Integrand size = 19, antiderivative size = 125 \[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=-\frac {1}{2 a^2 x^2}+\frac {2 b \left (c x^n\right )^{\frac {1}{n}}}{a^3 x^2}+\frac {b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {3 b^2 \left (c x^n\right )^{2/n} \log (x)}{a^4 x^2}-\frac {3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^4 x^2} \]
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Time = 0.04 (sec) , antiderivative size = 125, 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, 46} \[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {3 b^2 \log (x) \left (c x^n\right )^{2/n}}{a^4 x^2}-\frac {3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^4 x^2}+\frac {b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {2 b \left (c x^n\right )^{\frac {1}{n}}}{a^3 x^2}-\frac {1}{2 a^2 x^2} \]
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Rule 46
Rule 375
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int \frac {1}{x^3 (a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{x^2} \\ & = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int \left (\frac {1}{a^2 x^3}-\frac {2 b}{a^3 x^2}+\frac {3 b^2}{a^4 x}-\frac {b^3}{a^3 (a+b x)^2}-\frac {3 b^3}{a^4 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{x^2} \\ & = -\frac {1}{2 a^2 x^2}+\frac {2 b \left (c x^n\right )^{\frac {1}{n}}}{a^3 x^2}+\frac {b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {3 b^2 \left (c x^n\right )^{2/n} \log (x)}{a^4 x^2}-\frac {3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^4 x^2} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {\left (c x^n\right )^{2/n} \left (a \left (-a \left (c x^n\right )^{-2/n}+4 b \left (c x^n\right )^{-1/n}+\frac {2 b^2}{a+b \left (c x^n\right )^{\frac {1}{n}}}\right )+6 b^2 \log (x)-6 b^2 \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{2 a^4 x^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.56 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.03
method | result | size |
risch | \(\frac {1}{a \,x^{2} \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}-\frac {3 \ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) b^{2} c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{a^{4} x^{2}}+\frac {3 b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{a^{3} x^{2}}-\frac {3}{2 a^{2} x^{2}}+\frac {3 c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}} b^{2} \ln \left (x \right )}{a^{4} x^{2}}\) | \(379\) |
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Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {6 \, b^{3} c^{\frac {3}{n}} x^{3} \log \left (x\right ) + 3 \, a^{2} b c^{\left (\frac {1}{n}\right )} x - a^{3} + 6 \, {\left (a b^{2} x^{2} \log \left (x\right ) + a b^{2} x^{2}\right )} c^{\frac {2}{n}} - 6 \, {\left (b^{3} c^{\frac {3}{n}} x^{3} + a b^{2} c^{\frac {2}{n}} x^{2}\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{2 \, {\left (a^{4} b c^{\left (\frac {1}{n}\right )} x^{3} + a^{5} x^{2}\right )}} \]
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\[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {1}{x^{3} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{2}}\, dx \]
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\[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {1}{x^3\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \,d x \]
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