Optimal. Leaf size=34 \[ -\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {254, 32} \[ -\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]
Antiderivative was successfully verified.
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Rule 32
Rule 254
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x)^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=-\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 1.00 \[ -\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 43, normalized size = 1.26 \[ -\frac {1}{2 \, {\left (b^{3} c^{\frac {3}{n}} x^{2} + 2 \, a b^{2} c^{\frac {2}{n}} x + a^{2} b c^{\left (\frac {1}{n}\right )}\right )}} \]
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 \[ \int \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 143, normalized size = 4.21 \[ \frac {\left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+2 a \right ) x}{2 \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 69, normalized size = 2.03 \[ \frac {b c^{\left (\frac {1}{n}\right )} x {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + 2 \, a x}{2 \, {\left (a^{2} b^{2} c^{\frac {2}{n}} {\left (x^{n}\right )}^{\frac {2}{n}} + 2 \, a^{3} b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 36, normalized size = 1.06 \[ \frac {x\,\left (2\,a+b\,{\left (c\,x^n\right )}^{1/n}\right )}{2\,a^2\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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