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 = 0.0183645, antiderivative size = 44, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.030217, 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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Maple [A] time = 0.018, size = 40, normalized size = 0.9 \begin{align*}{\frac{4\,x \left ( a+b{x}^{n} \right ) }{5\,b{x}^{n}n}\sqrt [4]{{\frac{ \left ({x}^{n} \right ) ^{4} \left ( a+b{x}^{n} \right ) }{{x}^{4}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06904, size = 23, normalized size = 0.52 \begin{align*} \frac{4 \,{\left (b x^{n} + a\right )}^{\frac{5}{4}}}{5 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.93438, size = 90, normalized size = 2.05 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left ({\left (b x^{n} + a\right )} x^{4 \, n - 4}\right )^{\frac{1}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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