Optimal. Leaf size=114 \[ \frac{\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{4 b p^3}-\frac{a+b x^2}{4 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{a+b x^2}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )} \]
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Rubi [A] time = 0.0931416, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {2454, 2389, 2297, 2300, 2178} \[ \frac{\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{4 b p^3}-\frac{a+b x^2}{4 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{a+b x^2}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2389
Rule 2297
Rule 2300
Rule 2178
Rubi steps
\begin{align*} \int \frac{x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\log ^3\left (c (a+b x)^p\right )} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\log ^3\left (c x^p\right )} \, dx,x,a+b x^2\right )}{2 b}\\ &=-\frac{a+b x^2}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\log ^2\left (c x^p\right )} \, dx,x,a+b x^2\right )}{4 b p}\\ &=-\frac{a+b x^2}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac{a+b x^2}{4 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{4 b p^2}\\ &=-\frac{a+b x^2}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac{a+b x^2}{4 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}+\frac{\left (\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{p}}}{x} \, dx,x,\log \left (c \left (a+b x^2\right )^p\right )\right )}{4 b p^3}\\ &=\frac{\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{4 b p^3}-\frac{a+b x^2}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac{a+b x^2}{4 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}\\ \end{align*}
Mathematica [A] time = 0.0543933, size = 113, normalized size = 0.99 \[ -\frac{\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \left (p \left (c \left (a+b x^2\right )^p\right )^{\frac{1}{p}} \left (\log \left (c \left (a+b x^2\right )^p\right )+p\right )-\log ^2\left (c \left (a+b x^2\right )^p\right ) \text{Ei}\left (\frac{\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )\right )}{4 b p^3 \log ^2\left (c \left (a+b x^2\right )^p\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.2, size = 761, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b{\left (p + \log \left (c\right )\right )} x^{2} + a{\left (p + \log \left (c\right )\right )} +{\left (b x^{2} + a\right )} \log \left ({\left (b x^{2} + a\right )}^{p}\right )}{4 \,{\left (b p^{2} \log \left ({\left (b x^{2} + a\right )}^{p}\right )^{2} + 2 \, b p^{2} \log \left ({\left (b x^{2} + a\right )}^{p}\right ) \log \left (c\right ) + b p^{2} \log \left (c\right )^{2}\right )}} + \int \frac{x}{2 \,{\left (p^{2} \log \left ({\left (b x^{2} + a\right )}^{p}\right ) + p^{2} \log \left (c\right )\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25631, size = 378, normalized size = 3.32 \begin{align*} -\frac{{\left (b p^{2} x^{2} + a p^{2} +{\left (b p^{2} x^{2} + a p^{2}\right )} \log \left (b x^{2} + a\right ) +{\left (b p x^{2} + a p\right )} \log \left (c\right )\right )} c^{\left (\frac{1}{p}\right )} -{\left (p^{2} \log \left (b x^{2} + a\right )^{2} + 2 \, p \log \left (b x^{2} + a\right ) \log \left (c\right ) + \log \left (c\right )^{2}\right )} \logintegral \left ({\left (b x^{2} + a\right )} c^{\left (\frac{1}{p}\right )}\right )}{4 \,{\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac{1}{p}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\log{\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28209, size = 548, normalized size = 4.81 \begin{align*} -\frac{{\left (b x^{2} + a\right )} p^{2} \log \left (b x^{2} + a\right )}{4 \,{\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )}} + \frac{p^{2}{\rm Ei}\left (\frac{\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (b x^{2} + a\right )^{2}}{4 \,{\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac{1}{p}\right )}} - \frac{{\left (b x^{2} + a\right )} p^{2}}{4 \,{\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )}} - \frac{{\left (b x^{2} + a\right )} p \log \left (c\right )}{4 \,{\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )}} + \frac{p{\rm Ei}\left (\frac{\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (b x^{2} + a\right ) \log \left (c\right )}{2 \,{\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac{1}{p}\right )}} + \frac{{\rm Ei}\left (\frac{\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (c\right )^{2}}{4 \,{\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac{1}{p}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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