Optimal. Leaf size=75 \[ \frac {(f x)^{m+1} \log ^2\left (c \left (d+e x^2\right )^p\right )}{f (m+1)}-\frac {4 e p \text {Int}\left (\frac {(f x)^{m+2} \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{f^2 (m+1)} \]
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Rubi [A] time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \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) \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*}
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Mathematica [A] time = 1.08, size = 466, normalized size = 6.21 \[ \frac {(f x)^m \left (\frac {4 d (m+1) p^2 \left (\frac {e x^2}{d+e x^2}\right )^{\frac {1}{2}-\frac {m}{2}} \left ((m-1) \log \left (d+e x^2\right ) \, _2F_1\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2};\frac {d}{e x^2+d}\right )-2 \, _3F_2\left (\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2};\frac {d}{e x^2+d}\right )\right )}{e (m-1)^2 x}+\frac {2 p \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right ) \left (2 e x^3 \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )-d (m+3) x \log \left (d+e x^2\right )\right )}{d (m+3)}-\frac {2 m p \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right ) \left (d (m+3) x \log \left (d+e x^2\right )-2 e x^3 \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )\right )}{d (m+3)}+m x \left (\log \left (c \left (d+e x^2\right )^p\right )-p \log \left (d+e x^2\right )\right )^2+x \left (\log \left (c \left (d+e x^2\right )^p\right )-p \log \left (d+e x^2\right )\right )^2+4 p^2 x \left (\frac {2 e x^2 \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};-\frac {e x^2}{d}\right )}{d (m+3)}-\log \left (d+e x^2\right )\right )+(m+1) p^2 x \log ^2\left (d+e x^2\right )\right )}{(m+1)^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.27, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {f^{m} p^{2} x x^{m} \log \left (e x^{2} + d\right )^{2}}{m + 1} + \int \frac {2 \, {\left ({\left (m p + p\right )} d f^{m} \log \relax (c) - {\left (2 \, e f^{m} p^{2} - {\left (m p + p\right )} e f^{m} \log \relax (c)\right )} x^{2}\right )} x^{m} \log \left (e x^{2} + d\right ) + {\left (e f^{m} {\left (m + 1\right )} x^{2} \log \relax (c)^{2} + d f^{m} {\left (m + 1\right )} \log \relax (c)^{2}\right )} x^{m}}{e {\left (m + 1\right )} x^{2} + d {\left (m + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (f\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f x\right )^{m} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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