Integrand size = 17, antiderivative size = 53 \[ \int \frac {x}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {x^2 \left (c x^n\right )^{-1/n}}{b}-\frac {a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {375, 45} \[ \int \frac {x}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {x^2 \left (c x^n\right )^{-1/n}}{b}-\frac {a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]
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Rule 45
Rule 375
Rubi steps \begin{align*} \text {integral}& = \left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {x}{a+b x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {x^2 \left (c x^n\right )^{-1/n}}{b}-\frac {a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92 \[ \int \frac {x}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=x^2 \left (c x^n\right )^{-2/n} \left (\frac {\left (c x^n\right )^{\frac {1}{n}}}{b}-\frac {a \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.16 (sec) , antiderivative size = 233, normalized size of antiderivative = 4.40
method | result | size |
risch | \(\frac {c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} x^{2} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{b}-\frac {\ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) \left (c^{-\frac {1}{n}}\right )^{2} {\left (\left (x^{n}\right )^{-\frac {1}{n}}\right )}^{2} a \,x^{2} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{b^{2}}\) | \(233\) |
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none
Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.68 \[ \int \frac {x}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {b c^{\left (\frac {1}{n}\right )} x - a \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{b^{2} c^{\frac {2}{n}}} \]
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\[ \int \frac {x}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int \frac {x}{a + b \left (c x^{n}\right )^{\frac {1}{n}}}\, dx \]
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\[ \int \frac {x}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int { \frac {x}{\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a} \,d x } \]
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\[ \int \frac {x}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int { \frac {x}{\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a} \,d x } \]
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Timed out. \[ \int \frac {x}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int \frac {x}{a+b\,{\left (c\,x^n\right )}^{1/n}} \,d x \]
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