Optimal. Leaf size=294 \[ -\frac {4 a^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}-\frac {4 i a^{3/2} p^2 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}\right )}{3 b^{3/2}}-\frac {4 i a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{3/2}}-\frac {8 a^{3/2} p^2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2}}+\frac {4 a p x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {4}{9} p x^3 \log \left (c \left (a+b x^2\right )^p\right )-\frac {32 a p^2 x}{9 b}+\frac {8 p^2 x^3}{27} \]
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Rubi [A] time = 0.32, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {2457, 2476, 2448, 321, 205, 2455, 302, 2470, 12, 4920, 4854, 2402, 2315} \[ -\frac {4 i a^{3/2} p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{3/2}}-\frac {4 a^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}-\frac {4 i a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{3/2}}-\frac {8 a^{3/2} p^2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2}}+\frac {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {4}{9} p x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 a p x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}-\frac {32 a p^2 x}{9 b}+\frac {8 p^2 x^3}{27} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 2448
Rule 2455
Rule 2457
Rule 2470
Rule 2476
Rule 4854
Rule 4920
Rubi steps
\begin {align*} \int x^2 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{3} (4 b p) \int \frac {x^4 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=\frac {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{3} (4 b p) \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 {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{3} (4 p) \int x^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx+\frac {(4 a p) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{3 b}-\frac {\left (4 a^2 p\right ) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{3 b}\\ &=\frac {4 a p x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}-\frac {4}{9} p x^3 \log \left (c \left (a+b x^2\right )^p\right )-\frac {4 a^{3/2} p \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 {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{3} \left (8 a p^2\right ) \int \frac {x^2}{a+b x^2} \, dx+\frac {1}{3} \left (8 a^2 p^2\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 (8 b p^2\right ) \int \frac {x^4}{a+b x^2} \, dx\\ &=-\frac {8 a p^2 x}{3 b}+\frac {4 a p x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}-\frac {4}{9} p x^3 \log \left (c \left (a+b x^2\right )^p\right )-\frac {4 a^{3/2} p \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 {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {\left (8 a^2 p^2\right ) \int \frac {1}{a+b x^2} \, dx}{3 b}+\frac {\left (8 a^{3/2} p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a+b x^2} \, dx}{3 \sqrt {b}}+\frac {1}{9} \left (8 b p^2\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 {32 a p^2 x}{9 b}+\frac {8 p^2 x^3}{27}+\frac {8 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2}}-\frac {4 i a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}+\frac {4 a p x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}-\frac {4}{9} p x^3 \log \left (c \left (a+b x^2\right )^p\right )-\frac {4 a^{3/2} p \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 {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {\left (8 a p^2\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 (8 a^2 p^2\right ) \int \frac {1}{a+b x^2} \, dx}{9 b}\\ &=-\frac {32 a p^2 x}{9 b}+\frac {8 p^2 x^3}{27}+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{3/2}}-\frac {4 i a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}-\frac {8 a^{3/2} p^2 \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 {4 a p x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}-\frac {4}{9} p x^3 \log \left (c \left (a+b x^2\right )^p\right )-\frac {4 a^{3/2} p \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 {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {\left (8 a p^2\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 {32 a p^2 x}{9 b}+\frac {8 p^2 x^3}{27}+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{3/2}}-\frac {4 i a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}-\frac {8 a^{3/2} p^2 \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 {4 a p x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}-\frac {4}{9} p x^3 \log \left (c \left (a+b x^2\right )^p\right )-\frac {4 a^{3/2} p \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 {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {\left (8 i a^{3/2} p^2\right ) \operatorname {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 {32 a p^2 x}{9 b}+\frac {8 p^2 x^3}{27}+\frac {32 a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{3/2}}-\frac {4 i a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}-\frac {8 a^{3/2} p^2 \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 {4 a p x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}-\frac {4}{9} p x^3 \log \left (c \left (a+b x^2\right )^p\right )-\frac {4 a^{3/2} p \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 {1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {4 i a^{3/2} p^2 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 223, normalized size = 0.76 \[ \frac {-12 a^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 \log \left (c \left (a+b x^2\right )^p\right )+6 p \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )-8 p\right )-36 i a^{3/2} p^2 \text {Li}_2\left (\frac {\sqrt {b} x+i \sqrt {a}}{\sqrt {b} x-i \sqrt {a}}\right )-36 i a^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2+\sqrt {b} x \left (9 b x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )+12 p \left (3 a-b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+8 p^2 \left (b x^2-12 a\right )\right )}{27 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.96, size = 0, normalized size = 0.00 \[ \int x^{2} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, p^{2} x^{3} \log \left (b x^{2} + a\right )^{2} + \int \frac {3 \, b x^{4} \log \relax (c)^{2} + 3 \, a x^{2} \log \relax (c)^{2} - 2 \, {\left ({\left (2 \, p^{2} - 3 \, p \log \relax (c)\right )} b x^{4} - 3 \, a p x^{2} \log \relax (c)\right )} \log \left (b x^{2} + a\right )}{3 \, {\left (b x^{2} + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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