Optimal. Leaf size=58 \[ -\frac {2 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2}}+\frac {1}{3} x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {2 b p x}{3 a} \]
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Rubi [A] time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2455, 193, 321, 205} \[ -\frac {2 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2}}+\frac {1}{3} x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {2 b p x}{3 a} \]
Antiderivative was successfully verified.
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Rule 193
Rule 205
Rule 321
Rule 2455
Rubi steps
\begin {align*} \int x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx &=\frac {1}{3} x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {1}{3} (2 b p) \int \frac {1}{a+\frac {b}{x^2}} \, dx\\ &=\frac {1}{3} x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {1}{3} (2 b p) \int \frac {x^2}{b+a x^2} \, dx\\ &=\frac {2 b p x}{3 a}+\frac {1}{3} x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )-\frac {\left (2 b^2 p\right ) \int \frac {1}{b+a x^2} \, dx}{3 a}\\ &=\frac {2 b p x}{3 a}-\frac {2 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2}}+\frac {1}{3} x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 47, normalized size = 0.81 \[ \frac {1}{3} x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {2 b p x \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {b}{a x^2}\right )}{3 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 141, normalized size = 2.43 \[ \left [\frac {a p x^{3} \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + a x^{3} \log \relax (c) + b p \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right ) + 2 \, b p x}{3 \, a}, \frac {a p x^{3} \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + a x^{3} \log \relax (c) - 2 \, b p \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right ) + 2 \, b p x}{3 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 63, normalized size = 1.09 \[ \frac {1}{3} \, p x^{3} \log \left (a x^{2} + b\right ) - \frac {1}{3} \, p x^{3} \log \left (x^{2}\right ) + \frac {1}{3} \, x^{3} \log \relax (c) - \frac {2 \, b^{2} p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} a} + \frac {2 \, b p x}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int x^{2} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.50, size = 48, normalized size = 0.83 \[ \frac {1}{3} \, x^{3} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) - \frac {2}{3} \, b p {\left (\frac {b \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {x}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 44, normalized size = 0.76 \[ \frac {x^3\,\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{3}-\frac {2\,b^{3/2}\,p\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{3\,a^{3/2}}+\frac {2\,b\,p\,x}{3\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 33.14, size = 146, normalized size = 2.52 \[ \begin {cases} \frac {p x^{3} \log {\left (a + \frac {b}{x^{2}} \right )}}{3} + \frac {x^{3} \log {\relax (c )}}{3} + \frac {2 b p x}{3 a} + \frac {i b^{\frac {3}{2}} p \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + x \right )}}{3 a^{2} \sqrt {\frac {1}{a}}} - \frac {i b^{\frac {3}{2}} p \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + x \right )}}{3 a^{2} \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\\frac {p x^{3} \log {\relax (b )}}{3} - \frac {2 p x^{3} \log {\relax (x )}}{3} + \frac {2 p x^{3}}{9} + \frac {x^{3} \log {\relax (c )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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