Optimal. Leaf size=129 \[ \frac{b p q (h i-g j) \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{h^2}+\frac{(h i-g j) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^2}+\frac{a j x}{h}+\frac{b j (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac{b j p q x}{h} \]
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Rubi [A] time = 0.332362, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2418, 2389, 2295, 2394, 2393, 2391, 2445} \[ \frac{b p q (h i-g j) \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{h^2}+\frac{(h i-g j) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^2}+\frac{a j x}{h}+\frac{b j (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac{b j p q x}{h} \]
Antiderivative was successfully verified.
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Rule 2418
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rule 2445
Rubi steps
\begin{align*} \int \frac{(525+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx &=\operatorname{Subst}\left (\int \frac{(525+j x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{g+h 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{j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}+\frac{(525 h-g j) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h (g+h 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{j \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(525 h-g j) \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{a j x}{h}+\frac{(525 h-g j) \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^2}+\operatorname{Subst}\left (\frac{(b j) \int \log \left (c d^q (e+f x)^{p q}\right ) \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(b f (525 h-g j) p q) \int \frac{\log \left (\frac{f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{a j x}{h}+\frac{(525 h-g j) \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^2}+\operatorname{Subst}\left (\frac{(b j) \operatorname{Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(b (525 h-g j) 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 )}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{a j x}{h}-\frac{b j p q x}{h}+\frac{b j (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}+\frac{(525 h-g j) \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^2}+\frac{b (525 h-g j) p q \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^2}\\ \end{align*}
Mathematica [A] time = 0.119452, size = 120, normalized size = 0.93 \[ \frac{b p q (h i-g j) \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )+(h i-g j) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+a h j x+\frac{b h j (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-b h j p q x}{h^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.645, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( jx+i \right ) \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) }{hx+g}}\, 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*} a j{\left (\frac{x}{h} - \frac{g \log \left (h x + g\right )}{h^{2}}\right )} + \frac{a i \log \left (h x + g\right )}{h} + \int \frac{{\left (j \log \left (c\right ) + j \log \left (d^{q}\right )\right )} b x +{\left (i \log \left (c\right ) + i \log \left (d^{q}\right )\right )} b +{\left (b j x + b i\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h x + g}\,{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{a j x + a i +{\left (b j x + b i\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{h x + g}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right ) \left (i + j x\right )}{g + h x}\, dx \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{{\left (j x + i\right )}{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}}{h x + g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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