3.1515 \(\int \frac{(c+d x)^2}{(a-b x) (a+b x)} \, dx\)

Optimal. Leaf size=62 \[ -\frac{(a d+b c)^2 \log (a-b x)}{2 a b^3}+\frac{(b c-a d)^2 \log (a+b x)}{2 a b^3}-\frac{d^2 x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.106919, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{(a d+b c)^2 \log (a-b x)}{2 a b^3}+\frac{(b c-a d)^2 \log (a+b x)}{2 a b^3}-\frac{d^2 x}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^2/((a - b*x)*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - d^{2} \int \frac{1}{b^{2}}\, dx + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{2 a b^{3}} - \frac{\left (a d + b c\right )^{2} \log{\left (a - b x \right )}}{2 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**2*Integral(b**(-2), x) + (a*d - b*c)**2*log(a + b*x)/(2*a*b**3) - (a*d + b*c
)**2*log(a - b*x)/(2*a*b**3)

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Mathematica [A]  time = 0.0431151, size = 54, normalized size = 0.87 \[ \frac{-(a d+b c)^2 \log (a-b x)+(b c-a d)^2 \log (a+b x)-2 a b d^2 x}{2 a b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 107, normalized size = 1.7 \[ -{\frac{{d}^{2}x}{{b}^{2}}}+{\frac{a\ln \left ( bx+a \right ){d}^{2}}{2\,{b}^{3}}}-{\frac{\ln \left ( bx+a \right ) cd}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{2}}{2\,ab}}-{\frac{a\ln \left ( bx-a \right ){d}^{2}}{2\,{b}^{3}}}-{\frac{\ln \left ( bx-a \right ) cd}{{b}^{2}}}-{\frac{\ln \left ( bx-a \right ){c}^{2}}{2\,ab}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^2/(-b*x+a)/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.35469, size = 111, normalized size = 1.79 \[ -\frac{d^{2} x}{b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \, a b^{3}} - \frac{{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x - a\right )}{2 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.215527, size = 103, normalized size = 1.66 \[ -\frac{2 \, a b d^{2} x -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right ) +{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x - a\right )}{2 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.71343, size = 112, normalized size = 1.81 \[ - \frac{d^{2} x}{b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (x + \frac{2 a^{2} c d + \frac{a \left (a d - b c\right )^{2}}{b}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{2 a b^{3}} - \frac{\left (a d + b c\right )^{2} \log{\left (x + \frac{2 a^{2} c d - \frac{a \left (a d + b c\right )^{2}}{b}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{2 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208208, size = 113, normalized size = 1.82 \[ -\frac{d^{2} x}{b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{2 \, a b^{3}} - \frac{{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | b x - a \right |}\right )}{2 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(d*x + c)^2/((b*x + a)*(b*x - a)),x, algorithm="giac")

[Out]

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