Integrand size = 22, antiderivative size = 130 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=-\frac {2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {2 b d^3 n x}{9 e^3}-\frac {2 b d^2 n x^{5/3}}{15 e^2}+\frac {2 b d n x^{7/3}}{21 e}-\frac {2}{27} b n x^3+\frac {2 b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \] Output:
-2/3*b*d^4*n*x^(1/3)/e^4+2/9*b*d^3*n*x/e^3-2/15*b*d^2*n*x^(5/3)/e^2+2/21*b *d*n*x^(7/3)/e-2/27*b*n*x^3+2/3*b*d^(9/2)*n*arctan(e^(1/2)*x^(1/3)/d^(1/2) )/e^(9/2)+1/3*x^3*(a+b*ln(c*(d+e*x^(2/3))^n))
Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {a x^3}{3}-\frac {2}{9} b e n \left (\frac {3 d^4 \sqrt [3]{x}}{e^5}-\frac {d^3 x}{e^4}+\frac {3 d^2 x^{5/3}}{5 e^3}-\frac {3 d x^{7/3}}{7 e^2}+\frac {x^3}{3 e}-\frac {3 d^{9/2} \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{11/2}}\right )+\frac {1}{3} b x^3 \log \left (c \left (d+e x^{2/3}\right )^n\right ) \] Input:
Integrate[x^2*(a + b*Log[c*(d + e*x^(2/3))^n]),x]
Output:
(a*x^3)/3 - (2*b*e*n*((3*d^4*x^(1/3))/e^5 - (d^3*x)/e^4 + (3*d^2*x^(5/3))/ (5*e^3) - (3*d*x^(7/3))/(7*e^2) + x^3/(3*e) - (3*d^(9/2)*ArcTan[(Sqrt[e]*x ^(1/3))/Sqrt[d]])/e^(11/2)))/9 + (b*x^3*Log[c*(d + e*x^(2/3))^n])/3
Time = 0.47 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2905, 864, 254, 2009}
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.
\(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {2}{9} b e n \int \frac {x^{8/3}}{d+e x^{2/3}}dx\) |
\(\Big \downarrow \) 864 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {2}{3} b e n \int \frac {x^{10/3}}{d+e x^{2/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {2}{3} b e n \int \left (-\frac {d^5}{e^5 \left (d+e x^{2/3}\right )}+\frac {d^4}{e^5}-\frac {x^{2/3} d^3}{e^4}+\frac {x^{4/3} d^2}{e^3}-\frac {x^2 d}{e^2}+\frac {x^{8/3}}{e}\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {2}{3} b e n \left (-\frac {d^{9/2} \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{11/2}}+\frac {d^4 \sqrt [3]{x}}{e^5}-\frac {d^3 x}{3 e^4}+\frac {d^2 x^{5/3}}{5 e^3}-\frac {d x^{7/3}}{7 e^2}+\frac {x^3}{9 e}\right )\) |
Input:
Int[x^2*(a + b*Log[c*(d + e*x^(2/3))^n]),x]
Output:
(-2*b*e*n*((d^4*x^(1/3))/e^5 - (d^3*x)/(3*e^4) + (d^2*x^(5/3))/(5*e^3) - ( d*x^(7/3))/(7*e^2) + x^3/(9*e) - (d^(9/2)*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d] ])/e^(11/2)))/3 + (x^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/3
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x ^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^ (m_.), x_Symbol] :> Simp[(f*x)^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Simp[b*e*n*(p/(f*(m + 1))) Int[x^(n - 1)*((f*x)^(m + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]
\[\int x^{2} \left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )d x\]
Input:
int(x^2*(a+b*ln(c*(d+e*x^(2/3))^n)),x)
Output:
int(x^2*(a+b*ln(c*(d+e*x^(2/3))^n)),x)
Time = 0.10 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.59 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\left [\frac {315 \, b e^{4} n x^{3} \log \left (e x^{\frac {2}{3}} + d\right ) + 315 \, b e^{4} x^{3} \log \left (c\right ) - 126 \, b d^{2} e^{2} n x^{\frac {5}{3}} + 315 \, b d^{4} n \sqrt {-\frac {d}{e}} \log \left (\frac {e^{3} x^{2} - 2 \, d e^{2} x \sqrt {-\frac {d}{e}} - d^{3} + 2 \, {\left (e^{3} x \sqrt {-\frac {d}{e}} + d^{2} e\right )} x^{\frac {2}{3}} - 2 \, {\left (d e^{2} x - d^{2} e \sqrt {-\frac {d}{e}}\right )} x^{\frac {1}{3}}}{e^{3} x^{2} + d^{3}}\right ) + 210 \, b d^{3} e n x - 35 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{3} + 90 \, {\left (b d e^{3} n x^{2} - 7 \, b d^{4} n\right )} x^{\frac {1}{3}}}{945 \, e^{4}}, \frac {315 \, b e^{4} n x^{3} \log \left (e x^{\frac {2}{3}} + d\right ) + 315 \, b e^{4} x^{3} \log \left (c\right ) - 126 \, b d^{2} e^{2} n x^{\frac {5}{3}} + 630 \, b d^{4} n \sqrt {\frac {d}{e}} \arctan \left (\frac {e x^{\frac {1}{3}} \sqrt {\frac {d}{e}}}{d}\right ) + 210 \, b d^{3} e n x - 35 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{3} + 90 \, {\left (b d e^{3} n x^{2} - 7 \, b d^{4} n\right )} x^{\frac {1}{3}}}{945 \, e^{4}}\right ] \] Input:
integrate(x^2*(a+b*log(c*(d+e*x^(2/3))^n)),x, algorithm="fricas")
Output:
[1/945*(315*b*e^4*n*x^3*log(e*x^(2/3) + d) + 315*b*e^4*x^3*log(c) - 126*b* d^2*e^2*n*x^(5/3) + 315*b*d^4*n*sqrt(-d/e)*log((e^3*x^2 - 2*d*e^2*x*sqrt(- d/e) - d^3 + 2*(e^3*x*sqrt(-d/e) + d^2*e)*x^(2/3) - 2*(d*e^2*x - d^2*e*sqr t(-d/e))*x^(1/3))/(e^3*x^2 + d^3)) + 210*b*d^3*e*n*x - 35*(2*b*e^4*n - 9*a *e^4)*x^3 + 90*(b*d*e^3*n*x^2 - 7*b*d^4*n)*x^(1/3))/e^4, 1/945*(315*b*e^4* n*x^3*log(e*x^(2/3) + d) + 315*b*e^4*x^3*log(c) - 126*b*d^2*e^2*n*x^(5/3) + 630*b*d^4*n*sqrt(d/e)*arctan(e*x^(1/3)*sqrt(d/e)/d) + 210*b*d^3*e*n*x - 35*(2*b*e^4*n - 9*a*e^4)*x^3 + 90*(b*d*e^3*n*x^2 - 7*b*d^4*n)*x^(1/3))/e^4 ]
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(x**2*(a+b*ln(c*(d+e*x**(2/3))**n)),x)
Output:
Timed out
Exception generated. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(a+b*log(c*(d+e*x^(2/3))^n)),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.15 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {1}{3} \, b x^{3} \log \left (c\right ) + \frac {1}{3} \, a x^{3} + \frac {1}{945} \, {\left (315 \, x^{3} \log \left (e x^{\frac {2}{3}} + d\right ) + 2 \, e {\left (\frac {315 \, d^{5} \arctan \left (\frac {e x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} e^{5}} - \frac {35 \, e^{8} x^{3} - 45 \, d e^{7} x^{\frac {7}{3}} + 63 \, d^{2} e^{6} x^{\frac {5}{3}} - 105 \, d^{3} e^{5} x + 315 \, d^{4} e^{4} x^{\frac {1}{3}}}{e^{9}}\right )}\right )} b n \] Input:
integrate(x^2*(a+b*log(c*(d+e*x^(2/3))^n)),x, algorithm="giac")
Output:
1/3*b*x^3*log(c) + 1/3*a*x^3 + 1/945*(315*x^3*log(e*x^(2/3) + d) + 2*e*(31 5*d^5*arctan(e*x^(1/3)/sqrt(d*e))/(sqrt(d*e)*e^5) - (35*e^8*x^3 - 45*d*e^7 *x^(7/3) + 63*d^2*e^6*x^(5/3) - 105*d^3*e^5*x + 315*d^4*e^4*x^(1/3))/e^9)) *b*n
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\int x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right ) \,d x \] Input:
int(x^2*(a + b*log(c*(d + e*x^(2/3))^n)),x)
Output:
int(x^2*(a + b*log(c*(d + e*x^(2/3))^n)), x)
Time = 0.16 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.90 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {630 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) b \,d^{4} n -126 x^{\frac {5}{3}} b \,d^{2} e^{3} n -630 x^{\frac {1}{3}} b \,d^{4} e n +90 x^{\frac {7}{3}} b d \,e^{4} n +315 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b \,e^{5} x^{3}+315 a \,e^{5} x^{3}+210 b \,d^{3} e^{2} n x -70 b \,e^{5} n \,x^{3}}{945 e^{5}} \] Input:
int(x^2*(a+b*log(c*(d+e*x^(2/3))^n)),x)
Output:
(630*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*b*d**4*n - 126*x **(2/3)*b*d**2*e**3*n*x - 630*x**(1/3)*b*d**4*e*n + 90*x**(1/3)*b*d*e**4*n *x**2 + 315*log((x**(2/3)*e + d)**n*c)*b*e**5*x**3 + 315*a*e**5*x**3 + 210 *b*d**3*e**2*n*x - 70*b*e**5*n*x**3)/(945*e**5)