Optimal. Leaf size=44 \[ \frac {2 x^{3 (1-n)} \left (a x^{-2 (1-n)}+b x^{3 n-2}\right )^{3/2}}{3 b n} \]
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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} \begin {gather*} \frac {2 x^{3 (1-n)} \left (a x^{-2 (1-n)}+b x^{3 n-2}\right )^{3/2}}{3 b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 1979
Rule 2000
Rubi steps
\begin {align*} \int \sqrt {x^{2 (-1+n)} \left (a+b x^n\right )} \, dx &=\int \sqrt {a x^{2 (-1+n)}+b x^{2 (-1+n)+n}} \, dx\\ &=\frac {2 x^{3 (1-n)} \left (a x^{-2 (1-n)}+b x^{-2+3 n}\right )^{3/2}}{3 b n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 36, normalized size = 0.82 \begin {gather*} \frac {2 x^{3-3 n} \left (x^{2 n-2} \left (a+b x^n\right )\right )^{3/2}}{3 b n} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 43, normalized size = 0.98 \begin {gather*} \frac {2 x^{n-1} \left (a+b x^n\right )^2}{3 b n \sqrt {x^{2 n-2} \left (a+b x^n\right )}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 44, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (b x x^{n} + a x\right )} \sqrt {\frac {b x^{3 \, n} + a x^{2 \, n}}{x^{2}}}}{3 \, b n x^{n}} \end {gather*}
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 \begin {gather*} \int \sqrt {{\left (b x^{n} + a\right )} x^{2 \, n - 2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 40, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {\frac {\left (b \,x^{n}+a \right ) x^{2 n}}{x^{2}}}\, \left (b \,x^{n}+a \right ) x \,x^{-n}}{3 b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 17, normalized size = 0.39 \begin {gather*} \frac {2 \, {\left (b x^{n} + a\right )}^{\frac {3}{2}}}{3 \, b n} \end {gather*}
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 \begin {gather*} \int \sqrt {x^{2\,n-2}\,\left (a+b\,x^n\right )} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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