Optimal. Leaf size=78 \[ \frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-2 a b m n x-\frac{2 b^2 m n (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+2 b^2 m^2 n^2 x \]
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Rubi [A] time = 0.0944507, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2389, 2296, 2295, 2445} \[ \frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-2 a b m n x-\frac{2 b^2 m n (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+2 b^2 m^2 n^2 x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2295
Rule 2445
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2 \, dx &=\operatorname{Subst}\left (\int \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^2 \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right )^2 \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\operatorname{Subst}\left (\frac{(2 b m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right ) \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-2 a b m n x+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\operatorname{Subst}\left (\frac{\left (2 b^2 m n\right ) \operatorname{Subst}\left (\int \log \left (c d^n x^{m n}\right ) \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-2 a b m n x+2 b^2 m^2 n^2 x-\frac{2 b^2 m n (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}\\ \end{align*}
Mathematica [A] time = 0.0096565, size = 69, normalized size = 0.88 \[ \frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-2 b m n \left (a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-b m n x\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10344, size = 200, normalized size = 2.56 \begin{align*} -2 \, a b f m n{\left (\frac{x}{f} - \frac{e \log \left (f x + e\right )}{f^{2}}\right )} + b^{2} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{2} + 2 \, a b x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) -{\left (2 \, f m n{\left (\frac{x}{f} - \frac{e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + \frac{{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} m^{2} n^{2}}{f}\right )} b^{2} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.29019, size = 516, normalized size = 6.62 \begin{align*} \frac{b^{2} f n^{2} x \log \left (d\right )^{2} + b^{2} f x \log \left (c\right )^{2} +{\left (b^{2} f m^{2} n^{2} x + b^{2} e m^{2} n^{2}\right )} \log \left (f x + e\right )^{2} - 2 \,{\left (b^{2} f m n - a b f\right )} x \log \left (c\right ) +{\left (2 \, b^{2} f m^{2} n^{2} - 2 \, a b f m n + a^{2} f\right )} x - 2 \,{\left (b^{2} e m^{2} n^{2} - a b e m n +{\left (b^{2} f m^{2} n^{2} - a b f m n\right )} x -{\left (b^{2} f m n x + b^{2} e m n\right )} \log \left (c\right ) -{\left (b^{2} f m n^{2} x + b^{2} e m n^{2}\right )} \log \left (d\right )\right )} \log \left (f x + e\right ) + 2 \,{\left (b^{2} f n x \log \left (c\right ) -{\left (b^{2} f m n^{2} - a b f n\right )} x\right )} \log \left (d\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.27153, size = 343, normalized size = 4.4 \begin{align*} \begin{cases} a^{2} x + \frac{2 a b e m n \log{\left (e + f x \right )}}{f} + 2 a b m n x \log{\left (e + f x \right )} - 2 a b m n x + 2 a b n x \log{\left (d \right )} + 2 a b x \log{\left (c \right )} + \frac{b^{2} e m^{2} n^{2} \log{\left (e + f x \right )}^{2}}{f} - \frac{2 b^{2} e m^{2} n^{2} \log{\left (e + f x \right )}}{f} + \frac{2 b^{2} e m n^{2} \log{\left (d \right )} \log{\left (e + f x \right )}}{f} + \frac{2 b^{2} e m n \log{\left (c \right )} \log{\left (e + f x \right )}}{f} + b^{2} m^{2} n^{2} x \log{\left (e + f x \right )}^{2} - 2 b^{2} m^{2} n^{2} x \log{\left (e + f x \right )} + 2 b^{2} m^{2} n^{2} x + 2 b^{2} m n^{2} x \log{\left (d \right )} \log{\left (e + f x \right )} - 2 b^{2} m n^{2} x \log{\left (d \right )} + 2 b^{2} m n x \log{\left (c \right )} \log{\left (e + f x \right )} - 2 b^{2} m n x \log{\left (c \right )} + b^{2} n^{2} x \log{\left (d \right )}^{2} + 2 b^{2} n x \log{\left (c \right )} \log{\left (d \right )} + b^{2} x \log{\left (c \right )}^{2} & \text{for}\: f \neq 0 \\x \left (a + b \log{\left (c \left (d e^{m}\right )^{n} \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22781, size = 409, normalized size = 5.24 \begin{align*} \frac{{\left (f x + e\right )} b^{2} m^{2} n^{2} \log \left (f x + e\right )^{2}}{f} - \frac{2 \,{\left (f x + e\right )} b^{2} m^{2} n^{2} \log \left (f x + e\right )}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} m n^{2} \log \left (f x + e\right ) \log \left (d\right )}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} m^{2} n^{2}}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} m n \log \left (f x + e\right ) \log \left (c\right )}{f} - \frac{2 \,{\left (f x + e\right )} b^{2} m n^{2} \log \left (d\right )}{f} + \frac{{\left (f x + e\right )} b^{2} n^{2} \log \left (d\right )^{2}}{f} + \frac{2 \,{\left (f x + e\right )} a b m n \log \left (f x + e\right )}{f} - \frac{2 \,{\left (f x + e\right )} b^{2} m n \log \left (c\right )}{f} + \frac{2 \,{\left (f x + e\right )} b^{2} n \log \left (c\right ) \log \left (d\right )}{f} - \frac{2 \,{\left (f x + e\right )} a b m n}{f} + \frac{{\left (f x + e\right )} b^{2} \log \left (c\right )^{2}}{f} + \frac{2 \,{\left (f x + e\right )} a b n \log \left (d\right )}{f} + \frac{2 \,{\left (f x + e\right )} a b \log \left (c\right )}{f} + \frac{{\left (f x + e\right )} a^{2}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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