Optimal. Leaf size=294 \[ -\frac{4 i a^{3/2} p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{3 b^{3/2}}-\frac{4 a^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}-\frac{4 i a^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 b^{3/2}}+\frac{32 a^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{9 b^{3/2}}-\frac{8 a^{3/2} p^2 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 b^{3/2}}+\frac{1}{3} x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac{4}{9} p x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac{4 a p x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}-\frac{32 a p^2 x}{9 b}+\frac{8 p^2 x^3}{27} \]
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Rubi [A] time = 0.322765, antiderivative size = 294, normalized size of antiderivative = 1., 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 2457
Rule 2476
Rule 2448
Rule 321
Rule 205
Rule 2455
Rule 302
Rule 2470
Rule 12
Rule 4920
Rule 4854
Rule 2402
Rule 2315
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*}
Mathematica [A] time = 0.134917, size = 223, normalized size = 0.76 \[ \frac{-36 i a^{3/2} p^2 \text{PolyLog}\left (2,\frac{\sqrt{b} x+i \sqrt{a}}{\sqrt{b} x-i \sqrt{a}}\right )-12 a^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 \log \left (c \left (a+b x^2\right )^p\right )+6 p \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )-8 p\right )-36 i a^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2+\sqrt{b} x \left (9 b x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )+12 p \left (3 a-b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+8 p^2 \left (b x^2-12 a\right )\right )}{27 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.984, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}\, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \log{\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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