Optimal. Leaf size=17 \[ \frac {\left (\sqrt {b+x^2}+x\right )^a}{a} \]
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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} \[ \frac {\left (\sqrt {b+x^2}+x\right )^a}{a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2122
Rubi steps
\begin {align*} \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx &=\operatorname {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*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ \frac {\left (\sqrt {b+x^2}+x\right )^a}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 15, normalized size = 0.88 \[ \frac {{\left (x + \sqrt {x^{2} + b}\right )}^{a}}{a} \]
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 \[ \int \frac {{\left (x + \sqrt {x^{2} + b}\right )}^{a}}{\sqrt {x^{2} + b}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (x +\sqrt {x^{2}+b}\right )^{a}}{\sqrt {x^{2}+b}}\, dx \]
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 \[ \int \frac {{\left (x + \sqrt {x^{2} + b}\right )}^{a}}{\sqrt {x^{2} + b}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 15, normalized size = 0.88 \[ \frac {{\left (x+\sqrt {x^2+b}\right )}^a}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.58, size = 311, normalized size = 18.29 \[ \begin {cases} - \frac {\sqrt {b} b^{\frac {a}{2}} \sinh {\left (- a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a x \sqrt {\frac {b}{x^{2}} + 1}} + \frac {b^{\frac {a}{2}} x \cosh {\left (- a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {b}} - \frac {b^{\frac {a}{2}} x \sinh {\left (- a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {b} \sqrt {\frac {b}{x^{2}} + 1}} - \frac {2 b^{\frac {a}{2}} \cosh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )} \Gamma \left (1 - \frac {a}{2}\right )}{a^{2} \Gamma \left (- \frac {a}{2}\right )} & \text {for}\: \left |{\frac {x^{2}}{b}}\right | > 1 \\- \frac {b^{\frac {a}{2}} \sinh {\left (- a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {1 + \frac {x^{2}}{b}}} - \frac {b^{\frac {a}{2}} x^{2} \sinh {\left (- a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a b \sqrt {1 + \frac {x^{2}}{b}}} + \frac {b^{\frac {a}{2}} x \cosh {\left (- a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} + \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {b}} - \frac {2 b^{\frac {a}{2}} \cosh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )} \Gamma \left (1 - \frac {a}{2}\right )}{a^{2} \Gamma \left (- \frac {a}{2}\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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