Optimal. Leaf size=132 \[ \frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}+\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{9/2}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2505, 269, 348,
331, 211} \begin {gather*} -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}+\frac {2 b d^{9/2} n \text {ArcTan}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{9/2}}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b n}{27 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 269
Rule 331
Rule 348
Rule 2505
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx &=-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {1}{9} (2 b e n) \int \frac {1}{\left (d+\frac {e}{x^{2/3}}\right ) x^{14/3}} \, dx\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {1}{9} (2 b e n) \int \frac {1}{\left (e+d x^{2/3}\right ) x^4} \, dx\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {1}{3} (2 b e n) \text {Subst}\left (\int \frac {1}{x^{10} \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 b n}{27 x^3}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}+\frac {1}{3} (2 b d n) \text {Subst}\left (\int \frac {1}{x^8 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {\left (2 b d^2 n\right ) \text {Subst}\left (\int \frac {1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 e}\\ &=\frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}+\frac {\left (2 b d^3 n\right ) \text {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 e^2}\\ &=\frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d^3 n}{9 e^3 x}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {\left (2 b d^4 n\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}\\ &=\frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}\\ &=\frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}+\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{9/2}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 137, normalized size = 1.04 \begin {gather*} -\frac {a}{3 x^3}+\frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}-\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )}{3 e^{9/2}}-\frac {b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 97, normalized size = 0.73 \begin {gather*} \frac {2}{945} \, {\left (315 \, d^{\frac {9}{2}} \arctan \left (\sqrt {d} x^{\frac {1}{3}} e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {11}{2}\right )} + \frac {{\left (315 \, d^{4} x^{\frac {8}{3}} - 105 \, d^{3} x^{2} e + 63 \, d^{2} x^{\frac {4}{3}} e^{2} - 45 \, d x^{\frac {2}{3}} e^{3} + 35 \, e^{4}\right )} e^{\left (-5\right )}}{x^{3}}\right )} b n e - \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 317, normalized size = 2.40 \begin {gather*} \left [\frac {{\left (315 \, \sqrt {-d e^{\left (-1\right )}} b d^{4} n x^{3} \log \left (\frac {d^{3} x^{2} - 2 \, \sqrt {-d e^{\left (-1\right )}} d x e^{2} + 2 \, {\left (\sqrt {-d e^{\left (-1\right )}} d^{2} x e + d e^{2}\right )} x^{\frac {2}{3}} - 2 \, {\left (d^{2} x e - \sqrt {-d e^{\left (-1\right )}} e^{3}\right )} x^{\frac {1}{3}} - e^{3}}{d^{3} x^{2} + e^{3}}\right ) - 210 \, b d^{3} n x^{2} e + 126 \, b d^{2} n x^{\frac {4}{3}} e^{2} - 315 \, b n e^{4} \log \left (\frac {d x + x^{\frac {1}{3}} e}{x}\right ) - 315 \, b e^{4} \log \left (c\right ) + 35 \, {\left (2 \, b n - 9 \, a\right )} e^{4} + 90 \, {\left (7 \, b d^{4} n x^{2} - b d n e^{3}\right )} x^{\frac {2}{3}}\right )} e^{\left (-4\right )}}{945 \, x^{3}}, \frac {{\left (630 \, b d^{\frac {9}{2}} n x^{3} \arctan \left (\sqrt {d} x^{\frac {1}{3}} e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {1}{2}\right )} - 210 \, b d^{3} n x^{2} e + 126 \, b d^{2} n x^{\frac {4}{3}} e^{2} - 315 \, b n e^{4} \log \left (\frac {d x + x^{\frac {1}{3}} e}{x}\right ) - 315 \, b e^{4} \log \left (c\right ) + 35 \, {\left (2 \, b n - 9 \, a\right )} e^{4} + 90 \, {\left (7 \, b d^{4} n x^{2} - b d n e^{3}\right )} x^{\frac {2}{3}}\right )} e^{\left (-4\right )}}{945 \, x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.72, size = 103, normalized size = 0.78 \begin {gather*} \frac {1}{945} \, {\left (2 \, {\left (315 \, d^{\frac {9}{2}} \arctan \left (\sqrt {d} x^{\frac {1}{3}} e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {11}{2}\right )} + \frac {{\left (315 \, d^{4} x^{\frac {8}{3}} - 105 \, d^{3} x^{2} e + 63 \, d^{2} x^{\frac {4}{3}} e^{2} - 45 \, d x^{\frac {2}{3}} e^{3} + 35 \, e^{4}\right )} e^{\left (-5\right )}}{x^{3}}\right )} e - \frac {315 \, \log \left (d + \frac {e}{x^{\frac {2}{3}}}\right )}{x^{3}}\right )} b n - \frac {b \log \left (c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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