Optimal. Leaf size=87 \[ \frac {2 x \sqrt {a x^j+b x^n} \, _2F_1\left (-\frac {1}{2},\frac {n+2}{2 (j-n)};\frac {n+2}{2 j-2 n}+1;-\frac {a x^{j-n}}{b}\right )}{(n+2) \sqrt {\frac {a x^{j-n}}{b}+1}} \]
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Rubi [A] time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2011, 365, 364} \[ \frac {2 x \sqrt {a x^j+b x^n} \, _2F_1\left (-\frac {1}{2},\frac {n+2}{2 (j-n)};\frac {n+2}{2 j-2 n}+1;-\frac {a x^{j-n}}{b}\right )}{(n+2) \sqrt {\frac {a x^{j-n}}{b}+1}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 2011
Rubi steps
\begin {align*} \int \sqrt {a x^j+b x^n} \, dx &=\frac {\left (x^{-n/2} \sqrt {a x^j+b x^n}\right ) \int x^{n/2} \sqrt {b+a x^{j-n}} \, dx}{\sqrt {b+a x^{j-n}}}\\ &=\frac {\left (x^{-n/2} \sqrt {a x^j+b x^n}\right ) \int x^{n/2} \sqrt {1+\frac {a x^{j-n}}{b}} \, dx}{\sqrt {1+\frac {a x^{j-n}}{b}}}\\ &=\frac {2 x \sqrt {a x^j+b x^n} \, _2F_1\left (-\frac {1}{2},\frac {2+n}{2 (j-n)};1+\frac {2+n}{2 j-2 n};-\frac {a x^{j-n}}{b}\right )}{(2+n) \sqrt {1+\frac {a x^{j-n}}{b}}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 134, normalized size = 1.54 \[ \frac {2 x \left (a (j-n) x^j \sqrt {\frac {a x^{j-n}}{b}+1} \, _2F_1\left (\frac {1}{2},\frac {2 j-n+2}{2 j-2 n};\frac {4 j-3 n+2}{2 j-2 n};-\frac {a x^{j-n}}{b}\right )-(2 j-n+2) \left (a x^j+b x^n\right )\right )}{(n+2) (-2 j+n-2) \sqrt {a x^j+b x^n}} \]
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 \[ \int \sqrt {a x^{j} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.85, size = 0, normalized size = 0.00 \[ \int \sqrt {a \,x^{j}+b \,x^{n}}\, dx \]
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 \[ \int \sqrt {a x^{j} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.23, size = 82, normalized size = 0.94 \[ \frac {x\,\sqrt {a\,x^j+b\,x^n}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {\frac {n}{2}+1}{j-n};\ \frac {\frac {n}{2}+1}{j-n}+1;\ -\frac {a\,x^{j-n}}{b}\right )}{\left (\frac {n}{2}+1\right )\,\sqrt {\frac {a\,x^{j-n}}{b}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a x^{j} + b x^{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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