Integrand size = 18, antiderivative size = 18 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {208 a p^3 x}{9 b}-\frac {16 p^3 x^3}{27}-\frac {208 a^{3/2} p^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{3/2}}+\frac {32 i a^{3/2} p^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}+\frac {64 a^{3/2} p^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{3/2}}-\frac {32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac {8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac {32 a^{3/2} p^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )+\frac {32 i a^{3/2} p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{3/2}}-\frac {2 a^2 p \text {Int}\left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right )}{b} \]
[Out]
Not integrable
Time = 0.53 (sec) , antiderivative size = 18, 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 = {} \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-(2 b p) \int \frac {x^4 \log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx \\ & = \frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-(2 b p) \int \left (-\frac {a \log ^2\left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac {x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}+\frac {a^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{b^2 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-(2 p) \int x^2 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx+\frac {(2 a p) \int \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx}{b}-\frac {\left (2 a^2 p\right ) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b} \\ & = \frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {\left (2 a^2 p\right ) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}-\left (8 a p^2\right ) \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\frac {1}{3} \left (8 b p^2\right ) \int \frac {x^4 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx \\ & = \frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {\left (2 a^2 p\right ) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}-\left (8 a p^2\right ) \int \left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac {a \log \left (c \left (a+b x^2\right )^p\right )}{b \left (a+b x^2\right )}\right ) \, dx+\frac {1}{3} \left (8 b p^2\right ) \int \left (-\frac {a \log \left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{b}+\frac {a^2 \log \left (c \left (a+b x^2\right )^p\right )}{b^2 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {\left (2 a^2 p\right ) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}+\frac {1}{3} \left (8 p^2\right ) \int x^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx-\frac {\left (8 a p^2\right ) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{3 b}-\frac {\left (8 a p^2\right ) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{b}+\frac {\left (8 a^2 p^2\right ) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{3 b}+\frac {\left (8 a^2 p^2\right ) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b} \\ & = -\frac {32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac {8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {\left (2 a^2 p\right ) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}+\frac {1}{3} \left (16 a p^3\right ) \int \frac {x^2}{a+b x^2} \, dx+\left (16 a p^3\right ) \int \frac {x^2}{a+b x^2} \, dx-\frac {1}{3} \left (16 a^2 p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a+b x^2\right )} \, dx-\left (16 a^2 p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a+b x^2\right )} \, dx-\frac {1}{9} \left (16 b p^3\right ) \int \frac {x^4}{a+b x^2} \, dx \\ & = \frac {64 a p^3 x}{3 b}-\frac {32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac {8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {\left (2 a^2 p\right ) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}-\frac {\left (16 a^2 p^3\right ) \int \frac {1}{a+b x^2} \, dx}{3 b}-\frac {\left (16 a^2 p^3\right ) \int \frac {1}{a+b x^2} \, dx}{b}-\frac {\left (16 a^{3/2} p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a+b x^2} \, dx}{3 \sqrt {b}}-\frac {\left (16 a^{3/2} p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a+b x^2} \, dx}{\sqrt {b}}-\frac {1}{9} \left (16 b p^3\right ) \int \left (-\frac {a}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {208 a p^3 x}{9 b}-\frac {16 p^3 x^3}{27}-\frac {64 a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2}}+\frac {32 i a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}-\frac {32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac {8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {\left (2 a^2 p\right ) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}+\frac {\left (16 a p^3\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{i-\frac {\sqrt {b} x}{\sqrt {a}}} \, dx}{3 b}+\frac {\left (16 a p^3\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{i-\frac {\sqrt {b} x}{\sqrt {a}}} \, dx}{b}-\frac {\left (16 a^2 p^3\right ) \int \frac {1}{a+b x^2} \, dx}{9 b} \\ & = \frac {208 a p^3 x}{9 b}-\frac {16 p^3 x^3}{27}-\frac {208 a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{3/2}}+\frac {32 i a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}+\frac {64 a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{3/2}}-\frac {32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac {8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {\left (2 a^2 p\right ) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}-\frac {\left (16 a p^3\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{1+\frac {b x^2}{a}} \, dx}{3 b}-\frac {\left (16 a p^3\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{1+\frac {b x^2}{a}} \, dx}{b} \\ & = \frac {208 a p^3 x}{9 b}-\frac {16 p^3 x^3}{27}-\frac {208 a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{3/2}}+\frac {32 i a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}+\frac {64 a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{3/2}}-\frac {32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac {8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {\left (2 a^2 p\right ) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b}+\frac {\left (16 i a^{3/2} p^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{3 b^{3/2}}+\frac {\left (16 i a^{3/2} p^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{b^{3/2}} \\ & = \frac {208 a p^3 x}{9 b}-\frac {16 p^3 x^3}{27}-\frac {208 a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{3/2}}+\frac {32 i a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}+\frac {64 a^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{3/2}}-\frac {32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac {8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )+\frac {32 i a^{3/2} p^3 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{3/2}}-\frac {\left (2 a^2 p\right ) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{b} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(909\) vs. \(2(380)=760\).
Time = 3.13 (sec) , antiderivative size = 909, normalized size of antiderivative = 50.50 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {2 a p x \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{b}-\frac {2 a^{3/2} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{b^{3/2}}+p x^3 \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2+\frac {1}{3} x^3 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2 \left (-2 p-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )+3 p^2 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (\frac {1}{3} x^3 \log ^2\left (a+b x^2\right )-\frac {4 \left (9 i a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2+3 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-8+6 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )+3 \log \left (a+b x^2\right )\right )+\sqrt {b} x \left (24 a-2 b x^2+\left (-9 a+3 b x^2\right ) \log \left (a+b x^2\right )\right )+9 i a^{3/2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )}{27 b^{3/2}}\right )+\frac {p^3 \left (416 \sqrt {-a} a^{3/2} \sqrt {\frac {b x^2}{a+b x^2}} \sqrt {a+b x^2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right )+\frac {2}{3} \sqrt {-a} b x^2 \left (624 a-16 b x^2+\left (-288 a+24 b x^2\right ) \log \left (a+b x^2\right )+18 \left (3 a-b x^2\right ) \log ^2\left (a+b x^2\right )+9 b x^2 \log ^3\left (a+b x^2\right )\right )+36 \sqrt {-a} a^{3/2} \sqrt {\frac {b x^2}{a+b x^2}} \left (8 \sqrt {a} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+\log \left (a+b x^2\right ) \left (4 \sqrt {a} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+\sqrt {a+b x^2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right ) \log \left (a+b x^2\right )\right )\right )-48 a^2 \left (4 \sqrt {b x^2} \text {arctanh}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right ) \left (\log \left (a+b x^2\right )-\log \left (1+\frac {b x^2}{a}\right )\right )-\sqrt {-a} \sqrt {-\frac {b x^2}{a}} \left (\log ^2\left (1+\frac {b x^2}{a}\right )-4 \log \left (1+\frac {b x^2}{a}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {b x^2}{a}}\right )\right )\right )\right )}{18 \sqrt {-a} b^2 x} \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
\[\int x^{2} {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3}d x\]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int { x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} \,d x } \]
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Not integrable
Time = 3.88 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}\, dx \]
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Not integrable
Time = 1.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 6.83 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int { x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} \,d x } \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int { x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} \,d x } \]
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Not integrable
Time = 1.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int x^2\,{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3 \,d x \]
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