Integrand size = 20, antiderivative size = 20 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {(f x)^{1+m} \log ^2\left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {4 e p \text {Int}\left (\frac {(f x)^{2+m} \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{f^2 (1+m)} \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \log ^2\left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {(4 e p) \int \frac {(f x)^{2+m} \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{f^2 (1+m)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(466\) vs. \(2(75)=150\).
Time = 0.53 (sec) , antiderivative size = 466, normalized size of antiderivative = 23.30 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {(f x)^m \left (4 p^2 x \left (\frac {2 e x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )}{d (3+m)}-\log \left (d+e x^2\right )\right )+(1+m) p^2 x \log ^2\left (d+e x^2\right )+\frac {4 d (1+m) p^2 \left (\frac {e x^2}{d+e x^2}\right )^{\frac {1}{2}-\frac {m}{2}} \left (-2 \, _3F_2\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2};\frac {d}{d+e x^2}\right )+(-1+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2},\frac {d}{d+e x^2}\right ) \log \left (d+e x^2\right )\right )}{e (-1+m)^2 x}+\frac {2 p \left (2 e x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )-d (3+m) x \log \left (d+e x^2\right )\right ) \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right )}{d (3+m)}-\frac {2 m p \left (-2 e x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )+d (3+m) x \log \left (d+e x^2\right )\right ) \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right )}{d (3+m)}+x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2+m x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2\right )}{(1+m)^2} \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \left (f x \right )^{m} {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}^{2}d x\]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]
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Not integrable
Time = 46.41 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f x\right )^{m} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \]
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Not integrable
Time = 0.39 (sec) , antiderivative size = 126, normalized size of antiderivative = 6.30 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]
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Not integrable
Time = 1.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (f\,x\right )}^m \,d x \]
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