3.1043 \(\int \frac{(a+b x)^2}{(a c-b c x)^2} \, dx\)

Optimal. Leaf size=41 \[ \frac{4 a^2}{b c^2 (a-b x)}+\frac{4 a \log (a-b x)}{b c^2}+\frac{x}{c^2} \]

[Out]

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

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Rubi [A]  time = 0.0221189, antiderivative size = 41, 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} \[ \frac{4 a^2}{b c^2 (a-b x)}+\frac{4 a \log (a-b x)}{b c^2}+\frac{x}{c^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0295893, size = 35, normalized size = 0.85 \[ \frac{\frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}+x}{c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 44, normalized size = 1.1 \begin{align*}{\frac{x}{{c}^{2}}}-4\,{\frac{{a}^{2}}{{c}^{2}b \left ( bx-a \right ) }}+4\,{\frac{a\ln \left ( bx-a \right ) }{{c}^{2}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00933, size = 62, normalized size = 1.51 \begin{align*} -\frac{4 \, a^{2}}{b^{2} c^{2} x - a b c^{2}} + \frac{x}{c^{2}} + \frac{4 \, a \log \left (b x - a\right )}{b c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.51304, size = 108, normalized size = 2.63 \begin{align*} \frac{b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \,{\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} c^{2} x - a b c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.358625, size = 39, normalized size = 0.95 \begin{align*} - \frac{4 a^{2}}{- a b c^{2} + b^{2} c^{2} x} + \frac{4 a \log{\left (- a + b x \right )}}{b c^{2}} + \frac{x}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08186, size = 107, normalized size = 2.61 \begin{align*} -\frac{4 \, a^{2}}{{\left (b c x - a c\right )} b c} - \frac{4 \, a \log \left (\frac{{\left | b c x - a c \right |}}{{\left (b c x - a c\right )}^{2}{\left | b \right |}{\left | c \right |}}\right )}{b c^{2}} + \frac{b c x - a c}{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)^2,x, algorithm="giac")

[Out]

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