Optimal. Leaf size=51 \[ -\frac{(b c-a d)^2}{b^3 (a+b x)}+\frac{2 d (b c-a d) \log (a+b x)}{b^3}+\frac{d^2 x}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0346731, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{(b c-a d)^2}{b^3 (a+b x)}+\frac{2 d (b c-a d) \log (a+b x)}{b^3}+\frac{d^2 x}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{(a+b x)^2} \, dx &=\int \left (\frac{d^2}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)^2}+\frac{2 d (b c-a d)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{d^2 x}{b^2}-\frac{(b c-a d)^2}{b^3 (a+b x)}+\frac{2 d (b c-a d) \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0362308, size = 47, normalized size = 0.92 \[ \frac{-\frac{(b c-a d)^2}{a+b x}+2 d (b c-a d) \log (a+b x)+b d^2 x}{b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 86, normalized size = 1.7 \begin{align*}{\frac{{d}^{2}x}{{b}^{2}}}-{\frac{{a}^{2}{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}+2\,{\frac{acd}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{c}^{2}}{b \left ( bx+a \right ) }}-2\,{\frac{{d}^{2}\ln \left ( bx+a \right ) a}{{b}^{3}}}+2\,{\frac{d\ln \left ( bx+a \right ) c}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.960575, size = 90, normalized size = 1.76 \begin{align*} \frac{d^{2} x}{b^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{b^{4} x + a b^{3}} + \frac{2 \,{\left (b c d - a d^{2}\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.93322, size = 184, normalized size = 3.61 \begin{align*} \frac{b^{2} d^{2} x^{2} + a b d^{2} x - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} + 2 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.562278, size = 60, normalized size = 1.18 \begin{align*} - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{a b^{3} + b^{4} x} + \frac{d^{2} x}{b^{2}} - \frac{2 d \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.06726, size = 132, normalized size = 2.59 \begin{align*} \frac{{\left (b x + a\right )} d^{2}}{b^{3}} - \frac{2 \,{\left (b c d - a d^{2}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} - \frac{\frac{b^{3} c^{2}}{b x + a} - \frac{2 \, a b^{2} c d}{b x + a} + \frac{a^{2} b d^{2}}{b x + a}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]