Optimal. Leaf size=94 \[ -\frac {2 b \log (x) \left (c x^n\right )^{\frac {1}{n}}}{a^3 x}+\frac {2 b \left (c x^n\right )^{\frac {1}{n}} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x}-\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {1}{a^2 x} \]
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Rubi [A] time = 0.04, antiderivative size = 94, 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} \[ -\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 b \log (x) \left (c x^n\right )^{\frac {1}{n}}}{a^3 x}+\frac {2 b \left (c x^n\right )^{\frac {1}{n}} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x}-\frac {1}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 44
Rule 368
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx &=\frac {\left (c x^n\right )^{\frac {1}{n}} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{x}\\ &=\frac {\left (c x^n\right )^{\frac {1}{n}} \operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{x}\\ &=-\frac {1}{a^2 x}-\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 b \left (c x^n\right )^{\frac {1}{n}} \log (x)}{a^3 x}+\frac {2 b \left (c x^n\right )^{\frac {1}{n}} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 71, normalized size = 0.76 \[ -\frac {\left (c x^n\right )^{\frac {1}{n}} \left (a \left (\frac {b}{a+b \left (c x^n\right )^{\frac {1}{n}}}+\left (c x^n\right )^{-1/n}\right )-2 b \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )+2 b \log (x)\right )}{a^3 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 99, normalized size = 1.05 \[ -\frac {2 \, b^{2} c^{\frac {2}{n}} x^{2} \log \relax (x) + a^{2} + 2 \, {\left (a b x \log \relax (x) + a b x\right )} c^{\left (\frac {1}{n}\right )} - 2 \, {\left (b^{2} c^{\frac {2}{n}} x^{2} + a b c^{\left (\frac {1}{n}\right )} x\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{a^{3} b c^{\left (\frac {1}{n}\right )} x^{2} + a^{4} x} \]
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 \[ \int \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 296, normalized size = 3.15 \[ -\frac {2 b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}} \ln \relax (x )}{a^{3} x}+\frac {2 b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}} \ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )}{a^{3} x}+\frac {1}{\left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right ) a x}-\frac {2}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{a b c^{\left (\frac {1}{n}\right )} x {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2} x} + 2 \, \int \frac {1}{a b c^{\left (\frac {1}{n}\right )} x^{2} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^2\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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