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 = 0.0560585, 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 \[ \frac{4 a^2}{b c^2 (a-b x)}+\frac{4 a \log (a-b x)}{b c^2}+\frac{x}{c^2} \]

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)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 a^{2}}{b c^{2} \left (a - b x\right )} + \frac{4 a \log{\left (a - b x \right )}}{b c^{2}} + \int \frac{1}{c^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2/(-b*c*x+a*c)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0491715, 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 = 0.008, size = 44, normalized size = 1.1 \[{\frac{x}{{c}^{2}}}+4\,{\frac{a\ln \left ( bx-a \right ) }{{c}^{2}b}}-4\,{\frac{{a}^{2}}{{c}^{2}b \left ( bx-a \right ) }} \]

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

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Maxima [A]  time = 1.3402, size = 62, normalized size = 1.51 \[ -\frac{4 \, a^{2}}{b^{2} c^{2} x - a b c^{2}} + \frac{x}{c^{2}} + \frac{4 \, a \log \left (b x - a\right )}{b c^{2}} \]

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 = 0.198504, size = 77, normalized size = 1.88 \[ \frac{b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \,{\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} c^{2} x - a b c^{2}} \]

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 = 1.38417, size = 39, normalized size = 0.95 \[ - \frac{4 a^{2}}{- a b c^{2} + b^{2} c^{2} x} + \frac{4 a \log{\left (- a + b x \right )}}{b c^{2}} + \frac{x}{c^{2}} \]

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/XCAS [A]  time = 0.206212, size = 107, normalized size = 2.61 \[ -\frac{4 \, a^{2}}{{\left (b c x - a c\right )} b c} - \frac{4 \, a{\rm ln}\left (\frac{{\left | b c x - a c \right |}}{{\left (b c x - a c\right )}^{2}{\left | b \right |}{\left | c \right |}}\right )}{b c^{2}} + \frac{b c x - a c}{b c^{3}} \]

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*ln(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)