Optimal. Leaf size=17 \[ \frac {\left (\sqrt {a+x^2}+x\right )^n}{n} \]
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Rubi [A] time = 0.05, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2122, 30} \[ \frac {\left (\sqrt {a+x^2}+x\right )^n}{n} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2122
Rubi steps
\begin {align*} \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{\sqrt {a+x^2}} \, dx &=\operatorname {Subst}\left (\int x^{-1+n} \, dx,x,x+\sqrt {a+x^2}\right )\\ &=\frac {\left (x+\sqrt {a+x^2}\right )^n}{n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ \frac {\left (\sqrt {a+x^2}+x\right )^n}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 15, normalized size = 0.88 \[ \frac {{\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n} \]
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 \[ \int \frac {{\left (x + \sqrt {x^{2} + a}\right )}^{n}}{\sqrt {x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (x +\sqrt {x^{2}+a}\right )^{n}}{\sqrt {x^{2}+a}}\, dx \]
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 \[ \int \frac {{\left (x + \sqrt {x^{2} + a}\right )}^{n}}{\sqrt {x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.00, size = 15, normalized size = 0.88 \[ \frac {{\left (x+\sqrt {x^2+a}\right )}^n}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.64, size = 311, normalized size = 18.29 \[ \begin {cases} - \frac {\sqrt {a} a^{\frac {n}{2}} \sinh {\left (- n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{n x \sqrt {\frac {a}{x^{2}} + 1}} - \frac {2 a^{\frac {n}{2}} \cosh {\left (n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )} \Gamma \left (1 - \frac {n}{2}\right )}{n^{2} \Gamma \left (- \frac {n}{2}\right )} + \frac {a^{\frac {n}{2}} x \cosh {\left (- n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{\sqrt {a} n} - \frac {a^{\frac {n}{2}} x \sinh {\left (- n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{\sqrt {a} n \sqrt {\frac {a}{x^{2}} + 1}} & \text {for}\: \left |{\frac {x^{2}}{a}}\right | > 1 \\- \frac {a^{\frac {n}{2}} \sinh {\left (- n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{n \sqrt {1 + \frac {x^{2}}{a}}} - \frac {2 a^{\frac {n}{2}} \cosh {\left (n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )} \Gamma \left (1 - \frac {n}{2}\right )}{n^{2} \Gamma \left (- \frac {n}{2}\right )} - \frac {a^{\frac {n}{2}} x^{2} \sinh {\left (- n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{a n \sqrt {1 + \frac {x^{2}}{a}}} + \frac {a^{\frac {n}{2}} x \cosh {\left (- n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{\sqrt {a} n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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