Optimal. Leaf size=72 \[ -\frac{2 b^2 p x}{5 a^2}+\frac{2 b^{5/2} p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{5 a^{5/2}}+\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{2 b p x^3}{15 a} \]
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Rubi [A] time = 0.0375931, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 2455
Rule 263
Rule 302
Rule 205
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*}
Mathematica [C] time = 0.0060882, 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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Maple [F] time = 0.333, size = 0, normalized size = 0. \begin{align*} \int{x}^{4}\ln \left ( c \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \right ) \, 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 [A] time = 2.32446, size = 401, normalized size = 5.57 \begin{align*} \left [\frac{3 \, a^{2} p x^{5} \log \left (\frac{a x^{2} + b}{x^{2}}\right ) + 3 \, a^{2} x^{5} \log \left (c\right ) + 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 \left (c\right ) + 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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19816, size = 101, normalized size = 1.4 \begin{align*} \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 \left (c\right ) + \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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