Optimal. Leaf size=63 \[ \frac {a x \left (c x^n\right )^{-1/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {15, 368, 43} \[ \frac {a x \left (c x^n\right )^{-1/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rule 368
Rubi steps
\begin {align*} \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx &=\frac {\left (c x^n\right )^{\frac {1}{n}} \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {a x \left (c x^n\right )^{-1/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 0.76 \[ \frac {x \left (c x^n\right )^{-1/n} \left (\frac {a}{a+b \left (c x^n\right )^{\frac {1}{n}}}+\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 50, normalized size = 0.79 \[ \frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) + a}{b^{3} c^{\frac {2}{n}} x + a b^{2} c^{\left (\frac {1}{n}\right )}} \]
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 \[ \int \frac {\left (c x^{n}\right )^{\left (\frac {1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 214, normalized size = 3.40 \[ \frac {x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}} \ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )}{b^{2}}-\frac {x}{\left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right ) b} \]
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 \[ -\frac {x}{b^{2} c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a b} + \int \frac {1}{b^{2} c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x^n\right )}^{1/n}}{{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \,d x \]
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 \[ \int \frac {\left (c x^{n}\right )^{\frac {1}{n}}}{\left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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