Optimal. Leaf size=77 \[ \frac{a^2 \log (a+b x)}{b (b c-a d)^2}+\frac{c^2}{d^2 (c+d x) (b c-a d)}+\frac{c (b c-2 a d) \log (c+d x)}{d^2 (b c-a d)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0566586, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ \frac{a^2 \log (a+b x)}{b (b c-a d)^2}+\frac{c^2}{d^2 (c+d x) (b c-a d)}+\frac{c (b c-2 a d) \log (c+d x)}{d^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rubi steps
\begin{align*} \int \frac{x^2}{(a+b x) (c+d x)^2} \, dx &=\int \left (\frac{a^2}{(b c-a d)^2 (a+b x)}+\frac{c^2}{d (-b c+a d) (c+d x)^2}+\frac{c (b c-2 a d)}{d (-b c+a d)^2 (c+d x)}\right ) \, dx\\ &=\frac{c^2}{d^2 (b c-a d) (c+d x)}+\frac{a^2 \log (a+b x)}{b (b c-a d)^2}+\frac{c (b c-2 a d) \log (c+d x)}{d^2 (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.0441722, size = 77, normalized size = 1. \[ \frac{a^2 d^2 (c+d x) \log (a+b x)+b c ((c+d x) (b c-2 a d) \log (c+d x)+c (b c-a d))}{b d^2 (c+d x) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 97, normalized size = 1.3 \begin{align*} -{\frac{{c}^{2}}{{d}^{2} \left ( ad-bc \right ) \left ( dx+c \right ) }}-2\,{\frac{c\ln \left ( dx+c \right ) a}{d \left ( ad-bc \right ) ^{2}}}+{\frac{{c}^{2}\ln \left ( dx+c \right ) b}{ \left ( ad-bc \right ) ^{2}{d}^{2}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.13958, size = 162, normalized size = 2.1 \begin{align*} \frac{a^{2} \log \left (b x + a\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} + \frac{c^{2}}{b c^{2} d^{2} - a c d^{3} +{\left (b c d^{3} - a d^{4}\right )} x} + \frac{{\left (b c^{2} - 2 \, a c d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.3524, size = 301, normalized size = 3.91 \begin{align*} \frac{b^{2} c^{3} - a b c^{2} d +{\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \log \left (b x + a\right ) +{\left (b^{2} c^{3} - 2 \, a b c^{2} d +{\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{2} - 2 \, a b^{2} c^{2} d^{3} + a^{2} b c d^{4} +{\left (b^{3} c^{2} d^{3} - 2 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.85612, size = 333, normalized size = 4.32 \begin{align*} \frac{a^{2} \log{\left (x + \frac{\frac{a^{5} d^{4}}{b \left (a d - b c\right )^{2}} - \frac{3 a^{4} c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{3} b c^{2} d^{2}}{\left (a d - b c\right )^{2}} - \frac{a^{2} b^{2} c^{3} d}{\left (a d - b c\right )^{2}} + 3 a^{2} c d - a b c^{2}}{a^{2} d^{2} + 2 a b c d - b^{2} c^{2}} \right )}}{b \left (a d - b c\right )^{2}} - \frac{c^{2}}{a c d^{3} - b c^{2} d^{2} + x \left (a d^{4} - b c d^{3}\right )} - \frac{c \left (2 a d - b c\right ) \log{\left (x + \frac{- \frac{a^{3} c d^{2} \left (2 a d - b c\right )}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c^{2} d \left (2 a d - b c\right )}{\left (a d - b c\right )^{2}} + 3 a^{2} c d - \frac{3 a b^{2} c^{3} \left (2 a d - b c\right )}{\left (a d - b c\right )^{2}} - a b c^{2} + \frac{b^{3} c^{4} \left (2 a d - b c\right )}{d \left (a d - b c\right )^{2}}}{a^{2} d^{2} + 2 a b c d - b^{2} c^{2}} \right )}}{d^{2} \left (a d - b c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19136, size = 154, normalized size = 2. \begin{align*} \frac{a^{2} d \log \left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}} + \frac{c^{2} d}{{\left (b c d^{3} - a d^{4}\right )}{\left (d x + c\right )}} - \frac{\log \left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{b d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]