Optimal. Leaf size=204 \[ \frac{\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \text{Ei}\left (\frac{2 \log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{b^2 p^3}-\frac{a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{4 b^2 p^3}-\frac{a \left (a+b x^2\right )}{4 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )} \]
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Rubi [A] time = 0.288515, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2454, 2400, 2399, 2389, 2300, 2178, 2390, 2310, 2297} \[ \frac{\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \text{Ei}\left (\frac{2 \log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{b^2 p^3}-\frac{a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{4 b^2 p^3}-\frac{a \left (a+b x^2\right )}{4 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2400
Rule 2399
Rule 2389
Rule 2300
Rule 2178
Rule 2390
Rule 2310
Rule 2297
Rubi steps
\begin{align*} \int \frac{x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\log ^3\left (c (a+b x)^p\right )} \, dx,x,x^2\right )\\ &=-\frac{x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}+\frac{\operatorname{Subst}\left (\int \frac{x}{\log ^2\left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{2 p}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\log ^2\left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{4 b p}\\ &=-\frac{x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac{x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}+\frac{\operatorname{Subst}\left (\int \frac{x}{\log \left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{p^2}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\log \left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{2 b p^2}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\log ^2\left (c x^p\right )} \, dx,x,a+b x^2\right )}{4 b^2 p}\\ &=-\frac{x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac{a \left (a+b x^2\right )}{4 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}+\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b \log \left (c (a+b x)^p\right )}+\frac{a+b x}{b \log \left (c (a+b x)^p\right )}\right ) \, dx,x,x^2\right )}{p^2}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{4 b^2 p^2}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{2 b^2 p^2}\\ &=-\frac{x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac{a \left (a+b x^2\right )}{4 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}+\frac{\operatorname{Subst}\left (\int \frac{a+b x}{\log \left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{b p^2}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\log \left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{b p^2}+\frac{\left (a \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^2 p^3}+\frac{\left (a \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 )}{2 b^2 p^3}\\ &=\frac{3 a \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^2 p^3}-\frac{x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac{a \left (a+b x^2\right )}{4 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}+\frac{\operatorname{Subst}\left (\int \frac{x}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{b^2 p^2}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{b^2 p^2}\\ &=\frac{3 a \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^2 p^3}-\frac{x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac{a \left (a+b x^2\right )}{4 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}+\frac{\left (\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{p}}}{x} \, dx,x,\log \left (c \left (a+b x^2\right )^p\right )\right )}{b^2 p^3}-\frac{\left (a \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 )}{b^2 p^3}\\ &=-\frac{a \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^2 p^3}+\frac{\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \text{Ei}\left (\frac{2 \log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{b^2 p^3}-\frac{x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac{a \left (a+b x^2\right )}{4 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac{x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}\\ \end{align*}
Mathematica [A] time = 0.193008, size = 185, normalized size = 0.91 \[ -\frac{\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-2/p} \left (a \left (c \left (a+b x^2\right )^p\right )^{\frac{1}{p}} \log ^2\left (c \left (a+b x^2\right )^p\right ) \text{Ei}\left (\frac{\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )-4 \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right ) \text{Ei}\left (\frac{2 \log \left (c \left (b x^2+a\right )^p\right )}{p}\right )+p \left (c \left (a+b x^2\right )^p\right )^{2/p} \left (\left (a+2 b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+b p x^2\right )\right )}{4 b^2 p^3 \log ^2\left (c \left (a+b x^2\right )^p\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 5.264, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}}}\, dx \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^{2}{\left (p + 2 \, \log \left (c\right )\right )} x^{4} + a b{\left (p + 3 \, \log \left (c\right )\right )} x^{2} + a^{2} \log \left (c\right ) +{\left (2 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}\right )} \log \left ({\left (b x^{2} + a\right )}^{p}\right )}{4 \,{\left (b^{2} p^{2} \log \left ({\left (b x^{2} + a\right )}^{p}\right )^{2} + 2 \, b^{2} p^{2} \log \left ({\left (b x^{2} + a\right )}^{p}\right ) \log \left (c\right ) + b^{2} p^{2} \log \left (c\right )^{2}\right )}} + \int \frac{4 \, b x^{3} + 3 \, a x}{2 \,{\left (b p^{2} \log \left ({\left (b x^{2} + a\right )}^{p}\right ) + b 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 = 2.20534, size = 629, normalized size = 3.08 \begin{align*} -\frac{{\left (a p^{2} \log \left (b x^{2} + a\right )^{2} + 2 \, a p \log \left (b x^{2} + a\right ) \log \left (c\right ) + a \log \left (c\right )^{2}\right )} c^{\left (\frac{1}{p}\right )} \logintegral \left ({\left (b x^{2} + a\right )} c^{\left (\frac{1}{p}\right )}\right ) +{\left (b^{2} p^{2} x^{4} + a b p^{2} x^{2} +{\left (2 \, b^{2} p^{2} x^{4} + 3 \, a b p^{2} x^{2} + a^{2} p^{2}\right )} \log \left (b x^{2} + a\right ) +{\left (2 \, b^{2} p x^{4} + 3 \, a b p x^{2} + a^{2} p\right )} \log \left (c\right )\right )} c^{\frac{2}{p}} - 4 \,{\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^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} c^{\frac{2}{p}}\right )}{4 \,{\left (b^{2} p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b^{2} p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b^{2} p^{3} \log \left (c\right )^{2}\right )} c^{\frac{2}{p}}} \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^{3}}{\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 = 1.24803, size = 1134, normalized size = 5.56 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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