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 = 0.09, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2295
Rule 2296
Rule 2389
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*}
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Mathematica [A] time = 0.01, 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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fricas [B] time = 0.44, size = 231, normalized size = 2.96 \[ \frac {b^{2} f n^{2} x \log \relax (d)^{2} + b^{2} f x \log \relax (c)^{2} + {\left (b^{2} f m^{2} n^{2} x + b^{2} e m^{2} n^{2}\right )} \log \left (f x + e\right )^{2} - 2 \, {\left (b^{2} f m n - a b f\right )} x \log \relax (c) + {\left (2 \, b^{2} f m^{2} n^{2} - 2 \, a b f m n + a^{2} f\right )} x - 2 \, {\left (b^{2} e m^{2} n^{2} - a b e m n + {\left (b^{2} f m^{2} n^{2} - a b f m n\right )} x - {\left (b^{2} f m n x + b^{2} e m n\right )} \log \relax (c) - {\left (b^{2} f m n^{2} x + b^{2} e m n^{2}\right )} \log \relax (d)\right )} \log \left (f x + e\right ) + 2 \, {\left (b^{2} f n x \log \relax (c) - {\left (b^{2} f m n^{2} - a b f n\right )} x\right )} \log \relax (d)}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 303, normalized size = 3.88 \[ \frac {{\left (f x + e\right )} b^{2} m^{2} n^{2} \log \left (f x + e\right )^{2}}{f} - \frac {2 \, {\left (f x + e\right )} b^{2} m^{2} n^{2} \log \left (f x + e\right )}{f} + \frac {2 \, {\left (f x + e\right )} b^{2} m n^{2} \log \left (f x + e\right ) \log \relax (d)}{f} + \frac {2 \, {\left (f x + e\right )} b^{2} m^{2} n^{2}}{f} + \frac {2 \, {\left (f x + e\right )} b^{2} m n \log \left (f x + e\right ) \log \relax (c)}{f} - \frac {2 \, {\left (f x + e\right )} b^{2} m n^{2} \log \relax (d)}{f} + \frac {{\left (f x + e\right )} b^{2} n^{2} \log \relax (d)^{2}}{f} + \frac {2 \, {\left (f x + e\right )} a b m n \log \left (f x + e\right )}{f} - \frac {2 \, {\left (f x + e\right )} b^{2} m n \log \relax (c)}{f} + \frac {2 \, {\left (f x + e\right )} b^{2} n \log \relax (c) \log \relax (d)}{f} - \frac {2 \, {\left (f x + e\right )} a b m n}{f} + \frac {{\left (f x + e\right )} b^{2} \log \relax (c)^{2}}{f} + \frac {2 \, {\left (f x + e\right )} a b n \log \relax (d)}{f} + \frac {2 \, {\left (f x + e\right )} a b \log \relax (c)}{f} + \frac {{\left (f x + e\right )} a^{2}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )+a \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 148, normalized size = 1.90 \[ -2 \, a b f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + b^{2} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{2} + 2 \, a b x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) - {\left (2 \, f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + \frac {{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} m^{2} n^{2}}{f}\right )} b^{2} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 111, normalized size = 1.42 \[ {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^2\,\left (b^2\,x+\frac {b^2\,e}{f}\right )+x\,\left (a^2-2\,a\,b\,m\,n+2\,b^2\,m^2\,n^2\right )-\frac {\ln \left (e+f\,x\right )\,\left (2\,b^2\,e\,m^2\,n^2-2\,a\,b\,e\,m\,n\right )}{f}+2\,b\,x\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\,\left (a-b\,m\,n\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.59, size = 343, normalized size = 4.40 \[ \begin {cases} a^{2} x + \frac {2 a b e m n \log {\left (e + f x \right )}}{f} + 2 a b m n x \log {\left (e + f x \right )} - 2 a b m n x + 2 a b n x \log {\relax (d )} + 2 a b x \log {\relax (c )} + \frac {b^{2} e m^{2} n^{2} \log {\left (e + f x \right )}^{2}}{f} - \frac {2 b^{2} e m^{2} n^{2} \log {\left (e + f x \right )}}{f} + \frac {2 b^{2} e m n^{2} \log {\relax (d )} \log {\left (e + f x \right )}}{f} + \frac {2 b^{2} e m n \log {\relax (c )} \log {\left (e + f x \right )}}{f} + b^{2} m^{2} n^{2} x \log {\left (e + f x \right )}^{2} - 2 b^{2} m^{2} n^{2} x \log {\left (e + f x \right )} + 2 b^{2} m^{2} n^{2} x + 2 b^{2} m n^{2} x \log {\relax (d )} \log {\left (e + f x \right )} - 2 b^{2} m n^{2} x \log {\relax (d )} + 2 b^{2} m n x \log {\relax (c )} \log {\left (e + f x \right )} - 2 b^{2} m n x \log {\relax (c )} + b^{2} n^{2} x \log {\relax (d )}^{2} + 2 b^{2} n x \log {\relax (c )} \log {\relax (d )} + b^{2} x \log {\relax (c )}^{2} & \text {for}\: f \neq 0 \\x \left (a + b \log {\left (c \left (d e^{m}\right )^{n} \right )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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