Optimal. Leaf size=75 \[ -\frac {2 \sqrt {a^2-b^2 x} \tan ^{-1}\left (\frac {\sqrt {a^2-b^2 x}}{\sqrt {a^2+b^2 x}}\right )}{b^2 \sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}} \]
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Rubi [A] time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {519, 63, 217, 203} \[ -\frac {2 \sqrt {a^2-b^2 x} \tan ^{-1}\left (\frac {\sqrt {a^2-b^2 x}}{\sqrt {a^2+b^2 x}}\right )}{b^2 \sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 217
Rule 519
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}} \sqrt {a^2+b^2 x}} \, dx &=\frac {\sqrt {a^2-b^2 x} \int \frac {1}{\sqrt {a^2-b^2 x} \sqrt {a^2+b^2 x}} \, dx}{\sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}}\\ &=-\frac {\left (2 \sqrt {a^2-b^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 a^2-x^2}} \, dx,x,\sqrt {a^2-b^2 x}\right )}{b^2 \sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}}\\ &=-\frac {\left (2 \sqrt {a^2-b^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {a^2-b^2 x}}{\sqrt {a^2+b^2 x}}\right )}{b^2 \sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}}\\ &=-\frac {2 \sqrt {a^2-b^2 x} \tan ^{-1}\left (\frac {\sqrt {a^2-b^2 x}}{\sqrt {a^2+b^2 x}}\right )}{b^2 \sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 75, normalized size = 1.00 \[ -\frac {2 \sqrt {a^2-b^2 x} \tan ^{-1}\left (\frac {\sqrt {a^2-b^2 x}}{\sqrt {a^2+b^2 x}}\right )}{b^2 \sqrt {a-b \sqrt {x}} \sqrt {a+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 50, normalized size = 0.67 \[ -\frac {2 \, \arctan \left (-\frac {a^{2} - \sqrt {b^{2} x + a^{2}} \sqrt {b \sqrt {x} + a} \sqrt {-b \sqrt {x} + a}}{b^{2} x}\right )}{b^{2}} \]
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 \[ \int \frac {1}{\sqrt {b^{2} x + a^{2}} \sqrt {b \sqrt {x} + a} \sqrt {-b \sqrt {x} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.88, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b^{2} x +a^{2}}\, \sqrt {-b \sqrt {x}+a}\, \sqrt {b \sqrt {x}+a}}\, dx \]
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 \[ \int \frac {1}{\sqrt {b^{2} x + a^{2}} \sqrt {b \sqrt {x} + a} \sqrt {-b \sqrt {x} + a}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {a+b\,\sqrt {x}}\,\sqrt {a-b\,\sqrt {x}}\,\sqrt {a^2+x\,b^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a - b \sqrt {x}} \sqrt {a + b \sqrt {x}} \sqrt {a^{2} + b^{2} x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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