Optimal. Leaf size=44 \[ \frac {4 x^{5 (1-n)} \left (a x^{-4 (1-n)}+b x^{5 n-4}\right )^{5/4}}{5 b n} \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1979, 2000} \[ \frac {4 x^{5 (1-n)} \left (a x^{-4 (1-n)}+b x^{5 n-4}\right )^{5/4}}{5 b n} \]
Antiderivative was successfully verified.
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Rule 1979
Rule 2000
Rubi steps
\begin {align*} \int \sqrt [4]{x^{4 (-1+n)} \left (a+b x^n\right )} \, dx &=\int \sqrt [4]{a x^{4 (-1+n)}+b x^{4 (-1+n)+n}} \, dx\\ &=\frac {4 x^{5 (1-n)} \left (a x^{-4 (1-n)}+b x^{-4+5 n}\right )^{5/4}}{5 b n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 36, normalized size = 0.82 \[ \frac {4 x^{5-5 n} \left (x^{4 n-4} \left (a+b x^n\right )\right )^{5/4}}{5 b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 44, normalized size = 1.00 \[ \frac {4 \, {\left (b x x^{n} + a x\right )} \left (\frac {b x^{5 \, n} + a x^{4 \, n}}{x^{4}}\right )^{\frac {1}{4}}}{5 \, b n x^{n}} \]
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 \[ \int \left ({\left (b x^{n} + a\right )} x^{4 \, n - 4}\right )^{\frac {1}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 40, normalized size = 0.91 \[ \frac {4 \left (\frac {\left (b \,x^{n}+a \right ) x^{4 n}}{x^{4}}\right )^{\frac {1}{4}} \left (b \,x^{n}+a \right ) x \,x^{-n}}{5 b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.58, size = 17, normalized size = 0.39 \[ \frac {4 \, {\left (b x^{n} + a\right )}^{\frac {5}{4}}}{5 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (x^{4\,n-4}\,\left (a+b\,x^n\right )\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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