\(\int \frac {a+b \log (c (d+e \sqrt [3]{x})^n)}{x^4} \, dx\) [449]

Optimal result

Integrand size = 22, antiderivative size = 192 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^4} \, dx=-\frac {b e n}{24 d x^{8/3}}+\frac {b e^2 n}{21 d^2 x^{7/3}}-\frac {b e^3 n}{18 d^3 x^2}+\frac {b e^4 n}{15 d^4 x^{5/3}}-\frac {b e^5 n}{12 d^5 x^{4/3}}+\frac {b e^6 n}{9 d^6 x}-\frac {b e^7 n}{6 d^7 x^{2/3}}+\frac {b e^8 n}{3 d^8 \sqrt [3]{x}}-\frac {b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {b e^9 n \log (x)}{9 d^9} \]

[Out]

-1/24*b*e*n/d/x^(8/3)+1/21*b*e^2*n/d^2/x^(7/3)-1/18*b*e^3*n/d^3/x^2+1/15*b*e^4*n/d^4/x^(5/3)-1/12*b*e^5*n/d^5/
x^(4/3)+1/9*b*e^6*n/d^6/x-1/6*b*e^7*n/d^7/x^(2/3)+1/3*b*e^8*n/d^8/x^(1/3)-1/3*b*e^9*n*ln(d+e*x^(1/3))/d^9+1/3*
(-a-b*ln(c*(d+e*x^(1/3))^n))/x^3+1/9*b*e^9*n*ln(x)/d^9

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 192, 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, 46} \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^4} \, dx=-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}-\frac {b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}+\frac {b e^9 n \log (x)}{9 d^9}+\frac {b e^8 n}{3 d^8 \sqrt [3]{x}}-\frac {b e^7 n}{6 d^7 x^{2/3}}+\frac {b e^6 n}{9 d^6 x}-\frac {b e^5 n}{12 d^5 x^{4/3}}+\frac {b e^4 n}{15 d^4 x^{5/3}}-\frac {b e^3 n}{18 d^3 x^2}+\frac {b e^2 n}{21 d^2 x^{7/3}}-\frac {b e n}{24 d x^{8/3}} \]

[In]

Int[(a + b*Log[c*(d + e*x^(1/3))^n])/x^4,x]

[Out]

-1/24*(b*e*n)/(d*x^(8/3)) + (b*e^2*n)/(21*d^2*x^(7/3)) - (b*e^3*n)/(18*d^3*x^2) + (b*e^4*n)/(15*d^4*x^(5/3)) -
(b*e^5*n)/(12*d^5*x^(4/3)) + (b*e^6*n)/(9*d^6*x) - (b*e^7*n)/(6*d^7*x^(2/3)) + (b*e^8*n)/(3*d^8*x^(1/3)) - (b
*e^9*n*Log[d + e*x^(1/3)])/(3*d^9) - (a + b*Log[c*(d + e*x^(1/3))^n])/(3*x^3) + (b*e^9*n*Log[x])/(9*d^9)

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*} \text {integral}& = 3 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^{10}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {1}{3} (b e n) \text {Subst}\left (\int \frac {1}{x^9 (d+e x)} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {1}{3} (b e n) \text {Subst}\left (\int \left (\frac {1}{d x^9}-\frac {e}{d^2 x^8}+\frac {e^2}{d^3 x^7}-\frac {e^3}{d^4 x^6}+\frac {e^4}{d^5 x^5}-\frac {e^5}{d^6 x^4}+\frac {e^6}{d^7 x^3}-\frac {e^7}{d^8 x^2}+\frac {e^8}{d^9 x}-\frac {e^9}{d^9 (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {b e n}{24 d x^{8/3}}+\frac {b e^2 n}{21 d^2 x^{7/3}}-\frac {b e^3 n}{18 d^3 x^2}+\frac {b e^4 n}{15 d^4 x^{5/3}}-\frac {b e^5 n}{12 d^5 x^{4/3}}+\frac {b e^6 n}{9 d^6 x}-\frac {b e^7 n}{6 d^7 x^{2/3}}+\frac {b e^8 n}{3 d^8 \sqrt [3]{x}}-\frac {b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {b e^9 n \log (x)}{9 d^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {1}{9} b e n \left (-\frac {3}{8 d x^{8/3}}+\frac {3 e}{7 d^2 x^{7/3}}-\frac {e^2}{2 d^3 x^2}+\frac {3 e^3}{5 d^4 x^{5/3}}-\frac {3 e^4}{4 d^5 x^{4/3}}+\frac {e^5}{d^6 x}-\frac {3 e^6}{2 d^7 x^{2/3}}+\frac {3 e^7}{d^8 \sqrt [3]{x}}-\frac {3 e^8 \log \left (d+e \sqrt [3]{x}\right )}{d^9}+\frac {e^8 \log (x)}{d^9}\right ) \]

[In]

Integrate[(a + b*Log[c*(d + e*x^(1/3))^n])/x^4,x]

[Out]

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

Maple [F]

\[\int \frac {a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )}{x^{4}}d x\]

[In]

int((a+b*ln(c*(d+e*x^(1/3))^n))/x^4,x)

[Out]

int((a+b*ln(c*(d+e*x^(1/3))^n))/x^4,x)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^4} \, dx=\frac {840 \, b e^{9} n x^{3} \log \left (x^{\frac {1}{3}}\right ) + 280 \, b d^{3} e^{6} n x^{2} - 140 \, b d^{6} e^{3} n x - 840 \, b d^{9} \log \left (c\right ) - 840 \, a d^{9} - 840 \, {\left (b e^{9} n x^{3} + b d^{9} n\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 30 \, {\left (28 \, b d e^{8} n x^{2} - 7 \, b d^{4} e^{5} n x + 4 \, b d^{7} e^{2} n\right )} x^{\frac {2}{3}} - 21 \, {\left (20 \, b d^{2} e^{7} n x^{2} - 8 \, b d^{5} e^{4} n x + 5 \, b d^{8} e n\right )} x^{\frac {1}{3}}}{2520 \, d^{9} x^{3}} \]

[In]

integrate((a+b*log(c*(d+e*x^(1/3))^n))/x^4,x, algorithm="fricas")

[Out]

1/2520*(840*b*e^9*n*x^3*log(x^(1/3)) + 280*b*d^3*e^6*n*x^2 - 140*b*d^6*e^3*n*x - 840*b*d^9*log(c) - 840*a*d^9
- 840*(b*e^9*n*x^3 + b*d^9*n)*log(e*x^(1/3) + d) + 30*(28*b*d*e^8*n*x^2 - 7*b*d^4*e^5*n*x + 4*b*d^7*e^2*n)*x^(
2/3) - 21*(20*b*d^2*e^7*n*x^2 - 8*b*d^5*e^4*n*x + 5*b*d^8*e*n)*x^(1/3))/(d^9*x^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^4} \, dx=\text {Timed out} \]

[In]

integrate((a+b*ln(c*(d+e*x**(1/3))**n))/x**4,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^4} \, dx=-\frac {1}{2520} \, b e n {\left (\frac {840 \, e^{8} \log \left (e x^{\frac {1}{3}} + d\right )}{d^{9}} - \frac {280 \, e^{8} \log \left (x\right )}{d^{9}} - \frac {840 \, e^{7} x^{\frac {7}{3}} - 420 \, d e^{6} x^{2} + 280 \, d^{2} e^{5} x^{\frac {5}{3}} - 210 \, d^{3} e^{4} x^{\frac {4}{3}} + 168 \, d^{4} e^{3} x - 140 \, d^{5} e^{2} x^{\frac {2}{3}} + 120 \, d^{6} e x^{\frac {1}{3}} - 105 \, d^{7}}{d^{8} x^{\frac {8}{3}}}\right )} - \frac {b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \]

[In]

integrate((a+b*log(c*(d+e*x^(1/3))^n))/x^4,x, algorithm="maxima")

[Out]

-1/2520*b*e*n*(840*e^8*log(e*x^(1/3) + d)/d^9 - 280*e^8*log(x)/d^9 - (840*e^7*x^(7/3) - 420*d*e^6*x^2 + 280*d^
2*e^5*x^(5/3) - 210*d^3*e^4*x^(4/3) + 168*d^4*e^3*x - 140*d^5*e^2*x^(2/3) + 120*d^6*e*x^(1/3) - 105*d^7)/(d^8*
x^(8/3))) - 1/3*b*log((e*x^(1/3) + d)^n*c)/x^3 - 1/3*a/x^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (154) = 308\).

Time = 0.31 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.55 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^4} \, dx=-\frac {\frac {840 \, b e^{10} n \log \left (e x^{\frac {1}{3}} + d\right )}{{\left (e x^{\frac {1}{3}} + d\right )}^{9} - 9 \, {\left (e x^{\frac {1}{3}} + d\right )}^{8} d + 36 \, {\left (e x^{\frac {1}{3}} + d\right )}^{7} d^{2} - 84 \, {\left (e x^{\frac {1}{3}} + d\right )}^{6} d^{3} + 126 \, {\left (e x^{\frac {1}{3}} + d\right )}^{5} d^{4} - 126 \, {\left (e x^{\frac {1}{3}} + d\right )}^{4} d^{5} + 84 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} d^{6} - 36 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d^{7} + 9 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{8} - d^{9}} + \frac {840 \, b e^{10} n \log \left (e x^{\frac {1}{3}} + d\right )}{d^{9}} - \frac {840 \, b e^{10} n \log \left (e x^{\frac {1}{3}}\right )}{d^{9}} - \frac {840 \, {\left (e x^{\frac {1}{3}} + d\right )}^{8} b e^{10} n - 7140 \, {\left (e x^{\frac {1}{3}} + d\right )}^{7} b d e^{10} n + 26740 \, {\left (e x^{\frac {1}{3}} + d\right )}^{6} b d^{2} e^{10} n - 57750 \, {\left (e x^{\frac {1}{3}} + d\right )}^{5} b d^{3} e^{10} n + 78918 \, {\left (e x^{\frac {1}{3}} + d\right )}^{4} b d^{4} e^{10} n - 70252 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} b d^{5} e^{10} n + 40188 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} b d^{6} e^{10} n - 13827 \, {\left (e x^{\frac {1}{3}} + d\right )} b d^{7} e^{10} n + 2283 \, b d^{8} e^{10} n - 840 \, b d^{8} e^{10} \log \left (c\right ) - 840 \, a d^{8} e^{10}}{{\left (e x^{\frac {1}{3}} + d\right )}^{9} d^{8} - 9 \, {\left (e x^{\frac {1}{3}} + d\right )}^{8} d^{9} + 36 \, {\left (e x^{\frac {1}{3}} + d\right )}^{7} d^{10} - 84 \, {\left (e x^{\frac {1}{3}} + d\right )}^{6} d^{11} + 126 \, {\left (e x^{\frac {1}{3}} + d\right )}^{5} d^{12} - 126 \, {\left (e x^{\frac {1}{3}} + d\right )}^{4} d^{13} + 84 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} d^{14} - 36 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d^{15} + 9 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{16} - d^{17}}}{2520 \, e} \]

[In]

integrate((a+b*log(c*(d+e*x^(1/3))^n))/x^4,x, algorithm="giac")

[Out]

-1/2520*(840*b*e^10*n*log(e*x^(1/3) + d)/((e*x^(1/3) + d)^9 - 9*(e*x^(1/3) + d)^8*d + 36*(e*x^(1/3) + d)^7*d^2
- 84*(e*x^(1/3) + d)^6*d^3 + 126*(e*x^(1/3) + d)^5*d^4 - 126*(e*x^(1/3) + d)^4*d^5 + 84*(e*x^(1/3) + d)^3*d^6
- 36*(e*x^(1/3) + d)^2*d^7 + 9*(e*x^(1/3) + d)*d^8 - d^9) + 840*b*e^10*n*log(e*x^(1/3) + d)/d^9 - 840*b*e^10*
n*log(e*x^(1/3))/d^9 - (840*(e*x^(1/3) + d)^8*b*e^10*n - 7140*(e*x^(1/3) + d)^7*b*d*e^10*n + 26740*(e*x^(1/3)
+ d)^6*b*d^2*e^10*n - 57750*(e*x^(1/3) + d)^5*b*d^3*e^10*n + 78918*(e*x^(1/3) + d)^4*b*d^4*e^10*n - 70252*(e*x
^(1/3) + d)^3*b*d^5*e^10*n + 40188*(e*x^(1/3) + d)^2*b*d^6*e^10*n - 13827*(e*x^(1/3) + d)*b*d^7*e^10*n + 2283*
b*d^8*e^10*n - 840*b*d^8*e^10*log(c) - 840*a*d^8*e^10)/((e*x^(1/3) + d)^9*d^8 - 9*(e*x^(1/3) + d)^8*d^9 + 36*(
e*x^(1/3) + d)^7*d^10 - 84*(e*x^(1/3) + d)^6*d^11 + 126*(e*x^(1/3) + d)^5*d^12 - 126*(e*x^(1/3) + d)^4*d^13 +
84*(e*x^(1/3) + d)^3*d^14 - 36*(e*x^(1/3) + d)^2*d^15 + 9*(e*x^(1/3) + d)*d^16 - d^17))/e

Mupad [B] (verification not implemented)

Time = 1.76 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^4} \, dx=-\frac {\frac {a\,d^9}{3}+\frac {b\,d^9\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3}+\frac {b\,d^6\,e^3\,n\,x}{18}+\frac {b\,d^8\,e\,n\,x^{1/3}}{24}-\frac {b\,d\,e^8\,n\,x^{8/3}}{3}-\frac {b\,d^3\,e^6\,n\,x^2}{9}-\frac {b\,d^7\,e^2\,n\,x^{2/3}}{21}-\frac {b\,d^5\,e^4\,n\,x^{4/3}}{15}+\frac {b\,d^4\,e^5\,n\,x^{5/3}}{12}+\frac {b\,d^2\,e^7\,n\,x^{7/3}}{6}}{d^9\,x^3}-\frac {2\,b\,e^9\,n\,\mathrm {atanh}\left (\frac {2\,e\,x^{1/3}}{d}+1\right )}{3\,d^9} \]

[In]

int((a + b*log(c*(d + e*x^(1/3))^n))/x^4,x)

[Out]

- ((a*d^9)/3 + (b*d^9*log(c*(d + e*x^(1/3))^n))/3 + (b*d^6*e^3*n*x)/18 + (b*d^8*e*n*x^(1/3))/24 - (b*d*e^8*n*x
^(8/3))/3 - (b*d^3*e^6*n*x^2)/9 - (b*d^7*e^2*n*x^(2/3))/21 - (b*d^5*e^4*n*x^(4/3))/15 + (b*d^4*e^5*n*x^(5/3))/
12 + (b*d^2*e^7*n*x^(7/3))/6)/(d^9*x^3) - (2*b*e^9*n*atanh((2*e*x^(1/3))/d + 1))/(3*d^9)