Optimal. Leaf size=45 \[ -\frac {\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2}+\frac {\log (x)}{a^2}+\frac {1}{a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {368, 44} \[ -\frac {\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2}+\frac {\log (x)}{a^2}+\frac {1}{a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \]
Antiderivative was successfully verified.
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Rule 44
Rule 368
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {1}{a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 40, normalized size = 0.89 \[ \frac {\frac {a}{a+b \left (c x^n\right )^{\frac {1}{n}}}-\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )+\log (x)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 57, normalized size = 1.27 \[ \frac {b c^{\left (\frac {1}{n}\right )} x \log \relax (x) - {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) + a \log \relax (x) + a}{a^{2} b c^{\left (\frac {1}{n}\right )} x + a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 1.20 \[ \frac {1}{\left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right ) a}+\frac {\ln \left (\left (c \,x^{n}\right )^{\frac {1}{n}}\right )}{a^{2}}-\frac {\ln \left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 61, normalized size = 1.36 \[ \frac {1}{a b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2}} + \frac {\log \relax (x)}{a^{2}} - \frac {\log \left (\frac {b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a}{b c^{\left (\frac {1}{n}\right )}}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 44, normalized size = 0.98 \[ \frac {\ln \relax (x)}{a^2}-\frac {\ln \left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}{a^2}+\frac {1}{a^2+a\,b\,{\left (c\,x^n\right )}^{1/n}} \]
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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