3.1034 \(\int \frac{a+b x}{(a c-b c x)^3} \, dx\)

Optimal. Leaf size=13 \[ \frac{x}{c^3 (a-b x)^2} \]

[Out]

x/(c^3*(a - b*x)^2)

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Rubi [A]  time = 0.0100503, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{x}{c^3 (a-b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

x/(c^3*(a - b*x)^2)

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Rubi in Sympy [A]  time = 5.36207, size = 20, normalized size = 1.54 \[ \frac{\left (a + b x\right )^{2}}{4 a b c^{3} \left (a - b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0105498, size = 13, normalized size = 1. \[ \frac{x}{c^3 (a-b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

x/(c^3*(a - b*x)^2)

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Maple [B]  time = 0.007, size = 33, normalized size = 2.5 \[{\frac{1}{{c}^{3}} \left ({\frac{a}{b \left ( bx-a \right ) ^{2}}}+{\frac{1}{b \left ( bx-a \right ) }} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(-b*c*x+a*c)^3,x)

[Out]

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

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Maxima [A]  time = 1.34493, size = 41, normalized size = 3.15 \[ \frac{x}{b^{2} c^{3} x^{2} - 2 \, a b c^{3} x + a^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.202937, size = 41, normalized size = 3.15 \[ \frac{x}{b^{2} c^{3} x^{2} - 2 \, a b c^{3} x + a^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.51781, size = 27, normalized size = 2.08 \[ \frac{x}{a^{2} c^{3} - 2 a b c^{3} x + b^{2} c^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.202442, size = 19, normalized size = 1.46 \[ \frac{x}{{\left (b x - a\right )}^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/((b*x - a)^2*c^3)