Optimal. Leaf size=70 \[ -\frac{\sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2}}-\frac{\sqrt{c+d x^2}}{a x} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.145741, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2}}-\frac{\sqrt{c+d x^2}}{a x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^2]/(x^2*(a + b*x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 23.2525, size = 56, normalized size = 0.8 \[ - \frac{\sqrt{c + d x^{2}}}{a x} + \frac{\sqrt{a d - b c} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**(1/2)/x**2/(b*x**2+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.121519, size = 71, normalized size = 1.01 \[ -\frac{\frac{x \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2}}+\frac{\sqrt{c+d x^2}}{a}}{x} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x^2]/(x^2*(a + b*x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.024, size = 1017, normalized size = 14.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^(1/2)/x^2/(b*x^2+a),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.259548, size = 1, normalized size = 0.01 \[ \left [\frac{x \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \,{\left (a^{2} c x -{\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, \sqrt{d x^{2} + c}}{4 \, a x}, \frac{x \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c} a x \sqrt{\frac{b c - a d}{a}}}\right ) - 2 \, \sqrt{d x^{2} + c}}{2 \, a x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{2}}}{x^{2} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**(1/2)/x**2/(b*x**2+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.78806, size = 158, normalized size = 2.26 \[ \frac{{\left (b c \sqrt{d} - a d^{\frac{3}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} a} + \frac{2 \, c \sqrt{d}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x^2),x, algorithm="giac")
[Out]