Optimal. Leaf size=425 \[ \frac{b h^2 p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{(h i-g j)^3}-\frac{b h^2 p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right )}{(h i-g j)^3}+\frac{h^2 \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}-\frac{h^2 \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}+\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x) (h i-g j)^2}+\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (i+j x)^2 (h i-g j)}-\frac{b f^2 p q \log (e+f x)}{2 (f i-e j)^2 (h i-g j)}+\frac{b f^2 p q \log (i+j x)}{2 (f i-e j)^2 (h i-g j)}-\frac{b f p q}{2 (i+j x) (f i-e j) (h i-g j)}-\frac{b f h p q \log (e+f x)}{(f i-e j) (h i-g j)^2}+\frac{b f h p q \log (i+j x)}{(f i-e j) (h i-g j)^2} \]
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Rubi [A] time = 0.835042, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2418, 2394, 2393, 2391, 2395, 44, 36, 31, 2445} \[ \frac{b h^2 p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{(h i-g j)^3}-\frac{b h^2 p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right )}{(h i-g j)^3}+\frac{h^2 \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}-\frac{h^2 \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}+\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x) (h i-g j)^2}+\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (i+j x)^2 (h i-g j)}-\frac{b f^2 p q \log (e+f x)}{2 (f i-e j)^2 (h i-g j)}+\frac{b f^2 p q \log (i+j x)}{2 (f i-e j)^2 (h i-g j)}-\frac{b f p q}{2 (i+j x) (f i-e j) (h i-g j)}-\frac{b f h p q \log (e+f x)}{(f i-e j) (h i-g j)^2}+\frac{b f h p q \log (i+j x)}{(f i-e j) (h i-g j)^2} \]
Antiderivative was successfully verified.
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Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2395
Rule 44
Rule 36
Rule 31
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) (529+j x)^3} \, dx &=\operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x) (529+j x)^3} \, 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^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(529 h-g j)^3 (g+h x)}-\frac{j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(529 h-g j) (529+j x)^3}-\frac{h j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(529 h-g j)^2 (529+j x)^2}-\frac{h^2 j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(529 h-g j)^3 (529+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^3 \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x} \, dx}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (h^2 j\right ) \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{529+j x} \, dx}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(h j) \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{(529+j x)^2} \, dx}{(529 h-g j)^2},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 )}{(529+j x)^3} \, dx}{529 h-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 )}{2 (529 h-g j) (529+j x)^2}+\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(529 h-g j)^2 (529+j x)}+\frac{h^2 \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 )}{(529 h-g j)^3}-\frac{h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (529+j x)}{529 f-e j}\right )}{(529 h-g j)^3}-\operatorname{Subst}\left (\frac{\left (b f h^2 p q\right ) \int \frac{\log \left (\frac{f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (b f h^2 p q\right ) \int \frac{\log \left (\frac{f (529+j x)}{529 f-e j}\right )}{e+f x} \, dx}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(b f h p q) \int \frac{1}{(e+f x) (529+j x)} \, dx}{(529 h-g j)^2},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{1}{(e+f x) (529+j x)^2} \, dx}{2 (529 h-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 )}{2 (529 h-g j) (529+j x)^2}+\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(529 h-g j)^2 (529+j x)}+\frac{h^2 \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 )}{(529 h-g j)^3}-\frac{h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (529+j x)}{529 f-e j}\right )}{(529 h-g j)^3}-\operatorname{Subst}\left (\frac{\left (b h^2 p q\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (b h^2 p q\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{529 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (b f^2 h p q\right ) \int \frac{1}{e+f x} \, dx}{(529 f-e j) (529 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(b f h j p q) \int \frac{1}{529+j x} \, dx}{(529 f-e j) (529 h-g j)^2},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 \left (\frac{f^2}{(529 f-e j)^2 (e+f x)}-\frac{j}{(529 f-e j) (529+j x)^2}-\frac{f j}{(529 f-e j)^2 (529+j x)}\right ) \, dx}{2 (529 h-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}{2 (529 f-e j) (529 h-g j) (529+j x)}-\frac{b f h p q \log (e+f x)}{(529 f-e j) (529 h-g j)^2}-\frac{b f^2 p q \log (e+f x)}{2 (529 f-e j)^2 (529 h-g j)}+\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (529 h-g j) (529+j x)^2}+\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(529 h-g j)^2 (529+j x)}+\frac{h^2 \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 )}{(529 h-g j)^3}+\frac{b f h p q \log (529+j x)}{(529 f-e j) (529 h-g j)^2}+\frac{b f^2 p q \log (529+j x)}{2 (529 f-e j)^2 (529 h-g j)}-\frac{h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (529+j x)}{529 f-e j}\right )}{(529 h-g j)^3}+\frac{b h^2 p q \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{(529 h-g j)^3}-\frac{b h^2 p q \text{Li}_2\left (-\frac{j (e+f x)}{529 f-e j}\right )}{(529 h-g j)^3}\\ \end{align*}
Mathematica [A] time = 0.536791, size = 363, normalized size = 0.85 \[ \frac{2 b h^2 p q \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )-2 b h^2 p q \text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right )+2 h^2 \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )-2 h^2 \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+\frac{2 a h (h i-g j)}{i+j x}+\frac{a (h i-g j)^2}{(i+j x)^2}+\frac{2 b h (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{i+j x}+\frac{b (h i-g j)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{(i+j x)^2}-\frac{2 b f h p q (h i-g j) (\log (e+f x)-\log (i+j x))}{f i-e j}-\frac{b f p q (h i-g j)^2 (f (i+j x) \log (e+f x)-e j-f (i+j x) \log (i+j x)+f i)}{(i+j x) (f i-e j)^2}}{2 (h i-g j)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.024, 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 ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (\frac{2 \, h^{2} \log \left (h x + g\right )}{h^{3} i^{3} - 3 \, g h^{2} i^{2} j + 3 \, g^{2} h i j^{2} - g^{3} j^{3}} - \frac{2 \, h^{2} \log \left (j x + i\right )}{h^{3} i^{3} - 3 \, g h^{2} i^{2} j + 3 \, g^{2} h i j^{2} - g^{3} j^{3}} + \frac{2 \, h j x + 3 \, h i - g j}{h^{2} i^{4} - 2 \, g h i^{3} j + g^{2} i^{2} j^{2} +{\left (h^{2} i^{2} j^{2} - 2 \, g h i j^{3} + g^{2} j^{4}\right )} x^{2} + 2 \,{\left (h^{2} i^{3} j - 2 \, g h i^{2} j^{2} + g^{2} i j^{3}\right )} x}\right )} a + 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^{3} x^{4} + g i^{3} +{\left (3 \, h i j^{2} + g j^{3}\right )} x^{3} + 3 \,{\left (h i^{2} j + g i j^{2}\right )} x^{2} +{\left (h i^{3} + 3 \, g i^{2} j\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{3} x^{4} + g i^{3} +{\left (3 \, h i j^{2} + g j^{3}\right )} x^{3} + 3 \,{\left (h i^{2} j + g i j^{2}\right )} x^{2} +{\left (h i^{3} + 3 \, g i^{2} j\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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