Optimal. Leaf size=46 \[ \frac {x^m \sqrt {a+b x^2} \, _2F_1\left (1,\frac {1+m}{2};\frac {2+m}{2};-\frac {b x^2}{a}\right )}{a m} \]
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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.24, 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 \sqrt {\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {m}{2};\frac {m+2}{2};-\frac {b x^2}{a}\right )}{m \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^{-1+m}}{\sqrt {a+b x^2}} \, dx &=\frac {\sqrt {1+\frac {b x^2}{a}} \int \frac {x^{-1+m}}{\sqrt {1+\frac {b x^2}{a}}} \, dx}{\sqrt {a+b x^2}}\\ &=\frac {x^m \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{2},\frac {m}{2};\frac {2+m}{2};-\frac {b x^2}{a}\right )}{m \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 57, normalized size = 1.24 \begin {gather*} \frac {x^m \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{2},\frac {m}{2};1+\frac {m}{2};-\frac {b x^2}{a}\right )}{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 \[\int \frac {x^{-1+m}}{\sqrt {b \,x^{2}+a}}\, dx\]
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 = 1.46, size = 41, normalized size = 0.89 \begin {gather*} \frac {x^{m} \Gamma \left (\frac {m}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{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 )} \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-1}}{\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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