Optimal. Leaf size=84 \[ \frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{\sqrt{a} \sqrt{b} \sqrt{a+b x^2} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]
[Out]
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Rubi [A] time = 0.0573714, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{\sqrt{a} \sqrt{b} \sqrt{a+b x^2} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^2]/(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.98429, size = 71, normalized size = 0.85 \[ \frac{\sqrt{c + d x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{\sqrt{a} \sqrt{b} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**(1/2)/(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [C] time = 0.529476, size = 133, normalized size = 1.58 \[ \frac{x \left (c+d x^2\right )+\frac{i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )\right )}{\sqrt{\frac{b}{a}}}}{a \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x^2]/(a + b*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.049, size = 181, normalized size = 2.2 \[{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) a}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ({x}^{3}d\sqrt{-{\frac{b}{a}}}+{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) c\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) c\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+xc\sqrt{-{\frac{b}{a}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^(1/2)/(b*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{2}}}{\left (a + b x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**(1/2)/(b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]