3.199 \(\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=194 \[ \frac{x \sqrt{a+b x^2}}{\sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{c} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

(x*Sqrt[a + b*x^2])/Sqrt[c + d*x^2] - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[
(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*
x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/
Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[
c + d*x^2])

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Rubi [A]  time = 0.27356, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{x \sqrt{a+b x^2}}{\sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{c} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2],x]

[Out]

(x*Sqrt[a + b*x^2])/Sqrt[c + d*x^2] - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[
(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*
x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/
Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[
c + d*x^2])

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Rubi in Sympy [A]  time = 39.1481, size = 172, normalized size = 0.89 \[ - \frac{\sqrt{a} \sqrt{b} \sqrt{c + d x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{d \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{b x \sqrt{c + d x^{2}}}{d \sqrt{a + b x^{2}}} + \frac{\sqrt{c} \sqrt{a + b x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{\sqrt{d} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

-sqrt(a)*sqrt(b)*sqrt(c + d*x**2)*elliptic_e(atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c)
+ 1)/(d*sqrt(a*(c + d*x**2)/(c*(a + b*x**2)))*sqrt(a + b*x**2)) + b*x*sqrt(c +
d*x**2)/(d*sqrt(a + b*x**2)) + sqrt(c)*sqrt(a + b*x**2)*elliptic_f(atan(sqrt(d)*
x/sqrt(c)), 1 - b*c/(a*d))/(sqrt(d)*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(c
+ d*x**2))

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Mathematica [A]  time = 0.0810411, size = 86, normalized size = 0.44 \[ \frac{\sqrt{a+b x^2} \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|\frac{b c}{a d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^2]/Sqrt[c + d*x^2],x]

[Out]

(Sqrt[a + b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (b*c)/(a*
d)])/(Sqrt[-(d/c)]*Sqrt[(a + b*x^2)/a]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.018, size = 158, normalized size = 0.8 \[{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) d}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}} \left ( a{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) d-bc{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) +bc{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)

[Out]

(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*(a*Ellip
ticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*d-b*c*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1
/2))+b*c*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2)))/(b*d*x^4+a*d*x^2+b*c*x^2+a*c
)/(-b/a)^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)/sqrt(d*x^2 + c),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^2 + a)/sqrt(d*x^2 + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)/sqrt(d*x^2 + c),x, algorithm="fricas")

[Out]

integral(sqrt(b*x^2 + a)/sqrt(d*x^2 + c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{2}}}{\sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(sqrt(a + b*x**2)/sqrt(c + d*x**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)/sqrt(d*x^2 + c),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^2 + a)/sqrt(d*x^2 + c), x)