Integrand size = 23, antiderivative size = 17 \[ \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx=\frac {\left (x+\sqrt {b+x^2}\right )^a}{a} \]
[Out]
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} \[ \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx=\frac {\left (\sqrt {b+x^2}+x\right )^a}{a} \]
[In]
[Out]
Rule 30
Rule 2147
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^{-1+a} \, dx,x,x+\sqrt {b+x^2}\right ) \\ & = \frac {\left (x+\sqrt {b+x^2}\right )^a}{a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx=\frac {\left (x+\sqrt {b+x^2}\right )^a}{a} \]
[In]
[Out]
Time = 0.37 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\left (x +\sqrt {x^{2}+b}\right )^{a}}{a}\) | \(16\) |
default | \(\frac {\left (x +\sqrt {x^{2}+b}\right )^{a}}{a}\) | \(16\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx=\frac {{\left (x + \sqrt {x^{2} + b}\right )}^{a}}{a} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (12) = 24\).
Time = 1.25 (sec) , antiderivative size = 311, normalized size of antiderivative = 18.29 \[ \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx=\begin {cases} \frac {\sqrt {b} b^{\frac {a}{2}} \sinh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a x \sqrt {\frac {b}{x^{2}} + 1}} + \frac {b^{\frac {a}{2}} x \cosh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {b}} + \frac {b^{\frac {a}{2}} x \sinh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {b} \sqrt {\frac {b}{x^{2}} + 1}} - \frac {2 b^{\frac {a}{2}} \cosh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )} \Gamma \left (1 - \frac {a}{2}\right )}{a^{2} \Gamma \left (- \frac {a}{2}\right )} & \text {for}\: \left |{\frac {x^{2}}{b}}\right | > 1 \\\frac {b^{\frac {a}{2}} \sinh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {1 + \frac {x^{2}}{b}}} + \frac {b^{\frac {a}{2}} x^{2} \sinh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a b \sqrt {1 + \frac {x^{2}}{b}}} + \frac {b^{\frac {a}{2}} x \cosh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {b}} - \frac {2 b^{\frac {a}{2}} \cosh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )} \Gamma \left (1 - \frac {a}{2}\right )}{a^{2} \Gamma \left (- \frac {a}{2}\right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
\[ \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx=\int { \frac {{\left (x + \sqrt {x^{2} + b}\right )}^{a}}{\sqrt {x^{2} + b}} \,d x } \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx=\frac {{\left (x + \sqrt {x^{2} + b}\right )}^{a}}{a} \]
[In]
[Out]
Time = 0.35 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx=\frac {{\left (x+\sqrt {x^2+b}\right )}^a}{a} \]
[In]
[Out]