Optimal. Leaf size=45 \[ \frac{\text{Ei}\left (2 \log \left (c \left (b x^2+a\right )\right )\right )}{2 b^2 c^2}-\frac{a \text{li}\left (c \left (b x^2+a\right )\right )}{2 b^2 c} \]
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Rubi [A] time = 0.0954564, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {2454, 2399, 2389, 2298, 2390, 2309, 2178} \[ \frac{\text{Ei}\left (2 \log \left (c \left (b x^2+a\right )\right )\right )}{2 b^2 c^2}-\frac{a \text{li}\left (c \left (b x^2+a\right )\right )}{2 b^2 c} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2399
Rule 2389
Rule 2298
Rule 2390
Rule 2309
Rule 2178
Rubi steps
\begin{align*} \int \frac{x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\log (c (a+b x))} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b \log (c (a+b x))}+\frac{a+b x}{b \log (c (a+b x))}\right ) \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{a+b x}{\log (c (a+b x))} \, dx,x,x^2\right )}{2 b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\log (c (a+b x))} \, dx,x,x^2\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b^2}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b^2}\\ &=-\frac{a \text{li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c}+\frac{\operatorname{Subst}\left (\int \frac{e^{2 x}}{x} \, dx,x,\log \left (c \left (a+b x^2\right )\right )\right )}{2 b^2 c^2}\\ &=\frac{\text{Ei}\left (2 \log \left (c \left (a+b x^2\right )\right )\right )}{2 b^2 c^2}-\frac{a \text{li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c}\\ \end{align*}
Mathematica [A] time = 0.0851304, size = 41, normalized size = 0.91 \[ \frac{\text{Ei}\left (2 \log \left (b c x^2+a c\right )\right )-a c \text{Ei}\left (\log \left (b c x^2+a c\right )\right )}{2 b^2 c^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.352, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{\ln \left ( c \left ( b{x}^{2}+a \right ) \right ) }}\, 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*} \int \frac{x^{3}}{\log \left ({\left (b x^{2} + a\right )} c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87321, size = 140, normalized size = 3.11 \begin{align*} -\frac{a c \logintegral \left (b c x^{2} + a c\right ) - \logintegral \left (b^{2} c^{2} x^{4} + 2 \, a b c^{2} x^{2} + a^{2} c^{2}\right )}{2 \, b^{2} c^{2}} \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 (a c + b c x^{2} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18719, size = 51, normalized size = 1.13 \begin{align*} -\frac{a c{\rm Ei}\left (\log \left ({\left (b x^{2} + a\right )} c\right )\right ) -{\rm Ei}\left (2 \, \log \left ({\left (b x^{2} + a\right )} c\right )\right )}{2 \, b^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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