Optimal. Leaf size=72 \[ \frac {2 b^{5/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{5 a^{5/2}}-\frac {2 b^2 p x}{5 a^2}+\frac {1}{5} x^5 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {2 b p x^3}{15 a} \]
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Rubi [A] time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2455, 263, 302, 205} \[ -\frac {2 b^2 p x}{5 a^2}+\frac {2 b^{5/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{5 a^{5/2}}+\frac {1}{5} x^5 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {2 b p x^3}{15 a} \]
Antiderivative was successfully verified.
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Rule 205
Rule 263
Rule 302
Rule 2455
Rubi steps
\begin {align*} \int x^4 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx &=\frac {1}{5} x^5 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {1}{5} (2 b p) \int \frac {x^2}{a+\frac {b}{x^2}} \, dx\\ &=\frac {1}{5} x^5 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {1}{5} (2 b p) \int \frac {x^4}{b+a x^2} \, dx\\ &=\frac {1}{5} x^5 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {1}{5} (2 b p) \int \left (-\frac {b}{a^2}+\frac {x^2}{a}+\frac {b^2}{a^2 \left (b+a x^2\right )}\right ) \, dx\\ &=-\frac {2 b^2 p x}{5 a^2}+\frac {2 b p x^3}{15 a}+\frac {1}{5} x^5 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {\left (2 b^3 p\right ) \int \frac {1}{b+a x^2} \, dx}{5 a^2}\\ &=-\frac {2 b^2 p x}{5 a^2}+\frac {2 b p x^3}{15 a}+\frac {2 b^{5/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{5 a^{5/2}}+\frac {1}{5} x^5 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 49, normalized size = 0.68 \[ \frac {1}{5} x^5 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {2 b p x^3 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {b}{a x^2}\right )}{15 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 178, normalized size = 2.47 \[ \left [\frac {3 \, a^{2} p x^{5} \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + 3 \, a^{2} x^{5} \log \relax (c) + 2 \, a b p x^{3} + 3 \, b^{2} p \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right ) - 6 \, b^{2} p x}{15 \, a^{2}}, \frac {3 \, a^{2} p x^{5} \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + 3 \, a^{2} x^{5} \log \relax (c) + 2 \, a b p x^{3} + 6 \, b^{2} p \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right ) - 6 \, b^{2} p x}{15 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 75, normalized size = 1.04 \[ \frac {1}{5} \, p x^{5} \log \left (a x^{2} + b\right ) - \frac {1}{5} \, p x^{5} \log \left (x^{2}\right ) + \frac {1}{5} \, x^{5} \log \relax (c) + \frac {2 \, b p x^{3}}{15 \, a} + \frac {2 \, b^{3} p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{5 \, \sqrt {a b} a^{2}} - \frac {2 \, b^{2} p x}{5 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int x^{4} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 59, normalized size = 0.82 \[ \frac {1}{5} \, x^{5} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) + \frac {2}{15} \, b p {\left (\frac {3 \, b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {a x^{3} - 3 \, b x}{a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 56, normalized size = 0.78 \[ \frac {x^5\,\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{5}+\frac {2\,b^{5/2}\,p\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{5\,a^{5/2}}+\frac {2\,b\,p\,x^3}{15\,a}-\frac {2\,b^2\,p\,x}{5\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 92.91, size = 162, normalized size = 2.25 \[ \begin {cases} \frac {p x^{5} \log {\left (a + \frac {b}{x^{2}} \right )}}{5} + \frac {x^{5} \log {\relax (c )}}{5} + \frac {2 b p x^{3}}{15 a} - \frac {2 b^{2} p x}{5 a^{2}} - \frac {i b^{\frac {5}{2}} p \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + x \right )}}{5 a^{3} \sqrt {\frac {1}{a}}} + \frac {i b^{\frac {5}{2}} p \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + x \right )}}{5 a^{3} \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\\frac {p x^{5} \log {\relax (b )}}{5} - \frac {2 p x^{5} \log {\relax (x )}}{5} + \frac {2 p x^{5}}{25} + \frac {x^{5} \log {\relax (c )}}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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