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 = 0.0021911, 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, Rules used = {34} \[ \frac{x}{c^3 (a-b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(a*c - b*c*x)^3,x]

[Out]

x/(c^3*(a - b*x)^2)

Rule 34

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_)), x_Symbol] :> Simp[(d*x*(a + b*x)^(m + 1))/(b*(m + 2)), x] /
; FreeQ[{a, b, c, d, m}, x] && EqQ[a*d - b*c*(m + 2), 0]

Rubi steps

\begin{align*} \int \frac{a+b x}{(a c-b c x)^3} \, dx &=\frac{x}{c^3 (a-b x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0078797, 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 = 0.003, size = 33, normalized size = 2.5 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{1}{b \left ( bx-a \right ) }}+{\frac{a}{b \left ( bx-a \right ) ^{2}}} \right ) } \end{align*}

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

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Maxima [B]  time = 0.989695, size = 41, normalized size = 3.15 \begin{align*} \frac{x}{b^{2} c^{3} x^{2} - 2 \, a b c^{3} x + a^{2} c^{3}} \end{align*}

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 [B]  time = 1.48136, size = 55, normalized size = 4.23 \begin{align*} \frac{x}{b^{2} c^{3} x^{2} - 2 \, a b c^{3} x + a^{2} c^{3}} \end{align*}

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 [B]  time = 0.368923, size = 27, normalized size = 2.08 \begin{align*} \frac{x}{a^{2} c^{3} - 2 a b c^{3} x + b^{2} c^{3} x^{2}} \end{align*}

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 [A]  time = 1.05118, size = 19, normalized size = 1.46 \begin{align*} \frac{x}{{\left (b x - a\right )}^{2} c^{3}} \end{align*}

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)