\(\int \frac {a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n)}{x^4} \, dx\) [496]

Optimal result

Integrand size = 22, antiderivative size = 187 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^4} \, dx=\frac {b n}{27 x^3}-\frac {b d n}{24 e x^{8/3}}+\frac {b d^2 n}{21 e^2 x^{7/3}}-\frac {b d^3 n}{18 e^3 x^2}+\frac {b d^4 n}{15 e^4 x^{5/3}}-\frac {b d^5 n}{12 e^5 x^{4/3}}+\frac {b d^6 n}{9 e^6 x}-\frac {b d^7 n}{6 e^7 x^{2/3}}+\frac {b d^8 n}{3 e^8 \sqrt [3]{x}}-\frac {b d^9 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{3 e^9}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3} \]

[Out]

1/27*b*n/x^3-1/24*b*d*n/e/x^(8/3)+1/21*b*d^2*n/e^2/x^(7/3)-1/18*b*d^3*n/e^3/x^2+1/15*b*d^4*n/e^4/x^(5/3)-1/12*
b*d^5*n/e^5/x^(4/3)+1/9*b*d^6*n/e^6/x-1/6*b*d^7*n/e^7/x^(2/3)+1/3*b*d^8*n/e^8/x^(1/3)-1/3*b*d^9*n*ln(d+e/x^(1/
3))/e^9+1/3*(-a-b*ln(c*(d+e/x^(1/3))^n))/x^3

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 187, 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} \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^4} \, dx=-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3}-\frac {b d^9 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{3 e^9}+\frac {b d^8 n}{3 e^8 \sqrt [3]{x}}-\frac {b d^7 n}{6 e^7 x^{2/3}}+\frac {b d^6 n}{9 e^6 x}-\frac {b d^5 n}{12 e^5 x^{4/3}}+\frac {b d^4 n}{15 e^4 x^{5/3}}-\frac {b d^3 n}{18 e^3 x^2}+\frac {b d^2 n}{21 e^2 x^{7/3}}-\frac {b d n}{24 e x^{8/3}}+\frac {b n}{27 x^3} \]

[In]

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

[Out]

(b*n)/(27*x^3) - (b*d*n)/(24*e*x^(8/3)) + (b*d^2*n)/(21*e^2*x^(7/3)) - (b*d^3*n)/(18*e^3*x^2) + (b*d^4*n)/(15*
e^4*x^(5/3)) - (b*d^5*n)/(12*e^5*x^(4/3)) + (b*d^6*n)/(9*e^6*x) - (b*d^7*n)/(6*e^7*x^(2/3)) + (b*d^8*n)/(3*e^8
*x^(1/3)) - (b*d^9*n*Log[d + e/x^(1/3)])/(3*e^9) - (a + b*Log[c*(d + e/x^(1/3))^n])/(3*x^3)

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}& = -\left (3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3}+\frac {1}{3} (b e n) \text {Subst}\left (\int \frac {x^9}{d+e x} \, dx,x,\frac {1}{\sqrt [3]{x}}\right ) \\ & = -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3}+\frac {1}{3} (b e n) \text {Subst}\left (\int \left (\frac {d^8}{e^9}-\frac {d^7 x}{e^8}+\frac {d^6 x^2}{e^7}-\frac {d^5 x^3}{e^6}+\frac {d^4 x^4}{e^5}-\frac {d^3 x^5}{e^4}+\frac {d^2 x^6}{e^3}-\frac {d x^7}{e^2}+\frac {x^8}{e}-\frac {d^9}{e^9 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right ) \\ & = \frac {b n}{27 x^3}-\frac {b d n}{24 e x^{8/3}}+\frac {b d^2 n}{21 e^2 x^{7/3}}-\frac {b d^3 n}{18 e^3 x^2}+\frac {b d^4 n}{15 e^4 x^{5/3}}-\frac {b d^5 n}{12 e^5 x^{4/3}}+\frac {b d^6 n}{9 e^6 x}-\frac {b d^7 n}{6 e^7 x^{2/3}}+\frac {b d^8 n}{3 e^8 \sqrt [3]{x}}-\frac {b d^9 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{3 e^9}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {1}{9} b e n \left (-\frac {1}{3 e x^3}+\frac {3 d}{8 e^2 x^{8/3}}-\frac {3 d^2}{7 e^3 x^{7/3}}+\frac {d^3}{2 e^4 x^2}-\frac {3 d^4}{5 e^5 x^{5/3}}+\frac {3 d^5}{4 e^6 x^{4/3}}-\frac {d^6}{e^7 x}+\frac {3 d^7}{2 e^8 x^{2/3}}-\frac {3 d^8}{e^9 \sqrt [3]{x}}+\frac {3 d^9 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^{10}}\right )-\frac {b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3} \]

[In]

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

[Out]

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

Maple [F]

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

[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 = 0.41 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.14 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^4} \, dx=\frac {840 \, b d^{6} e^{3} n x^{2} - 420 \, b d^{3} e^{6} n x + 280 \, b e^{9} n - 2520 \, a e^{9} + 140 \, {\left (18 \, a e^{9} - {\left (6 \, b d^{6} e^{3} - 3 \, b d^{3} e^{6} + 2 \, b e^{9}\right )} n\right )} x^{3} + 2520 \, {\left (b e^{9} x^{3} - b e^{9}\right )} \log \left (c\right ) - 2520 \, {\left (b d^{9} n x^{3} + b e^{9} n\right )} \log \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right ) + 90 \, {\left (28 \, b d^{8} e n x^{2} - 7 \, b d^{5} e^{4} n x + 4 \, b d^{2} e^{7} n\right )} x^{\frac {2}{3}} - 63 \, {\left (20 \, b d^{7} e^{2} n x^{2} - 8 \, b d^{4} e^{5} n x + 5 \, b d e^{8} n\right )} x^{\frac {1}{3}}}{7560 \, e^{9} x^{3}} \]

[In]

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

[Out]

1/7560*(840*b*d^6*e^3*n*x^2 - 420*b*d^3*e^6*n*x + 280*b*e^9*n - 2520*a*e^9 + 140*(18*a*e^9 - (6*b*d^6*e^3 - 3*
b*d^3*e^6 + 2*b*e^9)*n)*x^3 + 2520*(b*e^9*x^3 - b*e^9)*log(c) - 2520*(b*d^9*n*x^3 + b*e^9*n)*log((d*x + e*x^(2
/3))/x) + 90*(28*b*d^8*e*n*x^2 - 7*b*d^5*e^4*n*x + 4*b*d^2*e^7*n)*x^(2/3) - 63*(20*b*d^7*e^2*n*x^2 - 8*b*d^4*e
^5*n*x + 5*b*d*e^8*n)*x^(1/3))/(e^9*x^3)

Sympy [F(-1)]

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

[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 = 0.20 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^4} \, dx=-\frac {1}{7560} \, b e n {\left (\frac {2520 \, d^{9} \log \left (d x^{\frac {1}{3}} + e\right )}{e^{10}} - \frac {840 \, d^{9} \log \left (x\right )}{e^{10}} - \frac {2520 \, d^{8} x^{\frac {8}{3}} - 1260 \, d^{7} e x^{\frac {7}{3}} + 840 \, d^{6} e^{2} x^{2} - 630 \, d^{5} e^{3} x^{\frac {5}{3}} + 504 \, d^{4} e^{4} x^{\frac {4}{3}} - 420 \, d^{3} e^{5} x + 360 \, d^{2} e^{6} x^{\frac {2}{3}} - 315 \, d e^{7} x^{\frac {1}{3}} + 280 \, e^{8}}{e^{9} x^{3}}\right )} - \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \]

[In]

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

[Out]

-1/7560*b*e*n*(2520*d^9*log(d*x^(1/3) + e)/e^10 - 840*d^9*log(x)/e^10 - (2520*d^8*x^(8/3) - 1260*d^7*e*x^(7/3)
+ 840*d^6*e^2*x^2 - 630*d^5*e^3*x^(5/3) + 504*d^4*e^4*x^(4/3) - 420*d^3*e^5*x + 360*d^2*e^6*x^(2/3) - 315*d*e
^7*x^(1/3) + 280*e^8)/(e^9*x^3)) - 1/3*b*log(c*(d + e/x^(1/3))^n)/x^3 - 1/3*a/x^3

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^4} \, dx=-\frac {1}{7560} \, {\left (e {\left (\frac {2520 \, d^{9} \log \left ({\left | d x^{\frac {1}{3}} + e \right |}\right )}{e^{10}} - \frac {840 \, d^{9} \log \left ({\left | x \right |}\right )}{e^{10}} - \frac {2520 \, d^{8} e x^{\frac {8}{3}} - 1260 \, d^{7} e^{2} x^{\frac {7}{3}} + 840 \, d^{6} e^{3} x^{2} - 630 \, d^{5} e^{4} x^{\frac {5}{3}} + 504 \, d^{4} e^{5} x^{\frac {4}{3}} - 420 \, d^{3} e^{6} x + 360 \, d^{2} e^{7} x^{\frac {2}{3}} - 315 \, d e^{8} x^{\frac {1}{3}} + 280 \, e^{9}}{e^{10} x^{3}}\right )} + \frac {2520 \, \log \left (d + \frac {e}{x^{\frac {1}{3}}}\right )}{x^{3}}\right )} b n - \frac {b \log \left (c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \]

[In]

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

[Out]

-1/7560*(e*(2520*d^9*log(abs(d*x^(1/3) + e))/e^10 - 840*d^9*log(abs(x))/e^10 - (2520*d^8*e*x^(8/3) - 1260*d^7*
e^2*x^(7/3) + 840*d^6*e^3*x^2 - 630*d^5*e^4*x^(5/3) + 504*d^4*e^5*x^(4/3) - 420*d^3*e^6*x + 360*d^2*e^7*x^(2/3
) - 315*d*e^8*x^(1/3) + 280*e^9)/(e^10*x^3)) + 2520*log(d + e/x^(1/3))/x^3)*b*n - 1/3*b*log(c)/x^3 - 1/3*a/x^3

Mupad [B] (verification not implemented)

Time = 1.73 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.81 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^4} \, dx=\frac {b\,n}{27\,x^3}-\frac {a}{3\,x^3}-\frac {b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{3\,x^3}-\frac {b\,d\,n}{24\,e\,x^{8/3}}-\frac {b\,d^9\,n\,\ln \left (d+\frac {e}{x^{1/3}}\right )}{3\,e^9}-\frac {b\,d^3\,n}{18\,e^3\,x^2}+\frac {b\,d^6\,n}{9\,e^6\,x}+\frac {b\,d^2\,n}{21\,e^2\,x^{7/3}}+\frac {b\,d^4\,n}{15\,e^4\,x^{5/3}}-\frac {b\,d^5\,n}{12\,e^5\,x^{4/3}}-\frac {b\,d^7\,n}{6\,e^7\,x^{2/3}}+\frac {b\,d^8\,n}{3\,e^8\,x^{1/3}} \]

[In]

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

[Out]

(b*n)/(27*x^3) - a/(3*x^3) - (b*log(c*(d + e/x^(1/3))^n))/(3*x^3) - (b*d*n)/(24*e*x^(8/3)) - (b*d^9*n*log(d +
e/x^(1/3)))/(3*e^9) - (b*d^3*n)/(18*e^3*x^2) + (b*d^6*n)/(9*e^6*x) + (b*d^2*n)/(21*e^2*x^(7/3)) + (b*d^4*n)/(1
5*e^4*x^(5/3)) - (b*d^5*n)/(12*e^5*x^(4/3)) - (b*d^7*n)/(6*e^7*x^(2/3)) + (b*d^8*n)/(3*e^8*x^(1/3))