Optimal. Leaf size=15 \[ \frac{x^m}{\sqrt{a+b x^n}} \]
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Rubi [A] time = 0.0181481, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {12, 449} \[ \frac{x^m}{\sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 449
Rubi steps
\begin{align*} \int \frac{x^{-1+m} \left (2 a m+b (2 m-n) x^n\right )}{2 \left (a+b x^n\right )^{3/2}} \, dx &=\frac{1}{2} \int \frac{x^{-1+m} \left (2 a m+b (2 m-n) x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx\\ &=\frac{x^m}{\sqrt{a+b x^n}}\\ \end{align*}
Mathematica [C] time = 0.177732, size = 111, normalized size = 7.4 \[ \frac{x^m \sqrt{\frac{b x^n}{a}+1} \left (b (2 m-n) x^n \, _2F_1\left (\frac{3}{2},\frac{m+n}{n};\frac{m}{n}+2;-\frac{b x^n}{a}\right )+2 a (m+n) \, _2F_1\left (\frac{3}{2},\frac{m}{n};\frac{m+n}{n};-\frac{b x^n}{a}\right )\right )}{2 a (m+n) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{-1+m} \left ( 2\,am+b \left ( 2\,m-n \right ){x}^{n} \right ) }{2} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24732, size = 18, normalized size = 1.2 \begin{align*} \frac{x^{m}}{\sqrt{b x^{n} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53544, size = 39, normalized size = 2.6 \begin{align*} \frac{x x^{m - 1}}{\sqrt{b x^{n} + a}} \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 \frac{{\left (b{\left (2 \, m - n\right )} x^{n} + 2 \, a m\right )} x^{m - 1}}{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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