Integrand size = 33, antiderivative size = 268 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=-\frac {b f p q \log (e+f x)}{(f i-e j) (h i-g j)}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(h i-g j) (i+j x)}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {b f p q \log (i+j x)}{(f i-e j) (h i-g j)}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^2}+\frac {b h p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac {b h p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2} \]
[Out]
Time = 0.41 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2465, 2441, 2440, 2438, 2442, 36, 31, 2495} \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(i+j x) (h i-g j)}+\frac {h \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}-\frac {h \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac {b h p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac {b h p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}-\frac {b f p q \log (e+f x)}{(f i-e j) (h i-g j)}+\frac {b f p q \log (i+j x)}{(f i-e j) (h i-g j)} \]
[In]
[Out]
Rule 31
Rule 36
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2465
Rule 2495
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x) (i+j x)^2} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\int \left (\frac {h^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(h i-g j)^2 (g+h x)}-\frac {j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(h i-g j) (i+j x)^2}-\frac {h j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(h i-g j)^2 (i+j x)}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {h^2 \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x} \, dx}{(h i-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(h j) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{i+j x} \, dx}{(h i-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {j \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(i+j x)^2} \, dx}{h i-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(h i-g j) (i+j x)}+\frac {h \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 )}{(h i-g j)^2}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^2}-\text {Subst}\left (\frac {(b f h p q) \int \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{(h i-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(b f h p q) \int \frac {\log \left (\frac {f (i+j x)}{f i-e j}\right )}{e+f x} \, dx}{(h i-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(b f p q) \int \frac {1}{(e+f x) (i+j x)} \, dx}{h i-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(h i-g j) (i+j x)}+\frac {h \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 )}{(h i-g j)^2}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^2}-\text {Subst}\left (\frac {(b h p q) \text {Subst}\left (\int \frac {\log \left (1+\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{(h i-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(b h p q) \text {Subst}\left (\int \frac {\log \left (1+\frac {j x}{f i-e j}\right )}{x} \, dx,x,e+f x\right )}{(h i-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (b f^2 p q\right ) \int \frac {1}{e+f x} \, dx}{(f i-e j) (h i-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(b f j p q) \int \frac {1}{i+j x} \, dx}{(f i-e j) (h i-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {b f p q \log (e+f x)}{(f i-e j) (h i-g j)}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(h i-g j) (i+j x)}+\frac {h \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 )}{(h i-g j)^2}+\frac {b f p q \log (i+j x)}{(f i-e j) (h i-g j)}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^2}+\frac {b h p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac {b h p q \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=\frac {\frac {a (h i-g j)}{i+j x}+\frac {b (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{i+j x}+h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-\frac {b f (h i-g j) p q (\log (e+f x)-\log (i+j x))}{f i-e j}-h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )+b h p q \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-b h p q \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )}{(h i-g j)^2} \]
[In]
[Out]
\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right ) \left (j x +i \right )^{2}}d x\]
[In]
[Out]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\left (g + h x\right ) \left (i + j x\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^2} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\left (g+h\,x\right )\,{\left (i+j\,x\right )}^2} \,d x \]
[In]
[Out]