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]
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]
[Out]
Rule 46
Rule 2442
Rule 2504
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*}
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]
[Out]
\[\int \frac {a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )}{x^{4}}d x\]
[In]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]