3.34 \(\int \frac{(a+b x) (a c-b c x)^5}{x^4} \, dx\)

Optimal. Leaf size=18 \[ -\frac{c^5 (a-b x)^6}{3 x^3} \]

[Out]

-(c^5*(a - b*x)^6)/(3*x^3)

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Rubi [A]  time = 0.0176333, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{c^5 (a-b x)^6}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^5)/x^4,x]

[Out]

-(c^5*(a - b*x)^6)/(3*x^3)

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Rubi in Sympy [A]  time = 8.11383, size = 15, normalized size = 0.83 \[ - \frac{c^{5} \left (a - b x\right )^{6}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**5/x**4,x)

[Out]

-c**5*(a - b*x)**6/(3*x**3)

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Mathematica [B]  time = 0.010506, size = 63, normalized size = 3.5 \[ c^5 \left (-\frac{a^6}{3 x^3}+\frac{2 a^5 b}{x^2}-\frac{5 a^4 b^2}{x}-5 a^2 b^4 x+2 a b^5 x^2-\frac{b^6 x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^4,x]

[Out]

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

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Maple [B]  time = 0.007, size = 60, normalized size = 3.3 \[{c}^{5} \left ( -{\frac{{b}^{6}{x}^{3}}{3}}+2\,a{b}^{5}{x}^{2}-5\,{a}^{2}{b}^{4}x+2\,{\frac{{a}^{5}b}{{x}^{2}}}-5\,{\frac{{a}^{4}{b}^{2}}{x}}-{\frac{{a}^{6}}{3\,{x}^{3}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^5/x^4,x)

[Out]

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

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Maxima [A]  time = 1.34962, size = 99, normalized size = 5.5 \[ -\frac{1}{3} \, b^{6} c^{5} x^{3} + 2 \, a b^{5} c^{5} x^{2} - 5 \, a^{2} b^{4} c^{5} x - \frac{15 \, a^{4} b^{2} c^{5} x^{2} - 6 \, a^{5} b c^{5} x + a^{6} c^{5}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^4,x, algorithm="maxima")

[Out]

-1/3*b^6*c^5*x^3 + 2*a*b^5*c^5*x^2 - 5*a^2*b^4*c^5*x - 1/3*(15*a^4*b^2*c^5*x^2 -
6*a^5*b*c^5*x + a^6*c^5)/x^3

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Fricas [A]  time = 0.196608, size = 99, normalized size = 5.5 \[ -\frac{b^{6} c^{5} x^{6} - 6 \, a b^{5} c^{5} x^{5} + 15 \, a^{2} b^{4} c^{5} x^{4} + 15 \, a^{4} b^{2} c^{5} x^{2} - 6 \, a^{5} b c^{5} x + a^{6} c^{5}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^4,x, algorithm="fricas")

[Out]

-1/3*(b^6*c^5*x^6 - 6*a*b^5*c^5*x^5 + 15*a^2*b^4*c^5*x^4 + 15*a^4*b^2*c^5*x^2 -
6*a^5*b*c^5*x + a^6*c^5)/x^3

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Sympy [A]  time = 1.68087, size = 76, normalized size = 4.22 \[ - 5 a^{2} b^{4} c^{5} x + 2 a b^{5} c^{5} x^{2} - \frac{b^{6} c^{5} x^{3}}{3} - \frac{a^{6} c^{5} - 6 a^{5} b c^{5} x + 15 a^{4} b^{2} c^{5} x^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**5/x**4,x)

[Out]

-5*a**2*b**4*c**5*x + 2*a*b**5*c**5*x**2 - b**6*c**5*x**3/3 - (a**6*c**5 - 6*a**
5*b*c**5*x + 15*a**4*b**2*c**5*x**2)/(3*x**3)

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GIAC/XCAS [A]  time = 0.223388, size = 99, normalized size = 5.5 \[ -\frac{1}{3} \, b^{6} c^{5} x^{3} + 2 \, a b^{5} c^{5} x^{2} - 5 \, a^{2} b^{4} c^{5} x - \frac{15 \, a^{4} b^{2} c^{5} x^{2} - 6 \, a^{5} b c^{5} x + a^{6} c^{5}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^4,x, algorithm="giac")

[Out]

-1/3*b^6*c^5*x^3 + 2*a*b^5*c^5*x^2 - 5*a^2*b^4*c^5*x - 1/3*(15*a^4*b^2*c^5*x^2 -
6*a^5*b*c^5*x + a^6*c^5)/x^3