Optimal. Leaf size=66 \[ \frac {x^{1+m-2 n} \sqrt {a+b x^n} \, _2F_1\left (1,\frac {1}{2} \left (-3+\frac {2 (1+m)}{n}\right );\frac {1+m-n}{n};-\frac {b x^n}{a}\right )}{a (1+m-2 n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.14, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {372, 371}
\begin {gather*} \frac {x^{m-2 n+1} \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {m-2 n+1}{n};\frac {m-n+1}{n};-\frac {b x^n}{a}\right )}{(m-2 n+1) \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rubi steps
\begin {align*} \int \frac {x^{m-2 n}}{\sqrt {a+b x^n}} \, dx &=\frac {\sqrt {1+\frac {b x^n}{a}} \int \frac {x^{m-2 n}}{\sqrt {1+\frac {b x^n}{a}}} \, dx}{\sqrt {a+b x^n}}\\ &=\frac {x^{1+m-2 n} \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {1}{2},\frac {1+m-2 n}{n};\frac {1+m-n}{n};-\frac {b x^n}{a}\right )}{(1+m-2 n) \sqrt {a+b x^n}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 77, normalized size = 1.17 \begin {gather*} \frac {x^{1+m-2 n} \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {1}{2},\frac {1+m-2 n}{n};1+\frac {1+m-2 n}{n};-\frac {b x^n}{a}\right )}{(1+m-2 n) \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{m -2 n}}{\sqrt {a +b \,x^{n}}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = 7.65, size = 65, normalized size = 0.98 \begin {gather*} \frac {x x^{m} x^{- 2 n} \Gamma \left (\frac {m}{n} - 2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} - 2 + \frac {1}{n} \\ \frac {m}{n} - 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (\frac {m}{n} - 1 + \frac {1}{n}\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\,n}}{\sqrt {a+b\,x^n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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