Optimal. Leaf size=39 \[ \frac {x \sqrt {a+b x^n} \, _2F_1\left (1,\frac {1}{2}+\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {246, 245} \[ \frac {x \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{\sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x^n}} \, dx &=\frac {\sqrt {1+\frac {b x^n}{a}} \int \frac {1}{\sqrt {1+\frac {b x^n}{a}}} \, dx}{\sqrt {a+b x^n}}\\ &=\frac {x \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {1}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{\sqrt {a+b x^n}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 48, normalized size = 1.23 \[ \frac {x \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{\sqrt {a+b x^n}} \]
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 \[ \int \frac {1}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \,x^{n}+a}}\, dx \]
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 \[ \int \frac {1}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 43, normalized size = 1.10 \[ \frac {x\,\sqrt {\frac {b\,x^n}{a}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{\sqrt {a+b\,x^n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.09, size = 39, normalized size = 1.00 \[ \frac {x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (1 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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