Optimal. Leaf size=75 \[ \frac {(f x)^{1+m} \log ^2\left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac {4 e p \text {Int}\left (\frac {(f x)^{2+m} \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{f^2 (1+m)} \]
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Rubi [A]
time = 0.06, 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 = {}
\begin {gather*} \int (f x)^m \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx \end {gather*}
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] Leaf count is larger than twice the leaf count of optimal. \(466\) vs. \(2(75)=150\).
time = 1.06, size = 466, normalized size = 6.21 \begin {gather*} \frac {(f x)^m \left (4 p^2 x \left (\frac {2 e x^2 \, _2F_1\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) \, _2F_1\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 \, _2F_1\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 \, _2F_1\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (f x\right )^{m} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (f\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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