Optimal. Leaf size=165 \[ \frac{b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac{b p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right )}{h i-g j}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac{\log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.467889, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {2418, 2394, 2393, 2391, 2445} \[ \frac{b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac{b p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right )}{h i-g j}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac{\log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2445
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (527+j x)} \, dx &=\operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x) (527+j x)} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{h \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(527 h-g j) (g+h x)}-\frac{j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(527 h-g j) (527+j x)}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{h \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x} \, dx}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{j \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{527+j x} \, dx}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right )}{527 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (527+j x)}{527 f-e j}\right )}{527 h-g j}-\operatorname{Subst}\left (\frac{(b f p q) \int \frac{\log \left (\frac{f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(b f p q) \int \frac{\log \left (\frac{f (527+j x)}{527 f-e j}\right )}{e+f x} \, dx}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right )}{527 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (527+j x)}{527 f-e j}\right )}{527 h-g j}-\operatorname{Subst}\left (\frac{(b p q) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(b p q) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{527 f-e j}\right )}{x} \, dx,x,e+f x\right )}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right )}{527 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (527+j x)}{527 f-e j}\right )}{527 h-g j}+\frac{b p q \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{527 h-g j}-\frac{b p q \text{Li}_2\left (-\frac{j (e+f x)}{527 f-e j}\right )}{527 h-g j}\\ \end{align*}
Mathematica [A] time = 0.0676646, size = 117, normalized size = 0.71 \[ \frac{b p q \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )-b p q \text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right )+\left (\log \left (\frac{f (g+h x)}{f g-e h}\right )-\log \left (\frac{f (i+j x)}{f i-e j}\right )\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.122, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) }{ \left ( hx+g \right ) \left ( jx+i \right ) }}\, dx \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*} a{\left (\frac{\log \left (h x + g\right )}{h i - g j} - \frac{\log \left (j x + i\right )}{h i - g j}\right )} + b \int \frac{\log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right ) + \log \left (c\right ) + \log \left (d^{q}\right )}{h j x^{2} + g i +{\left (h i + g j\right )} x}\,{d x} \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 (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h j x^{2} + g i +{\left (h i + g j\right )} x}, 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 (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )}{\left (j x + i\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]