Optimal. Leaf size=234 \[ \frac {b d^{11} n \sqrt [3]{x}}{4 e^{11}}-\frac {b d^{10} n x^{2/3}}{8 e^{10}}+\frac {b d^9 n x}{12 e^9}-\frac {b d^8 n x^{4/3}}{16 e^8}+\frac {b d^7 n x^{5/3}}{20 e^7}-\frac {b d^6 n x^2}{24 e^6}+\frac {b d^5 n x^{7/3}}{28 e^5}-\frac {b d^4 n x^{8/3}}{32 e^4}+\frac {b d^3 n x^3}{36 e^3}-\frac {b d^2 n x^{10/3}}{40 e^2}+\frac {b d n x^{11/3}}{44 e}-\frac {1}{48} b n x^4-\frac {b d^{12} n \log \left (d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 234, 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, 45}
\begin {gather*} \frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {b d^{12} n \log \left (d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac {b d^{11} n \sqrt [3]{x}}{4 e^{11}}-\frac {b d^{10} n x^{2/3}}{8 e^{10}}+\frac {b d^9 n x}{12 e^9}-\frac {b d^8 n x^{4/3}}{16 e^8}+\frac {b d^7 n x^{5/3}}{20 e^7}-\frac {b d^6 n x^2}{24 e^6}+\frac {b d^5 n x^{7/3}}{28 e^5}-\frac {b d^4 n x^{8/3}}{32 e^4}+\frac {b d^3 n x^3}{36 e^3}-\frac {b d^2 n x^{10/3}}{40 e^2}+\frac {b d n x^{11/3}}{44 e}-\frac {1}{48} b n x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx &=3 \text {Subst}\left (\int x^{11} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{4} (b e n) \text {Subst}\left (\int \frac {x^{12}}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{4} (b e n) \text {Subst}\left (\int \left (-\frac {d^{11}}{e^{12}}+\frac {d^{10} x}{e^{11}}-\frac {d^9 x^2}{e^{10}}+\frac {d^8 x^3}{e^9}-\frac {d^7 x^4}{e^8}+\frac {d^6 x^5}{e^7}-\frac {d^5 x^6}{e^6}+\frac {d^4 x^7}{e^5}-\frac {d^3 x^8}{e^4}+\frac {d^2 x^9}{e^3}-\frac {d x^{10}}{e^2}+\frac {x^{11}}{e}+\frac {d^{12}}{e^{12} (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {b d^{11} n \sqrt [3]{x}}{4 e^{11}}-\frac {b d^{10} n x^{2/3}}{8 e^{10}}+\frac {b d^9 n x}{12 e^9}-\frac {b d^8 n x^{4/3}}{16 e^8}+\frac {b d^7 n x^{5/3}}{20 e^7}-\frac {b d^6 n x^2}{24 e^6}+\frac {b d^5 n x^{7/3}}{28 e^5}-\frac {b d^4 n x^{8/3}}{32 e^4}+\frac {b d^3 n x^3}{36 e^3}-\frac {b d^2 n x^{10/3}}{40 e^2}+\frac {b d n x^{11/3}}{44 e}-\frac {1}{48} b n x^4-\frac {b d^{12} n \log \left (d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 219, normalized size = 0.94 \begin {gather*} \frac {a x^4}{4}-\frac {1}{4} b e n \left (-\frac {d^{11} \sqrt [3]{x}}{e^{12}}+\frac {d^{10} x^{2/3}}{2 e^{11}}-\frac {d^9 x}{3 e^{10}}+\frac {d^8 x^{4/3}}{4 e^9}-\frac {d^7 x^{5/3}}{5 e^8}+\frac {d^6 x^2}{6 e^7}-\frac {d^5 x^{7/3}}{7 e^6}+\frac {d^4 x^{8/3}}{8 e^5}-\frac {d^3 x^3}{9 e^4}+\frac {d^2 x^{10/3}}{10 e^3}-\frac {d x^{11/3}}{11 e^2}+\frac {x^4}{12 e}+\frac {d^{12} \log \left (d+e \sqrt [3]{x}\right )}{e^{13}}\right )+\frac {1}{4} b x^4 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 164, normalized size = 0.70 \begin {gather*} \frac {1}{4} \, b x^{4} \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right ) + \frac {1}{4} \, a x^{4} - \frac {1}{110880} \, {\left (27720 \, d^{12} e^{\left (-13\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (13860 \, d^{10} x^{\frac {2}{3}} e - 27720 \, d^{11} x^{\frac {1}{3}} - 9240 \, d^{9} x e^{2} + 6930 \, d^{8} x^{\frac {4}{3}} e^{3} - 5544 \, d^{7} x^{\frac {5}{3}} e^{4} + 4620 \, d^{6} x^{2} e^{5} - 3960 \, d^{5} x^{\frac {7}{3}} e^{6} + 3465 \, d^{4} x^{\frac {8}{3}} e^{7} - 3080 \, d^{3} x^{3} e^{8} + 2772 \, d^{2} x^{\frac {10}{3}} e^{9} - 2520 \, d x^{\frac {11}{3}} e^{10} + 2310 \, x^{4} e^{11}\right )} e^{\left (-12\right )}\right )} b n e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 186, normalized size = 0.79 \begin {gather*} \frac {1}{110880} \, {\left (9240 \, b d^{9} n x e^{3} - 4620 \, b d^{6} n x^{2} e^{6} + 3080 \, b d^{3} n x^{3} e^{9} + 27720 \, b x^{4} e^{12} \log \left (c\right ) - 2310 \, {\left (b n - 12 \, a\right )} x^{4} e^{12} - 27720 \, {\left (b d^{12} n - b n x^{4} e^{12}\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 63 \, {\left (220 \, b d^{10} n e^{2} - 88 \, b d^{7} n x e^{5} + 55 \, b d^{4} n x^{2} e^{8} - 40 \, b d n x^{3} e^{11}\right )} x^{\frac {2}{3}} + 198 \, {\left (140 \, b d^{11} n e - 35 \, b d^{8} n x e^{4} + 20 \, b d^{5} n x^{2} e^{7} - 14 \, b d^{2} n x^{3} e^{10}\right )} x^{\frac {1}{3}}\right )} e^{\left (-12\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 44.14, size = 216, normalized size = 0.92 \begin {gather*} \frac {a x^{4}}{4} + b \left (- \frac {e n \left (\frac {3 d^{12} \left (\begin {cases} \frac {\sqrt [3]{x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt [3]{x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{12}} - \frac {3 d^{11} \sqrt [3]{x}}{e^{12}} + \frac {3 d^{10} x^{\frac {2}{3}}}{2 e^{11}} - \frac {d^{9} x}{e^{10}} + \frac {3 d^{8} x^{\frac {4}{3}}}{4 e^{9}} - \frac {3 d^{7} x^{\frac {5}{3}}}{5 e^{8}} + \frac {d^{6} x^{2}}{2 e^{7}} - \frac {3 d^{5} x^{\frac {7}{3}}}{7 e^{6}} + \frac {3 d^{4} x^{\frac {8}{3}}}{8 e^{5}} - \frac {d^{3} x^{3}}{3 e^{4}} + \frac {3 d^{2} x^{\frac {10}{3}}}{10 e^{3}} - \frac {3 d x^{\frac {11}{3}}}{11 e^{2}} + \frac {x^{4}}{4 e}\right )}{12} + \frac {x^{4} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 529 vs.
\(2 (176) = 352\).
time = 3.65, size = 529, normalized size = 2.26 \begin {gather*} \frac {1}{110880} \, {\left (27720 \, b x^{4} e \log \left (c\right ) + 27720 \, a x^{4} e + {\left (27720 \, {\left (x^{\frac {1}{3}} e + d\right )}^{12} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 332640 \, {\left (x^{\frac {1}{3}} e + d\right )}^{11} d e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 1829520 \, {\left (x^{\frac {1}{3}} e + d\right )}^{10} d^{2} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 6098400 \, {\left (x^{\frac {1}{3}} e + d\right )}^{9} d^{3} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 13721400 \, {\left (x^{\frac {1}{3}} e + d\right )}^{8} d^{4} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 21954240 \, {\left (x^{\frac {1}{3}} e + d\right )}^{7} d^{5} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 25613280 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} d^{6} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 21954240 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d^{7} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 13721400 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{8} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 6098400 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{9} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 1829520 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{10} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 332640 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{11} e^{\left (-11\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 2310 \, {\left (x^{\frac {1}{3}} e + d\right )}^{12} e^{\left (-11\right )} + 30240 \, {\left (x^{\frac {1}{3}} e + d\right )}^{11} d e^{\left (-11\right )} - 182952 \, {\left (x^{\frac {1}{3}} e + d\right )}^{10} d^{2} e^{\left (-11\right )} + 677600 \, {\left (x^{\frac {1}{3}} e + d\right )}^{9} d^{3} e^{\left (-11\right )} - 1715175 \, {\left (x^{\frac {1}{3}} e + d\right )}^{8} d^{4} e^{\left (-11\right )} + 3136320 \, {\left (x^{\frac {1}{3}} e + d\right )}^{7} d^{5} e^{\left (-11\right )} - 4268880 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} d^{6} e^{\left (-11\right )} + 4390848 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d^{7} e^{\left (-11\right )} - 3430350 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{8} e^{\left (-11\right )} + 2032800 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{9} e^{\left (-11\right )} - 914760 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{10} e^{\left (-11\right )} + 332640 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{11} e^{\left (-11\right )}\right )} b n\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.65, size = 189, normalized size = 0.81 \begin {gather*} \frac {a\,x^4}{4}-\frac {b\,n\,x^4}{48}+\frac {b\,x^4\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{4}+\frac {b\,d\,n\,x^{11/3}}{44\,e}+\frac {b\,d^9\,n\,x}{12\,e^9}-\frac {b\,d^{12}\,n\,\ln \left (d+e\,x^{1/3}\right )}{4\,e^{12}}+\frac {b\,d^3\,n\,x^3}{36\,e^3}-\frac {b\,d^6\,n\,x^2}{24\,e^6}-\frac {b\,d^2\,n\,x^{10/3}}{40\,e^2}-\frac {b\,d^4\,n\,x^{8/3}}{32\,e^4}+\frac {b\,d^5\,n\,x^{7/3}}{28\,e^5}+\frac {b\,d^7\,n\,x^{5/3}}{20\,e^7}-\frac {b\,d^8\,n\,x^{4/3}}{16\,e^8}-\frac {b\,d^{10}\,n\,x^{2/3}}{8\,e^{10}}+\frac {b\,d^{11}\,n\,x^{1/3}}{4\,e^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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