3.5.89 \(\int x^3 (a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n)) \, dx\) [489]

Optimal. Leaf size=239 \[ \frac {b e^{11} n \sqrt [3]{x}}{4 d^{11}}-\frac {b e^{10} n x^{2/3}}{8 d^{10}}+\frac {b e^9 n x}{12 d^9}-\frac {b e^8 n x^{4/3}}{16 d^8}+\frac {b e^7 n x^{5/3}}{20 d^7}-\frac {b e^6 n x^2}{24 d^6}+\frac {b e^5 n x^{7/3}}{28 d^5}-\frac {b e^4 n x^{8/3}}{32 d^4}+\frac {b e^3 n x^3}{36 d^3}-\frac {b e^2 n x^{10/3}}{40 d^2}+\frac {b e n x^{11/3}}{44 d}-\frac {b e^{12} n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{4 d^{12}}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {b e^{12} n \log (x)}{12 d^{12}} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2442, 46} \begin {gather*} \frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {b e^{12} n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{4 d^{12}}-\frac {b e^{12} n \log (x)}{12 d^{12}}+\frac {b e^{11} n \sqrt [3]{x}}{4 d^{11}}-\frac {b e^{10} n x^{2/3}}{8 d^{10}}+\frac {b e^9 n x}{12 d^9}-\frac {b e^8 n x^{4/3}}{16 d^8}+\frac {b e^7 n x^{5/3}}{20 d^7}-\frac {b e^6 n x^2}{24 d^6}+\frac {b e^5 n x^{7/3}}{28 d^5}-\frac {b e^4 n x^{8/3}}{32 d^4}+\frac {b e^3 n x^3}{36 d^3}-\frac {b e^2 n x^{10/3}}{40 d^2}+\frac {b e n x^{11/3}}{44 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 46

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] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[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*} \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx &=-\left (3 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^{13}} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {1}{4} (b e n) \text {Subst}\left (\int \frac {1}{x^{12} (d+e x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\\ &=\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {1}{4} (b e n) \text {Subst}\left (\int \left (\frac {1}{d x^{12}}-\frac {e}{d^2 x^{11}}+\frac {e^2}{d^3 x^{10}}-\frac {e^3}{d^4 x^9}+\frac {e^4}{d^5 x^8}-\frac {e^5}{d^6 x^7}+\frac {e^6}{d^7 x^6}-\frac {e^7}{d^8 x^5}+\frac {e^8}{d^9 x^4}-\frac {e^9}{d^{10} x^3}+\frac {e^{10}}{d^{11} x^2}-\frac {e^{11}}{d^{12} x}+\frac {e^{12}}{d^{12} (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\\ &=\frac {b e^{11} n \sqrt [3]{x}}{4 d^{11}}-\frac {b e^{10} n x^{2/3}}{8 d^{10}}+\frac {b e^9 n x}{12 d^9}-\frac {b e^8 n x^{4/3}}{16 d^8}+\frac {b e^7 n x^{5/3}}{20 d^7}-\frac {b e^6 n x^2}{24 d^6}+\frac {b e^5 n x^{7/3}}{28 d^5}-\frac {b e^4 n x^{8/3}}{32 d^4}+\frac {b e^3 n x^3}{36 d^3}-\frac {b e^2 n x^{10/3}}{40 d^2}+\frac {b e n x^{11/3}}{44 d}-\frac {b e^{12} n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{4 d^{12}}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {b e^{12} n \log (x)}{12 d^{12}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 218, normalized size = 0.91 \begin {gather*} \frac {a x^4}{4}+\frac {1}{4} b x^4 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-\frac {1}{4} b e n \left (-\frac {e^{10} \sqrt [3]{x}}{d^{11}}+\frac {e^9 x^{2/3}}{2 d^{10}}-\frac {e^8 x}{3 d^9}+\frac {e^7 x^{4/3}}{4 d^8}-\frac {e^6 x^{5/3}}{5 d^7}+\frac {e^5 x^2}{6 d^6}-\frac {e^4 x^{7/3}}{7 d^5}+\frac {e^3 x^{8/3}}{8 d^4}-\frac {e^2 x^3}{9 d^3}+\frac {e x^{10/3}}{10 d^2}-\frac {x^{11/3}}{11 d}+\frac {e^{11} \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^{12}}+\frac {e^{11} \log (x)}{3 d^{12}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.28, size = 155, normalized size = 0.65 \begin {gather*} \frac {1}{4} \, b x^{4} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + \frac {1}{4} \, a x^{4} + \frac {1}{110880} \, b n {\left (\frac {2520 \, d^{10} x^{\frac {11}{3}} - 2772 \, d^{9} x^{\frac {10}{3}} e + 3080 \, d^{8} x^{3} e^{2} - 3465 \, d^{7} x^{\frac {8}{3}} e^{3} + 3960 \, d^{6} x^{\frac {7}{3}} e^{4} - 4620 \, d^{5} x^{2} e^{5} + 5544 \, d^{4} x^{\frac {5}{3}} e^{6} - 6930 \, d^{3} x^{\frac {4}{3}} e^{7} + 9240 \, d^{2} x e^{8} - 13860 \, d x^{\frac {2}{3}} e^{9} + 27720 \, x^{\frac {1}{3}} e^{10}}{d^{11}} - \frac {27720 \, e^{11} \log \left (d x^{\frac {1}{3}} + e\right )}{d^{12}}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4*log(c*(d + e/x^(1/3))^n) + 1/4*a*x^4 + 1/110880*b*n*((2520*d^10*x^(11/3) - 2772*d^9*x^(10/3)*e + 308
0*d^8*x^3*e^2 - 3465*d^7*x^(8/3)*e^3 + 3960*d^6*x^(7/3)*e^4 - 4620*d^5*x^2*e^5 + 5544*d^4*x^(5/3)*e^6 - 6930*d
^3*x^(4/3)*e^7 + 9240*d^2*x*e^8 - 13860*d*x^(2/3)*e^9 + 27720*x^(1/3)*e^10)/d^11 - 27720*e^11*log(d*x^(1/3) +
e)/d^12)*e

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Fricas [A]
time = 0.38, size = 225, normalized size = 0.94 \begin {gather*} \frac {27720 \, b d^{12} x^{4} \log \left (c\right ) + 27720 \, a d^{12} x^{4} + 3080 \, b d^{9} n x^{3} e^{3} - 27720 \, b d^{12} n \log \left (x^{\frac {1}{3}}\right ) - 4620 \, b d^{6} n x^{2} e^{6} + 9240 \, b d^{3} n x e^{9} + 27720 \, {\left (b d^{12} n - b n e^{12}\right )} \log \left (d x^{\frac {1}{3}} + e\right ) + 27720 \, {\left (b d^{12} n x^{4} - b d^{12} n\right )} \log \left (\frac {d x + x^{\frac {2}{3}} e}{x}\right ) + 63 \, {\left (40 \, b d^{11} n x^{3} e - 55 \, b d^{8} n x^{2} e^{4} + 88 \, b d^{5} n x e^{7} - 220 \, b d^{2} n e^{10}\right )} x^{\frac {2}{3}} - 198 \, {\left (14 \, b d^{10} n x^{3} e^{2} - 20 \, b d^{7} n x^{2} e^{5} + 35 \, b d^{4} n x e^{8} - 140 \, b d n e^{11}\right )} x^{\frac {1}{3}}}{110880 \, d^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/110880*(27720*b*d^12*x^4*log(c) + 27720*a*d^12*x^4 + 3080*b*d^9*n*x^3*e^3 - 27720*b*d^12*n*log(x^(1/3)) - 46
20*b*d^6*n*x^2*e^6 + 9240*b*d^3*n*x*e^9 + 27720*(b*d^12*n - b*n*e^12)*log(d*x^(1/3) + e) + 27720*(b*d^12*n*x^4
- b*d^12*n)*log((d*x + x^(2/3)*e)/x) + 63*(40*b*d^11*n*x^3*e - 55*b*d^8*n*x^2*e^4 + 88*b*d^5*n*x*e^7 - 220*b*
d^2*n*e^10)*x^(2/3) - 198*(14*b*d^10*n*x^3*e^2 - 20*b*d^7*n*x^2*e^5 + 35*b*d^4*n*x*e^8 - 140*b*d*n*e^11)*x^(1/
3))/d^12

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3878 deep

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Giac [A]
time = 4.40, size = 161, normalized size = 0.67 \begin {gather*} \frac {1}{4} \, b x^{4} \log \left (c\right ) + \frac {1}{4} \, a x^{4} + \frac {1}{110880} \, {\left (27720 \, x^{4} \log \left (d + \frac {e}{x^{\frac {1}{3}}}\right ) + {\left (\frac {2520 \, d^{10} x^{\frac {11}{3}} - 2772 \, d^{9} x^{\frac {10}{3}} e + 3080 \, d^{8} x^{3} e^{2} - 3465 \, d^{7} x^{\frac {8}{3}} e^{3} + 3960 \, d^{6} x^{\frac {7}{3}} e^{4} - 4620 \, d^{5} x^{2} e^{5} + 5544 \, d^{4} x^{\frac {5}{3}} e^{6} - 6930 \, d^{3} x^{\frac {4}{3}} e^{7} + 9240 \, d^{2} x e^{8} - 13860 \, d x^{\frac {2}{3}} e^{9} + 27720 \, x^{\frac {1}{3}} e^{10}}{d^{11}} - \frac {27720 \, e^{11} \log \left ({\left | d x^{\frac {1}{3}} + e \right |}\right )}{d^{12}}\right )} e\right )} b n \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.80, size = 191, normalized size = 0.80 \begin {gather*} \frac {\frac {a\,d^{12}\,x^4}{4}-\frac {b\,e^{12}\,n\,\mathrm {atanh}\left (\frac {2\,e}{d\,x^{1/3}}+1\right )}{2}+\frac {b\,d^{12}\,x^4\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{4}+\frac {b\,d^3\,e^9\,n\,x}{12}+\frac {b\,d\,e^{11}\,n\,x^{1/3}}{4}+\frac {b\,d^{11}\,e\,n\,x^{11/3}}{44}-\frac {b\,d^6\,e^6\,n\,x^2}{24}+\frac {b\,d^9\,e^3\,n\,x^3}{36}-\frac {b\,d^2\,e^{10}\,n\,x^{2/3}}{8}-\frac {b\,d^4\,e^8\,n\,x^{4/3}}{16}+\frac {b\,d^5\,e^7\,n\,x^{5/3}}{20}+\frac {b\,d^7\,e^5\,n\,x^{7/3}}{28}-\frac {b\,d^8\,e^4\,n\,x^{8/3}}{32}-\frac {b\,d^{10}\,e^2\,n\,x^{10/3}}{40}}{d^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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