Optimal. Leaf size=63 \[ -\frac {\sqrt {x} \sqrt {1-a x}}{4 a}+\frac {1}{2} x^{3/2} \sqrt {1-a x}+\frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2}} \]
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Rubi [A]
time = 0.02, 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 = {862, 52, 56,
222} \begin {gather*} \frac {\text {ArcSin}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 222
Rule 862
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 {\text {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 = 1.47, size = 49, normalized size = 0.78 \begin {gather*} \frac {\sqrt {a} \sqrt {x} \sqrt {1-a x} (-1+2 a x)+\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs.
\(2(43)=86\).
time = 0.07, size = 92, normalized size = 1.46
method | result | size |
default | \(\frac {\sqrt {x}\, \sqrt {-a^{2} x^{2}+1}\, \left (4 a^{\frac {3}{2}} x \sqrt {-x \left (a x -1\right )}-2 \sqrt {a}\, \sqrt {-x \left (a x -1\right )}+\arctan \left (\frac {2 a x -1}{2 \sqrt {a}\, \sqrt {-x \left (a x -1\right )}}\right )\right )}{8 a^{\frac {3}{2}} \sqrt {a x +1}\, \sqrt {-x \left (a x -1\right )}}\) | \(92\) |
risch | \(-\frac {\left (2 a x -1\right ) \sqrt {x}\, \left (a x -1\right ) \sqrt {\frac {x \left (-a^{2} x^{2}+1\right )}{a x +1}}\, \sqrt {a x +1}}{4 a \sqrt {-x \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a \,x^{2}+x}}\right ) \sqrt {\frac {x \left (-a^{2} x^{2}+1\right )}{a x +1}}\, \sqrt {a x +1}}{8 a^{\frac {3}{2}} \sqrt {x}\, \sqrt {-a^{2} x^{2}+1}}\) | \(141\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs.
\(2 (43) = 86\).
time = 2.48, size = 221, normalized size = 3.51 \begin {gather*} \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 ] \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 {x} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {a x + 1}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 {x}\,\sqrt {1-a^2\,x^2}}{\sqrt {a\,x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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