Optimal. Leaf size=83 \[ \frac{\left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e p^2}-\frac{d+e x^3}{3 e p \log \left (c \left (d+e x^3\right )^p\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0820962, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {2454, 2389, 2297, 2300, 2178} \[ \frac{\left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e p^2}-\frac{d+e x^3}{3 e p \log \left (c \left (d+e x^3\right )^p\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2389
Rule 2297
Rule 2300
Rule 2178
Rubi steps
\begin{align*} \int \frac{x^2}{\log ^2\left (c \left (d+e x^3\right )^p\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\log ^2\left (c (d+e x)^p\right )} \, dx,x,x^3\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\log ^2\left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e}\\ &=-\frac{d+e x^3}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e p}\\ &=-\frac{d+e x^3}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac{\left (\left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{p}}}{x} \, dx,x,\log \left (c \left (d+e x^3\right )^p\right )\right )}{3 e p^2}\\ &=\frac{\left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{3 e p^2}-\frac{d+e x^3}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}\\ \end{align*}
Mathematica [A] time = 0.0454005, size = 97, normalized size = 1.17 \[ -\frac{\left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \left (p \left (c \left (d+e x^3\right )^p\right )^{\frac{1}{p}}-\log \left (c \left (d+e x^3\right )^p\right ) \text{Ei}\left (\frac{\log \left (c \left (e x^3+d\right )^p\right )}{p}\right )\right )}{3 e p^2 \log \left (c \left (d+e x^3\right )^p\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 1.288, size = 466, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{e x^{3} + d}{3 \,{\left (e p \log \left ({\left (e x^{3} + d\right )}^{p}\right ) + e p \log \left (c\right )\right )}} + \int \frac{x^{2}}{p \log \left ({\left (e x^{3} + d\right )}^{p}\right ) + p \log \left (c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.98641, size = 194, normalized size = 2.34 \begin{align*} -\frac{{\left (e p x^{3} + d p\right )} c^{\left (\frac{1}{p}\right )} -{\left (p \log \left (e x^{3} + d\right ) + \log \left (c\right )\right )} \logintegral \left ({\left (e x^{3} + d\right )} c^{\left (\frac{1}{p}\right )}\right )}{3 \,{\left (e p^{3} \log \left (e x^{3} + d\right ) + e p^{2} \log \left (c\right )\right )} c^{\left (\frac{1}{p}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\log{\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.28646, size = 208, normalized size = 2.51 \begin{align*} -\frac{{\left (x^{3} e + d\right )} p}{3 \,{\left (p^{3} e \log \left (x^{3} e + d\right ) + p^{2} e \log \left (c\right )\right )}} + \frac{p{\rm Ei}\left (\frac{\log \left (c\right )}{p} + \log \left (x^{3} e + d\right )\right ) \log \left (x^{3} e + d\right )}{3 \,{\left (p^{3} e \log \left (x^{3} e + d\right ) + p^{2} e \log \left (c\right )\right )} c^{\left (\frac{1}{p}\right )}} + \frac{{\rm Ei}\left (\frac{\log \left (c\right )}{p} + \log \left (x^{3} e + d\right )\right ) \log \left (c\right )}{3 \,{\left (p^{3} e \log \left (x^{3} e + d\right ) + p^{2} e \log \left (c\right )\right )} c^{\left (\frac{1}{p}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]