Optimal. Leaf size=215 \[ -\frac{a^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}+\frac{a^3 p \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^3}+\frac{a^2 p^2 x^2}{b^2}-\frac{a^3 p^2 \log ^2\left (a+b x^2\right )}{6 b^3}-\frac{p \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}+\frac{a p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac{p^2 \left (a+b x^2\right )^3}{27 b^3}-\frac{a p^2 \left (a+b x^2\right )^2}{4 b^3}+\frac{1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right ) \]
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Rubi [A] time = 0.298564, antiderivative size = 175, normalized size of antiderivative = 0.81, number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ -\frac{1}{18} p \left (\frac{18 a^2 \left (a+b x^2\right )}{b^3}-\frac{6 a^3 \log \left (a+b x^2\right )}{b^3}-\frac{9 a \left (a+b x^2\right )^2}{b^3}+\frac{2 \left (a+b x^2\right )^3}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac{a^2 p^2 x^2}{b^2}-\frac{a^3 p^2 \log ^2\left (a+b x^2\right )}{6 b^3}+\frac{p^2 \left (a+b x^2\right )^3}{27 b^3}-\frac{a p^2 \left (a+b x^2\right )^2}{4 b^3}+\frac{1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right ) \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2398
Rule 2411
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rubi steps
\begin{align*} \int x^5 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac{1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac{1}{3} (b p) \operatorname{Subst}\left (\int \frac{x^3 \log \left (c (a+b x)^p\right )}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac{1}{3} p \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^3 \log \left (c x^p\right )}{x} \, dx,x,a+b x^2\right )\\ &=-\frac{1}{18} p \left (\frac{18 a^2 \left (a+b x^2\right )}{b^3}-\frac{9 a \left (a+b x^2\right )^2}{b^3}+\frac{2 \left (a+b x^2\right )^3}{b^3}-\frac{6 a^3 \log \left (a+b x^2\right )}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac{1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{1}{3} p^2 \operatorname{Subst}\left (\int \frac{18 a^2 x-9 a x^2+2 x^3-6 a^3 \log (x)}{6 b^3 x} \, dx,x,a+b x^2\right )\\ &=-\frac{1}{18} p \left (\frac{18 a^2 \left (a+b x^2\right )}{b^3}-\frac{9 a \left (a+b x^2\right )^2}{b^3}+\frac{2 \left (a+b x^2\right )^3}{b^3}-\frac{6 a^3 \log \left (a+b x^2\right )}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac{1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{p^2 \operatorname{Subst}\left (\int \frac{18 a^2 x-9 a x^2+2 x^3-6 a^3 \log (x)}{x} \, dx,x,a+b x^2\right )}{18 b^3}\\ &=-\frac{1}{18} p \left (\frac{18 a^2 \left (a+b x^2\right )}{b^3}-\frac{9 a \left (a+b x^2\right )^2}{b^3}+\frac{2 \left (a+b x^2\right )^3}{b^3}-\frac{6 a^3 \log \left (a+b x^2\right )}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac{1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{p^2 \operatorname{Subst}\left (\int \left (18 a^2-9 a x+2 x^2-\frac{6 a^3 \log (x)}{x}\right ) \, dx,x,a+b x^2\right )}{18 b^3}\\ &=\frac{a^2 p^2 x^2}{b^2}-\frac{a p^2 \left (a+b x^2\right )^2}{4 b^3}+\frac{p^2 \left (a+b x^2\right )^3}{27 b^3}-\frac{1}{18} p \left (\frac{18 a^2 \left (a+b x^2\right )}{b^3}-\frac{9 a \left (a+b x^2\right )^2}{b^3}+\frac{2 \left (a+b x^2\right )^3}{b^3}-\frac{6 a^3 \log \left (a+b x^2\right )}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac{1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac{\left (a^3 p^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x^2\right )}{3 b^3}\\ &=\frac{a^2 p^2 x^2}{b^2}-\frac{a p^2 \left (a+b x^2\right )^2}{4 b^3}+\frac{p^2 \left (a+b x^2\right )^3}{27 b^3}-\frac{a^3 p^2 \log ^2\left (a+b x^2\right )}{6 b^3}-\frac{1}{18} p \left (\frac{18 a^2 \left (a+b x^2\right )}{b^3}-\frac{9 a \left (a+b x^2\right )^2}{b^3}+\frac{2 \left (a+b x^2\right )^3}{b^3}-\frac{6 a^3 \log \left (a+b x^2\right )}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac{1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )\\ \end{align*}
Mathematica [A] time = 0.0665617, size = 200, normalized size = 0.93 \[ \frac{a^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac{a^2 p x^2 \log \left (c \left (a+b x^2\right )^p\right )}{3 b^2}-\frac{a^3 p \log \left (c \left (a+b x^2\right )^p\right )}{3 b^3}+\frac{11 a^2 p^2 x^2}{18 b^2}-\frac{5 a^3 p^2 \log \left (a+b x^2\right )}{18 b^3}+\frac{1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac{1}{9} p x^6 \log \left (c \left (a+b x^2\right )^p\right )+\frac{a p x^4 \log \left (c \left (a+b x^2\right )^p\right )}{6 b}-\frac{5 a p^2 x^4}{36 b}+\frac{p^2 x^6}{27} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.519, size = 1436, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06111, size = 196, normalized size = 0.91 \begin{align*} \frac{1}{6} \, x^{6} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} + \frac{1}{18} \, b p{\left (\frac{6 \, a^{3} \log \left (b x^{2} + a\right )}{b^{4}} - \frac{2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{b^{3}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) + \frac{{\left (4 \, b^{3} x^{6} - 15 \, a b^{2} x^{4} + 66 \, a^{2} b x^{2} - 18 \, a^{3} \log \left (b x^{2} + a\right )^{2} - 66 \, a^{3} \log \left (b x^{2} + a\right )\right )} p^{2}}{108 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08301, size = 412, normalized size = 1.92 \begin{align*} \frac{4 \, b^{3} p^{2} x^{6} + 18 \, b^{3} x^{6} \log \left (c\right )^{2} - 15 \, a b^{2} p^{2} x^{4} + 66 \, a^{2} b p^{2} x^{2} + 18 \,{\left (b^{3} p^{2} x^{6} + a^{3} p^{2}\right )} \log \left (b x^{2} + a\right )^{2} - 6 \,{\left (2 \, b^{3} p^{2} x^{6} - 3 \, a b^{2} p^{2} x^{4} + 6 \, a^{2} b p^{2} x^{2} + 11 \, a^{3} p^{2} - 6 \,{\left (b^{3} p x^{6} + a^{3} p\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right ) - 6 \,{\left (2 \, b^{3} p x^{6} - 3 \, a b^{2} p x^{4} + 6 \, a^{2} b p x^{2}\right )} \log \left (c\right )}{108 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 87.5571, size = 267, normalized size = 1.24 \begin{align*} \begin{cases} \frac{a^{3} p^{2} \log{\left (a + b x^{2} \right )}^{2}}{6 b^{3}} - \frac{11 a^{3} p^{2} \log{\left (a + b x^{2} \right )}}{18 b^{3}} + \frac{a^{3} p \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{3 b^{3}} - \frac{a^{2} p^{2} x^{2} \log{\left (a + b x^{2} \right )}}{3 b^{2}} + \frac{11 a^{2} p^{2} x^{2}}{18 b^{2}} - \frac{a^{2} p x^{2} \log{\left (c \right )}}{3 b^{2}} + \frac{a p^{2} x^{4} \log{\left (a + b x^{2} \right )}}{6 b} - \frac{5 a p^{2} x^{4}}{36 b} + \frac{a p x^{4} \log{\left (c \right )}}{6 b} + \frac{p^{2} x^{6} \log{\left (a + b x^{2} \right )}^{2}}{6} - \frac{p^{2} x^{6} \log{\left (a + b x^{2} \right )}}{9} + \frac{p^{2} x^{6}}{27} + \frac{p x^{6} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{3} - \frac{p x^{6} \log{\left (c \right )}}{9} + \frac{x^{6} \log{\left (c \right )}^{2}}{6} & \text{for}\: b \neq 0 \\\frac{x^{6} \log{\left (a^{p} c \right )}^{2}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27299, size = 439, normalized size = 2.04 \begin{align*} \frac{18 \, b x^{6} \log \left (c\right )^{2} +{\left (\frac{18 \,{\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )^{2}}{b^{2}} - \frac{54 \,{\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )^{2}}{b^{2}} + \frac{54 \,{\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )^{2}}{b^{2}} - \frac{12 \,{\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )}{b^{2}} + \frac{54 \,{\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )}{b^{2}} - \frac{108 \,{\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )}{b^{2}} + \frac{4 \,{\left (b x^{2} + a\right )}^{3}}{b^{2}} - \frac{27 \,{\left (b x^{2} + a\right )}^{2} a}{b^{2}} + \frac{108 \,{\left (b x^{2} + a\right )} a^{2}}{b^{2}}\right )} p^{2} + 6 \,{\left (\frac{6 \,{\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )}{b^{2}} - \frac{18 \,{\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )}{b^{2}} + \frac{18 \,{\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )}{b^{2}} - \frac{2 \,{\left (b x^{2} + a\right )}^{3}}{b^{2}} + \frac{9 \,{\left (b x^{2} + a\right )}^{2} a}{b^{2}} - \frac{18 \,{\left (b x^{2} + a\right )} a^{2}}{b^{2}}\right )} p \log \left (c\right )}{108 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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