Integrand size = 20, antiderivative size = 136 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {b d^5 n \sqrt [3]{x}}{2 e^5}-\frac {b d^4 n x^{2/3}}{4 e^4}+\frac {b d^3 n x}{6 e^3}-\frac {b d^2 n x^{4/3}}{8 e^2}+\frac {b d n x^{5/3}}{10 e}-\frac {1}{12} b n x^2-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \] Output:
1/2*b*d^5*n*x^(1/3)/e^5-1/4*b*d^4*n*x^(2/3)/e^4+1/6*b*d^3*n*x/e^3-1/8*b*d^ 2*n*x^(4/3)/e^2+1/10*b*d*n*x^(5/3)/e-1/12*b*n*x^2-1/2*b*d^6*n*ln(d+e*x^(1/ 3))/e^6+1/2*x^2*(a+b*ln(c*(d+e*x^(1/3))^n))
Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.97 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {a x^2}{2}-\frac {1}{6} b e n \left (-\frac {3 d^5 \sqrt [3]{x}}{e^6}+\frac {3 d^4 x^{2/3}}{2 e^5}-\frac {d^3 x}{e^4}+\frac {3 d^2 x^{4/3}}{4 e^3}-\frac {3 d x^{5/3}}{5 e^2}+\frac {x^2}{2 e}+\frac {3 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^7}\right )+\frac {1}{2} b x^2 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \] Input:
Integrate[x*(a + b*Log[c*(d + e*x^(1/3))^n]),x]
Output:
(a*x^2)/2 - (b*e*n*((-3*d^5*x^(1/3))/e^6 + (3*d^4*x^(2/3))/(2*e^5) - (d^3* x)/e^4 + (3*d^2*x^(4/3))/(4*e^3) - (3*d*x^(5/3))/(5*e^2) + x^2/(2*e) + (3* d^6*Log[d + e*x^(1/3)])/e^7))/6 + (b*x^2*Log[c*(d + e*x^(1/3))^n])/2
Time = 0.52 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2904, 2842, 49, 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.
\(\displaystyle \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle 3 \int x^{5/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle 3 \left (\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{6} b e n \int \frac {x^2}{d+e \sqrt [3]{x}}d\sqrt [3]{x}\right )\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 3 \left (\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{6} b e n \int \left (\frac {d^6}{e^6 \left (d+e \sqrt [3]{x}\right )}-\frac {d^5}{e^6}+\frac {\sqrt [3]{x} d^4}{e^5}-\frac {x^{2/3} d^3}{e^4}+\frac {x d^2}{e^3}-\frac {x^{4/3} d}{e^2}+\frac {x^{5/3}}{e}\right )d\sqrt [3]{x}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{6} b e n \left (\frac {d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^7}-\frac {d^5 \sqrt [3]{x}}{e^6}+\frac {d^4 x^{2/3}}{2 e^5}-\frac {d^3 x}{3 e^4}+\frac {d^2 x^{4/3}}{4 e^3}-\frac {d x^{5/3}}{5 e^2}+\frac {x^2}{6 e}\right )\right )\) |
Input:
Int[x*(a + b*Log[c*(d + e*x^(1/3))^n]),x]
Output:
3*(-1/6*(b*e*n*(-((d^5*x^(1/3))/e^6) + (d^4*x^(2/3))/(2*e^5) - (d^3*x)/(3* e^4) + (d^2*x^(4/3))/(4*e^3) - (d*x^(5/3))/(5*e^2) + x^2/(6*e) + (d^6*Log[ d + e*x^(1/3)])/e^7)) + (x^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/6)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ ))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1))) Int[(f + g*x)^(q + 1)/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
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])
\[\int x \left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )d x\]
Input:
int(x*(a+b*ln(c*(d+e*x^(1/3))^n)),x)
Output:
int(x*(a+b*ln(c*(d+e*x^(1/3))^n)),x)
Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.90 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {60 \, b e^{6} x^{2} \log \left (c\right ) + 20 \, b d^{3} e^{3} n x - 10 \, {\left (b e^{6} n - 6 \, a e^{6}\right )} x^{2} + 60 \, {\left (b e^{6} n x^{2} - b d^{6} n\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 6 \, {\left (2 \, b d e^{5} n x - 5 \, b d^{4} e^{2} n\right )} x^{\frac {2}{3}} - 15 \, {\left (b d^{2} e^{4} n x - 4 \, b d^{5} e n\right )} x^{\frac {1}{3}}}{120 \, e^{6}} \] Input:
integrate(x*(a+b*log(c*(d+e*x^(1/3))^n)),x, algorithm="fricas")
Output:
1/120*(60*b*e^6*x^2*log(c) + 20*b*d^3*e^3*n*x - 10*(b*e^6*n - 6*a*e^6)*x^2 + 60*(b*e^6*n*x^2 - b*d^6*n)*log(e*x^(1/3) + d) + 6*(2*b*d*e^5*n*x - 5*b* d^4*e^2*n)*x^(2/3) - 15*(b*d^2*e^4*n*x - 4*b*d^5*e*n)*x^(1/3))/e^6
Time = 2.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {a x^{2}}{2} + b \left (- \frac {e n \left (\frac {3 d^{6} \left (\begin {cases} \frac {\sqrt [3]{x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt [3]{x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{6}} - \frac {3 d^{5} \sqrt [3]{x}}{e^{6}} + \frac {3 d^{4} x^{\frac {2}{3}}}{2 e^{5}} - \frac {d^{3} x}{e^{4}} + \frac {3 d^{2} x^{\frac {4}{3}}}{4 e^{3}} - \frac {3 d x^{\frac {5}{3}}}{5 e^{2}} + \frac {x^{2}}{2 e}\right )}{6} + \frac {x^{2} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{2}\right ) \] Input:
integrate(x*(a+b*ln(c*(d+e*x**(1/3))**n)),x)
Output:
a*x**2/2 + b*(-e*n*(3*d**6*Piecewise((x**(1/3)/d, Eq(e, 0)), (log(d + e*x* *(1/3))/e, True))/e**6 - 3*d**5*x**(1/3)/e**6 + 3*d**4*x**(2/3)/(2*e**5) - d**3*x/e**4 + 3*d**2*x**(4/3)/(4*e**3) - 3*d*x**(5/3)/(5*e**2) + x**2/(2* e))/6 + x**2*log(c*(d + e*x**(1/3))**n)/2)
Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.78 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=-\frac {1}{120} \, b e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} + \frac {1}{2} \, b x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + \frac {1}{2} \, a x^{2} \] Input:
integrate(x*(a+b*log(c*(d+e*x^(1/3))^n)),x, algorithm="maxima")
Output:
-1/120*b*e*n*(60*d^6*log(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 - 12*d*e^4*x^(5/ 3) + 15*d^2*e^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) - 60*d^5*x^(1/3) )/e^6) + 1/2*b*x^2*log((e*x^(1/3) + d)^n*c) + 1/2*a*x^2
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (108) = 216\).
Time = 0.13 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.94 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {60 \, b e x^{2} \log \left (c\right ) + 60 \, a e x^{2} + {\left (\frac {60 \, {\left (e x^{\frac {1}{3}} + d\right )}^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{5}} - \frac {360 \, {\left (e x^{\frac {1}{3}} + d\right )}^{5} d \log \left (e x^{\frac {1}{3}} + d\right )}{e^{5}} + \frac {900 \, {\left (e x^{\frac {1}{3}} + d\right )}^{4} d^{2} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{5}} - \frac {1200 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{5}} + \frac {900 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d^{4} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{5}} - \frac {360 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{5} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{5}} - \frac {10 \, {\left (e x^{\frac {1}{3}} + d\right )}^{6}}{e^{5}} + \frac {72 \, {\left (e x^{\frac {1}{3}} + d\right )}^{5} d}{e^{5}} - \frac {225 \, {\left (e x^{\frac {1}{3}} + d\right )}^{4} d^{2}}{e^{5}} + \frac {400 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} d^{3}}{e^{5}} - \frac {450 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d^{4}}{e^{5}} + \frac {360 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{5}}{e^{5}}\right )} b n}{120 \, e} \] Input:
integrate(x*(a+b*log(c*(d+e*x^(1/3))^n)),x, algorithm="giac")
Output:
1/120*(60*b*e*x^2*log(c) + 60*a*e*x^2 + (60*(e*x^(1/3) + d)^6*log(e*x^(1/3 ) + d)/e^5 - 360*(e*x^(1/3) + d)^5*d*log(e*x^(1/3) + d)/e^5 + 900*(e*x^(1/ 3) + d)^4*d^2*log(e*x^(1/3) + d)/e^5 - 1200*(e*x^(1/3) + d)^3*d^3*log(e*x^ (1/3) + d)/e^5 + 900*(e*x^(1/3) + d)^2*d^4*log(e*x^(1/3) + d)/e^5 - 360*(e *x^(1/3) + d)*d^5*log(e*x^(1/3) + d)/e^5 - 10*(e*x^(1/3) + d)^6/e^5 + 72*( e*x^(1/3) + d)^5*d/e^5 - 225*(e*x^(1/3) + d)^4*d^2/e^5 + 400*(e*x^(1/3) + d)^3*d^3/e^5 - 450*(e*x^(1/3) + d)^2*d^4/e^5 + 360*(e*x^(1/3) + d)*d^5/e^5 )*b*n)/e
Time = 14.59 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.82 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {a\,x^2}{2}-\frac {b\,n\,x^2}{12}+\frac {b\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2}+\frac {b\,d^3\,n\,x}{6\,e^3}+\frac {b\,d\,n\,x^{5/3}}{10\,e}-\frac {b\,d^6\,n\,\ln \left (d+e\,x^{1/3}\right )}{2\,e^6}-\frac {b\,d^2\,n\,x^{4/3}}{8\,e^2}-\frac {b\,d^4\,n\,x^{2/3}}{4\,e^4}+\frac {b\,d^5\,n\,x^{1/3}}{2\,e^5} \] Input:
int(x*(a + b*log(c*(d + e*x^(1/3))^n)),x)
Output:
(a*x^2)/2 - (b*n*x^2)/12 + (b*x^2*log(c*(d + e*x^(1/3))^n))/2 + (b*d^3*n*x )/(6*e^3) + (b*d*n*x^(5/3))/(10*e) - (b*d^6*n*log(d + e*x^(1/3)))/(2*e^6) - (b*d^2*n*x^(4/3))/(8*e^2) - (b*d^4*n*x^(2/3))/(4*e^4) + (b*d^5*n*x^(1/3) )/(2*e^5)
Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {-30 x^{\frac {2}{3}} b \,d^{4} e^{2} n +12 x^{\frac {5}{3}} b d \,e^{5} n +60 x^{\frac {1}{3}} b \,d^{5} e n -15 x^{\frac {4}{3}} b \,d^{2} e^{4} n -60 \,\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) b \,d^{6}+60 \,\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) b \,e^{6} x^{2}+60 a \,e^{6} x^{2}+20 b \,d^{3} e^{3} n x -10 b \,e^{6} n \,x^{2}}{120 e^{6}} \] Input:
int(x*(a+b*log(c*(d+e*x^(1/3))^n)),x)
Output:
( - 30*x**(2/3)*b*d**4*e**2*n + 12*x**(2/3)*b*d*e**5*n*x + 60*x**(1/3)*b*d **5*e*n - 15*x**(1/3)*b*d**2*e**4*n*x - 60*log((x**(1/3)*e + d)**n*c)*b*d* *6 + 60*log((x**(1/3)*e + d)**n*c)*b*e**6*x**2 + 60*a*e**6*x**2 + 20*b*d** 3*e**3*n*x - 10*b*e**6*n*x**2)/(120*e**6)