Optimal. Leaf size=209 \[ \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(f g-e h) (g+h x)}-\frac {3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}-\frac {6 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}+\frac {6 b^3 f p^3 q^3 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)} \]
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Rubi [A]
time = 0.26, 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 = {2444, 2443,
2481, 2421, 6724, 2495} \begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2443
Rule 2444
Rule 2481
Rule 2495
Rule 6724
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 &=\text {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)}-\text {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)}+\text {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)}+\text {Subst}\left (\frac {\left (6 b^2 f p^2 q^2\right ) \text {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)}+\text {Subst}\left (\frac {\left (6 b^3 f p^3 q^3\right ) \text {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] Leaf count is larger than twice the leaf count of optimal. \(444\) vs. \(2(209)=418\).
time = 0.32, size = 444, normalized size = 2.12 \begin {gather*} \frac {-3 b (f g-e h) p q \log (e+f x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+3 b f p q (g+h x) \log (e+f x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-(f g-e h) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3-3 b f p q (g+h x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log (g+h x)+3 b^2 p^2 q^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \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)}{-f g+e h}\right )\right )+b^3 p^3 q^3 \left (\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 )-6 f (g+h x) \log (e+f x) \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )+6 f (g+h x) \text {Li}_3\left (\frac {h (e+f x)}{-f g+e h}\right )\right )}{h (f g-e h) (g+h x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{3}}{\left (h x +g \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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