Optimal. Leaf size=61 \[ \frac {2 x \sqrt {\frac {a}{x^2}+b x^n}}{n+2}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{n+2} \]
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Rubi [A] time = 0.08, antiderivative size = 61, 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 = {2007, 2029, 206} \begin {gather*} \frac {2 x \sqrt {\frac {a}{x^2}+b x^n}}{n+2}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{n+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2007
Rule 2029
Rubi steps
\begin {align*} \int \sqrt {\frac {a}{x^2}+b x^n} \, dx &=\frac {2 x \sqrt {\frac {a}{x^2}+b x^n}}{2+n}+a \int \frac {1}{x^2 \sqrt {\frac {a}{x^2}+b x^n}} \, dx\\ &=\frac {2 x \sqrt {\frac {a}{x^2}+b x^n}}{2+n}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {1}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{2+n}\\ &=\frac {2 x \sqrt {\frac {a}{x^2}+b x^n}}{2+n}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{2+n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 78, normalized size = 1.28 \begin {gather*} \frac {x \sqrt {\frac {a}{x^2}+b x^n} \left (2 \sqrt {a+b x^{n+2}}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+2}}}{\sqrt {a}}\right )\right )}{(n+2) \sqrt {a+b x^{n+2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 83, normalized size = 1.36 \begin {gather*} \frac {x \sqrt {\frac {a}{x^2}+b x^n} \left (\frac {2 \sqrt {a+b x^{n+2}}}{n+2}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+2}}}{\sqrt {a}}\right )}{n+2}\right )}{\sqrt {a+b x^{n+2}}} \end {gather*}
Antiderivative was successfully verified.
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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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {b x^{n} + \frac {a}{x^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {b \,x^{n}+\frac {a}{x^{2}}}\, dx \end {gather*}
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*} \int \sqrt {b x^{n} + \frac {a}{x^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.17, size = 97, normalized size = 1.59 \begin {gather*} \frac {x\,\sqrt {b\,x^n+\frac {a}{x^2}}}{\frac {n}{2}+1}+\frac {\sqrt {a}\,x\,\mathrm {asin}\left (\frac {\sqrt {a}\,1{}\mathrm {i}}{\sqrt {b}\,x^{\frac {n}{2}+1}}\right )\,\sqrt {b\,x^n+\frac {a}{x^2}}\,1{}\mathrm {i}}{\sqrt {b}\,x^{\frac {n}{2}+1}\,\left (\frac {n}{2}+1\right )\,\sqrt {\frac {a}{b\,x^{n+2}}+1}} \end {gather*}
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 \begin {gather*} \int \sqrt {\frac {a}{x^{2}} + b x^{n}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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