Optimal. Leaf size=132 \[ -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}+\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{9/2}}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b n}{27 x^3} \]
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Rubi [A] time = 0.09, 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 = {2455, 263, 341, 325, 205} \[ -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{9/2}}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b n}{27 x^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 263
Rule 325
Rule 341
Rule 2455
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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.07, size = 137, normalized size = 1.04 \[ -\frac {a}{3 x^3}-\frac {b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )}{3 e^{9/2}}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b n}{27 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 339, normalized size = 2.57 \[ \left [\frac {315 \, b d^{4} n x^{3} \sqrt {-\frac {d}{e}} \log \left (\frac {d^{3} x^{2} - 2 \, d e^{2} x \sqrt {-\frac {d}{e}} - e^{3} + 2 \, {\left (d^{2} e x \sqrt {-\frac {d}{e}} + d e^{2}\right )} x^{\frac {2}{3}} - 2 \, {\left (d^{2} e x - e^{3} \sqrt {-\frac {d}{e}}\right )} x^{\frac {1}{3}}}{d^{3} x^{2} + e^{3}}\right ) - 210 \, b d^{3} e n x^{2} + 126 \, b d^{2} e^{2} n x^{\frac {4}{3}} - 315 \, b e^{4} n \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) + 70 \, b e^{4} n - 315 \, b e^{4} \log \relax (c) - 315 \, a e^{4} + 90 \, {\left (7 \, b d^{4} n x^{2} - b d e^{3} n\right )} x^{\frac {2}{3}}}{945 \, e^{4} x^{3}}, \frac {630 \, b d^{4} n x^{3} \sqrt {\frac {d}{e}} \arctan \left (x^{\frac {1}{3}} \sqrt {\frac {d}{e}}\right ) - 210 \, b d^{3} e n x^{2} + 126 \, b d^{2} e^{2} n x^{\frac {4}{3}} - 315 \, b e^{4} n \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) + 70 \, b e^{4} n - 315 \, b e^{4} \log \relax (c) - 315 \, a e^{4} + 90 \, {\left (7 \, b d^{4} n x^{2} - b d e^{3} n\right )} x^{\frac {2}{3}}}{945 \, e^{4} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 103, normalized size = 0.78 \[ \frac {1}{945} \, {\left (2 \, {\left (315 \, d^{\frac {9}{2}} \arctan \left (\sqrt {d} x^{\frac {1}{3}} e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {11}{2}\right )} + \frac {{\left (315 \, d^{4} x^{\frac {8}{3}} - 105 \, d^{3} x^{2} e + 63 \, d^{2} x^{\frac {4}{3}} e^{2} - 45 \, d x^{\frac {2}{3}} e^{3} + 35 \, e^{4}\right )} e^{\left (-5\right )}}{x^{3}}\right )} e - \frac {315 \, \log \left (d + \frac {e}{x^{\frac {2}{3}}}\right )}{x^{3}}\right )} b n - \frac {b \log \relax (c)}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \]
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 \[ \int \frac {b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )+a}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 105, normalized size = 0.80 \[ \frac {2}{945} \, b e n {\left (\frac {315 \, d^{5} \arctan \left (\frac {d x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} e^{5}} + \frac {315 \, d^{4} x^{\frac {8}{3}} - 105 \, d^{3} e x^{2} + 63 \, d^{2} e^{2} x^{\frac {4}{3}} - 45 \, d e^{3} x^{\frac {2}{3}} + 35 \, e^{4}}{e^{5} x^{3}}\right )} - \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{x^4} \,d x \]
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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