Optimal. Leaf size=76 \[ -\frac{B \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d i}-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.215483, antiderivative size = 122, normalized size of antiderivative = 1.61, number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2524, 12, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{B \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d i}+\frac{\log (c i+d i x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i}-\frac{B \log (c i+d i x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d i}+\frac{B \log ^2(i (c+d x))}{2 d i} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2524
Rule 12
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{34 c+34 d x} \, dx &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}-\frac{B \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (34 c+34 d x)}{e (a+b x)} \, dx}{34 d}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}-\frac{B \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (34 c+34 d x)}{a+b x} \, dx}{34 d e}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}-\frac{B \int \left (\frac{b e \log (34 c+34 d x)}{a+b x}-\frac{d e \log (34 c+34 d x)}{c+d x}\right ) \, dx}{34 d e}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}+\frac{1}{34} B \int \frac{\log (34 c+34 d x)}{c+d x} \, dx-\frac{(b B) \int \frac{\log (34 c+34 d x)}{a+b x} \, dx}{34 d}\\ &=-\frac{B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (34 c+34 d x)}{34 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}+B \int \frac{\log \left (\frac{34 d (a+b x)}{-34 b c+34 a d}\right )}{34 c+34 d x} \, dx+\frac{B \operatorname{Subst}\left (\int \frac{34 \log (x)}{x} \, dx,x,34 c+34 d x\right )}{1156 d}\\ &=-\frac{B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (34 c+34 d x)}{34 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}+\frac{B \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,34 c+34 d x\right )}{34 d}+\frac{B \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-34 b c+34 a d}\right )}{x} \, dx,x,34 c+34 d x\right )}{34 d}\\ &=\frac{B \log ^2(34 (c+d x))}{68 d}-\frac{B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (34 c+34 d x)}{34 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}-\frac{B \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{34 d}\\ \end{align*}
Mathematica [A] time = 0.0317992, size = 95, normalized size = 1.25 \[ \frac{\log (i (c+d x)) \left (2 B \log \left (\frac{e (a+b x)}{c+d x}\right )-2 B \log \left (\frac{d (a+b x)}{a d-b c}\right )+2 A+B \log (i (c+d x))\right )-2 B \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{2 d i} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.056, size = 411, normalized size = 5.4 \begin{align*} -{\frac{Aa}{i \left ( ad-bc \right ) }\ln \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) }+{\frac{Abc}{di \left ( ad-bc \right ) }\ln \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) }-{\frac{Ba}{i \left ( ad-bc \right ) }{\it dilog} \left ( -{\frac{1}{be} \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) } \right ) }+{\frac{Bbc}{di \left ( ad-bc \right ) }{\it dilog} \left ( -{\frac{1}{be} \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) } \right ) }-{\frac{Ba}{i \left ( ad-bc \right ) }\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \ln \left ( -{\frac{1}{be} \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) } \right ) }+{\frac{Bbc}{di \left ( ad-bc \right ) }\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \ln \left ( -{\frac{1}{be} \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, B{\left (\frac{\log \left (d x + c\right )^{2}}{d i} - 2 \, \int \frac{\log \left (b x + a\right ) + \log \left (e\right )}{d i x + c i}\,{d x}\right )} + \frac{A \log \left (d i x + c i\right )}{d i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B \log \left (\frac{b e x + a e}{d x + c}\right ) + A}{d i x + c i}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A}{d i x + c i}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]