Integrand size = 19, antiderivative size = 77 \[ \int \frac {x^2}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=-\frac {a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac {x^3 \left (c x^n\right )^{-1/n}}{2 b}+\frac {a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3} \]
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Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {375, 45} \[ \int \frac {x^2}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3}-\frac {a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac {x^3 \left (c x^n\right )^{-1/n}}{2 b} \]
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Rule 45
Rule 375
Rubi steps \begin{align*} \text {integral}& = \left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^2}{a+b x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = -\frac {a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac {x^3 \left (c x^n\right )^{-1/n}}{2 b}+\frac {a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {x^3 \left (c x^n\right )^{-3/n} \left (b \left (c x^n\right )^{\frac {1}{n}} \left (-2 a+b \left (c x^n\right )^{\frac {1}{n}}\right )+2 a^2 \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{2 b^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.03 (sec) , antiderivative size = 310, normalized size of antiderivative = 4.03
method | result | size |
risch | \(\frac {c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} x^{3} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{2 b}-\frac {c^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {2}{n}} x^{3} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}} a}{b^{2}}+\frac {\ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) \left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} a^{2} c^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {2}{n}} x^{3} {\mathrm e}^{-\frac {3 i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{b^{3}}\) | \(310\) |
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Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {b^{2} c^{\frac {2}{n}} x^{2} - 2 \, a b c^{\left (\frac {1}{n}\right )} x + 2 \, a^{2} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{2 \, b^{3} c^{\frac {3}{n}}} \]
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\[ \int \frac {x^2}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int \frac {x^{2}}{a + b \left (c x^{n}\right )^{\frac {1}{n}}}\, dx \]
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\[ \int \frac {x^2}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int { \frac {x^{2}}{\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a} \,d x } \]
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\[ \int \frac {x^2}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int { \frac {x^{2}}{\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a} \,d x } \]
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Timed out. \[ \int \frac {x^2}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int \frac {x^2}{a+b\,{\left (c\,x^n\right )}^{1/n}} \,d x \]
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