Optimal. Leaf size=119 \[ -\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 h (g+h x)^2}+\frac{b f^2 p q \log (e+f x)}{2 h (f g-e h)^2}-\frac{b f^2 p q \log (g+h x)}{2 h (f g-e h)^2}+\frac{b f p q}{2 h (g+h x) (f g-e h)} \]
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Rubi [A] time = 0.129398, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2395, 44, 2445} \[ -\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 h (g+h x)^2}+\frac{b f^2 p q \log (e+f x)}{2 h (f g-e h)^2}-\frac{b f^2 p q \log (g+h x)}{2 h (f g-e h)^2}+\frac{b f p q}{2 h (g+h x) (f g-e h)} \]
Antiderivative was successfully verified.
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Rule 2395
Rule 44
Rule 2445
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^3} \, dx &=\operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x)^3} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 h (g+h x)^2}+\operatorname{Subst}\left (\frac{(b f p q) \int \frac{1}{(e+f x) (g+h x)^2} \, dx}{2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 h (g+h x)^2}+\operatorname{Subst}\left (\frac{(b f p q) \int \left (\frac{f^2}{(f g-e h)^2 (e+f x)}-\frac{h}{(f g-e h) (g+h x)^2}-\frac{f h}{(f g-e h)^2 (g+h x)}\right ) \, dx}{2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{b f p q}{2 h (f g-e h) (g+h x)}+\frac{b f^2 p q \log (e+f x)}{2 h (f g-e h)^2}-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 h (g+h x)^2}-\frac{b f^2 p q \log (g+h x)}{2 h (f g-e h)^2}\\ \end{align*}
Mathematica [A] time = 0.147197, size = 88, normalized size = 0.74 \[ -\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-\frac{b f p q (g+h x) (f (g+h x) \log (e+f x)-e h-f (g+h x) \log (g+h x)+f g)}{(f g-e h)^2}}{2 h (g+h x)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.664, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) }{ \left ( hx+g \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06033, size = 232, normalized size = 1.95 \begin{align*} \frac{1}{2} \, b f p q{\left (\frac{f \log \left (f x + e\right )}{f^{2} g^{2} h - 2 \, e f g h^{2} + e^{2} h^{3}} - \frac{f \log \left (h x + g\right )}{f^{2} g^{2} h - 2 \, e f g h^{2} + e^{2} h^{3}} + \frac{1}{f g^{2} h - e g h^{2} +{\left (f g h^{2} - e h^{3}\right )} x}\right )} - \frac{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{2 \,{\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} - \frac{a}{2 \,{\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.23678, size = 667, normalized size = 5.61 \begin{align*} -\frac{a f^{2} g^{2} - 2 \, a e f g h + a e^{2} h^{2} -{\left (b f^{2} g h - b e f h^{2}\right )} p q x -{\left (b f^{2} g^{2} - b e f g h\right )} p q +{\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} q \log \left (d\right ) -{\left (b f^{2} h^{2} p q x^{2} + 2 \, b f^{2} g h p q x +{\left (2 \, b e f g h - b e^{2} h^{2}\right )} p q\right )} \log \left (f x + e\right ) +{\left (b f^{2} h^{2} p q x^{2} + 2 \, b f^{2} g h p q x + b f^{2} g^{2} p q\right )} \log \left (h x + g\right ) +{\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} \log \left (c\right )}{2 \,{\left (f^{2} g^{4} h - 2 \, e f g^{3} h^{2} + e^{2} g^{2} h^{3} +{\left (f^{2} g^{2} h^{3} - 2 \, e f g h^{4} + e^{2} h^{5}\right )} x^{2} + 2 \,{\left (f^{2} g^{3} h^{2} - 2 \, e f g^{2} h^{3} + e^{2} g h^{4}\right )} 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 [B] time = 1.1509, size = 485, normalized size = 4.08 \begin{align*} \frac{b f^{2} h^{2} p q x^{2} \log \left (f x + e\right ) - b f^{2} h^{2} p q x^{2} \log \left (h x + g\right ) + 2 \, b f^{2} g h p q x \log \left (f x + e\right ) - 2 \, b f^{2} g h p q x \log \left (h x + g\right ) + b f^{2} g h p q x - b f h^{2} p q x e + 2 \, b f g h p q e \log \left (f x + e\right ) - b f^{2} g^{2} p q \log \left (h x + g\right ) + b f^{2} g^{2} p q - b f g h p q e - b h^{2} p q e^{2} \log \left (f x + e\right ) - b f^{2} g^{2} q \log \left (d\right ) + 2 \, b f g h q e \log \left (d\right ) - b f^{2} g^{2} \log \left (c\right ) + 2 \, b f g h e \log \left (c\right ) - b h^{2} q e^{2} \log \left (d\right ) - a f^{2} g^{2} + 2 \, a f g h e - b h^{2} e^{2} \log \left (c\right ) - a h^{2} e^{2}}{2 \,{\left (f^{2} g^{2} h^{3} x^{2} - 2 \, f g h^{4} x^{2} e + 2 \, f^{2} g^{3} h^{2} x + h^{5} x^{2} e^{2} - 4 \, f g^{2} h^{3} x e + f^{2} g^{4} h + 2 \, g h^{4} x e^{2} - 2 \, f g^{3} h^{2} e + g^{2} h^{3} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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