Optimal. Leaf size=141 \[ \frac{2 \left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p} \text{Ei}\left (\frac{2 \log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e^2 p^2}-\frac{d \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e^2 p^2}-\frac{x^3 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.207805, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {2454, 2400, 2399, 2389, 2300, 2178, 2390, 2310} \[ \frac{2 \left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p} \text{Ei}\left (\frac{2 \log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e^2 p^2}-\frac{d \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e^2 p^2}-\frac{x^3 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2400
Rule 2399
Rule 2389
Rule 2300
Rule 2178
Rule 2390
Rule 2310
Rubi steps
\begin{align*} \int \frac{x^5}{\log ^2\left (c \left (d+e x^3\right )^p\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{\log ^2\left (c (d+e x)^p\right )} \, dx,x,x^3\right )\\ &=-\frac{x^3 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{x}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 p}+\frac{d \operatorname{Subst}\left (\int \frac{1}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 e p}\\ &=-\frac{x^3 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{d}{e \log \left (c (d+e x)^p\right )}+\frac{d+e x}{e \log \left (c (d+e x)^p\right )}\right ) \, dx,x,x^3\right )}{3 p}+\frac{d \operatorname{Subst}\left (\int \frac{1}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e^2 p}\\ &=-\frac{x^3 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{d+e x}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 e p}-\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 e p}+\frac{\left (d \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^2 p^2}\\ &=\frac{d \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^2 p^2}-\frac{x^3 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{x}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e^2 p}-\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e^2 p}\\ &=\frac{d \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^2 p^2}-\frac{x^3 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac{\left (2 \left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{p}}}{x} \, dx,x,\log \left (c \left (d+e x^3\right )^p\right )\right )}{3 e^2 p^2}-\frac{\left (2 d \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^2 p^2}\\ &=-\frac{d \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^2 p^2}+\frac{2 \left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p} \text{Ei}\left (\frac{2 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{3 e^2 p^2}-\frac{x^3 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}\\ \end{align*}
Mathematica [A] time = 0.134023, size = 157, normalized size = 1.11 \[ -\frac{\left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-2/p} \left (d \left (c \left (d+e x^3\right )^p\right )^{\frac{1}{p}} \log \left (c \left (d+e x^3\right )^p\right ) \text{Ei}\left (\frac{\log \left (c \left (e x^3+d\right )^p\right )}{p}\right )-2 \left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right ) \text{Ei}\left (\frac{2 \log \left (c \left (e x^3+d\right )^p\right )}{p}\right )+e p x^3 \left (c \left (d+e x^3\right )^p\right )^{2/p}\right )}{3 e^2 p^2 \log \left (c \left (d+e x^3\right )^p\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 5.191, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{ \left ( \ln \left ( c \left ( e{x}^{3}+d \right ) ^{p} \right ) \right ) ^{2}}}\, dx \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^{6} + d x^{3}}{3 \,{\left (e p \log \left ({\left (e x^{3} + d\right )}^{p}\right ) + e p \log \left (c\right )\right )}} + \int \frac{2 \, e x^{5} + d x^{2}}{e p \log \left ({\left (e x^{3} + d\right )}^{p}\right ) + e p \log \left (c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.97112, size = 338, normalized size = 2.4 \begin{align*} -\frac{{\left (d p \log \left (e x^{3} + d\right ) + d \log \left (c\right )\right )} c^{\left (\frac{1}{p}\right )} \logintegral \left ({\left (e x^{3} + d\right )} c^{\left (\frac{1}{p}\right )}\right ) +{\left (e^{2} p x^{6} + d e p x^{3}\right )} c^{\frac{2}{p}} - 2 \,{\left (p \log \left (e x^{3} + d\right ) + \log \left (c\right )\right )} \logintegral \left ({\left (e^{2} x^{6} + 2 \, d e x^{3} + d^{2}\right )} c^{\frac{2}{p}}\right )}{3 \,{\left (e^{2} p^{3} \log \left (e x^{3} + d\right ) + e^{2} p^{2} \log \left (c\right )\right )} c^{\frac{2}{p}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.31127, size = 436, normalized size = 3.09 \begin{align*} -\frac{1}{3} \,{\left (\frac{{\left (x^{3} e + d\right )}^{2} p}{p^{3} e \log \left (x^{3} e + d\right ) + p^{2} e \log \left (c\right )} - \frac{{\left (x^{3} e + d\right )} d p}{p^{3} e \log \left (x^{3} e + d\right ) + p^{2} e \log \left (c\right )} + \frac{d 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 )}{{\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{2 \, p{\rm Ei}\left (\frac{2 \, \log \left (c\right )}{p} + 2 \, \log \left (x^{3} e + d\right )\right ) \log \left (x^{3} e + d\right )}{{\left (p^{3} e \log \left (x^{3} e + d\right ) + p^{2} e \log \left (c\right )\right )} c^{\frac{2}{p}}} + \frac{d{\rm Ei}\left (\frac{\log \left (c\right )}{p} + \log \left (x^{3} e + d\right )\right ) \log \left (c\right )}{{\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{2 \,{\rm Ei}\left (\frac{2 \, \log \left (c\right )}{p} + 2 \, \log \left (x^{3} e + d\right )\right ) \log \left (c\right )}{{\left (p^{3} e \log \left (x^{3} e + d\right ) + p^{2} e \log \left (c\right )\right )} c^{\frac{2}{p}}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]