Optimal. Leaf size=131 \[ \frac {2 \sqrt {x^2+1} \sqrt {a x}}{x+1}+\frac {\sqrt {a} (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a x}}{\sqrt {a}}\right )|\frac {1}{2}\right )}{\sqrt {x^2+1}}-\frac {2 \sqrt {a} (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {a x}}{\sqrt {a}}\right )|\frac {1}{2}\right )}{\sqrt {x^2+1}} \]
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Rubi [A] time = 0.08, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {329, 305, 220, 1196} \[ \frac {2 \sqrt {x^2+1} \sqrt {a x}}{x+1}+\frac {\sqrt {a} (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a x}}{\sqrt {a}}\right )|\frac {1}{2}\right )}{\sqrt {x^2+1}}-\frac {2 \sqrt {a} (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {a x}}{\sqrt {a}}\right )|\frac {1}{2}\right )}{\sqrt {x^2+1}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rubi steps
\begin {align*} \int \frac {\sqrt {a x}}{\sqrt {1+x^2}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{a^2}}} \, dx,x,\sqrt {a x}\right )}{a}\\ &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{a^2}}} \, dx,x,\sqrt {a x}\right )-2 \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{a}}{\sqrt {1+\frac {x^4}{a^2}}} \, dx,x,\sqrt {a x}\right )\\ &=\frac {2 \sqrt {a x} \sqrt {1+x^2}}{1+x}-\frac {2 \sqrt {a} (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {a x}}{\sqrt {a}}\right )|\frac {1}{2}\right )}{\sqrt {1+x^2}}+\frac {\sqrt {a} (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a x}}{\sqrt {a}}\right )|\frac {1}{2}\right )}{\sqrt {1+x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 27, normalized size = 0.21 \[ \frac {2}{3} x \sqrt {a x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-x^2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a x}}{\sqrt {x^{2} + 1}}, x\right ) \]
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 \[ \int \frac {\sqrt {a x}}{\sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 81, normalized size = 0.62 \[ \frac {\sqrt {a x}\, \sqrt {-i \left (x +i\right )}\, \sqrt {2}\, \sqrt {-i \left (-x +i\right )}\, \sqrt {i x}\, \left (2 \EllipticE \left (\sqrt {-i \left (x +i\right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {-i \left (x +i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {x^{2}+1}\, x} \]
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 \[ \int \frac {\sqrt {a x}}{\sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a\,x}}{\sqrt {x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.02, size = 36, normalized size = 0.27 \[ \frac {\sqrt {a} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {x^{2} e^{i \pi }} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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