Integrand size = 25, antiderivative size = 63 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {a x \left (c x^n\right )^{-1/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]
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Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {15, 375, 45} \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {a x \left (c x^n\right )^{-1/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]
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Rule 15
Rule 45
Rule 375
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{\frac {1}{n}} \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx}{x} \\ & = \left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {a x \left (c x^n\right )^{-1/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {x \left (c x^n\right )^{-1/n} \left (\frac {a}{a+b \left (c x^n\right )^{\frac {1}{n}}}+\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{b^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.30 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.40
method | result | size |
risch | \(-\frac {x}{b \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}+\frac {\ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) \left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} x \,{\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{b^{2}}\) | \(214\) |
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Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) + a}{b^{3} c^{\frac {2}{n}} x + a b^{2} c^{\left (\frac {1}{n}\right )}} \]
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\[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {\left (c x^{n}\right )^{\frac {1}{n}}}{\left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{2}}\, dx \]
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\[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {\left (c x^{n}\right )^{\left (\frac {1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {\left (c x^{n}\right )^{\left (\frac {1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {{\left (c\,x^n\right )}^{1/n}}{{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \,d x \]
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