Optimal. Leaf size=63 \[ \frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2}}+\frac {1}{2} x^{3/2} \sqrt {1-a x}-\frac {\sqrt {x} \sqrt {1-a x}}{4 a} \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {848, 50, 54, 216} \[ \frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2}}+\frac {1}{2} x^{3/2} \sqrt {1-a x}-\frac {\sqrt {x} \sqrt {1-a x}}{4 a} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rule 848
Rubi steps
\begin {align*} \int \frac {\sqrt {x} \sqrt {1-a^2 x^2}}{\sqrt {1+a x}} \, dx &=\int \sqrt {x} \sqrt {1-a x} \, dx\\ &=\frac {1}{2} x^{3/2} \sqrt {1-a x}+\frac {1}{4} \int \frac {\sqrt {x}}{\sqrt {1-a x}} \, dx\\ &=-\frac {\sqrt {x} \sqrt {1-a x}}{4 a}+\frac {1}{2} x^{3/2} \sqrt {1-a x}+\frac {\int \frac {1}{\sqrt {x} \sqrt {1-a x}} \, dx}{8 a}\\ &=-\frac {\sqrt {x} \sqrt {1-a x}}{4 a}+\frac {1}{2} x^{3/2} \sqrt {1-a x}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-a x^2}} \, dx,x,\sqrt {x}\right )}{4 a}\\ &=-\frac {\sqrt {x} \sqrt {1-a x}}{4 a}+\frac {1}{2} x^{3/2} \sqrt {1-a x}+\frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 0.78 \[ \frac {\sqrt {a} \sqrt {x} \sqrt {1-a x} (2 a x-1)+\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 221, normalized size = 3.51 \[ \left [\frac {4 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a^{2} x - a\right )} \sqrt {a x + 1} \sqrt {x} - {\left (a x + 1\right )} \sqrt {-a} \log \left (-\frac {8 \, a^{3} x^{3} - 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - 1\right )} \sqrt {a x + 1} \sqrt {-a} \sqrt {x} - 7 \, a x + 1}{a x + 1}\right )}{16 \, {\left (a^{3} x + a^{2}\right )}}, \frac {2 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a^{2} x - a\right )} \sqrt {a x + 1} \sqrt {x} - {\left (a x + 1\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {a x + 1} \sqrt {a} \sqrt {x}}{2 \, a^{2} x^{2} + a x - 1}\right )}{8 \, {\left (a^{3} x + a^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 92, normalized size = 1.46 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 \sqrt {-\left (a x -1\right ) x}\, a^{\frac {3}{2}} x +\arctan \left (\frac {2 a x -1}{2 \sqrt {-\left (a x -1\right ) x}\, \sqrt {a}}\right )-2 \sqrt {-\left (a x -1\right ) x}\, \sqrt {a}\right ) \sqrt {x}}{8 \sqrt {a x +1}\, \sqrt {-\left (a x -1\right ) x}\, a^{\frac {3}{2}}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {x}}{\sqrt {a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {x}\,\sqrt {1-a^2\,x^2}}{\sqrt {a\,x+1}} \,d x \]
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 \[ \int \frac {\sqrt {x} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {a x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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