Integrand size = 24, antiderivative size = 24 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=-48 f^2 p^3 x+\frac {351136 d^3 g^2 p^3 x}{25725 e^3}+\frac {6 d f g p^3 x^2}{e}-\frac {55456 d^2 g^2 p^3 x^3}{77175 e^2}+\frac {5232 d g^2 p^3 x^5}{42875 e}-\frac {48 g^2 p^3 x^7}{2401}-\frac {3 f g p^3 \left (d+e x^2\right )^2}{8 e^2}+\frac {48 \sqrt {d} f^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {351136 d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{25725 e^{7/2}}-\frac {24 i \sqrt {d} f^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {1408 i d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{245 e^{7/2}}-\frac {48 \sqrt {d} f^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {2816 d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{245 e^{7/2}}+24 f^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {1408 d^3 g^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{245 e^3}+\frac {568 d^2 g^2 p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )}{735 e^2}-\frac {288 d g^2 p^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )}{1225 e}+\frac {24}{343} g^2 p^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {6 d f g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {3 f g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac {24 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{245 e^{7/2}}-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3 d f g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {3 f g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac {24 i \sqrt {d} f^2 p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {1408 i d^{7/2} g^2 p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{245 e^{7/2}}+6 d f^2 p \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )-\frac {6 d^4 g^2 p \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{7 e^3} \]
[Out]
Not integrable
Time = 1.66 (sec) , antiderivative size = 24, 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 = {} \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (f^2 \log ^3\left (c \left (d+e x^2\right )^p\right )+2 f g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \log ^3\left (c \left (d+e x^2\right )^p\right )\right ) \, dx \\ & = f^2 \int \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^3 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^6 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \\ & = f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )+(f g) \text {Subst}\left (\int x \log ^3\left (c (d+e x)^p\right ) \, dx,x,x^2\right )-\left (6 e f^2 p\right ) \int \frac {x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{7} \left (6 e g^2 p\right ) \int \frac {x^8 \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx \\ & = f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )+(f g) \text {Subst}\left (\int \left (-\frac {d \log ^3\left (c (d+e x)^p\right )}{e}+\frac {(d+e x) \log ^3\left (c (d+e x)^p\right )}{e}\right ) \, dx,x,x^2\right )-\left (6 e f^2 p\right ) \int \left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {d \log ^2\left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx-\frac {1}{7} \left (6 e g^2 p\right ) \int \left (-\frac {d^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^4}+\frac {d^2 x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac {d x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^6 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac {d^4 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx \\ & = f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {(f g) \text {Subst}\left (\int (d+e x) \log ^3\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{e}-\frac {(d f g) \text {Subst}\left (\int \log ^3\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{e}-\left (6 f^2 p\right ) \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\left (6 d f^2 p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{7} \left (6 g^2 p\right ) \int x^6 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\frac {\left (6 d^3 g^2 p\right ) \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}-\frac {\left (6 d^4 g^2 p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}-\frac {\left (6 d^2 g^2 p\right ) \int x^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^2}+\frac {\left (6 d g^2 p\right ) \int x^4 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e} \\ & = -6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {(f g) \text {Subst}\left (\int x \log ^3\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\frac {(d f g) \text {Subst}\left (\int \log ^3\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}+\left (6 d f^2 p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (6 d^4 g^2 p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\left (24 e f^2 p^2\right ) \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{35} \left (24 d g^2 p^2\right ) \int \frac {x^6 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (24 d^3 g^2 p^2\right ) \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^2}+\frac {\left (8 d^2 g^2 p^2\right ) \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e}+\frac {1}{49} \left (24 e g^2 p^2\right ) \int \frac {x^8 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx \\ & = -6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\left (6 d f^2 p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {(3 f g p) \text {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}+\frac {(3 d f g p) \text {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\frac {\left (6 d^4 g^2 p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\left (24 e f^2 p^2\right ) \int \left (\frac {\log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac {d \log \left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx-\frac {1}{35} \left (24 d g^2 p^2\right ) \int \left (\frac {d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac {d^3 \log \left (c \left (d+e x^2\right )^p\right )}{e^3 \left (d+e x^2\right )}\right ) \, dx-\frac {\left (24 d^3 g^2 p^2\right ) \int \left (\frac {\log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac {d \log \left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx}{7 e^2}+\frac {\left (8 d^2 g^2 p^2\right ) \int \left (-\frac {d \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx}{7 e}+\frac {1}{49} \left (24 e g^2 p^2\right ) \int \left (-\frac {d^3 \log \left (c \left (d+e x^2\right )^p\right )}{e^4}+\frac {d^2 x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac {d x^4 \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^6 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {d^4 \log \left (c \left (d+e x^2\right )^p\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx \\ & = -6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3 d f g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {3 f g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\left (6 d f^2 p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (6 d^4 g^2 p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\left (24 f^2 p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\left (24 d f^2 p^2\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\frac {\left (3 f g p^2\right ) \text {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}-\frac {\left (6 d f g p^2\right ) \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}+\frac {1}{49} \left (24 g^2 p^2\right ) \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\frac {\left (24 d^3 g^2 p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{49 e^3}-\frac {\left (24 d^3 g^2 p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{35 e^3}-\frac {\left (8 d^3 g^2 p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}-\frac {\left (24 d^3 g^2 p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}+\frac {\left (24 d^4 g^2 p^2\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{49 e^3}+\frac {\left (24 d^4 g^2 p^2\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{35 e^3}+\frac {\left (8 d^4 g^2 p^2\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\frac {\left (24 d^4 g^2 p^2\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\frac {\left (24 d^2 g^2 p^2\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{49 e^2}+\frac {\left (24 d^2 g^2 p^2\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{35 e^2}+\frac {\left (8 d^2 g^2 p^2\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^2}-\frac {\left (24 d g^2 p^2\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{49 e}-\frac {\left (24 d g^2 p^2\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{35 e} \\ & = \text {Too large to display} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2385\) vs. \(2(1126)=2252\).
Time = 8.70 (sec) , antiderivative size = 2385, normalized size of antiderivative = 99.38 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Result too large to show} \]
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Not integrable
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
\[\int \left (g \,x^{3}+f \right )^{2} {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}^{3}d x\]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]
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Not integrable
Time = 32.81 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f + g x^{3}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{3}\, dx \]
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Exception generated. \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]
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[Out]
Not integrable
Time = 1.57 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^3\,{\left (g\,x^3+f\right )}^2 \,d x \]
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