Optimal. Leaf size=17 \[ \frac {\left (x+\sqrt {b+x^2}\right )^a}{a} \]
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Rubi [A]
time = 0.03, 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 = {2147, 30}
\begin {gather*} \frac {\left (\sqrt {b+x^2}+x\right )^a}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2147
Rubi steps
\begin {align*} \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx &=\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*}
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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} \frac {\left (x+\sqrt {b+x^2}\right )^a}{a} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 7.54, size = 269, normalized size = 15.82 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {x b^{\frac {1}{2}+\frac {a}{2}} \sqrt {\frac {b+x^2}{x^2}} \text {Sinh}\left [\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ] \left (-1+a\right )\right ]+x^3 b^{-\frac {1}{2}+\frac {a}{2}} \sqrt {\frac {b+x^2}{x^2}} \text {Sinh}\left [\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ] \left (-1+a\right )\right ]+\left (b+x^2\right ) \left (\frac {x \text {Cosh}\left [\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ] \left (-1+a\right )\right ]}{\sqrt {b}}+\text {Cosh}\left [a \text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]\right ]\right ) b^{\frac {a}{2}}}{a \left (b+x^2\right )},\text {Abs}\left [\frac {x^2}{b}\right ]>1\right \}\right \},\frac {-2 \text {Cosh}\left [a \text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]\right ] \text {Gamma}\left [1-\frac {a}{2}\right ] b^{\frac {a}{2}}}{a^2 \text {Gamma}\left [-\frac {a}{2}\right ]}+\frac {b^{\frac {a}{2}} \text {Sinh}\left [a \text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]-\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]\right ]}{a \sqrt {1+\frac {x^2}{b}}}+\frac {x \text {Cosh}\left [a \text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]-\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]\right ] b^{\frac {a}{2}}}{a \sqrt {b}}+\frac {x^2 b^{\frac {a}{2}} \text {Sinh}\left [a \text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]-\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]\right ]}{a b \sqrt {1+\frac {x^2}{b}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 15, normalized size = 0.88 \begin {gather*} \frac {{\left (x + \sqrt {x^{2} + b}\right )}^{a}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 311 vs.
\(2 (12) = 24\)
time = 1.46, size = 311, normalized size = 18.29 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (x+\sqrt {b+x^{2}}\right )^{a}}{a} \end {gather*}
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 \begin {gather*} \frac {{\left (x+\sqrt {x^2+b}\right )}^a}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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