, size = 60, normalized size = 0.92 \[ - \frac{c^{5} \left (a - b x\right )^{6}}{8 x^{8}} - \frac{5 b c^{5} \left (a - b x\right )^{6}}{28 a x^{7}} - \frac{5 b^{2} c^{5} \left (a - b x\right )^{6}}{168 a^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**5/x**9,x)

[Out]

-c**5*(a - b*x)**6/(8*x**8) - 5*b*c**5*(a - b*x)**6/(28*a*x**7) - 5*b**2*c**5*(a
- b*x)**6/(168*a**2*x**6)

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Mathematica [A]  time = 0.0105354, size = 73, normalized size = 1.12 \[ c^5 \left (-\frac{a^6}{8 x^8}+\frac{4 a^5 b}{7 x^7}-\frac{5 a^4 b^2}{6 x^6}+\frac{5 a^2 b^4}{4 x^4}-\frac{4 a b^5}{3 x^3}+\frac{b^6}{2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^9,x]

[Out]

c^5*(-a^6/(8*x^8) + (4*a^5*b)/(7*x^7) - (5*a^4*b^2)/(6*x^6) + (5*a^2*b^4)/(4*x^4
) - (4*a*b^5)/(3*x^3) + b^6/(2*x^2))

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Maple [A]  time = 0.008, size = 62, normalized size = 1. \[{c}^{5} \left ( -{\frac{{a}^{6}}{8\,{x}^{8}}}+{\frac{4\,{a}^{5}b}{7\,{x}^{7}}}+{\frac{{b}^{6}}{2\,{x}^{2}}}-{\frac{4\,a{b}^{5}}{3\,{x}^{3}}}+{\frac{5\,{a}^{2}{b}^{4}}{4\,{x}^{4}}}-{\frac{5\,{a}^{4}{b}^{2}}{6\,{x}^{6}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^5/x^9,x)

[Out]

c^5*(-1/8*a^6/x^8+4/7*a^5*b/x^7+1/2*b^6/x^2-4/3*a*b^5/x^3+5/4*a^2*b^4/x^4-5/6*a^
4*b^2/x^6)

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Maxima [A]  time = 1.34215, size = 101, normalized size = 1.55 \[ \frac{84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^9,x, algorithm="maxima")

[Out]

1/168*(84*b^6*c^5*x^6 - 224*a*b^5*c^5*x^5 + 210*a^2*b^4*c^5*x^4 - 140*a^4*b^2*c^
5*x^2 + 96*a^5*b*c^5*x - 21*a^6*c^5)/x^8

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Fricas [A]  time = 0.204573, size = 101, normalized size = 1.55 \[ \frac{84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^9,x, algorithm="fricas")

[Out]

1/168*(84*b^6*c^5*x^6 - 224*a*b^5*c^5*x^5 + 210*a^2*b^4*c^5*x^4 - 140*a^4*b^2*c^
5*x^2 + 96*a^5*b*c^5*x - 21*a^6*c^5)/x^8

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Sympy [A]  time = 2.69636, size = 80, normalized size = 1.23 \[ \frac{- 21 a^{6} c^{5} + 96 a^{5} b c^{5} x - 140 a^{4} b^{2} c^{5} x^{2} + 210 a^{2} b^{4} c^{5} x^{4} - 224 a b^{5} c^{5} x^{5} + 84 b^{6} c^{5} x^{6}}{168 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**5/x**9,x)

[Out]

(-21*a**6*c**5 + 96*a**5*b*c**5*x - 140*a**4*b**2*c**5*x**2 + 210*a**2*b**4*c**5
*x**4 - 224*a*b**5*c**5*x**5 + 84*b**6*c**5*x**6)/(168*x**8)

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GIAC/XCAS [A]  time = 0.239899, size = 101, normalized size = 1.55 \[ \frac{84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^9,x, algorithm="giac")

[Out]

1/168*(84*b^6*c^5*x^6 - 224*a*b^5*c^5*x^5 + 210*a^2*b^4*c^5*x^4 - 140*a^4*b^2*c^
5*x^2 + 96*a^5*b*c^5*x - 21*a^6*c^5)/x^8