Optimal. Leaf size=54 \[ \frac {\sqrt {\frac {a}{x}} \sqrt {x} (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {1+x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {15, 335, 226}
\begin {gather*} \frac {\sqrt {x} (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} \sqrt {\frac {a}{x}} F\left (2 \text {ArcTan}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 226
Rule 335
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {a}{x}}}{\sqrt {1+x^2}} \, dx &=\left (\sqrt {\frac {a}{x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x^2}} \, dx\\ &=\left (2 \sqrt {\frac {a}{x}} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {\frac {a}{x}} \sqrt {x} (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {1+x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 27, normalized size = 0.50 \begin {gather*} 2 \sqrt {\frac {a}{x}} x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.20, size = 62, normalized size = 1.15
method | result | size |
meijerg | \(2 \sqrt {\frac {a}{x}}\, x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], -x^{2}\right )\) | \(22\) |
default | \(\frac {i \sqrt {\frac {a}{x}}\, \sqrt {-i \left (x +i\right )}\, \sqrt {2}\, \sqrt {-i \left (-x +i\right )}\, \sqrt {i x}\, \EllipticF \left (\sqrt {-i \left (x +i\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{2}+1}}\) | \(62\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 9, normalized size = 0.17 \begin {gather*} 2 \, \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a}{x}}}{\sqrt {x^{2} + 1}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {\frac {a}{x}}}{\sqrt {x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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