Optimal. Leaf size=59 \[ \frac {x^{2 (n+2)} \sqrt {a+b x^n} \, _2F_1\left (1,\frac {1}{2} \left (5+\frac {8}{n}\right );3+\frac {4}{n};-\frac {b x^n}{a}\right )}{2 a (n+2)} \]
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Rubi [A] time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.19, 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 = {365, 364} \[ \frac {x^{2 (n+2)} \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {1}{2},2 \left (1+\frac {2}{n}\right );3+\frac {4}{n};-\frac {b x^n}{a}\right )}{2 (n+2) \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {x^{3+2 n}}{\sqrt {a+b x^n}} \, dx &=\frac {\sqrt {1+\frac {b x^n}{a}} \int \frac {x^{3+2 n}}{\sqrt {1+\frac {b x^n}{a}}} \, dx}{\sqrt {a+b x^n}}\\ &=\frac {x^{2 (2+n)} \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {1}{2},2 \left (1+\frac {2}{n}\right );3+\frac {4}{n};-\frac {b x^n}{a}\right )}{2 (2+n) \sqrt {a+b x^n}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 68, normalized size = 1.15 \[ \frac {x^{2 n+4} \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {1}{2},2+\frac {4}{n};3+\frac {4}{n};-\frac {b x^n}{a}\right )}{2 (n+2) \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {x^{2 \, n + 3}}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {x^{2 n +3}}{\sqrt {b \,x^{n}+a}}\, dx \]
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 \[ \int \frac {x^{2 \, n + 3}}{\sqrt {b x^{n} + a}}\,{d x} \]
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 \[ \int \frac {x^{2\,n+3}}{\sqrt {a+b\,x^n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 29.26, size = 49, normalized size = 0.83 \[ \frac {x^{4} x^{2 n} \Gamma \left (2 + \frac {4}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, 2 + \frac {4}{n} \\ 3 + \frac {4}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (3 + \frac {4}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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