Optimal. Leaf size=56 \[ \frac{1}{(c+d x) (b c-a d)}+\frac{b \log (a+b x)}{(b c-a d)^2}-\frac{b \log (c+d x)}{(b c-a d)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0316406, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {626, 44} \[ \frac{1}{(c+d x) (b c-a d)}+\frac{b \log (a+b x)}{(b c-a d)^2}-\frac{b \log (c+d x)}{(b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b x}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx &=\int \frac{1}{(a+b x) (c+d x)^2} \, dx\\ &=\int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx\\ &=\frac{1}{(b c-a d) (c+d x)}+\frac{b \log (a+b x)}{(b c-a d)^2}-\frac{b \log (c+d x)}{(b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.0237145, size = 53, normalized size = 0.95 \[ \frac{b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c}{(c+d x) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.055, size = 58, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( ad-bc \right ) \left ( dx+c \right ) }}-{\frac{b\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{2}}}+{\frac{b\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.14036, size = 122, normalized size = 2.18 \begin{align*} \frac{b \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac{b \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac{1}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65618, size = 198, normalized size = 3.54 \begin{align*} \frac{b c - a d +{\left (b d x + b c\right )} \log \left (b x + a\right ) -{\left (b d x + b c\right )} \log \left (d x + c\right )}{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.19887, size = 233, normalized size = 4.16 \begin{align*} - \frac{b \log{\left (x + \frac{- \frac{a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac{b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{\left (a d - b c\right )^{2}} + \frac{b \log{\left (x + \frac{\frac{a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac{b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{\left (a d - b c\right )^{2}} - \frac{1}{a c d - b c^{2} + x \left (a d^{2} - b c d\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18335, size = 126, normalized size = 2.25 \begin{align*} \frac{b^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} - \frac{b d \log \left ({\left | d x + c \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac{1}{{\left (b c - a d\right )}{\left (d x + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]