Optimal. Leaf size=15 \[ \frac {x^m}{\sqrt {a+b x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {449} \[ \frac {x^m}{\sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin {align*} \int \frac {x^{-1+m} \left (a m+b (-1+m) x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {x^m}{\sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 103, normalized size = 6.87 \[ \frac {x^m \sqrt {\frac {b x^2}{a}+1} \left (b (m-1) x^2 \, _2F_1\left (\frac {3}{2},\frac {m+2}{2};\frac {m+4}{2};-\frac {b x^2}{a}\right )+a (m+2) \, _2F_1\left (\frac {3}{2},\frac {m}{2};\frac {m+2}{2};-\frac {b x^2}{a}\right )\right )}{a (m+2) \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 16, normalized size = 1.07 \[ \frac {x x^{m - 1}}{\sqrt {b x^{2} + a}} \]
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 \[ \int \frac {{\left (b {\left (m - 1\right )} x^{2} + a m\right )} x^{m - 1}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 14, normalized size = 0.93 \[ \frac {x^{m}}{\sqrt {b \,x^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.91, size = 13, normalized size = 0.87 \[ \frac {x^{m}}{\sqrt {b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.43, size = 13, normalized size = 0.87 \[ \frac {x^m}{\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 52.96, size = 97, normalized size = 6.47 \[ \frac {m x^{m} \Gamma \left (\frac {m}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{2} \\ \frac {m}{2} + 1 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {m}{2} + 1\right )} + \frac {b x^{2} x^{m} \left (m - 1\right ) \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {m}{2} + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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