Optimal. Leaf size=116 \[ \frac{x \sqrt{a+b x^2}}{b \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.169276, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x \sqrt{a+b x^2}}{b \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 22.4463, size = 97, normalized size = 0.84 \[ - \frac{\sqrt{a} \sqrt{c + d x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{\sqrt{b} d \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{x \sqrt{c + d x^{2}}}{d \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.112398, size = 122, normalized size = 1.05 \[ -\frac{i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )\right )}{d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.027, size = 129, normalized size = 1.1 \[{\frac{c}{d \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) } \left ( -{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) +{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) \right ) \sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="giac")
[Out]