\(\int x^3 (a+b \log (c (d+e \sqrt [3]{x})^n)) \, dx\) [442]

Optimal result

Integrand size = 22, antiderivative size = 234 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {b d^{11} n \sqrt [3]{x}}{4 e^{11}}-\frac {b d^{10} n x^{2/3}}{8 e^{10}}+\frac {b d^9 n x}{12 e^9}-\frac {b d^8 n x^{4/3}}{16 e^8}+\frac {b d^7 n x^{5/3}}{20 e^7}-\frac {b d^6 n x^2}{24 e^6}+\frac {b d^5 n x^{7/3}}{28 e^5}-\frac {b d^4 n x^{8/3}}{32 e^4}+\frac {b d^3 n x^3}{36 e^3}-\frac {b d^2 n x^{10/3}}{40 e^2}+\frac {b d n x^{11/3}}{44 e}-\frac {1}{48} b n x^4-\frac {b d^{12} n \log \left (d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \]

[Out]

1/4*b*d^11*n*x^(1/3)/e^11-1/8*b*d^10*n*x^(2/3)/e^10+1/12*b*d^9*n*x/e^9-1/16*b*d^8*n*x^(4/3)/e^8+1/20*b*d^7*n*x
^(5/3)/e^7-1/24*b*d^6*n*x^2/e^6+1/28*b*d^5*n*x^(7/3)/e^5-1/32*b*d^4*n*x^(8/3)/e^4+1/36*b*d^3*n*x^3/e^3-1/40*b*
d^2*n*x^(10/3)/e^2+1/44*b*d*n*x^(11/3)/e-1/48*b*n*x^4-1/4*b*d^12*n*ln(d+e*x^(1/3))/e^12+1/4*x^4*(a+b*ln(c*(d+e
*x^(1/3))^n))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2442, 45} \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {b d^{12} n \log \left (d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac {b d^{11} n \sqrt [3]{x}}{4 e^{11}}-\frac {b d^{10} n x^{2/3}}{8 e^{10}}+\frac {b d^9 n x}{12 e^9}-\frac {b d^8 n x^{4/3}}{16 e^8}+\frac {b d^7 n x^{5/3}}{20 e^7}-\frac {b d^6 n x^2}{24 e^6}+\frac {b d^5 n x^{7/3}}{28 e^5}-\frac {b d^4 n x^{8/3}}{32 e^4}+\frac {b d^3 n x^3}{36 e^3}-\frac {b d^2 n x^{10/3}}{40 e^2}+\frac {b d n x^{11/3}}{44 e}-\frac {1}{48} b n x^4 \]

[In]

Int[x^3*(a + b*Log[c*(d + e*x^(1/3))^n]),x]

[Out]

(b*d^11*n*x^(1/3))/(4*e^11) - (b*d^10*n*x^(2/3))/(8*e^10) + (b*d^9*n*x)/(12*e^9) - (b*d^8*n*x^(4/3))/(16*e^8)
+ (b*d^7*n*x^(5/3))/(20*e^7) - (b*d^6*n*x^2)/(24*e^6) + (b*d^5*n*x^(7/3))/(28*e^5) - (b*d^4*n*x^(8/3))/(32*e^4
) + (b*d^3*n*x^3)/(36*e^3) - (b*d^2*n*x^(10/3))/(40*e^2) + (b*d*n*x^(11/3))/(44*e) - (b*n*x^4)/48 - (b*d^12*n*
Log[d + e*x^(1/3)])/(4*e^12) + (x^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/4

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2442

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] - Dist[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]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[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])

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^{11} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{4} (b e n) \text {Subst}\left (\int \frac {x^{12}}{d+e x} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{4} (b e n) \text {Subst}\left (\int \left (-\frac {d^{11}}{e^{12}}+\frac {d^{10} x}{e^{11}}-\frac {d^9 x^2}{e^{10}}+\frac {d^8 x^3}{e^9}-\frac {d^7 x^4}{e^8}+\frac {d^6 x^5}{e^7}-\frac {d^5 x^6}{e^6}+\frac {d^4 x^7}{e^5}-\frac {d^3 x^8}{e^4}+\frac {d^2 x^9}{e^3}-\frac {d x^{10}}{e^2}+\frac {x^{11}}{e}+\frac {d^{12}}{e^{12} (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {b d^{11} n \sqrt [3]{x}}{4 e^{11}}-\frac {b d^{10} n x^{2/3}}{8 e^{10}}+\frac {b d^9 n x}{12 e^9}-\frac {b d^8 n x^{4/3}}{16 e^8}+\frac {b d^7 n x^{5/3}}{20 e^7}-\frac {b d^6 n x^2}{24 e^6}+\frac {b d^5 n x^{7/3}}{28 e^5}-\frac {b d^4 n x^{8/3}}{32 e^4}+\frac {b d^3 n x^3}{36 e^3}-\frac {b d^2 n x^{10/3}}{40 e^2}+\frac {b d n x^{11/3}}{44 e}-\frac {1}{48} b n x^4-\frac {b d^{12} n \log \left (d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {a x^4}{4}-\frac {1}{12} b e n \left (-\frac {3 d^{11} \sqrt [3]{x}}{e^{12}}+\frac {3 d^{10} x^{2/3}}{2 e^{11}}-\frac {d^9 x}{e^{10}}+\frac {3 d^8 x^{4/3}}{4 e^9}-\frac {3 d^7 x^{5/3}}{5 e^8}+\frac {d^6 x^2}{2 e^7}-\frac {3 d^5 x^{7/3}}{7 e^6}+\frac {3 d^4 x^{8/3}}{8 e^5}-\frac {d^3 x^3}{3 e^4}+\frac {3 d^2 x^{10/3}}{10 e^3}-\frac {3 d x^{11/3}}{11 e^2}+\frac {x^4}{4 e}+\frac {3 d^{12} \log \left (d+e \sqrt [3]{x}\right )}{e^{13}}\right )+\frac {1}{4} b x^4 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \]

[In]

Integrate[x^3*(a + b*Log[c*(d + e*x^(1/3))^n]),x]

[Out]

(a*x^4)/4 - (b*e*n*((-3*d^11*x^(1/3))/e^12 + (3*d^10*x^(2/3))/(2*e^11) - (d^9*x)/e^10 + (3*d^8*x^(4/3))/(4*e^9
) - (3*d^7*x^(5/3))/(5*e^8) + (d^6*x^2)/(2*e^7) - (3*d^5*x^(7/3))/(7*e^6) + (3*d^4*x^(8/3))/(8*e^5) - (d^3*x^3
)/(3*e^4) + (3*d^2*x^(10/3))/(10*e^3) - (3*d*x^(11/3))/(11*e^2) + x^4/(4*e) + (3*d^12*Log[d + e*x^(1/3)])/e^13
))/12 + (b*x^4*Log[c*(d + e*x^(1/3))^n])/4

Maple [F]

\[\int x^{3} \left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )d x\]

[In]

int(x^3*(a+b*ln(c*(d+e*x^(1/3))^n)),x)

[Out]

int(x^3*(a+b*ln(c*(d+e*x^(1/3))^n)),x)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.86 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {27720 \, b e^{12} x^{4} \log \left (c\right ) + 3080 \, b d^{3} e^{9} n x^{3} - 4620 \, b d^{6} e^{6} n x^{2} + 9240 \, b d^{9} e^{3} n x - 2310 \, {\left (b e^{12} n - 12 \, a e^{12}\right )} x^{4} + 27720 \, {\left (b e^{12} n x^{4} - b d^{12} n\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 63 \, {\left (40 \, b d e^{11} n x^{3} - 55 \, b d^{4} e^{8} n x^{2} + 88 \, b d^{7} e^{5} n x - 220 \, b d^{10} e^{2} n\right )} x^{\frac {2}{3}} - 198 \, {\left (14 \, b d^{2} e^{10} n x^{3} - 20 \, b d^{5} e^{7} n x^{2} + 35 \, b d^{8} e^{4} n x - 140 \, b d^{11} e n\right )} x^{\frac {1}{3}}}{110880 \, e^{12}} \]

[In]

integrate(x^3*(a+b*log(c*(d+e*x^(1/3))^n)),x, algorithm="fricas")

[Out]

1/110880*(27720*b*e^12*x^4*log(c) + 3080*b*d^3*e^9*n*x^3 - 4620*b*d^6*e^6*n*x^2 + 9240*b*d^9*e^3*n*x - 2310*(b
*e^12*n - 12*a*e^12)*x^4 + 27720*(b*e^12*n*x^4 - b*d^12*n)*log(e*x^(1/3) + d) + 63*(40*b*d*e^11*n*x^3 - 55*b*d
^4*e^8*n*x^2 + 88*b*d^7*e^5*n*x - 220*b*d^10*e^2*n)*x^(2/3) - 198*(14*b*d^2*e^10*n*x^3 - 20*b*d^5*e^7*n*x^2 +
35*b*d^8*e^4*n*x - 140*b*d^11*e*n)*x^(1/3))/e^12

Sympy [A] (verification not implemented)

Time = 45.06 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.92 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {a x^{4}}{4} + b \left (- \frac {e n \left (\frac {3 d^{12} \left (\begin {cases} \frac {\sqrt [3]{x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt [3]{x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{12}} - \frac {3 d^{11} \sqrt [3]{x}}{e^{12}} + \frac {3 d^{10} x^{\frac {2}{3}}}{2 e^{11}} - \frac {d^{9} x}{e^{10}} + \frac {3 d^{8} x^{\frac {4}{3}}}{4 e^{9}} - \frac {3 d^{7} x^{\frac {5}{3}}}{5 e^{8}} + \frac {d^{6} x^{2}}{2 e^{7}} - \frac {3 d^{5} x^{\frac {7}{3}}}{7 e^{6}} + \frac {3 d^{4} x^{\frac {8}{3}}}{8 e^{5}} - \frac {d^{3} x^{3}}{3 e^{4}} + \frac {3 d^{2} x^{\frac {10}{3}}}{10 e^{3}} - \frac {3 d x^{\frac {11}{3}}}{11 e^{2}} + \frac {x^{4}}{4 e}\right )}{12} + \frac {x^{4} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{4}\right ) \]

[In]

integrate(x**3*(a+b*ln(c*(d+e*x**(1/3))**n)),x)

[Out]

a*x**4/4 + b*(-e*n*(3*d**12*Piecewise((x**(1/3)/d, Eq(e, 0)), (log(d + e*x**(1/3))/e, True))/e**12 - 3*d**11*x
**(1/3)/e**12 + 3*d**10*x**(2/3)/(2*e**11) - d**9*x/e**10 + 3*d**8*x**(4/3)/(4*e**9) - 3*d**7*x**(5/3)/(5*e**8
) + d**6*x**2/(2*e**7) - 3*d**5*x**(7/3)/(7*e**6) + 3*d**4*x**(8/3)/(8*e**5) - d**3*x**3/(3*e**4) + 3*d**2*x**
(10/3)/(10*e**3) - 3*d*x**(11/3)/(11*e**2) + x**4/(4*e))/12 + x**4*log(c*(d + e*x**(1/3))**n)/4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.74 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {1}{4} \, b x^{4} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + \frac {1}{4} \, a x^{4} - \frac {1}{110880} \, b e n {\left (\frac {27720 \, d^{12} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{13}} + \frac {2310 \, e^{11} x^{4} - 2520 \, d e^{10} x^{\frac {11}{3}} + 2772 \, d^{2} e^{9} x^{\frac {10}{3}} - 3080 \, d^{3} e^{8} x^{3} + 3465 \, d^{4} e^{7} x^{\frac {8}{3}} - 3960 \, d^{5} e^{6} x^{\frac {7}{3}} + 4620 \, d^{6} e^{5} x^{2} - 5544 \, d^{7} e^{4} x^{\frac {5}{3}} + 6930 \, d^{8} e^{3} x^{\frac {4}{3}} - 9240 \, d^{9} e^{2} x + 13860 \, d^{10} e x^{\frac {2}{3}} - 27720 \, d^{11} x^{\frac {1}{3}}}{e^{12}}\right )} \]

[In]

integrate(x^3*(a+b*log(c*(d+e*x^(1/3))^n)),x, algorithm="maxima")

[Out]

1/4*b*x^4*log((e*x^(1/3) + d)^n*c) + 1/4*a*x^4 - 1/110880*b*e*n*(27720*d^12*log(e*x^(1/3) + d)/e^13 + (2310*e^
11*x^4 - 2520*d*e^10*x^(11/3) + 2772*d^2*e^9*x^(10/3) - 3080*d^3*e^8*x^3 + 3465*d^4*e^7*x^(8/3) - 3960*d^5*e^6
*x^(7/3) + 4620*d^6*e^5*x^2 - 5544*d^7*e^4*x^(5/3) + 6930*d^8*e^3*x^(4/3) - 9240*d^9*e^2*x + 13860*d^10*e*x^(2
/3) - 27720*d^11*x^(1/3))/e^12)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (186) = 372\).

Time = 0.30 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.21 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {27720 \, b e x^{4} \log \left (c\right ) + 27720 \, a e x^{4} + {\left (\frac {27720 \, {\left (e x^{\frac {1}{3}} + d\right )}^{12} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} - \frac {332640 \, {\left (e x^{\frac {1}{3}} + d\right )}^{11} d \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} + \frac {1829520 \, {\left (e x^{\frac {1}{3}} + d\right )}^{10} d^{2} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} - \frac {6098400 \, {\left (e x^{\frac {1}{3}} + d\right )}^{9} d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} + \frac {13721400 \, {\left (e x^{\frac {1}{3}} + d\right )}^{8} d^{4} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} - \frac {21954240 \, {\left (e x^{\frac {1}{3}} + d\right )}^{7} d^{5} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} + \frac {25613280 \, {\left (e x^{\frac {1}{3}} + d\right )}^{6} d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} - \frac {21954240 \, {\left (e x^{\frac {1}{3}} + d\right )}^{5} d^{7} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} + \frac {13721400 \, {\left (e x^{\frac {1}{3}} + d\right )}^{4} d^{8} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} - \frac {6098400 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} d^{9} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} + \frac {1829520 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d^{10} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} - \frac {332640 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{11} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{11}} - \frac {2310 \, {\left (e x^{\frac {1}{3}} + d\right )}^{12}}{e^{11}} + \frac {30240 \, {\left (e x^{\frac {1}{3}} + d\right )}^{11} d}{e^{11}} - \frac {182952 \, {\left (e x^{\frac {1}{3}} + d\right )}^{10} d^{2}}{e^{11}} + \frac {677600 \, {\left (e x^{\frac {1}{3}} + d\right )}^{9} d^{3}}{e^{11}} - \frac {1715175 \, {\left (e x^{\frac {1}{3}} + d\right )}^{8} d^{4}}{e^{11}} + \frac {3136320 \, {\left (e x^{\frac {1}{3}} + d\right )}^{7} d^{5}}{e^{11}} - \frac {4268880 \, {\left (e x^{\frac {1}{3}} + d\right )}^{6} d^{6}}{e^{11}} + \frac {4390848 \, {\left (e x^{\frac {1}{3}} + d\right )}^{5} d^{7}}{e^{11}} - \frac {3430350 \, {\left (e x^{\frac {1}{3}} + d\right )}^{4} d^{8}}{e^{11}} + \frac {2032800 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} d^{9}}{e^{11}} - \frac {914760 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d^{10}}{e^{11}} + \frac {332640 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{11}}{e^{11}}\right )} b n}{110880 \, e} \]

[In]

integrate(x^3*(a+b*log(c*(d+e*x^(1/3))^n)),x, algorithm="giac")

[Out]

1/110880*(27720*b*e*x^4*log(c) + 27720*a*e*x^4 + (27720*(e*x^(1/3) + d)^12*log(e*x^(1/3) + d)/e^11 - 332640*(e
*x^(1/3) + d)^11*d*log(e*x^(1/3) + d)/e^11 + 1829520*(e*x^(1/3) + d)^10*d^2*log(e*x^(1/3) + d)/e^11 - 6098400*
(e*x^(1/3) + d)^9*d^3*log(e*x^(1/3) + d)/e^11 + 13721400*(e*x^(1/3) + d)^8*d^4*log(e*x^(1/3) + d)/e^11 - 21954
240*(e*x^(1/3) + d)^7*d^5*log(e*x^(1/3) + d)/e^11 + 25613280*(e*x^(1/3) + d)^6*d^6*log(e*x^(1/3) + d)/e^11 - 2
1954240*(e*x^(1/3) + d)^5*d^7*log(e*x^(1/3) + d)/e^11 + 13721400*(e*x^(1/3) + d)^4*d^8*log(e*x^(1/3) + d)/e^11
- 6098400*(e*x^(1/3) + d)^3*d^9*log(e*x^(1/3) + d)/e^11 + 1829520*(e*x^(1/3) + d)^2*d^10*log(e*x^(1/3) + d)/e
^11 - 332640*(e*x^(1/3) + d)*d^11*log(e*x^(1/3) + d)/e^11 - 2310*(e*x^(1/3) + d)^12/e^11 + 30240*(e*x^(1/3) +
d)^11*d/e^11 - 182952*(e*x^(1/3) + d)^10*d^2/e^11 + 677600*(e*x^(1/3) + d)^9*d^3/e^11 - 1715175*(e*x^(1/3) + d
)^8*d^4/e^11 + 3136320*(e*x^(1/3) + d)^7*d^5/e^11 - 4268880*(e*x^(1/3) + d)^6*d^6/e^11 + 4390848*(e*x^(1/3) +
d)^5*d^7/e^11 - 3430350*(e*x^(1/3) + d)^4*d^8/e^11 + 2032800*(e*x^(1/3) + d)^3*d^9/e^11 - 914760*(e*x^(1/3) +
d)^2*d^10/e^11 + 332640*(e*x^(1/3) + d)*d^11/e^11)*b*n)/e

Mupad [B] (verification not implemented)

Time = 1.76 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.81 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {a\,x^4}{4}-\frac {b\,n\,x^4}{48}+\frac {b\,x^4\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{4}+\frac {b\,d\,n\,x^{11/3}}{44\,e}+\frac {b\,d^9\,n\,x}{12\,e^9}-\frac {b\,d^{12}\,n\,\ln \left (d+e\,x^{1/3}\right )}{4\,e^{12}}+\frac {b\,d^3\,n\,x^3}{36\,e^3}-\frac {b\,d^6\,n\,x^2}{24\,e^6}-\frac {b\,d^2\,n\,x^{10/3}}{40\,e^2}-\frac {b\,d^4\,n\,x^{8/3}}{32\,e^4}+\frac {b\,d^5\,n\,x^{7/3}}{28\,e^5}+\frac {b\,d^7\,n\,x^{5/3}}{20\,e^7}-\frac {b\,d^8\,n\,x^{4/3}}{16\,e^8}-\frac {b\,d^{10}\,n\,x^{2/3}}{8\,e^{10}}+\frac {b\,d^{11}\,n\,x^{1/3}}{4\,e^{11}} \]

[In]

int(x^3*(a + b*log(c*(d + e*x^(1/3))^n)),x)

[Out]

(a*x^4)/4 - (b*n*x^4)/48 + (b*x^4*log(c*(d + e*x^(1/3))^n))/4 + (b*d*n*x^(11/3))/(44*e) + (b*d^9*n*x)/(12*e^9)
- (b*d^12*n*log(d + e*x^(1/3)))/(4*e^12) + (b*d^3*n*x^3)/(36*e^3) - (b*d^6*n*x^2)/(24*e^6) - (b*d^2*n*x^(10/3
))/(40*e^2) - (b*d^4*n*x^(8/3))/(32*e^4) + (b*d^5*n*x^(7/3))/(28*e^5) + (b*d^7*n*x^(5/3))/(20*e^7) - (b*d^8*n*
x^(4/3))/(16*e^8) - (b*d^10*n*x^(2/3))/(8*e^10) + (b*d^11*n*x^(1/3))/(4*e^11)