Integrand size = 18, antiderivative size = 204 \[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{4 b^2 p^3}+\frac {\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{b^2 p^3}-\frac {x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac {a \left (a+b x^2\right )}{4 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac {x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )} \]
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Time = 0.18 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2504, 2447, 2446, 2436, 2337, 2209, 2437, 2347, 2334} \[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\frac {\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{b^2 p^3}-\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{4 b^2 p^3}-\frac {a \left (a+b x^2\right )}{4 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac {x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac {x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )} \]
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Rule 2209
Rule 2334
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2446
Rule 2447
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {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 {\text {Subst}\left (\int \frac {x}{\log ^2\left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{2 p}+\frac {a \text {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 {\text {Subst}\left (\int \frac {x}{\log \left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{p^2}+\frac {a \text {Subst}\left (\int \frac {1}{\log \left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{2 b p^2}+\frac {a \text {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 {\text {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 \text {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 \text {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 {\text {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 \text {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 ) \text {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 ) \text {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 {\text {Subst}\left (\int \frac {x}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{b^2 p^2}-\frac {a \text {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 ) \text {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 ) \text {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*}
Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-2/p} \left (a \left (c \left (a+b x^2\right )^p\right )^{\frac {1}{p}} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )-4 \left (a+b x^2\right ) \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (a+b x^2\right )^p\right )}{p}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p \left (c \left (a+b x^2\right )^p\right )^{2/p} \left (b p x^2+\left (a+2 b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )\right )\right )}{4 b^2 p^3 \log ^2\left (c \left (a+b x^2\right )^p\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.90 (sec) , antiderivative size = 1969, normalized size of antiderivative = 9.65
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Time = 0.34 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.32 \[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (a p^{2} \log \left (b x^{2} + a\right )^{2} + 2 \, a p \log \left (b x^{2} + a\right ) \log \left (c\right ) + a \log \left (c\right )^{2}\right )} c^{\left (\frac {1}{p}\right )} \operatorname {log\_integral}\left ({\left (b x^{2} + a\right )} c^{\left (\frac {1}{p}\right )}\right ) + {\left (b^{2} p^{2} x^{4} + a b p^{2} x^{2} + {\left (2 \, b^{2} p^{2} x^{4} + 3 \, a b p^{2} x^{2} + a^{2} p^{2}\right )} \log \left (b x^{2} + a\right ) + {\left (2 \, b^{2} p x^{4} + 3 \, a b p x^{2} + a^{2} p\right )} \log \left (c\right )\right )} c^{\frac {2}{p}} - 4 \, {\left (p^{2} \log \left (b x^{2} + a\right )^{2} + 2 \, p \log \left (b x^{2} + a\right ) \log \left (c\right ) + \log \left (c\right )^{2}\right )} \operatorname {log\_integral}\left ({\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} c^{\frac {2}{p}}\right )}{4 \, {\left (b^{2} p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b^{2} p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b^{2} p^{3} \log \left (c\right )^{2}\right )} c^{\frac {2}{p}}} \]
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\[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x^{3}}{\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}\, dx \]
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\[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {x^{3}}{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (198) = 396\).
Time = 0.38 (sec) , antiderivative size = 874, normalized size of antiderivative = 4.28 \[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x^3}{{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3} \,d x \]
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