Optimal. Leaf size=69 \[ -\frac{2 b n (f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m^2}+\frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )^2}{f m}+\frac{2 b^2 n^2 (f x)^m}{f m^3} \]
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Rubi [A] time = 0.0495456, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2305, 2304} \[ -\frac{2 b n (f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m^2}+\frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )^2}{f m}+\frac{2 b^2 n^2 (f x)^m}{f m^3} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )^2}{f m}-\frac{(2 b n) \int (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \, dx}{m}\\ &=\frac{2 b^2 n^2 (f x)^m}{f m^3}-\frac{2 b n (f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m^2}+\frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )^2}{f m}\\ \end{align*}
Mathematica [A] time = 0.0221567, size = 67, normalized size = 0.97 \[ \frac{(f x)^m \left (a^2 m^2+2 b m (a m-b n) \log \left (c x^n\right )-2 a b m n+b^2 m^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )}{f m^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.148, size = 1008, normalized size = 14.6 \begin{align*} \text{result too large to display} \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 [A] time = 1.36628, size = 294, normalized size = 4.26 \begin{align*} \frac{{\left (b^{2} m^{2} n^{2} x \log \left (x\right )^{2} + b^{2} m^{2} x \log \left (c\right )^{2} + 2 \,{\left (a b m^{2} - b^{2} m n\right )} x \log \left (c\right ) +{\left (a^{2} m^{2} - 2 \, a b m n + 2 \, b^{2} n^{2}\right )} x + 2 \,{\left (b^{2} m^{2} n x \log \left (c\right ) +{\left (a b m^{2} n - b^{2} m n^{2}\right )} x\right )} \log \left (x\right )\right )} e^{\left ({\left (m - 1\right )} \log \left (f\right ) +{\left (m - 1\right )} \log \left (x\right )\right )}}{m^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.40299, size = 267, normalized size = 3.87 \begin{align*} \frac{b^{2} f^{m} n^{2} x^{m} \log \left (x\right )^{2}}{f m} + \frac{2 \, b^{2} f^{m} n x^{m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac{b^{2} f^{m} x^{m} \log \left (c\right )^{2}}{f m} + \frac{2 \, a b f^{m} n x^{m} \log \left (x\right )}{f m} - \frac{2 \, b^{2} f^{m} n^{2} x^{m} \log \left (x\right )}{f m^{2}} + \frac{2 \, a b f^{m} x^{m} \log \left (c\right )}{f m} - \frac{2 \, b^{2} f^{m} n x^{m} \log \left (c\right )}{f m^{2}} + \frac{a^{2} f^{m} x^{m}}{f m} - \frac{2 \, a b f^{m} n x^{m}}{f m^{2}} + \frac{2 \, b^{2} f^{m} n^{2} x^{m}}{f m^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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