3.10 \(\int \frac{(a+b x) (a c-b c x)^3}{x^7} \, dx\)

Optimal. Leaf size=55 \[ -\frac{a^4 c^3}{6 x^6}+\frac{2 a^3 b c^3}{5 x^5}-\frac{2 a b^3 c^3}{3 x^3}+\frac{b^4 c^3}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0679135, 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 \[ -\frac{a^4 c^3}{6 x^6}+\frac{2 a^3 b c^3}{5 x^5}-\frac{2 a b^3 c^3}{3 x^3}+\frac{b^4 c^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^3)/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 19.6266, size = 53, normalized size = 0.96 \[ - \frac{a^{4} c^{3}}{6 x^{6}} + \frac{2 a^{3} b c^{3}}{5 x^{5}} - \frac{2 a b^{3} c^{3}}{3 x^{3}} + \frac{b^{4} c^{3}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**3/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.00965933, size = 47, normalized size = 0.85 \[ c^3 \left (-\frac{a^4}{6 x^6}+\frac{2 a^3 b}{5 x^5}-\frac{2 a b^3}{3 x^3}+\frac{b^4}{2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^7,x]

[Out]

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

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Maple [A]  time = 0.007, size = 40, normalized size = 0.7 \[{c}^{3} \left ({\frac{{b}^{4}}{2\,{x}^{2}}}+{\frac{2\,{a}^{3}b}{5\,{x}^{5}}}-{\frac{2\,a{b}^{3}}{3\,{x}^{3}}}-{\frac{{a}^{4}}{6\,{x}^{6}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^3/x^7,x)

[Out]

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

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Maxima [A]  time = 1.34674, size = 63, normalized size = 1.15 \[ \frac{15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)/x^7,x, algorithm="maxima")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6

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Fricas [A]  time = 0.199403, size = 63, normalized size = 1.15 \[ \frac{15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)/x^7,x, algorithm="fricas")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6

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Sympy [A]  time = 0.928871, size = 49, normalized size = 0.89 \[ \frac{- 5 a^{4} c^{3} + 12 a^{3} b c^{3} x - 20 a b^{3} c^{3} x^{3} + 15 b^{4} c^{3} x^{4}}{30 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**3/x**7,x)

[Out]

(-5*a**4*c**3 + 12*a**3*b*c**3*x - 20*a*b**3*c**3*x**3 + 15*b**4*c**3*x**4)/(30*
x**6)

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GIAC/XCAS [A]  time = 0.2405, size = 63, normalized size = 1.15 \[ \frac{15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^3*(b*x + a)/x^7,x, algorithm="giac")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6