Optimal. Leaf size=20 \[ \frac {x}{a^2+a b \left (c x^n\right )^{\frac {1}{n}}} \]
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Rubi [A] time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.60, 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} \[ -\frac {x \left (c x^n\right )^{-1/n}}{b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \]
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 )^2} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=-\frac {x \left (c x^n\right )^{-1/n}}{b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.65 \[ -\frac {x \left (c x^n\right )^{-1/n}}{a b+b^2 \left (c x^n\right )^{\frac {1}{n}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 25, normalized size = 1.25 \[ -\frac {1}{b^{2} c^{\frac {2}{n}} x + a b c^{\left (\frac {1}{n}\right )}} \]
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 \[ \int \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 74, normalized size = 3.70 \[ \frac {x}{\left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 23, normalized size = 1.15 \[ \frac {x}{a b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 20, normalized size = 1.00 \[ \frac {x}{a\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 169.22, size = 80, normalized size = 4.00 \[ \begin {cases} \tilde {\infty } c^{- \frac {2}{n}} x \left (x^{n}\right )^{- \frac {2}{n}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {c^{- \frac {2}{n}} x \left (x^{n}\right )^{- \frac {2}{n}}}{b^{2}} & \text {for}\: a = 0 \\\tilde {\infty } c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}} & \text {for}\: b = - a c^{- \frac {1}{n}} \left (x^{n}\right )^{- \frac {1}{n}} \\\frac {x}{a^{2} + a b c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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