Optimal. Leaf size=145 \[ \frac{\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac{a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}-\frac{p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^2}+\frac{a p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac{p^2 \left (a+b x^2\right )^2}{8 b^2}-\frac{a p^2 x^2}{b} \]
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Rubi [A] time = 0.153345, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac{a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}-\frac{p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^2}+\frac{a p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac{p^2 \left (a+b x^2\right )^2}{8 b^2}-\frac{a p^2 x^2}{b} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2401
Rule 2389
Rule 2296
Rule 2295
Rule 2390
Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x^3 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a \log ^2\left (c (a+b x)^p\right )}{b}+\frac{(a+b x) \log ^2\left (c (a+b x)^p\right )}{b}\right ) \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int (a+b x) \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b}-\frac{a \operatorname{Subst}\left (\int \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}-\frac{a \operatorname{Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}\\ &=-\frac{a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac{\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac{p \operatorname{Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}+\frac{(a p) \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{b^2}\\ &=-\frac{a p^2 x^2}{b}+\frac{p^2 \left (a+b x^2\right )^2}{8 b^2}+\frac{a p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}-\frac{p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac{a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac{\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.056058, size = 105, normalized size = 0.72 \[ \frac{-2 \left (a^2-b^2 x^4\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+2 p \left (2 a^2+2 a b x^2-b^2 x^4\right ) \log \left (c \left (a+b x^2\right )^p\right )+2 a^2 p^2 \log \left (a+b x^2\right )+b p^2 x^2 \left (b x^2-6 a\right )}{8 b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.492, size = 1242, normalized size = 8.6 \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 = 1.05056, size = 162, normalized size = 1.12 \begin{align*} \frac{1}{4} \, x^{4} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} - \frac{1}{4} \, b p{\left (\frac{2 \, a^{2} \log \left (b x^{2} + a\right )}{b^{3}} + \frac{b x^{4} - 2 \, a x^{2}}{b^{2}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) + \frac{{\left (b^{2} x^{4} - 6 \, a b x^{2} + 2 \, a^{2} \log \left (b x^{2} + a\right )^{2} + 6 \, a^{2} \log \left (b x^{2} + a\right )\right )} p^{2}}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90911, size = 316, normalized size = 2.18 \begin{align*} \frac{b^{2} p^{2} x^{4} + 2 \, b^{2} x^{4} \log \left (c\right )^{2} - 6 \, a b p^{2} x^{2} + 2 \,{\left (b^{2} p^{2} x^{4} - a^{2} p^{2}\right )} \log \left (b x^{2} + a\right )^{2} - 2 \,{\left (b^{2} p^{2} x^{4} - 2 \, a b p^{2} x^{2} - 3 \, a^{2} p^{2} - 2 \,{\left (b^{2} p x^{4} - a^{2} p\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (b^{2} p x^{4} - 2 \, a b p x^{2}\right )} \log \left (c\right )}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.3503, size = 209, normalized size = 1.44 \begin{align*} \begin{cases} - \frac{a^{2} p^{2} \log{\left (a + b x^{2} \right )}^{2}}{4 b^{2}} + \frac{3 a^{2} p^{2} \log{\left (a + b x^{2} \right )}}{4 b^{2}} - \frac{a^{2} p \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{a p^{2} x^{2} \log{\left (a + b x^{2} \right )}}{2 b} - \frac{3 a p^{2} x^{2}}{4 b} + \frac{a p x^{2} \log{\left (c \right )}}{2 b} + \frac{p^{2} x^{4} \log{\left (a + b x^{2} \right )}^{2}}{4} - \frac{p^{2} x^{4} \log{\left (a + b x^{2} \right )}}{4} + \frac{p^{2} x^{4}}{8} + \frac{p x^{4} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{2} - \frac{p x^{4} \log{\left (c \right )}}{4} + \frac{x^{4} \log{\left (c \right )}^{2}}{4} & \text{for}\: b \neq 0 \\\frac{x^{4} \log{\left (a^{p} c \right )}^{2}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23159, size = 279, normalized size = 1.92 \begin{align*} \frac{\frac{{\left (2 \,{\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right )^{2} - 4 \,{\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right )^{2} - 2 \,{\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right ) + 8 \,{\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right ) +{\left (b x^{2} + a\right )}^{2} - 8 \,{\left (b x^{2} + a\right )} a\right )} p^{2}}{b} + \frac{2 \,{\left (2 \,{\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right ) - 4 \,{\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right ) -{\left (b x^{2} + a\right )}^{2} + 4 \,{\left (b x^{2} + a\right )} a\right )} p \log \left (c\right )}{b} + \frac{2 \,{\left ({\left (b x^{2} + a\right )}^{2} - 2 \,{\left (b x^{2} + a\right )} a\right )} \log \left (c\right )^{2}}{b}}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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