3.1532 \(\int \frac{(c+d x)^2}{(a-b x) (a+b x)} \, dx\)

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 = 0.0455141, 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, Rules used = {72} \[ -\frac{(a d+b c)^2 \log (a-b x)}{2 a b^3}+\frac{(b c-a d)^2 \log (a+b x)}{2 a b^3}-\frac{d^2 x}{b^2} \]

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)

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.025909, 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 = 0.006, size = 107, normalized size = 1.7 \begin{align*} -{\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}} \end{align*}

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 = 1.77663, size = 111, normalized size = 1.79 \begin{align*} -\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}} \end{align*}

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)*lo
g(b*x - a)/(a*b^3)

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Fricas [A]  time = 1.19701, size = 165, normalized size = 2.66 \begin{align*} -\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}} \end{align*}

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 [B]  time = 0.691094, size = 112, normalized size = 1.81 \begin{align*} - \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}} \end{align*}

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 [A]  time = 1.76339, size = 113, normalized size = 1.82 \begin{align*} -\frac{d^{2} x}{b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \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 )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{3}} \end{align*}

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