Optimal. Leaf size=250 \[ \frac{b f^2 \text{PolyLog}\left (2,-\frac{f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}-\frac{f^2 \log \left (\frac{f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac{f i (e+f x) (a+b \log (c (e+f x)))}{d (h+i x) (f h-e i)^3}+\frac{a+b \log (c (e+f x))}{2 d (h+i x)^2 (f h-e i)}-\frac{b f^2 \log (e+f x)}{2 d (f h-e i)^3}+\frac{3 b f^2 \log (h+i x)}{2 d (f h-e i)^3}-\frac{b f}{2 d (h+i x) (f h-e i)^2} \]
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Rubi [A] time = 0.571978, antiderivative size = 282, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367, Rules used = {2411, 12, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac{b f^2 \text{PolyLog}\left (2,-\frac{i (e+f x)}{f h-e i}\right )}{d (f h-e i)^3}+\frac{f^2 (a+b \log (c (e+f x)))^2}{2 b d (f h-e i)^3}-\frac{f^2 \log \left (\frac{f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac{f i (e+f x) (a+b \log (c (e+f x)))}{d (h+i x) (f h-e i)^3}+\frac{a+b \log (c (e+f x))}{2 d (h+i x)^2 (f h-e i)}-\frac{b f^2 \log (e+f x)}{2 d (f h-e i)^3}+\frac{3 b f^2 \log (h+i x)}{2 d (f h-e i)^3}-\frac{b f}{2 d (h+i x) (f h-e i)^2} \]
Antiderivative was successfully verified.
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Rule 2411
Rule 12
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 2319
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \log (c (e+f x))}{(h+182 x)^3 (d e+d f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \log (c x)}{d x \left (\frac{-182 e+f h}{f}+\frac{182 x}{f}\right )^3} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{a+b \log (c x)}{x \left (\frac{-182 e+f h}{f}+\frac{182 x}{f}\right )^3} \, dx,x,e+f x\right )}{d f}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \log (c x)}{x \left (\frac{-182 e+f h}{f}+\frac{182 x}{f}\right )^2} \, dx,x,e+f x\right )}{d (182 e-f h)}+\frac{182 \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{\left (\frac{-182 e+f h}{f}+\frac{182 x}{f}\right )^3} \, dx,x,e+f x\right )}{d f (182 e-f h)}\\ &=-\frac{a+b \log (c (e+f x))}{2 d (182 e-f h) (h+182 x)^2}-\frac{182 \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{\left (\frac{-182 e+f h}{f}+\frac{182 x}{f}\right )^2} \, dx,x,e+f x\right )}{d (182 e-f h)^2}+\frac{f \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{x \left (\frac{-182 e+f h}{f}+\frac{182 x}{f}\right )} \, dx,x,e+f x\right )}{d (182 e-f h)^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{-182 e+f h}{f}+\frac{182 x}{f}\right )^2} \, dx,x,e+f x\right )}{2 d (182 e-f h)}\\ &=-\frac{a+b \log (c (e+f x))}{2 d (182 e-f h) (h+182 x)^2}+\frac{182 f (e+f x) (a+b \log (c (e+f x)))}{d (182 e-f h)^3 (h+182 x)}+\frac{(182 f) \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{\frac{-182 e+f h}{f}+\frac{182 x}{f}} \, dx,x,e+f x\right )}{d (182 e-f h)^3}-\frac{(182 b f) \operatorname{Subst}\left (\int \frac{1}{\frac{-182 e+f h}{f}+\frac{182 x}{f}} \, dx,x,e+f x\right )}{d (182 e-f h)^3}-\frac{f^2 \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{x} \, dx,x,e+f x\right )}{d (182 e-f h)^3}+\frac{b \operatorname{Subst}\left (\int \left (\frac{182 f^2}{(182 e-f h) (182 e-f h-182 x)^2}+\frac{182 f^2}{(182 e-f h)^2 (182 e-f h-182 x)}+\frac{f^2}{(182 e-f h)^2 x}\right ) \, dx,x,e+f x\right )}{2 d (182 e-f h)}\\ &=-\frac{b f}{2 d (182 e-f h)^2 (h+182 x)}-\frac{3 b f^2 \log (h+182 x)}{2 d (182 e-f h)^3}+\frac{b f^2 \log (e+f x)}{2 d (182 e-f h)^3}-\frac{a+b \log (c (e+f x))}{2 d (182 e-f h) (h+182 x)^2}+\frac{182 f (e+f x) (a+b \log (c (e+f x)))}{d (182 e-f h)^3 (h+182 x)}+\frac{f^2 \log \left (-\frac{f (h+182 x)}{182 e-f h}\right ) (a+b \log (c (e+f x)))}{d (182 e-f h)^3}-\frac{f^2 (a+b \log (c (e+f x)))^2}{2 b d (182 e-f h)^3}-\frac{\left (b f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{182 x}{-182 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (182 e-f h)^3}\\ &=-\frac{b f}{2 d (182 e-f h)^2 (h+182 x)}-\frac{3 b f^2 \log (h+182 x)}{2 d (182 e-f h)^3}+\frac{b f^2 \log (e+f x)}{2 d (182 e-f h)^3}-\frac{a+b \log (c (e+f x))}{2 d (182 e-f h) (h+182 x)^2}+\frac{182 f (e+f x) (a+b \log (c (e+f x)))}{d (182 e-f h)^3 (h+182 x)}+\frac{f^2 \log \left (-\frac{f (h+182 x)}{182 e-f h}\right ) (a+b \log (c (e+f x)))}{d (182 e-f h)^3}-\frac{f^2 (a+b \log (c (e+f x)))^2}{2 b d (182 e-f h)^3}+\frac{b f^2 \text{Li}_2\left (\frac{182 (e+f x)}{182 e-f h}\right )}{d (182 e-f h)^3}\\ \end{align*}
Mathematica [A] time = 0.228339, size = 226, normalized size = 0.9 \[ \frac{-2 b f^2 \text{PolyLog}\left (2,\frac{i (e+f x)}{e i-f h}\right )-2 f^2 \log \left (\frac{f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))+\frac{f^2 (a+b \log (c (e+f x)))^2}{b}+\frac{2 f (f h-e i) (a+b \log (c (e+f x)))}{h+i x}+\frac{(f h-e i)^2 (a+b \log (c (e+f x)))}{(h+i x)^2}-2 b f^2 (\log (e+f x)-\log (h+i x))-\frac{b f (f (h+i x) \log (e+f x)-e i-f (h+i x) \log (h+i x)+f h)}{h+i x}}{2 d (f h-e i)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.476, size = 656, normalized size = 2.6 \begin{align*} -{\frac{a{f}^{2}\ln \left ( cfx+ce \right ) }{d \left ( ei-fh \right ) ^{3}}}+{\frac{a{f}^{2}\ln \left ( -cei+hcf+ \left ( cfx+ce \right ) i \right ) }{d \left ( ei-fh \right ) ^{3}}}-{\frac{{f}^{2}{c}^{2}a}{2\,d \left ( ei-fh \right ) \left ( cfix+hcf \right ) ^{2}}}+{\frac{c{f}^{2}a}{d \left ( ei-fh \right ) ^{2} \left ( cfix+hcf \right ) }}-{\frac{b{f}^{2} \left ( \ln \left ( cfx+ce \right ) \right ) ^{2}}{2\,d \left ( ei-fh \right ) ^{3}}}-{\frac{3\,b{f}^{2}\ln \left ( -cei+hcf+ \left ( cfx+ce \right ) i \right ) }{2\,d \left ( ei-fh \right ) ^{3}}}-{\frac{c{f}^{2}bie}{2\,d \left ( ei-fh \right ) ^{3} \left ( cfix+hcf \right ) }}+{\frac{c{f}^{3}bh}{2\,d \left ( ei-fh \right ) ^{3} \left ( cfix+hcf \right ) }}-{\frac{{f}^{2}{c}^{2}b{i}^{2}\ln \left ( cfx+ce \right ){e}^{2}}{2\,d \left ( ei-fh \right ) ^{3} \left ( cfix+hcf \right ) ^{2}}}+{\frac{{c}^{2}{f}^{4}bi\ln \left ( cfx+ce \right ) hx}{d \left ( ei-fh \right ) ^{3} \left ( cfix+hcf \right ) ^{2}}}+{\frac{{c}^{2}{f}^{3}bi\ln \left ( cfx+ce \right ) he}{d \left ( ei-fh \right ) ^{3} \left ( cfix+hcf \right ) ^{2}}}+{\frac{{c}^{2}{f}^{4}b{i}^{2}\ln \left ( cfx+ce \right ){x}^{2}}{2\,d \left ( ei-fh \right ) ^{3} \left ( cfix+hcf \right ) ^{2}}}+{\frac{b{f}^{2}}{d \left ( ei-fh \right ) ^{3}}{\it dilog} \left ({\frac{-cei+hcf+ \left ( cfx+ce \right ) i}{-cei+hcf}} \right ) }+{\frac{b{f}^{2}\ln \left ( cfx+ce \right ) }{d \left ( ei-fh \right ) ^{3}}\ln \left ({\frac{-cei+hcf+ \left ( cfx+ce \right ) i}{-cei+hcf}} \right ) }+{\frac{c{f}^{3}bi\ln \left ( cfx+ce \right ) x}{d \left ( ei-fh \right ) ^{3} \left ( cfix+hcf \right ) }}+{\frac{c{f}^{2}bi\ln \left ( cfx+ce \right ) e}{d \left ( ei-fh \right ) ^{3} \left ( cfix+hcf \right ) }} \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 \, f^{2} \log \left (f x + e\right )}{d f^{3} h^{3} - 3 \, d e f^{2} h^{2} i + 3 \, d e^{2} f h i^{2} - d e^{3} i^{3}} - \frac{2 \, f^{2} \log \left (i x + h\right )}{d f^{3} h^{3} - 3 \, d e f^{2} h^{2} i + 3 \, d e^{2} f h i^{2} - d e^{3} i^{3}} + \frac{2 \, f i x + 3 \, f h - e i}{d f^{2} h^{4} - 2 \, d e f h^{3} i + d e^{2} h^{2} i^{2} +{\left (d f^{2} h^{2} i^{2} - 2 \, d e f h i^{3} + d e^{2} i^{4}\right )} x^{2} + 2 \,{\left (d f^{2} h^{3} i - 2 \, d e f h^{2} i^{2} + d e^{2} h i^{3}\right )} x}\right )} a + b \int \frac{\log \left (f x + e\right ) + \log \left (c\right )}{d f i^{3} x^{4} + d e h^{3} +{\left (3 \, f h i^{2} + e i^{3}\right )} d x^{3} + 3 \,{\left (f h^{2} i + e h i^{2}\right )} d x^{2} +{\left (f h^{3} + 3 \, e h^{2} i\right )} d 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 (c f x + c e\right ) + a}{d f i^{3} x^{4} + d e h^{3} +{\left (3 \, d f h i^{2} + d e i^{3}\right )} x^{3} + 3 \,{\left (d f h^{2} i + d e h i^{2}\right )} x^{2} +{\left (d f h^{3} + 3 \, d e h^{2} i\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 (f x + e\right )} c\right ) + a}{{\left (d f x + d e\right )}{\left (i x + h\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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