3.1044 \(\int \frac{(a+b x)^2}{(a c-b c x)^3} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 a^2}{b c^3 (a-b x)^2}-\frac{4 a}{b c^3 (a-b x)}-\frac{\log (a-b x)}{b c^3} \]

[Out]

(2*a^2)/(b*c^3*(a - b*x)^2) - (4*a)/(b*c^3*(a - b*x)) - Log[a - b*x]/(b*c^3)

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Rubi [A]  time = 0.0280616, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {43} \[ \frac{2 a^2}{b c^3 (a-b x)^2}-\frac{4 a}{b c^3 (a-b x)}-\frac{\log (a-b x)}{b c^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^3,x]

[Out]

(2*a^2)/(b*c^3*(a - b*x)^2) - (4*a)/(b*c^3*(a - b*x)) - Log[a - b*x]/(b*c^3)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^2}{(a c-b c x)^3} \, dx &=\int \left (\frac{4 a^2}{c^3 (a-b x)^3}-\frac{4 a}{c^3 (a-b x)^2}+\frac{1}{c^3 (a-b x)}\right ) \, dx\\ &=\frac{2 a^2}{b c^3 (a-b x)^2}-\frac{4 a}{b c^3 (a-b x)}-\frac{\log (a-b x)}{b c^3}\\ \end{align*}

Mathematica [A]  time = 0.0244032, size = 33, normalized size = 0.63 \[ -\frac{\frac{2 a (a-2 b x)}{(a-b x)^2}+\log (a-b x)}{b c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(a*c - b*c*x)^3,x]

[Out]

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

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Maple [A]  time = 0.005, size = 56, normalized size = 1.1 \begin{align*} 4\,{\frac{a}{{c}^{3}b \left ( bx-a \right ) }}+2\,{\frac{{a}^{2}}{{c}^{3}b \left ( bx-a \right ) ^{2}}}-{\frac{\ln \left ( bx-a \right ) }{{c}^{3}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03558, size = 82, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (2 \, a b x - a^{2}\right )}}{b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{3} x + a^{2} b c^{3}} - \frac{\log \left (b x - a\right )}{b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(-b*c*x+a*c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.57548, size = 138, normalized size = 2.65 \begin{align*} \frac{4 \, a b x - 2 \, a^{2} -{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{3} x + a^{2} b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(-b*c*x+a*c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.447033, size = 53, normalized size = 1.02 \begin{align*} \frac{- 2 a^{2} + 4 a b x}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} - \frac{\log{\left (- a + b x \right )}}{b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05647, size = 62, normalized size = 1.19 \begin{align*} -\frac{\log \left ({\left | b x - a \right |}\right )}{b c^{3}} + \frac{2 \,{\left (2 \, a b x - a^{2}\right )}}{{\left (b x - a\right )}^{2} b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(-b*c*x+a*c)^3,x, algorithm="giac")

[Out]

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