Optimal. Leaf size=35 \[ -\frac {x f^a \left (-b \log (f) x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n \log (f)\right )}{n} \]
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Rubi [A] time = 0.00, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2208} \[ -\frac {x f^a \left (-b \log (f) x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-b \log (f) x^n\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2208
Rubi steps
\begin {align*} \int f^{a+b x^n} \, dx &=-\frac {f^a x \Gamma \left (\frac {1}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-1/n}}{n}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 35, normalized size = 1.00 \[ -\frac {x f^a \left (-b \log (f) x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n \log (f)\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (f^{b x^{n} + a}, x\right ) \]
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 \[ \int f^{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int f^{b \,x^{n}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 35, normalized size = 1.00 \[ -\frac {f^{a} x \Gamma \left (\frac {1}{n}, -b x^{n} \log \relax (f)\right )}{\left (-b x^{n} \log \relax (f)\right )^{\left (\frac {1}{n}\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int f^{a+b\,x^n} \,d x \]
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 \[ \begin {cases} - \frac {b f^{a} f^{b x^{n}} n x x^{n} \log {\relax (f )}}{n + 1} + \frac {f^{a} f^{b x^{n}} n x}{n + 1} + \frac {f^{a} f^{b x^{n}} x}{n + 1} & \text {for}\: n \neq -1 \\\int f^{a + \frac {b}{x}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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