Optimal. Leaf size=53 \[ \frac {x^2 \left (c x^n\right )^{-1/n}}{b}-\frac {a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {368, 43} \[ \frac {x^2 \left (c x^n\right )^{-1/n}}{b}-\frac {a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 368
Rubi steps
\begin {align*} \int \frac {x}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx &=\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {x^2 \left (c x^n\right )^{-1/n}}{b}-\frac {a x^2 \left (c x^n\right )^{-2/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 = 0.02, size = 49, normalized size = 0.92 \[ x^2 \left (c x^n\right )^{-2/n} \left (\frac {\left (c x^n\right )^{\frac {1}{n}}}{b}-\frac {a \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 36, normalized size = 0.68 \[ \frac {b c^{\left (\frac {1}{n}\right )} x - a \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{b^{2} c^{\frac {2}{n}}} \]
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 \[ \int \frac {x}{\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 233, normalized size = 4.40 \[ -\frac {a \,x^{2} \left (c^{-\frac {1}{n}}\right )^{2} \left (\left (x^{n}\right )^{-\frac {1}{n}}\right )^{2} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}} \ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )}{b^{2}}+\frac {x^{2} c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}}{b} \]
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 \[ \int \frac {x}{\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a}\,{d x} \]
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 \[ \int \frac {x}{a+b\,{\left (c\,x^n\right )}^{1/n}} \,d x \]
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 \[ \int \frac {x}{a + b \left (c x^{n}\right )^{\frac {1}{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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