Integrand size = 22, antiderivative size = 239 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\frac {b^2 e^2 n^2 x^{2/3}}{2 d^2}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{x^{2/3}}\right )}{2 d^3}-\frac {b e^2 n \left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 d}-\frac {b e^3 n \log \left (1-\frac {d}{d+\frac {e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{d^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2-\frac {b^2 e^3 n^2 \log (x)}{d^3}+\frac {b^2 e^3 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{d^3} \] Output:
1/2*b^2*e^2*n^2*x^(2/3)/d^2-1/2*b^2*e^3*n^2*ln(d+e/x^(2/3))/d^3-b*e^2*n*(d +e/x^(2/3))*x^(2/3)*(a+b*ln(c*(d+e/x^(2/3))^n))/d^3+1/2*b*e*n*x^(4/3)*(a+b *ln(c*(d+e/x^(2/3))^n))/d-b*e^3*n*ln(1-d/(d+e/x^(2/3)))*(a+b*ln(c*(d+e/x^( 2/3))^n))/d^3+1/2*x^2*(a+b*ln(c*(d+e/x^(2/3))^n))^2-b^2*e^3*n^2*ln(x)/d^3+ b^2*e^3*n^2*polylog(2,d/(d+e/x^(2/3)))/d^3
Leaf count is larger than twice the leaf count of optimal. \(542\) vs. \(2(239)=478\).
Time = 0.47 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.27 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2-\frac {b e n \left (6 d e x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-3 d^2 x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-6 e^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )-6 e^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+2 b e^2 n \left (3 \log \left (d+\frac {e}{x^{2/3}}\right )+2 \log (x)\right )+b e n \left (-3 d x^{2/3}+3 e \log \left (d+\frac {e}{x^{2/3}}\right )+2 e \log (x)\right )+3 b e^2 n \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \operatorname {PolyLog}\left (2,1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )\right )+3 b e^2 n \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )-4 \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )\right )}{6 d^3} \] Input:
Integrate[x*(a + b*Log[c*(d + e/x^(2/3))^n])^2,x]
Output:
(x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/2 - (b*e*n*(6*d*e*x^(2/3)*(a + b* Log[c*(d + e/x^(2/3))^n]) - 3*d^2*x^(4/3)*(a + b*Log[c*(d + e/x^(2/3))^n]) - 6*e^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] - 6*e^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 2*b*e^2*n*(3*Log[d + e/x^(2/3)] + 2*Log[x]) + b*e*n*(-3*d*x^(2/3) + 3*e*L og[d + e/x^(2/3)] + 2*e*Log[x]) + 3*b*e^2*n*(Log[Sqrt[e] - Sqrt[-d]*x^(1/3 )]*(Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] + 2*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e ])/2] - 4*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]]) - 4*PolyLog[2, 1 - (Sqrt[-d]*x^ (1/3))/Sqrt[e]] + 2*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])]) + 3* b*e^2*n*(Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*(Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 2*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] - 4*Log[-((Sqrt[-d]*x^(1/3)) /Sqrt[e])]) + 2*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 4*PolyLog [2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]])))/(6*d^3)
Time = 1.62 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {2904, 2845, 2858, 25, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}
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.
| \(\displaystyle \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -\frac {3}{2} \int x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2d\frac {1}{x^{2/3}}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle -\frac {3}{2} \left (\frac {2}{3} b e n \int \frac {x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{d+\frac {e}{x^{2/3}}}d\frac {1}{x^{2/3}}-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -\frac {3}{2} \left (\frac {2}{3} b n \int x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )d\left (d+\frac {e}{x^{2/3}}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b n \int -x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )d\left (d+\frac {e}{x^{2/3}}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b e^3 n \int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b e^3 n \left (\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \frac {x^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \left (\frac {x^{4/3}}{d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {x^{2/3}}{d^2}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b e^3 n \left (\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {-\frac {b n \int -\frac {x^{2/3}}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2\right )\) |
Input:
Int[x*(a + b*Log[c*(d + e/x^(2/3))^n])^2,x]
Output:
(-3*(-1/3*(x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^2) - (2*b*e^3*n*((-1/2*(b* n*(-(x^(2/3)/(d*e)) + Log[d + e/x^(2/3)]/d^2 - Log[-(e/x^(2/3))]/d^2)) + ( x^(4/3)*(a + b*Log[c/x^((2*n)/3)]))/(2*e^2))/d + (((b*n*Log[-(e/x^(2/3))]) /d - ((d + e/x^(2/3))*x^(2/3)*(a + b*Log[c/x^((2*n)/3)]))/(d*e))/d + (-((L og[1 - d*x^(2/3)]*(a + b*Log[c/x^((2*n)/3)]))/d) + (b*n*PolyLog[2, d*x^(2/ 3)])/d)/d)/d))/3))/2
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
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])
\[\int x {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )}^{2}d x\]
Input:
int(x*(a+b*ln(c*(d+e/x^(2/3))^n))^2,x)
Output:
int(x*(a+b*ln(c*(d+e/x^(2/3))^n))^2,x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e/x^(2/3))^n))^2,x, algorithm="fricas")
Output:
integral(b^2*x*log(c*((d*x + e*x^(1/3))/x)^n)^2 + 2*a*b*x*log(c*((d*x + e* x^(1/3))/x)^n) + a^2*x, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\text {Timed out} \] Input:
integrate(x*(a+b*ln(c*(d+e/x**(2/3))**n))**2,x)
Output:
Timed out
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e/x^(2/3))^n))^2,x, algorithm="maxima")
Output:
1/2*b^2*x^2*log((d*x^(2/3) + e)^n)^2 - integrate(-1/3*(3*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^2 + 12*(b^2*d*x^2 + b^2*e*x^(4/3))*log(x^(1/3*n ))^2 + 3*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^(4/3) - 2*(b^2*d*n*x^ 2 - 3*(b^2*d*log(c) + a*b*d)*x^2 - 3*(b^2*e*log(c) + a*b*e)*x^(4/3) + 6*(b ^2*d*x^2 + b^2*e*x^(4/3))*log(x^(1/3*n)))*log((d*x^(2/3) + e)^n) - 12*((b^ 2*d*log(c) + a*b*d)*x^2 + (b^2*e*log(c) + a*b*e)*x^(4/3))*log(x^(1/3*n)))/ (d*x + e*x^(1/3)), x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e/x^(2/3))^n))^2,x, algorithm="giac")
Output:
integrate((b*log(c*(d + e/x^(2/3))^n) + a)^2*x, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^2 \,d x \] Input:
int(x*(a + b*log(c*(d + e/x^(2/3))^n))^2,x)
Output:
int(x*(a + b*log(c*(d + e/x^(2/3))^n))^2, x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\frac {-6 x^{\frac {2}{3}} a b d \,e^{2} n +3 x^{\frac {2}{3}} b^{2} d \,e^{2} n^{2}+3 x^{\frac {4}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d^{2} e n +3 x^{\frac {4}{3}} a b \,d^{2} e n -4 \left (\int \frac {x^{\frac {1}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}{x^{\frac {2}{3}} d +e}d x \right ) b^{2} d^{2} e^{2} n +12 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) a b \,e^{3} n -6 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) b^{2} e^{3} n^{2}+3 {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} b^{2} d^{3} x^{2}+6 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a b \,d^{3} x^{2}+6 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a b \,e^{3}-3 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} e^{3} n +3 a^{2} d^{3} x^{2}}{6 d^{3}} \] Input:
int(x*(a+b*log(c*(d+e/x^(2/3))^n))^2,x)
Output:
( - 6*x**(2/3)*a*b*d*e**2*n + 3*x**(2/3)*b**2*d*e**2*n**2 + 3*x**(1/3)*log (((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d**2*e*n*x + 3*x**(1/3)*a*b*d* *2*e*n*x - 4*int((x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3)))/(x** (2/3)*d + e),x)*b**2*d**2*e**2*n + 12*log(x**(1/3))*a*b*e**3*n - 6*log(x** (1/3))*b**2*e**3*n**2 + 3*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*b** 2*d**3*x**2 + 6*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*a*b*d**3*x**2 + 6*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*a*b*e**3 - 3*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*e**3*n + 3*a**2*d**3*x**2)/(6*d**3)