Optimal. Leaf size=35 \[ \frac{(d+e f+e g x) \log \left (c (d+e (f+g x))^q\right )}{e g}-q x \]
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Rubi [A] time = 0.0160286, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2444, 2389, 2295} \[ \frac{(d+e f+e g x) \log \left (c (d+e (f+g x))^q\right )}{e g}-q x \]
Antiderivative was successfully verified.
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Rule 2444
Rule 2389
Rule 2295
Rubi steps
\begin{align*} \int \log \left (c (d+e (f+g x))^q\right ) \, dx &=\int \log \left (c (d+e f+e g x)^q\right ) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \log \left (c x^q\right ) \, dx,x,d+e f+e g x\right )}{e g}\\ &=-q x+\frac{(d+e f+e g x) \log \left (c (d+e (f+g x))^q\right )}{e g}\\ \end{align*}
Mathematica [A] time = 0.031426, size = 47, normalized size = 1.34 \[ \frac{(f+g x) \log \left (c (d+e (f+g x))^q\right )}{g}+\frac{d q \log (d+e f+e g x)}{e g}-q x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 57, normalized size = 1.6 \begin{align*} \ln \left ( c \left ( egx+fe+d \right ) ^{q} \right ) x-qx+{\frac{q\ln \left ( egx+fe+d \right ) f}{g}}+{\frac{q\ln \left ( egx+fe+d \right ) d}{eg}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01673, size = 73, normalized size = 2.09 \begin{align*} -e g q{\left (\frac{x}{e g} - \frac{{\left (e f + d\right )} \log \left (e g x + e f + d\right )}{e^{2} g^{2}}\right )} + x \log \left ({\left ({\left (g x + f\right )} e + d\right )}^{q} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50005, size = 108, normalized size = 3.09 \begin{align*} -\frac{e g q x - e g x \log \left (c\right ) -{\left (e g q x +{\left (e f + d\right )} q\right )} \log \left (e g x + e f + d\right )}{e g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.51682, size = 85, normalized size = 2.43 \begin{align*} \begin{cases} x \log{\left (c d^{q} \right )} & \text{for}\: e = 0 \wedge g = 0 \\x \log{\left (c \left (d + e f\right )^{q} \right )} & \text{for}\: g = 0 \\x \log{\left (c d^{q} \right )} & \text{for}\: e = 0 \\\frac{d q \log{\left (d + e f + e g x \right )}}{e g} + \frac{f q \log{\left (d + e f + e g x \right )}}{g} + q x \log{\left (d + e f + e g x \right )} - q x + x \log{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17378, size = 93, normalized size = 2.66 \begin{align*} \frac{{\left (g x e + f e + d\right )} q e^{\left (-1\right )} \log \left (g x e + f e + d\right )}{g} - \frac{{\left (g x e + f e + d\right )} q e^{\left (-1\right )}}{g} + \frac{{\left (g x e + f e + d\right )} e^{\left (-1\right )} \log \left (c\right )}{g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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