Optimal. Leaf size=46 \[ \frac {\sqrt {2} b \sin ^{-1}\left (\frac {a x-b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.76, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2156, 2155,
222} \begin {gather*} \frac {\sqrt {2} b \text {ArcSin}\left (\frac {a x-b \sqrt {\frac {a^2 x^2}{b^2}+\frac {a}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 222
Rule 2155
Rule 2156
Rubi steps
\begin {align*} \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx &=\int \frac {\sqrt {-a x^2+b x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\\ &=-\frac {\left (\sqrt {2} b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}{a}\\ &=\frac {\sqrt {2} b \sin ^{-1}\left (\frac {a x-b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(46)=92\).
time = 0.01, size = 114, normalized size = 2.48 \begin {gather*} \frac {\sqrt {2} b \sqrt {x \left (-a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )} \sqrt {a x \left (a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a x \left (a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )}}{\sqrt {a}}\right )}{a^{3/2} x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x \left (-a x +b \sqrt {\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\right )}}{x \sqrt {\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}\, dx\]
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 [A]
time = 5.27, size = 161, normalized size = 3.50 \begin {gather*} \left [\frac {1}{2} \, \sqrt {2} b \sqrt {-\frac {1}{a}} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}} + 2 \, \sqrt {-a x^{2} + b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}} {\left (\sqrt {2} a x \sqrt {-\frac {1}{a}} - \sqrt {2} b \sqrt {-\frac {1}{a}} \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}\right )} + 1\right ), -\frac {\sqrt {2} b \arctan \left (\frac {\sqrt {2} \sqrt {-a x^{2} + b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}}}{2 \, \sqrt {a} x}\right )}{\sqrt {a}}\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 {- x \left (a x - b \sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}}\right )}}{x \sqrt {\frac {a \left (a x^{2} + 1\right )}{b^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {-x\,\left (a\,x-b\,\sqrt {\frac {a}{b^2}+\frac {a^2\,x^2}{b^2}}\right )}}{x\,\sqrt {\frac {a}{b^2}+\frac {a^2\,x^2}{b^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________