3.1037 \(\int \frac{a+b x}{(a c-b c x)^6} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 a}{5 b c^6 (a-b x)^5}-\frac{1}{4 b c^6 (a-b x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0456257, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 a}{5 b c^6 (a-b x)^5}-\frac{1}{4 b c^6 (a-b x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(a*c - b*c*x)^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.3774, size = 31, normalized size = 0.82 \[ \frac{2 a}{5 b c^{6} \left (a - b x\right )^{5}} - \frac{1}{4 b c^{6} \left (a - b x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(-b*c*x+a*c)**6,x)

[Out]

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

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Mathematica [A]  time = 0.0185107, size = 27, normalized size = 0.71 \[ -\frac{3 a+5 b x}{20 b c^6 (b x-a)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/(a*c - b*c*x)^6,x]

[Out]

-(3*a + 5*b*x)/(20*b*c^6*(-a + b*x)^5)

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Maple [A]  time = 0.007, size = 35, normalized size = 0.9 \[{\frac{1}{{c}^{6}} \left ( -{\frac{2\,a}{5\,b \left ( bx-a \right ) ^{5}}}-{\frac{1}{4\,b \left ( bx-a \right ) ^{4}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(-b*c*x+a*c)^6,x)

[Out]

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

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Maxima [A]  time = 1.35479, size = 113, normalized size = 2.97 \[ -\frac{5 \, b x + 3 \, a}{20 \,{\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(b*c*x - a*c)^6,x, algorithm="maxima")

[Out]

-1/20*(5*b*x + 3*a)/(b^6*c^6*x^5 - 5*a*b^5*c^6*x^4 + 10*a^2*b^4*c^6*x^3 - 10*a^3
*b^3*c^6*x^2 + 5*a^4*b^2*c^6*x - a^5*b*c^6)

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Fricas [A]  time = 0.198317, size = 113, normalized size = 2.97 \[ -\frac{5 \, b x + 3 \, a}{20 \,{\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(b*c*x - a*c)^6,x, algorithm="fricas")

[Out]

-1/20*(5*b*x + 3*a)/(b^6*c^6*x^5 - 5*a*b^5*c^6*x^4 + 10*a^2*b^4*c^6*x^3 - 10*a^3
*b^3*c^6*x^2 + 5*a^4*b^2*c^6*x - a^5*b*c^6)

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Sympy [A]  time = 2.2953, size = 88, normalized size = 2.32 \[ - \frac{3 a + 5 b x}{- 20 a^{5} b c^{6} + 100 a^{4} b^{2} c^{6} x - 200 a^{3} b^{3} c^{6} x^{2} + 200 a^{2} b^{4} c^{6} x^{3} - 100 a b^{5} c^{6} x^{4} + 20 b^{6} c^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(-b*c*x+a*c)**6,x)

[Out]

-(3*a + 5*b*x)/(-20*a**5*b*c**6 + 100*a**4*b**2*c**6*x - 200*a**3*b**3*c**6*x**2
+ 200*a**2*b**4*c**6*x**3 - 100*a*b**5*c**6*x**4 + 20*b**6*c**6*x**5)

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GIAC/XCAS [A]  time = 0.201707, size = 34, normalized size = 0.89 \[ -\frac{5 \, b x + 3 \, a}{20 \,{\left (b x - a\right )}^{5} b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(b*c*x - a*c)^6,x, algorithm="giac")

[Out]

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