Integrand size = 18, antiderivative size = 66 \[ \int x^2 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {2 p^2 x^3}{3}-\frac {2 p \left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e}+\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 e} \] Output:
2/3*p^2*x^3-2/3*p*(e*x^3+d)*ln(c*(e*x^3+d)^p)/e+1/3*(e*x^3+d)*ln(c*(e*x^3+ d)^p)^2/e
Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int x^2 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {1}{3} \left (\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{e}-2 p \left (-p x^3+\frac {\left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e}\right )\right ) \] Input:
Integrate[x^2*Log[c*(d + e*x^3)^p]^2,x]
Output:
(((d + e*x^3)*Log[c*(d + e*x^3)^p]^2)/e - 2*p*(-(p*x^3) + ((d + e*x^3)*Log [c*(d + e*x^3)^p])/e))/3
Time = 0.43 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2904, 2836, 2733, 2732}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {1}{3} \int \log ^2\left (c \left (e x^3+d\right )^p\right )dx^3\) |
\(\Big \downarrow \) 2836 |
\(\displaystyle \frac {\int \log ^2\left (c \left (e x^3+d\right )^p\right )d\left (e x^3+d\right )}{3 e}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )-2 p \int \log \left (c \left (e x^3+d\right )^p\right )d\left (e x^3+d\right )}{3 e}\) |
\(\Big \downarrow \) 2732 |
\(\displaystyle \frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )-2 p \left (\left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right )-p \left (d+e x^3\right )\right )}{3 e}\) |
Input:
Int[x^2*Log[c*(d + e*x^3)^p]^2,x]
Output:
((d + e*x^3)*Log[c*(d + e*x^3)^p]^2 - 2*p*(-(p*(d + e*x^3)) + (d + e*x^3)* Log[c*(d + e*x^3)^p]))/(3*e)
Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x ] /; FreeQ[{c, n}, x]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b *Log[c*x^n])^p, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : > Simp[1/e Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ a, b, c, d, e, n, p}, x]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
Time = 1.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.53
method | result | size |
parallelrisch | \(\frac {x^{3} {\ln \left (c \left (e \,x^{3}+d \right )^{p}\right )}^{2} e p -2 x^{3} \ln \left (c \left (e \,x^{3}+d \right )^{p}\right ) e \,p^{2}+2 x^{3} e \,p^{3}+{\ln \left (c \left (e \,x^{3}+d \right )^{p}\right )}^{2} d p -2 \ln \left (c \left (e \,x^{3}+d \right )^{p}\right ) d \,p^{2}-2 d \,p^{3}}{3 e p}\) | \(101\) |
risch | \(\text {Expression too large to display}\) | \(1036\) |
Input:
int(x^2*ln(c*(e*x^3+d)^p)^2,x,method=_RETURNVERBOSE)
Output:
1/3*(x^3*ln(c*(e*x^3+d)^p)^2*e*p-2*x^3*ln(c*(e*x^3+d)^p)*e*p^2+2*x^3*e*p^3 +ln(c*(e*x^3+d)^p)^2*d*p-2*ln(c*(e*x^3+d)^p)*d*p^2-2*d*p^3)/e/p
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.45 \[ \int x^2 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {2 \, e p^{2} x^{3} - 2 \, e p x^{3} \log \left (c\right ) + e x^{3} \log \left (c\right )^{2} + {\left (e p^{2} x^{3} + d p^{2}\right )} \log \left (e x^{3} + d\right )^{2} - 2 \, {\left (e p^{2} x^{3} + d p^{2} - {\left (e p x^{3} + d p\right )} \log \left (c\right )\right )} \log \left (e x^{3} + d\right )}{3 \, e} \] Input:
integrate(x^2*log(c*(e*x^3+d)^p)^2,x, algorithm="fricas")
Output:
1/3*(2*e*p^2*x^3 - 2*e*p*x^3*log(c) + e*x^3*log(c)^2 + (e*p^2*x^3 + d*p^2) *log(e*x^3 + d)^2 - 2*(e*p^2*x^3 + d*p^2 - (e*p*x^3 + d*p)*log(c))*log(e*x ^3 + d))/e
Time = 1.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.52 \[ \int x^2 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\begin {cases} - \frac {2 d p \log {\left (c \left (d + e x^{3}\right )^{p} \right )}}{3 e} + \frac {d \log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{3 e} + \frac {2 p^{2} x^{3}}{3} - \frac {2 p x^{3} \log {\left (c \left (d + e x^{3}\right )^{p} \right )}}{3} + \frac {x^{3} \log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{3} & \text {for}\: e \neq 0 \\\frac {x^{3} \log {\left (c d^{p} \right )}^{2}}{3} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*ln(c*(e*x**3+d)**p)**2,x)
Output:
Piecewise((-2*d*p*log(c*(d + e*x**3)**p)/(3*e) + d*log(c*(d + e*x**3)**p)* *2/(3*e) + 2*p**2*x**3/3 - 2*p*x**3*log(c*(d + e*x**3)**p)/3 + x**3*log(c* (d + e*x**3)**p)**2/3, Ne(e, 0)), (x**3*log(c*d**p)**2/3, True))
Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.47 \[ \int x^2 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {1}{3} \, x^{3} \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2} - \frac {2}{3} \, {\left (\frac {x^{3}}{e} - \frac {d \log \left (e x^{3} + d\right )}{e^{2}}\right )} e p \log \left ({\left (e x^{3} + d\right )}^{p} c\right ) + \frac {{\left (2 \, e x^{3} - d \log \left (e x^{3} + d\right )^{2} - 2 \, d \log \left (e x^{3} + d\right )\right )} p^{2}}{3 \, e} \] Input:
integrate(x^2*log(c*(e*x^3+d)^p)^2,x, algorithm="maxima")
Output:
1/3*x^3*log((e*x^3 + d)^p*c)^2 - 2/3*(x^3/e - d*log(e*x^3 + d)/e^2)*e*p*lo g((e*x^3 + d)^p*c) + 1/3*(2*e*x^3 - d*log(e*x^3 + d)^2 - 2*d*log(e*x^3 + d ))*p^2/e
Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.45 \[ \int x^2 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {{\left (2 \, e x^{3} + {\left (e x^{3} + d\right )} \log \left (e x^{3} + d\right )^{2} - 2 \, {\left (e x^{3} + d\right )} \log \left (e x^{3} + d\right ) + 2 \, d\right )} p^{2} - 2 \, {\left (e x^{3} - {\left (e x^{3} + d\right )} \log \left (e x^{3} + d\right ) + d\right )} p \log \left (c\right ) + {\left (e x^{3} + d\right )} \log \left (c\right )^{2}}{3 \, e} \] Input:
integrate(x^2*log(c*(e*x^3+d)^p)^2,x, algorithm="giac")
Output:
1/3*((2*e*x^3 + (e*x^3 + d)*log(e*x^3 + d)^2 - 2*(e*x^3 + d)*log(e*x^3 + d ) + 2*d)*p^2 - 2*(e*x^3 - (e*x^3 + d)*log(e*x^3 + d) + d)*p*log(c) + (e*x^ 3 + d)*log(c)^2)/e
Time = 25.56 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08 \[ \int x^2 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {2\,p^2\,x^3}{3}+{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2\,\left (\frac {d}{3\,e}+\frac {x^3}{3}\right )-\frac {2\,p\,x^3\,\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}{3}-\frac {2\,d\,p^2\,\ln \left (e\,x^3+d\right )}{3\,e} \] Input:
int(x^2*log(c*(d + e*x^3)^p)^2,x)
Output:
(2*p^2*x^3)/3 + log(c*(d + e*x^3)^p)^2*(d/(3*e) + x^3/3) - (2*p*x^3*log(c* (d + e*x^3)^p))/3 - (2*d*p^2*log(d + e*x^3))/(3*e)
Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.29 \[ \int x^2 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {{\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}^{2} d +{\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}^{2} e \,x^{3}-2 \,\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) d p -2 \,\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) e p \,x^{3}+2 e \,p^{2} x^{3}}{3 e} \] Input:
int(x^2*log(c*(e*x^3+d)^p)^2,x)
Output:
(log((d + e*x**3)**p*c)**2*d + log((d + e*x**3)**p*c)**2*e*x**3 - 2*log((d + e*x**3)**p*c)*d*p - 2*log((d + e*x**3)**p*c)*e*p*x**3 + 2*e*p**2*x**3)/ (3*e)