Optimal. Leaf size=24 \[ \frac {(c x)^n}{a c n \left (a+b x^n\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {264} \[ \frac {(c x)^n}{a c n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {(c x)^{-1+n}}{\left (a+b x^n\right )^2} \, dx &=\frac {(c x)^n}{a c n \left (a+b x^n\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 1.29 \[ -\frac {x^{1-n} (c x)^{n-1}}{b n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 22, normalized size = 0.92 \[ -\frac {c^{n - 1}}{b^{2} n x^{n} + a b n} \]
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 \[ \int \frac {\left (c x\right )^{n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 99, normalized size = 4.12 \[ \frac {x \,{\mathrm e}^{\frac {\left (n -1\right ) \left (-i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i c x \right )+i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i c x \right )^{2}-i \pi \mathrm {csgn}\left (i c x \right )^{3}+2 \ln \relax (c )+2 \ln \relax (x )\right )}{2}}}{\left (b \,x^{n}+a \right ) a n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 22, normalized size = 0.92 \[ -\frac {c^{n}}{b^{2} c n x^{n} + a b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 29, normalized size = 1.21 \[ \frac {x\,{\left (c\,x\right )}^{n-1}}{a\,b\,n\,\left (x^n+\frac {a}{b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 43.30, size = 330, normalized size = 13.75 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \wedge c = 0 \wedge n = 0 \\- \frac {c^{n} x^{- n}}{b^{2} c n} & \text {for}\: a = 0 \\\frac {\tilde {\infty } c^{n} x^{n}}{c n} & \text {for}\: b = - a x^{- n} \\0^{n - 1} \left (\frac {n x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {n x \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {b n x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {b x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )}\right ) & \text {for}\: c = 0 \\\frac {\log {\relax (x )}}{c \left (a + b\right )^{2}} & \text {for}\: n = 0 \\\frac {c^{n} x^{n}}{a^{2} c n + a b c n x^{n}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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