3.41 \(\int \frac{(a+b x) (a c-b c x)^5}{x^{11}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{a^6 c^5}{10 x^{10}}+\frac{4 a^5 b c^5}{9 x^9}-\frac{5 a^4 b^2 c^5}{8 x^8}+\frac{5 a^2 b^4 c^5}{6 x^6}-\frac{4 a b^5 c^5}{5 x^5}+\frac{b^6 c^5}{4 x^4} \]

[Out]

-(a^6*c^5)/(10*x^10) + (4*a^5*b*c^5)/(9*x^9) - (5*a^4*b^2*c^5)/(8*x^8) + (5*a^2*
b^4*c^5)/(6*x^6) - (4*a*b^5*c^5)/(5*x^5) + (b^6*c^5)/(4*x^4)

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Rubi [A]  time = 0.0999413, antiderivative size = 87, 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 \[ -\frac{a^6 c^5}{10 x^{10}}+\frac{4 a^5 b c^5}{9 x^9}-\frac{5 a^4 b^2 c^5}{8 x^8}+\frac{5 a^2 b^4 c^5}{6 x^6}-\frac{4 a b^5 c^5}{5 x^5}+\frac{b^6 c^5}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^5)/x^11,x]

[Out]

-(a^6*c^5)/(10*x^10) + (4*a^5*b*c^5)/(9*x^9) - (5*a^4*b^2*c^5)/(8*x^8) + (5*a^2*
b^4*c^5)/(6*x^6) - (4*a*b^5*c^5)/(5*x^5) + (b^6*c^5)/(4*x^4)

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Rubi in Sympy [A]  time = 32.0793, size = 87, normalized size = 1. \[ - \frac{a^{6} c^{5}}{10 x^{10}} + \frac{4 a^{5} b c^{5}}{9 x^{9}} - \frac{5 a^{4} b^{2} c^{5}}{8 x^{8}} + \frac{5 a^{2} b^{4} c^{5}}{6 x^{6}} - \frac{4 a b^{5} c^{5}}{5 x^{5}} + \frac{b^{6} c^{5}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**5/x**11,x)

[Out]

-a**6*c**5/(10*x**10) + 4*a**5*b*c**5/(9*x**9) - 5*a**4*b**2*c**5/(8*x**8) + 5*a
**2*b**4*c**5/(6*x**6) - 4*a*b**5*c**5/(5*x**5) + b**6*c**5/(4*x**4)

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Mathematica [A]  time = 0.0105546, size = 73, normalized size = 0.84 \[ c^5 \left (-\frac{a^6}{10 x^{10}}+\frac{4 a^5 b}{9 x^9}-\frac{5 a^4 b^2}{8 x^8}+\frac{5 a^2 b^4}{6 x^6}-\frac{4 a b^5}{5 x^5}+\frac{b^6}{4 x^4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^11,x]

[Out]

c^5*(-a^6/(10*x^10) + (4*a^5*b)/(9*x^9) - (5*a^4*b^2)/(8*x^8) + (5*a^2*b^4)/(6*x
^6) - (4*a*b^5)/(5*x^5) + b^6/(4*x^4))

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Maple [A]  time = 0.007, size = 62, normalized size = 0.7 \[{c}^{5} \left ( -{\frac{5\,{a}^{4}{b}^{2}}{8\,{x}^{8}}}+{\frac{4\,{a}^{5}b}{9\,{x}^{9}}}-{\frac{4\,a{b}^{5}}{5\,{x}^{5}}}-{\frac{{a}^{6}}{10\,{x}^{10}}}+{\frac{{b}^{6}}{4\,{x}^{4}}}+{\frac{5\,{a}^{2}{b}^{4}}{6\,{x}^{6}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^5/x^11,x)

[Out]

c^5*(-5/8*a^4*b^2/x^8+4/9*a^5*b/x^9-4/5*a*b^5/x^5-1/10*a^6/x^10+1/4*b^6/x^4+5/6*
a^2*b^4/x^6)

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Maxima [A]  time = 1.36719, size = 101, normalized size = 1.16 \[ \frac{90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^11,x, algorithm="maxima")

[Out]

1/360*(90*b^6*c^5*x^6 - 288*a*b^5*c^5*x^5 + 300*a^2*b^4*c^5*x^4 - 225*a^4*b^2*c^
5*x^2 + 160*a^5*b*c^5*x - 36*a^6*c^5)/x^10

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Fricas [A]  time = 0.197007, size = 101, normalized size = 1.16 \[ \frac{90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^11,x, algorithm="fricas")

[Out]

1/360*(90*b^6*c^5*x^6 - 288*a*b^5*c^5*x^5 + 300*a^2*b^4*c^5*x^4 - 225*a^4*b^2*c^
5*x^2 + 160*a^5*b*c^5*x - 36*a^6*c^5)/x^10

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Sympy [A]  time = 3.05442, size = 80, normalized size = 0.92 \[ \frac{- 36 a^{6} c^{5} + 160 a^{5} b c^{5} x - 225 a^{4} b^{2} c^{5} x^{2} + 300 a^{2} b^{4} c^{5} x^{4} - 288 a b^{5} c^{5} x^{5} + 90 b^{6} c^{5} x^{6}}{360 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**5/x**11,x)

[Out]

(-36*a**6*c**5 + 160*a**5*b*c**5*x - 225*a**4*b**2*c**5*x**2 + 300*a**2*b**4*c**
5*x**4 - 288*a*b**5*c**5*x**5 + 90*b**6*c**5*x**6)/(360*x**10)

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GIAC/XCAS [A]  time = 0.246342, size = 101, normalized size = 1.16 \[ \frac{90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^11,x, algorithm="giac")

[Out]

1/360*(90*b^6*c^5*x^6 - 288*a*b^5*c^5*x^5 + 300*a^2*b^4*c^5*x^4 - 225*a^4*b^2*c^
5*x^2 + 160*a^5*b*c^5*x - 36*a^6*c^5)/x^10