Optimal. Leaf size=98 \[ \frac{A (a+b x)}{i^2 (c+d x) (b c-a d)}+\frac{B (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{i^2 (c+d x) (b c-a d)}-\frac{B (a+b x)}{i^2 (c+d x) (b c-a d)} \]
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Rubi [A] time = 0.0729925, antiderivative size = 101, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{d i^2 (c+d x)}+\frac{b B \log (a+b x)}{d i^2 (b c-a d)}-\frac{b B \log (c+d x)}{d i^2 (b c-a d)}+\frac{B}{d i^2 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(42 c+42 d x)^2} \, dx &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1764 d (c+d x)}+\frac{B \int \frac{b c-a d}{42 (a+b x) (c+d x)^2} \, dx}{42 d}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1764 d (c+d x)}+\frac{(B (b c-a d)) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{1764 d}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1764 d (c+d x)}+\frac{(B (b c-a d)) \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}{1764 d}\\ &=\frac{B}{1764 d (c+d x)}+\frac{b B \log (a+b x)}{1764 d (b c-a d)}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1764 d (c+d x)}-\frac{b B \log (c+d x)}{1764 d (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0482936, size = 104, normalized size = 1.06 \[ \frac{-a A d+B (b c-a d) \log \left (\frac{e (a+b x)}{c+d x}\right )-b B (c+d x) \log (a+b x)+a B d+A b c+b B c \log (c+d x)+b B d x \log (c+d x)-b B c}{d i^2 (c+d x) (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 515, normalized size = 5.3 \begin{align*} -{\frac{Aba}{ \left ( ad-bc \right ) ^{2}{i}^{2}}}+{\frac{A{b}^{2}c}{d \left ( ad-bc \right ) ^{2}{i}^{2}}}-{\frac{dA{a}^{2}}{ \left ( ad-bc \right ) ^{2}{i}^{2} \left ( dx+c \right ) }}+2\,{\frac{Abac}{ \left ( ad-bc \right ) ^{2}{i}^{2} \left ( dx+c \right ) }}-{\frac{A{b}^{2}{c}^{2}}{d \left ( ad-bc \right ) ^{2}{i}^{2} \left ( dx+c \right ) }}-{\frac{Bba}{ \left ( ad-bc \right ) ^{2}{i}^{2}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) }+{\frac{B{b}^{2}c}{d \left ( ad-bc \right ) ^{2}{i}^{2}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) }-{\frac{dB{a}^{2}}{ \left ( ad-bc \right ) ^{2}{i}^{2} \left ( dx+c \right ) }\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) }+2\,{\frac{Bbac}{ \left ( ad-bc \right ) ^{2}{i}^{2} \left ( dx+c \right ) }\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) }-{\frac{B{b}^{2}{c}^{2}}{d \left ( ad-bc \right ) ^{2}{i}^{2} \left ( dx+c \right ) }\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) }+{\frac{dB{a}^{2}}{ \left ( ad-bc \right ) ^{2}{i}^{2} \left ( dx+c \right ) }}-2\,{\frac{Bbac}{ \left ( ad-bc \right ) ^{2}{i}^{2} \left ( dx+c \right ) }}+{\frac{B{b}^{2}{c}^{2}}{d \left ( ad-bc \right ) ^{2}{i}^{2} \left ( dx+c \right ) }}+{\frac{Bba}{ \left ( ad-bc \right ) ^{2}{i}^{2}}}-{\frac{B{b}^{2}c}{d \left ( ad-bc \right ) ^{2}{i}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22939, size = 181, normalized size = 1.85 \begin{align*} -B{\left (\frac{\log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac{1}{d^{2} i^{2} x + c d i^{2}} - \frac{b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} + \frac{b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} - \frac{A}{d^{2} i^{2} x + c d i^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.477514, size = 177, normalized size = 1.81 \begin{align*} -\frac{{\left (A - B\right )} b c -{\left (A - B\right )} a d -{\left (B b d x + B a d\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} i^{2} x +{\left (b c^{2} d - a c d^{2}\right )} i^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.69012, size = 231, normalized size = 2.36 \begin{align*} \frac{B b \log{\left (x + \frac{- \frac{B a^{2} b d^{2}}{a d - b c} + \frac{2 B a b^{2} c d}{a d - b c} + B a b d - \frac{B b^{3} c^{2}}{a d - b c} + B b^{2} c}{2 B b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} - \frac{B b \log{\left (x + \frac{\frac{B a^{2} b d^{2}}{a d - b c} - \frac{2 B a b^{2} c d}{a d - b c} + B a b d + \frac{B b^{3} c^{2}}{a d - b c} + B b^{2} c}{2 B b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} - \frac{B \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{c d i^{2} + d^{2} i^{2} x} - \frac{A - B}{c d i^{2} + d^{2} i^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3797, size = 116, normalized size = 1.18 \begin{align*} -\frac{B b \log \left (b x + a\right )}{b c d - a d^{2}} + \frac{B b \log \left (d x + c\right )}{b c d - a d^{2}} + \frac{B \log \left (\frac{b x + a}{d x + c}\right )}{d^{2} x + c d} + \frac{A}{d^{2} x + c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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