Optimal. Leaf size=68 \[ -\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x}-\frac {2 b e^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 b e n}{d \sqrt [3]{x}} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2455, 341, 325, 205} \[ -\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x}-\frac {2 b e^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 b e n}{d \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 325
Rule 341
Rule 2455
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x^2} \, dx &=-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x}+\frac {1}{3} (2 b e n) \int \frac {1}{\left (d+e x^{2/3}\right ) x^{4/3}} \, dx\\ &=-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x}+(2 b e n) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 b e n}{d \sqrt [3]{x}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x}-\frac {\left (2 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=-\frac {2 b e n}{d \sqrt [3]{x}}-\frac {2 b e^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 59, normalized size = 0.87 \[ -\frac {a}{x}-\frac {b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x}-\frac {2 b e n \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {e x^{2/3}}{d}\right )}{d \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 208, normalized size = 3.06 \[ \left [\frac {b e n x \sqrt {-\frac {e}{d}} \log \left (\frac {e^{3} x^{2} + 2 \, d^{2} e x \sqrt {-\frac {e}{d}} - d^{3} - 2 \, {\left (d e^{2} x \sqrt {-\frac {e}{d}} - d^{2} e\right )} x^{\frac {2}{3}} - 2 \, {\left (d e^{2} x + d^{3} \sqrt {-\frac {e}{d}}\right )} x^{\frac {1}{3}}}{e^{3} x^{2} + d^{3}}\right ) - b d n \log \left (e x^{\frac {2}{3}} + d\right ) - 2 \, b e n x^{\frac {2}{3}} - b d \log \relax (c) - a d}{d x}, -\frac {2 \, b e n x \sqrt {\frac {e}{d}} \arctan \left (x^{\frac {1}{3}} \sqrt {\frac {e}{d}}\right ) + b d n \log \left (e x^{\frac {2}{3}} + d\right ) + 2 \, b e n x^{\frac {2}{3}} + b d \log \relax (c) + a d}{d x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 61, normalized size = 0.90 \[ -{\left (2 \, {\left (\frac {\arctan \left (\frac {x^{\frac {1}{3}} e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}}}{d^{\frac {3}{2}}} + \frac {1}{d x^{\frac {1}{3}}}\right )} e + \frac {\log \left (x^{\frac {2}{3}} e + d\right )}{x}\right )} b n - \frac {b \log \relax (c)}{x} - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 59, normalized size = 0.87 \[ -2 \, b e n {\left (\frac {e \arctan \left (\frac {e x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} d} + \frac {1}{d x^{\frac {1}{3}}}\right )} - \frac {b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )}{x} - \frac {a}{x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{x^2} \,d x \]
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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