Optimal. Leaf size=76 \[ \frac{(f x)^{m+1} \log ^3\left (c \left (d+e x^2\right )^p\right )}{f (m+1)}-\frac{6 e p \text{Unintegrable}\left (\frac{(f x)^{m+2} \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{f^2 (m+1)} \]
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Rubi [A] time = 0.118039, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (f x)^m \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx &=\frac{(f x)^{1+m} \log ^3\left (c \left (d+e x^2\right )^p\right )}{f (1+m)}-\frac{(6 e p) \int \frac{(f x)^{2+m} \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{f^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 2.27135, size = 994, normalized size = 13.08 \[ \frac{(f x)^m \left (\frac{6 p^3 \left (d \left (\left (-\frac{e x^2}{d}\right )^{\frac{m+1}{2}}-1\right ) \log ^2\left (e x^2+d\right )+(m+1) \left (e x^2+d\right ) \, _3F_2\left (1,1,\frac{1}{2}-\frac{m}{2};2,2;\frac{e x^2}{d}+1\right ) \log \left (e x^2+d\right )-(m+1) \left (e x^2+d\right ) \, _4F_3\left (1,1,1,\frac{1}{2}-\frac{m}{2};2,2,2;\frac{e x^2}{d}+1\right )\right ) \left (-\frac{e x^2}{d}\right )^{\frac{1}{2}-\frac{m}{2}}}{e}-\frac{3 m p^2 \left (d \left (\left (-\frac{e x^2}{d}\right )^{\frac{m+1}{2}}-1\right ) \log ^2\left (e x^2+d\right )+(m+1) \left (e x^2+d\right ) \, _3F_2\left (1,1,\frac{1}{2}-\frac{m}{2};2,2;\frac{e x^2}{d}+1\right ) \log \left (e x^2+d\right )-(m+1) \left (e x^2+d\right ) \, _4F_3\left (1,1,1,\frac{1}{2}-\frac{m}{2};2,2,2;\frac{e x^2}{d}+1\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right ) \left (-\frac{e x^2}{d}\right )^{\frac{1}{2}-\frac{m}{2}}}{e}-\frac{3 p^2 \left (d \left (\left (-\frac{e x^2}{d}\right )^{\frac{m+1}{2}}-1\right ) \log ^2\left (e x^2+d\right )+(m+1) \left (e x^2+d\right ) \, _3F_2\left (1,1,\frac{1}{2}-\frac{m}{2};2,2;\frac{e x^2}{d}+1\right ) \log \left (e x^2+d\right )-(m+1) \left (e x^2+d\right ) \, _4F_3\left (1,1,1,\frac{1}{2}-\frac{m}{2};2,2,2;\frac{e x^2}{d}+1\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right ) \left (-\frac{e x^2}{d}\right )^{\frac{1}{2}-\frac{m}{2}}}{e}+(m+1) p^3 x^2 \log ^3\left (e x^2+d\right )+m x^2 \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^3+x^2 \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^3+\frac{3 m p x^2 \left (d (m+3) \log \left (e x^2+d\right )-2 e x^2 \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{e x^2}{d}\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^2}{d (m+3)}+\frac{3 p x^2 \left (d (m+3) \log \left (e x^2+d\right )-2 e x^2 \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{e x^2}{d}\right )\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )^2}{d (m+3)}+\frac{6 d (m+1) p^3 \left (\frac{e x^2}{e x^2+d}\right )^{\frac{1}{2}-\frac{m}{2}} \left (8 \, _4F_3\left (\frac{1}{2}-\frac{m}{2},\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{3}{2}-\frac{m}{2};\frac{d}{e x^2+d}\right )+(m-1) \log \left (e x^2+d\right ) \left ((m-1) \, _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 ) \log \left (e x^2+d\right )-4 \, _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 )\right )}{e (m-1)^3}\right )}{(m+1)^2 x} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.897, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( \ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}\, dx \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 = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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