Optimal. Leaf size=138 \[ -\frac{B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A}{2 b g^3 (a+b x)^2}+\frac{B d^2 \log (a+b x)}{b g^3 (b c-a d)^2}-\frac{B d^2 \log (c+d x)}{b g^3 (b c-a d)^2}+\frac{B d}{b g^3 (a+b x) (b c-a d)}-\frac{B}{2 b g^3 (a+b x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0934478, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A}{2 b g^3 (a+b x)^2}+\frac{B d^2 \log (a+b x)}{b g^3 (b c-a d)^2}-\frac{B d^2 \log (c+d x)}{b g^3 (b c-a d)^2}+\frac{B d}{b g^3 (a+b x) (b c-a d)}-\frac{B}{2 b g^3 (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2525
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{2 b g^3 (a+b x)^2}+\frac{B \int \frac{2 (b c-a d)}{g^2 (a+b x)^3 (c+d x)} \, dx}{2 b g}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{2 b g^3 (a+b x)^2}+\frac{(B (b c-a d)) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b g^3}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{2 b g^3 (a+b x)^2}+\frac{(B (b c-a d)) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b g^3}\\ &=-\frac{B}{2 b g^3 (a+b x)^2}+\frac{B d}{b (b c-a d) g^3 (a+b x)}+\frac{B d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{2 b g^3 (a+b x)^2}-\frac{B d^2 \log (c+d x)}{b (b c-a d)^2 g^3}\\ \end{align*}
Mathematica [A] time = 0.136493, size = 109, normalized size = 0.79 \[ -\frac{\frac{B \left (2 d^2 (a+b x)^2 \log (c+d x)+(b c-a d) (b (c-2 d x)-3 a d)-2 d^2 (a+b x)^2 \log (a+b x)\right )}{(b c-a d)^2}+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A}{2 b g^3 (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.108, size = 355, normalized size = 2.6 \begin{align*} -{\frac{{d}^{2}Ab}{2\,{g}^{3} \left ( ad-bc \right ) ^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-2}}+{\frac{{d}^{2}A}{{g}^{3} \left ( ad-bc \right ) ^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}-{\frac{{d}^{2}B}{{g}^{3} \left ( ad-bc \right ) \left ( dx+c \right ) } \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-2}}-{\frac{3\,{d}^{2}B}{2\,{g}^{3}b \left ( dx+c \right ) ^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-2}}+{\frac{b{d}^{2}B}{2\,{g}^{3} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-2}}+{\frac{{d}^{2}B}{{g}^{3} \left ( ad-bc \right ) \left ( dx+c \right ) }\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.29435, size = 414, normalized size = 3. \begin{align*} \frac{1}{2} \, B{\left (\frac{2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \,{\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x +{\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac{\log \left (\frac{b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac{2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac{2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac{A}{2 \,{\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.05097, size = 495, normalized size = 3.59 \begin{align*} -\frac{{\left (A + B\right )} b^{2} c^{2} - 2 \,{\left (A + 2 \, B\right )} a b c d +{\left (A + 3 \, B\right )} a^{2} d^{2} - 2 \,{\left (B b^{2} c d - B a b d^{2}\right )} x -{\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x - B b^{2} c^{2} + 2 \, B a b c d\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \,{\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \,{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x +{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 3.17597, size = 418, normalized size = 3.03 \begin{align*} - \frac{B \log{\left (\frac{e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{2 a^{2} b g^{3} + 4 a b^{2} g^{3} x + 2 b^{3} g^{3} x^{2}} - \frac{B d^{2} \log{\left (x + \frac{- \frac{B a^{3} d^{5}}{\left (a d - b c\right )^{2}} + \frac{3 B a^{2} b c d^{4}}{\left (a d - b c\right )^{2}} - \frac{3 B a b^{2} c^{2} d^{3}}{\left (a d - b c\right )^{2}} + B a d^{3} + \frac{B b^{3} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + B b c d^{2}}{2 B b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} + \frac{B d^{2} \log{\left (x + \frac{\frac{B a^{3} d^{5}}{\left (a d - b c\right )^{2}} - \frac{3 B a^{2} b c d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 B a b^{2} c^{2} d^{3}}{\left (a d - b c\right )^{2}} + B a d^{3} - \frac{B b^{3} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + B b c d^{2}}{2 B b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} - \frac{A a d - A b c + 3 B a d - B b c + 2 B b d x}{2 a^{3} b d g^{3} - 2 a^{2} b^{2} c g^{3} + x^{2} \left (2 a b^{3} d g^{3} - 2 b^{4} c g^{3}\right ) + x \left (4 a^{2} b^{2} d g^{3} - 4 a b^{3} c g^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.31868, size = 356, normalized size = 2.58 \begin{align*} \frac{B d^{2} \log \left (b x + a\right )}{b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}} - \frac{B d^{2} \log \left (d x + c\right )}{b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}} - \frac{B \log \left (\frac{b^{2} x^{2} + 2 \, a b x + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \,{\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} + \frac{2 \, B b d x - A b c - 2 \, B b c + A a d + 4 \, B a d}{2 \,{\left (b^{4} c g^{3} x^{2} - a b^{3} d g^{3} x^{2} + 2 \, a b^{3} c g^{3} x - 2 \, a^{2} b^{2} d g^{3} x + a^{2} b^{2} c g^{3} - a^{3} b d g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]