Optimal. Leaf size=209 \[ -\frac {6 b^2 f p^2 q^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)}-\frac {3 b f p q \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h (f g-e h)}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x) (f g-e h)}+\frac {6 b^3 f p^3 q^3 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)} \]
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Rubi [A] time = 0.37, antiderivative size = 209, 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 = {2397, 2396, 2433, 2374, 6589, 2445} \[ -\frac {6 b^2 f p^2 q^2 \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)}+\frac {6 b^3 f p^3 q^3 \text {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}-\frac {3 b f p q \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h (f g-e h)}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x) (f g-e h)} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2396
Rule 2397
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)^2} \, 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)^2} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(f g-e h) (g+h x)}-\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}{g+h x} \, dx}{f g-e h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(f g-e h) (g+h x)}-\frac {3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}+\operatorname {Subst}\left (\frac {\left (6 b^2 f^2 p^2 q^2\right ) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(f g-e h) (g+h x)}-\frac {3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}+\operatorname {Subst}\left (\frac {\left (6 b^2 f p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \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 (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(f g-e h) (g+h x)}-\frac {3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}-\frac {6 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}+\operatorname {Subst}\left (\frac {\left (6 b^3 f p^3 q^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(f g-e h) (g+h x)}-\frac {3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}-\frac {6 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}+\frac {6 b^3 f p^3 q^3 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}\\ \end {align*}
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Mathematica [B] time = 0.55, size = 444, normalized size = 2.12 \[ \frac {3 b^2 p^2 q^2 \left (\log (e+f x) \left (h (e+f x) \log (e+f x)-2 f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )-2 f (g+h x) \text {Li}_2\left (\frac {h (e+f x)}{e h-f g}\right )\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )-3 b p q (f g-e h) \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2+3 b f p q (g+h x) \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2-3 b f p q (g+h x) \log (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2-(f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^3+b^3 p^3 q^3 \left (6 f (g+h x) \text {Li}_3\left (\frac {h (e+f x)}{e h-f g}\right )-6 f (g+h x) \log (e+f x) \text {Li}_2\left (\frac {h (e+f x)}{e h-f g}\right )+\log ^2(e+f x) \left (h (e+f x) \log (e+f x)-3 f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )\right )}{h (g+h x) (f g-e h)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 3 \, a b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 3 \, a^{2} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a^{3}}{h^{2} x^{2} + 2 \, g h x + g^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{{\left (h x + g\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{3}}{\left (h x +g \right )^{2}}\, dx \]
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 \[ 3 \, a^{2} b f p q {\left (\frac {\log \left (f x + e\right )}{f g h - e h^{2}} - \frac {\log \left (h x + g\right )}{f g h - e h^{2}}\right )} - \frac {b^{3} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{3}}{h^{2} x + g h} - \frac {3 \, a^{2} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{h^{2} x + g h} - \frac {a^{3}}{h^{2} x + g h} + \int \frac {3 \, {\left (e h q^{2} \log \relax (d)^{2} + 2 \, e h q \log \relax (c) \log \relax (d) + e h \log \relax (c)^{2}\right )} a b^{2} + {\left (e h q^{3} \log \relax (d)^{3} + 3 \, e h q^{2} \log \relax (c) \log \relax (d)^{2} + 3 \, e h q \log \relax (c)^{2} \log \relax (d) + e h \log \relax (c)^{3}\right )} b^{3} + 3 \, {\left (a b^{2} e h + {\left (f g p q + e h q \log \relax (d) + e h \log \relax (c)\right )} b^{3} + {\left (a b^{2} f h + {\left (f h p q + f h q \log \relax (d) + f h \log \relax (c)\right )} b^{3}\right )} x\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} + {\left (3 \, {\left (f h q^{2} \log \relax (d)^{2} + 2 \, f h q \log \relax (c) \log \relax (d) + f h \log \relax (c)^{2}\right )} a b^{2} + {\left (f h q^{3} \log \relax (d)^{3} + 3 \, f h q^{2} \log \relax (c) \log \relax (d)^{2} + 3 \, f h q \log \relax (c)^{2} \log \relax (d) + f h \log \relax (c)^{3}\right )} b^{3}\right )} x + 3 \, {\left (2 \, {\left (e h q \log \relax (d) + e h \log \relax (c)\right )} a b^{2} + {\left (e h q^{2} \log \relax (d)^{2} + 2 \, e h q \log \relax (c) \log \relax (d) + e h \log \relax (c)^{2}\right )} b^{3} + {\left (2 \, {\left (f h q \log \relax (d) + f h \log \relax (c)\right )} a b^{2} + {\left (f h q^{2} \log \relax (d)^{2} + 2 \, f h q \log \relax (c) \log \relax (d) + f h \log \relax (c)^{2}\right )} b^{3}\right )} x\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{f h^{3} x^{3} + e g^{2} h + {\left (2 \, f g h^{2} + e h^{3}\right )} x^{2} + {\left (f g^{2} h + 2 \, e g h^{2}\right )} x}\,{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 = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3}{{\left (g+h\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3}}{\left (g + h x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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