Integrand size = 22, antiderivative size = 138 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {b d^5 n x^{2/3}}{4 e^5}-\frac {b d^4 n x^{4/3}}{8 e^4}+\frac {b d^3 n x^2}{12 e^3}-\frac {b d^2 n x^{8/3}}{16 e^2}+\frac {b d n x^{10/3}}{20 e}-\frac {1}{24} b n x^4-\frac {b d^6 n \log \left (d+e x^{2/3}\right )}{4 e^6}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \] Output:
1/4*b*d^5*n*x^(2/3)/e^5-1/8*b*d^4*n*x^(4/3)/e^4+1/12*b*d^3*n*x^2/e^3-1/16* b*d^2*n*x^(8/3)/e^2+1/20*b*d*n*x^(10/3)/e-1/24*b*n*x^4-1/4*b*d^6*n*ln(d+e* x^(2/3))/e^6+1/4*x^4*(a+b*ln(c*(d+e*x^(2/3))^n))
Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.98 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {a x^4}{4}-\frac {1}{4} b e n \left (-\frac {d^5 x^{2/3}}{e^6}+\frac {d^4 x^{4/3}}{2 e^5}-\frac {d^3 x^2}{3 e^4}+\frac {d^2 x^{8/3}}{4 e^3}-\frac {d x^{10/3}}{5 e^2}+\frac {x^4}{6 e}+\frac {d^6 \log \left (d+e x^{2/3}\right )}{e^7}\right )+\frac {1}{4} b x^4 \log \left (c \left (d+e x^{2/3}\right )^n\right ) \] Input:
Integrate[x^3*(a + b*Log[c*(d + e*x^(2/3))^n]),x]
Output:
(a*x^4)/4 - (b*e*n*(-((d^5*x^(2/3))/e^6) + (d^4*x^(4/3))/(2*e^5) - (d^3*x^ 2)/(3*e^4) + (d^2*x^(8/3))/(4*e^3) - (d*x^(10/3))/(5*e^2) + x^4/(6*e) + (d ^6*Log[d + e*x^(2/3)])/e^7))/4 + (b*x^4*Log[c*(d + e*x^(2/3))^n])/4
Time = 0.54 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, 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.
\(\displaystyle \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {3}{2} \int x^{10/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )dx^{2/3}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle \frac {3}{2} \left (\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {1}{6} b e n \int \frac {x^4}{d+e x^{2/3}}dx^{2/3}\right )\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {3}{2} \left (\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {1}{6} b e n \int \left (\frac {d^6}{e^6 \left (d+e x^{2/3}\right )}-\frac {d^5}{e^6}+\frac {x^{2/3} d^4}{e^5}-\frac {x^{4/3} d^3}{e^4}+\frac {x^2 d^2}{e^3}-\frac {x^{8/3} d}{e^2}+\frac {x^{10/3}}{e}\right )dx^{2/3}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{2} \left (\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {1}{6} b e n \left (\frac {d^6 \log \left (d+e x^{2/3}\right )}{e^7}-\frac {d^5 x^{2/3}}{e^6}+\frac {d^4 x^{4/3}}{2 e^5}-\frac {d^3 x^2}{3 e^4}+\frac {d^2 x^{8/3}}{4 e^3}-\frac {d x^{10/3}}{5 e^2}+\frac {x^4}{6 e}\right )\right )\) |
Input:
Int[x^3*(a + b*Log[c*(d + e*x^(2/3))^n]),x]
Output:
(3*(-1/6*(b*e*n*(-((d^5*x^(2/3))/e^6) + (d^4*x^(4/3))/(2*e^5) - (d^3*x^2)/ (3*e^4) + (d^2*x^(8/3))/(4*e^3) - (d*x^(10/3))/(5*e^2) + x^4/(6*e) + (d^6* Log[d + e*x^(2/3)])/e^7)) + (x^4*(a + b*Log[c*(d + e*x^(2/3))^n]))/6))/2
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])
\[\int x^{3} \left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )d x\]
Input:
int(x^3*(a+b*ln(c*(d+e*x^(2/3))^n)),x)
Output:
int(x^3*(a+b*ln(c*(d+e*x^(2/3))^n)),x)
Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {60 \, b e^{6} x^{4} \log \left (c\right ) + 20 \, b d^{3} e^{3} n x^{2} - 10 \, {\left (b e^{6} n - 6 \, a e^{6}\right )} x^{4} + 60 \, {\left (b e^{6} n x^{4} - b d^{6} n\right )} \log \left (e x^{\frac {2}{3}} + d\right ) - 15 \, {\left (b d^{2} e^{4} n x^{2} - 4 \, b d^{5} e n\right )} x^{\frac {2}{3}} + 6 \, {\left (2 \, b d e^{5} n x^{3} - 5 \, b d^{4} e^{2} n x\right )} x^{\frac {1}{3}}}{240 \, e^{6}} \] Input:
integrate(x^3*(a+b*log(c*(d+e*x^(2/3))^n)),x, algorithm="fricas")
Output:
1/240*(60*b*e^6*x^4*log(c) + 20*b*d^3*e^3*n*x^2 - 10*(b*e^6*n - 6*a*e^6)*x ^4 + 60*(b*e^6*n*x^4 - b*d^6*n)*log(e*x^(2/3) + d) - 15*(b*d^2*e^4*n*x^2 - 4*b*d^5*e*n)*x^(2/3) + 6*(2*b*d*e^5*n*x^3 - 5*b*d^4*e^2*n*x)*x^(1/3))/e^6
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(x**3*(a+b*ln(c*(d+e*x**(2/3))**n)),x)
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {1}{4} \, b x^{4} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + \frac {1}{4} \, a x^{4} - \frac {1}{240} \, b e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac {10}{3}} + 15 \, d^{2} e^{3} x^{\frac {8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac {4}{3}} - 60 \, d^{5} x^{\frac {2}{3}}}{e^{6}}\right )} \] Input:
integrate(x^3*(a+b*log(c*(d+e*x^(2/3))^n)),x, algorithm="maxima")
Output:
1/4*b*x^4*log((e*x^(2/3) + d)^n*c) + 1/4*a*x^4 - 1/240*b*e*n*(60*d^6*log(e *x^(2/3) + d)/e^7 + (10*e^5*x^4 - 12*d*e^4*x^(10/3) + 15*d^2*e^3*x^(8/3) - 20*d^3*e^2*x^2 + 30*d^4*e*x^(4/3) - 60*d^5*x^(2/3))/e^6)
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (110) = 220\).
Time = 0.18 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.84 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {1}{4} \, b x^{4} \log \left (c\right ) + \frac {1}{4} \, a x^{4} + \frac {1}{240} \, b n {\left (\frac {60 \, {\left (e x^{\frac {2}{3}} + d\right )}^{6} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{6}} - \frac {360 \, {\left (e x^{\frac {2}{3}} + d\right )}^{5} d \log \left (e x^{\frac {2}{3}} + d\right )}{e^{6}} + \frac {900 \, {\left (e x^{\frac {2}{3}} + d\right )}^{4} d^{2} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{6}} - \frac {1200 \, {\left (e x^{\frac {2}{3}} + d\right )}^{3} d^{3} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{6}} + \frac {900 \, {\left (e x^{\frac {2}{3}} + d\right )}^{2} d^{4} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{6}} - \frac {10 \, {\left (e x^{\frac {2}{3}} + d\right )}^{6}}{e^{6}} + \frac {72 \, {\left (e x^{\frac {2}{3}} + d\right )}^{5} d}{e^{6}} - \frac {225 \, {\left (e x^{\frac {2}{3}} + d\right )}^{4} d^{2}}{e^{6}} + \frac {400 \, {\left (e x^{\frac {2}{3}} + d\right )}^{3} d^{3}}{e^{6}} - \frac {450 \, {\left (e x^{\frac {2}{3}} + d\right )}^{2} d^{4}}{e^{6}} - \frac {360 \, {\left ({\left (e x^{\frac {2}{3}} + d\right )} \log \left (e x^{\frac {2}{3}} + d\right ) - e x^{\frac {2}{3}} - d\right )} d^{5}}{e^{6}}\right )} \] Input:
integrate(x^3*(a+b*log(c*(d+e*x^(2/3))^n)),x, algorithm="giac")
Output:
1/4*b*x^4*log(c) + 1/4*a*x^4 + 1/240*b*n*(60*(e*x^(2/3) + d)^6*log(e*x^(2/ 3) + d)/e^6 - 360*(e*x^(2/3) + d)^5*d*log(e*x^(2/3) + d)/e^6 + 900*(e*x^(2 /3) + d)^4*d^2*log(e*x^(2/3) + d)/e^6 - 1200*(e*x^(2/3) + d)^3*d^3*log(e*x ^(2/3) + d)/e^6 + 900*(e*x^(2/3) + d)^2*d^4*log(e*x^(2/3) + d)/e^6 - 10*(e *x^(2/3) + d)^6/e^6 + 72*(e*x^(2/3) + d)^5*d/e^6 - 225*(e*x^(2/3) + d)^4*d ^2/e^6 + 400*(e*x^(2/3) + d)^3*d^3/e^6 - 450*(e*x^(2/3) + d)^2*d^4/e^6 - 3 60*((e*x^(2/3) + d)*log(e*x^(2/3) + d) - e*x^(2/3) - d)*d^5/e^6)
Time = 14.59 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.82 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {a\,x^4}{4}-\frac {b\,n\,x^4}{24}+\frac {b\,x^4\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{4}+\frac {b\,d\,n\,x^{10/3}}{20\,e}-\frac {b\,d^6\,n\,\ln \left (d+e\,x^{2/3}\right )}{4\,e^6}+\frac {b\,d^3\,n\,x^2}{12\,e^3}-\frac {b\,d^2\,n\,x^{8/3}}{16\,e^2}-\frac {b\,d^4\,n\,x^{4/3}}{8\,e^4}+\frac {b\,d^5\,n\,x^{2/3}}{4\,e^5} \] Input:
int(x^3*(a + b*log(c*(d + e*x^(2/3))^n)),x)
Output:
(a*x^4)/4 - (b*n*x^4)/24 + (b*x^4*log(c*(d + e*x^(2/3))^n))/4 + (b*d*n*x^( 10/3))/(20*e) - (b*d^6*n*log(d + e*x^(2/3)))/(4*e^6) + (b*d^3*n*x^2)/(12*e ^3) - (b*d^2*n*x^(8/3))/(16*e^2) - (b*d^4*n*x^(4/3))/(8*e^4) + (b*d^5*n*x^ (2/3))/(4*e^5)
Time = 0.18 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.91 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {60 x^{\frac {2}{3}} b \,d^{5} e n -15 x^{\frac {8}{3}} b \,d^{2} e^{4} n -30 x^{\frac {4}{3}} b \,d^{4} e^{2} n +12 x^{\frac {10}{3}} b d \,e^{5} n -60 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b \,d^{6}+60 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b \,e^{6} x^{4}+60 a \,e^{6} x^{4}+20 b \,d^{3} e^{3} n \,x^{2}-10 b \,e^{6} n \,x^{4}}{240 e^{6}} \] Input:
int(x^3*(a+b*log(c*(d+e*x^(2/3))^n)),x)
Output:
(60*x**(2/3)*b*d**5*e*n - 15*x**(2/3)*b*d**2*e**4*n*x**2 - 30*x**(1/3)*b*d **4*e**2*n*x + 12*x**(1/3)*b*d*e**5*n*x**3 - 60*log((x**(2/3)*e + d)**n*c) *b*d**6 + 60*log((x**(2/3)*e + d)**n*c)*b*e**6*x**4 + 60*a*e**6*x**4 + 20* b*d**3*e**3*n*x**2 - 10*b*e**6*n*x**4)/(240*e**6)