Optimal. Leaf size=177 \[ -\frac {6 b^2 p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {3 b p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h}+\frac {6 b^3 p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h} \]
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Rubi [A] time = 0.41, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2396, 2433, 2374, 2383, 6589, 2445} \[ -\frac {6 b^2 p^2 q^2 \text {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {3 b p q \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac {6 b^3 p^3 q^3 \text {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h}+\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2383
Rule 2396
Rule 2433
Rule 2445
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx &=\operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx,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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}-\operatorname {Subst}\left (\frac {(3 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h},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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}-\operatorname {Subst}\left (\frac {(3 b p q) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \log \left (\frac {f \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h},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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\operatorname {Subst}\left (\frac {\left (6 b^2 p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h},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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}+\operatorname {Subst}\left (\frac {\left (6 b^3 p^3 q^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h},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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}+\frac {6 b^3 p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}\\ \end {align*}
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Mathematica [B] time = 0.29, size = 646, normalized size = 3.65 \[ \frac {a^3 \log (g+h x)+3 a^2 b \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )-3 a^2 b p q \log (e+f x) \log (g+h x)+3 a^2 b p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-6 b^2 p^2 q^2 \text {Li}_3\left (\frac {h (e+f x)}{e h-f g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+3 a b^2 \log (g+h x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-6 a b^2 p q \log (e+f x) \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )+6 a b^2 p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+3 a b^2 p^2 q^2 \log ^2(e+f x) \log (g+h x)-3 a b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+3 b p q \text {Li}_2\left (\frac {h (e+f x)}{e h-f g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+3 b^3 p^2 q^2 \log ^2(e+f x) \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )-3 b^3 p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+b^3 \log (g+h x) \log ^3\left (c \left (d (e+f x)^p\right )^q\right )-3 b^3 p q \log (e+f x) \log (g+h x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+3 b^3 p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right ) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+6 b^3 p^3 q^3 \text {Li}_4\left (\frac {h (e+f x)}{e h-f g}\right )-b^3 p^3 q^3 \log ^3(e+f x) \log (g+h x)+b^3 p^3 q^3 \log ^3(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 3 \, a b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 3 \, a^{2} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a^{3}}{h x + g}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{h x + g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{3}}{h x +g}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{3} \log \left (h x + g\right )}{h} + \int \frac {b^{3} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{3} + 3 \, {\left (q \log \relax (d) + \log \relax (c)\right )} a^{2} b + 3 \, {\left (q^{2} \log \relax (d)^{2} + 2 \, q \log \relax (c) \log \relax (d) + \log \relax (c)^{2}\right )} a b^{2} + {\left (q^{3} \log \relax (d)^{3} + 3 \, q^{2} \log \relax (c) \log \relax (d)^{2} + 3 \, q \log \relax (c)^{2} \log \relax (d) + \log \relax (c)^{3}\right )} b^{3} + 3 \, {\left ({\left (q \log \relax (d) + \log \relax (c)\right )} b^{3} + a b^{2}\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} + 3 \, {\left (2 \, {\left (q \log \relax (d) + \log \relax (c)\right )} a b^{2} + {\left (q^{2} \log \relax (d)^{2} + 2 \, q \log \relax (c) \log \relax (d) + \log \relax (c)^{2}\right )} b^{3} + a^{2} b\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h x + g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3}{g+h\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3}}{g + h x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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