Optimal. Leaf size=35 \[ \sqrt {x} \sqrt {1-a x}+\frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 )}{\sqrt {a}}+\sqrt {x} \sqrt {1-a x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 56
Rule 222
Rule 862
Rubi steps
\begin {align*} \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1+a x}} \, dx &=\int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx\\ &=\sqrt {x} \sqrt {1-a x}+\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {1-a x}} \, dx\\ &=\sqrt {x} \sqrt {1-a x}+\text {Subst}\left (\int \frac {1}{\sqrt {1-a x^2}} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x} \sqrt {1-a x}+\frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.51, size = 35, normalized size = 1.00 \begin {gather*} \sqrt {x} \sqrt {1-a x}+\frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs.
\(2(25)=50\).
time = 0.07, size = 76, normalized size = 2.17
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {x}\, \left (2 \sqrt {a}\, \sqrt {-x \left (a x -1\right )}+\arctan \left (\frac {2 a x -1}{2 \sqrt {a}\, \sqrt {-x \left (a x -1\right )}}\right )\right )}{2 \sqrt {a x +1}\, \sqrt {-x \left (a x -1\right )}\, \sqrt {a}}\) | \(76\) |
risch | \(-\frac {\sqrt {x}\, \left (a x -1\right ) \sqrt {\frac {x \left (-a^{2} x^{2}+1\right )}{a x +1}}\, \sqrt {a x +1}}{\sqrt {-x \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a \,x^{2}+x}}\right ) \sqrt {\frac {x \left (-a^{2} x^{2}+1\right )}{a x +1}}\, \sqrt {a x +1}}{2 \sqrt {a}\, \sqrt {x}\, \sqrt {-a^{2} x^{2}+1}}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (25) = 50\).
time = 2.71, size = 199, normalized size = 5.69 \begin {gather*} \left [\frac {4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {a x + 1} a \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 )}{4 \, {\left (a^{2} x + a\right )}}, \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {a x + 1} a \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 )}{2 \, {\left (a^{2} x + a\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {x} \sqrt {a x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {1-a^2\,x^2}}{\sqrt {x}\,\sqrt {a\,x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________