Optimal. Leaf size=159 \[ \frac {2 \sqrt {x^2+1} x^2 \sqrt {\frac {a}{x^3}}}{x+1}-2 \sqrt {x^2+1} x \sqrt {\frac {a}{x^3}}+\frac {(x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} x^{3/2} \sqrt {\frac {a}{x^3}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+1}}-\frac {2 (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} x^{3/2} \sqrt {\frac {a}{x^3}} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+1}} \]
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Rubi [A] time = 0.05, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {15, 325, 329, 305, 220, 1196} \[ \frac {2 \sqrt {x^2+1} x^2 \sqrt {\frac {a}{x^3}}}{x+1}-2 \sqrt {x^2+1} x \sqrt {\frac {a}{x^3}}+\frac {(x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} x^{3/2} \sqrt {\frac {a}{x^3}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+1}}-\frac {2 (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} x^{3/2} \sqrt {\frac {a}{x^3}} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+1}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 220
Rule 305
Rule 325
Rule 329
Rule 1196
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {a}{x^3}}}{\sqrt {1+x^2}} \, dx &=\left (\sqrt {\frac {a}{x^3}} x^{3/2}\right ) \int \frac {1}{x^{3/2} \sqrt {1+x^2}} \, dx\\ &=-2 \sqrt {\frac {a}{x^3}} x \sqrt {1+x^2}+\left (\sqrt {\frac {a}{x^3}} x^{3/2}\right ) \int \frac {\sqrt {x}}{\sqrt {1+x^2}} \, dx\\ &=-2 \sqrt {\frac {a}{x^3}} x \sqrt {1+x^2}+\left (2 \sqrt {\frac {a}{x^3}} x^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {\frac {a}{x^3}} x \sqrt {1+x^2}+\left (2 \sqrt {\frac {a}{x^3}} x^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )-\left (2 \sqrt {\frac {a}{x^3}} x^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {\frac {a}{x^3}} x \sqrt {1+x^2}+\frac {2 \sqrt {\frac {a}{x^3}} x^2 \sqrt {1+x^2}}{1+x}-\frac {2 \sqrt {\frac {a}{x^3}} x^{3/2} (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {1+x^2}}+\frac {\sqrt {\frac {a}{x^3}} x^{3/2} (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] time = 0.01, size = 27, normalized size = 0.17 \[ -2 x \sqrt {\frac {a}{x^3}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-x^2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\frac {a}{x^{3}}}}{\sqrt {x^{2} + 1}}, x\right ) \]
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 \[ \int \frac {\sqrt {\frac {a}{x^{3}}}}{\sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 116, normalized size = 0.73 \[ \frac {\sqrt {\frac {a}{x^{3}}}\, \left (-2 x^{2}+2 \sqrt {-i \left (x +i\right )}\, \sqrt {2}\, \sqrt {-i \left (-x +i\right )}\, \sqrt {i x}\, \EllipticE \left (\sqrt {-i \left (x +i\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (x +i\right )}\, \sqrt {2}\, \sqrt {-i \left (-x +i\right )}\, \sqrt {i x}\, \EllipticF \left (\sqrt {-i \left (x +i\right )}, \frac {\sqrt {2}}{2}\right )-2\right ) x}{\sqrt {x^{2}+1}} \]
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 \[ \int \frac {\sqrt {\frac {a}{x^{3}}}}{\sqrt {x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {a}{x^3}}}{\sqrt {x^2+1}} \,d x \]
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 \[ \int \frac {\sqrt {\frac {a}{x^{3}}}}{\sqrt {x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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