Optimal. Leaf size=192 \[ -\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}-\frac {b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}+\frac {b e^9 n \log (x)}{9 d^9}+\frac {b e^8 n}{3 d^8 \sqrt [3]{x}}-\frac {b e^7 n}{6 d^7 x^{2/3}}+\frac {b e^6 n}{9 d^6 x}-\frac {b e^5 n}{12 d^5 x^{4/3}}+\frac {b e^4 n}{15 d^4 x^{5/3}}-\frac {b e^3 n}{18 d^3 x^2}+\frac {b e^2 n}{21 d^2 x^{7/3}}-\frac {b e n}{24 d x^{8/3}} \]
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Rubi [A] time = 0.13, antiderivative size = 192, 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 = {2454, 2395, 44} \[ -\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}-\frac {b e^7 n}{6 d^7 x^{2/3}}-\frac {b e^5 n}{12 d^5 x^{4/3}}+\frac {b e^4 n}{15 d^4 x^{5/3}}-\frac {b e^3 n}{18 d^3 x^2}+\frac {b e^2 n}{21 d^2 x^{7/3}}+\frac {b e^8 n}{3 d^8 \sqrt [3]{x}}+\frac {b e^6 n}{9 d^6 x}-\frac {b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}+\frac {b e^9 n \log (x)}{9 d^9}-\frac {b e n}{24 d x^{8/3}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^4} \, dx &=3 \operatorname {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) \operatorname {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) \operatorname {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*}
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Mathematica [A] time = 0.20, size = 177, normalized size = 0.92 \[ -\frac {a}{3 x^3}-\frac {b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {1}{3} b e n \left (-\frac {e^8 \log \left (d+e \sqrt [3]{x}\right )}{d^9}+\frac {e^8 \log (x)}{3 d^9}+\frac {e^7}{d^8 \sqrt [3]{x}}-\frac {e^6}{2 d^7 x^{2/3}}+\frac {e^5}{3 d^6 x}-\frac {e^4}{4 d^5 x^{4/3}}+\frac {e^3}{5 d^4 x^{5/3}}-\frac {e^2}{6 d^3 x^2}+\frac {e}{7 d^2 x^{7/3}}-\frac {1}{8 d x^{8/3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 163, normalized size = 0.85 \[ \frac {840 \, b e^{9} n x^{3} \log \left (x^{\frac {1}{3}}\right ) + 280 \, b d^{3} e^{6} n x^{2} - 140 \, b d^{6} e^{3} n x - 840 \, b d^{9} \log \relax (c) - 840 \, a d^{9} - 840 \, {\left (b e^{9} n x^{3} + b d^{9} n\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 30 \, {\left (28 \, b d e^{8} n x^{2} - 7 \, b d^{4} e^{5} n x + 4 \, b d^{7} e^{2} n\right )} x^{\frac {2}{3}} - 21 \, {\left (20 \, b d^{2} e^{7} n x^{2} - 8 \, b d^{5} e^{4} n x + 5 \, b d^{8} e n\right )} x^{\frac {1}{3}}}{2520 \, d^{9} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 808, normalized size = 4.21 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e \,x^{\frac {1}{3}}+d \right )^{n}\right )+a}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 139, normalized size = 0.72 \[ -\frac {1}{2520} \, b e n {\left (\frac {840 \, e^{8} \log \left (e x^{\frac {1}{3}} + d\right )}{d^{9}} - \frac {280 \, e^{8} \log \relax (x)}{d^{9}} - \frac {840 \, e^{7} x^{\frac {7}{3}} - 420 \, d e^{6} x^{2} + 280 \, d^{2} e^{5} x^{\frac {5}{3}} - 210 \, d^{3} e^{4} x^{\frac {4}{3}} + 168 \, d^{4} e^{3} x - 140 \, d^{5} e^{2} x^{\frac {2}{3}} + 120 \, d^{6} e x^{\frac {1}{3}} - 105 \, d^{7}}{d^{8} x^{\frac {8}{3}}}\right )} - \frac {b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 154, normalized size = 0.80 \[ -\frac {\frac {a\,d^9}{3}+\frac {b\,d^9\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3}+\frac {b\,d^6\,e^3\,n\,x}{18}+\frac {b\,d^8\,e\,n\,x^{1/3}}{24}-\frac {b\,d\,e^8\,n\,x^{8/3}}{3}-\frac {b\,d^3\,e^6\,n\,x^2}{9}-\frac {b\,d^7\,e^2\,n\,x^{2/3}}{21}-\frac {b\,d^5\,e^4\,n\,x^{4/3}}{15}+\frac {b\,d^4\,e^5\,n\,x^{5/3}}{12}+\frac {b\,d^2\,e^7\,n\,x^{7/3}}{6}}{d^9\,x^3}-\frac {2\,b\,e^9\,n\,\mathrm {atanh}\left (\frac {2\,e\,x^{1/3}}{d}+1\right )}{3\,d^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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