Optimal. Leaf size=17 \[ x^{m+2} \sqrt {a+b x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {449} \[ x^{m+2} \sqrt {a+b x^2} \]
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin {align*} \int \frac {x^{1+m} \left (a (2+m)+b (3+m) x^2\right )}{\sqrt {a+b x^2}} \, dx &=x^{2+m} \sqrt {a+b x^2}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 104, normalized size = 6.12 \[ \frac {x^{m+2} \sqrt {\frac {b x^2}{a}+1} \left (b (m+3) x^2 \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};-\frac {b x^2}{a}\right )+a (m+4) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};-\frac {b x^2}{a}\right )\right )}{(m+4) \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 16, normalized size = 0.94 \[ \sqrt {b x^{2} + a} x x^{m + 1} \]
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 \[ \int \frac {{\left (b {\left (m + 3\right )} x^{2} + a {\left (m + 2\right )}\right )} x^{m + 1}}{\sqrt {b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 16, normalized size = 0.94 \[ \sqrt {b \,x^{2}+a}\, x^{m +2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.88, size = 16, normalized size = 0.94 \[ \sqrt {b x^{2} + a} x^{2} x^{m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.19, size = 24, normalized size = 1.41 \[ \frac {x^{m+1}\,\left (b\,x^3+a\,x\right )}{\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 10.06, size = 202, normalized size = 11.88 \[ \frac {\sqrt {a} m x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {\sqrt {a} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {m}{2} + 2\right )} + \frac {b m x^{4} x^{m} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {m}{2} + 3\right )} + \frac {3 b x^{4} x^{m} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {m}{2} + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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