Integrand size = 23, antiderivative size = 17 \[ \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{\sqrt {a+x^2}} \, dx=\frac {\left (x+\sqrt {a+x^2}\right )^n}{n} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2147, 30} \[ \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{\sqrt {a+x^2}} \, dx=\frac {\left (\sqrt {a+x^2}+x\right )^n}{n} \]
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Rule 30
Rule 2147
Rubi steps \begin{align*} \text {integral}& = \text {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*}
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{\sqrt {a+x^2}} \, dx=\frac {\left (x+\sqrt {a+x^2}\right )^n}{n} \]
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Time = 0.85 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\left (x +\sqrt {x^{2}+a}\right )^{n}}{n}\) | \(16\) |
default | \(\frac {\left (x +\sqrt {x^{2}+a}\right )^{n}}{n}\) | \(16\) |
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Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{\sqrt {a+x^2}} \, dx=\frac {{\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (12) = 24\).
Time = 1.51 (sec) , antiderivative size = 311, normalized size of antiderivative = 18.29 \[ \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{\sqrt {a+x^2}} \, dx=\begin {cases} \frac {\sqrt {a} a^{\frac {n}{2}} \sinh {\left (n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{n x \sqrt {\frac {a}{x^{2}} + 1}} - \frac {2 a^{\frac {n}{2}} \cosh {\left (n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )} \Gamma \left (1 - \frac {n}{2}\right )}{n^{2} \Gamma \left (- \frac {n}{2}\right )} + \frac {a^{\frac {n}{2}} x \cosh {\left (n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{\sqrt {a} n} + \frac {a^{\frac {n}{2}} x \sinh {\left (n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{\sqrt {a} n \sqrt {\frac {a}{x^{2}} + 1}} & \text {for}\: \left |{\frac {x^{2}}{a}}\right | > 1 \\\frac {a^{\frac {n}{2}} \sinh {\left (n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{n \sqrt {1 + \frac {x^{2}}{a}}} - \frac {2 a^{\frac {n}{2}} \cosh {\left (n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )} \Gamma \left (1 - \frac {n}{2}\right )}{n^{2} \Gamma \left (- \frac {n}{2}\right )} + \frac {a^{\frac {n}{2}} x^{2} \sinh {\left (n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{a n \sqrt {1 + \frac {x^{2}}{a}}} + \frac {a^{\frac {n}{2}} x \cosh {\left (n \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{\sqrt {a} n} & \text {otherwise} \end {cases} \]
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\[ \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{\sqrt {a+x^2}} \, dx=\int { \frac {{\left (x + \sqrt {x^{2} + a}\right )}^{n}}{\sqrt {x^{2} + a}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{\sqrt {a+x^2}} \, dx=\frac {{\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n} \]
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Time = 16.97 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{\sqrt {a+x^2}} \, dx=\frac {{\left (x+\sqrt {x^2+a}\right )}^n}{n} \]
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