Optimal. Leaf size=59 \[ \frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^2}\right )^q\right )}{g}+\frac{2 \sqrt{e} q \tan ^{-1}\left (\frac{\sqrt{d} (f+g x)}{\sqrt{e}}\right )}{\sqrt{d} g} \]
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Rubi [A] time = 0.0356361, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2483, 2448, 263, 205} \[ \frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^2}\right )^q\right )}{g}+\frac{2 \sqrt{e} q \tan ^{-1}\left (\frac{\sqrt{d} (f+g x)}{\sqrt{e}}\right )}{\sqrt{d} g} \]
Antiderivative was successfully verified.
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Rule 2483
Rule 2448
Rule 263
Rule 205
Rubi steps
\begin{align*} \int \log \left (c \left (d+\frac{e}{(f+g x)^2}\right )^q\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x^2}\right )^q\right ) \, dx,x,f+g x\right )}{g}\\ &=\frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^2}\right )^q\right )}{g}+\frac{(2 e q) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^2} \, dx,x,f+g x\right )}{g}\\ &=\frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^2}\right )^q\right )}{g}+\frac{(2 e q) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,f+g x\right )}{g}\\ &=\frac{2 \sqrt{e} q \tan ^{-1}\left (\frac{\sqrt{d} (f+g x)}{\sqrt{e}}\right )}{\sqrt{d} g}+\frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^2}\right )^q\right )}{g}\\ \end{align*}
Mathematica [A] time = 0.0468646, size = 61, normalized size = 1.03 \[ \frac{(f+g x) \log \left (c \left (d+\frac{e}{(f+g x)^2}\right )^q\right )}{g}-\frac{2 \sqrt{e} q \tan ^{-1}\left (\frac{\sqrt{e}}{\sqrt{d} (f+g x)}\right )}{\sqrt{d} g} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.133, size = 115, normalized size = 2. \begin{align*} \ln \left ( c \left ({\frac{d{g}^{2}{x}^{2}+2\,dfgx+d{f}^{2}+e}{ \left ( gx+f \right ) ^{2}}} \right ) ^{q} \right ) x-2\,{\frac{qf\ln \left ( gx+f \right ) }{g}}+{\frac{qf\ln \left ( d{g}^{2}{x}^{2}+2\,dfgx+d{f}^{2}+e \right ) }{g}}+2\,{\frac{eq}{g\sqrt{de}}\arctan \left ( 1/2\,{\frac{2\,d{g}^{2}x+2\,dfg}{g\sqrt{de}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7431, size = 660, normalized size = 11.19 \begin{align*} \left [\frac{g q x \log \left (\frac{d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e}{g^{2} x^{2} + 2 \, f g x + f^{2}}\right ) + f q \log \left (d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e\right ) - 2 \, f q \log \left (g x + f\right ) + g x \log \left (c\right ) + q \sqrt{-\frac{e}{d}} \log \left (\frac{d g^{2} x^{2} + 2 \, d f g x + d f^{2} + 2 \,{\left (d g x + d f\right )} \sqrt{-\frac{e}{d}} - e}{d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e}\right )}{g}, \frac{g q x \log \left (\frac{d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e}{g^{2} x^{2} + 2 \, f g x + f^{2}}\right ) + f q \log \left (d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e\right ) - 2 \, f q \log \left (g x + f\right ) + g x \log \left (c\right ) + 2 \, q \sqrt{\frac{e}{d}} \arctan \left (\frac{{\left (d g x + d f\right )} \sqrt{\frac{e}{d}}}{e}\right )}{g}\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 = 4.57269, size = 185, normalized size = 3.14 \begin{align*} d g^{4} q{\left (\frac{f e^{\left (-1\right )} \log \left (d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e\right )}{d g^{5}} - \frac{2 \, f e^{\left (-1\right )} \log \left ({\left | g x + f \right |}\right )}{d g^{5}} + \frac{2 \, \arctan \left (\frac{{\left (d g x + d f\right )} e^{\left (-\frac{1}{2}\right )}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{d^{\frac{3}{2}} g^{5}}\right )} e + q x \log \left (d g^{2} x^{2} + 2 \, d f g x + d f^{2} + e\right ) - q x \log \left (g^{2} x^{2} + 2 \, f g x + f^{2}\right ) + x \log \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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