Optimal. Leaf size=50 \[ \frac {x^{3+m} \sqrt {a+b x^2} \, _2F_1\left (1,\frac {4+m}{2};\frac {5+m}{2};-\frac {b x^2}{a}\right )}{a (3+m)} \]
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Rubi [A]
time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.26, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {372, 371}
\begin {gather*} \frac {x^{m+3} \sqrt {\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};-\frac {b x^2}{a}\right )}{(m+3) \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rubi steps
\begin {align*} \int \frac {x^{2+m}}{\sqrt {a+b x^2}} \, dx &=\frac {\sqrt {1+\frac {b x^2}{a}} \int \frac {x^{2+m}}{\sqrt {1+\frac {b x^2}{a}}} \, dx}{\sqrt {a+b x^2}}\\ &=\frac {x^{3+m} \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};-\frac {b x^2}{a}\right )}{(3+m) \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 65, normalized size = 1.30 \begin {gather*} \frac {x^{3+m} \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};1+\frac {3+m}{2};-\frac {b x^2}{a}\right )}{(3+m) \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{2+m}}{\sqrt {b \,x^{2}+a}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.29, size = 54, normalized size = 1.08 \begin {gather*} \frac {x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \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*} \text {could not integrate} \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 \frac {x^{m+2}}{\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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