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

Optimal. Leaf size=232 \[ \frac{d x \sqrt{a+b x^2}}{c \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{c x}+\frac{b \sqrt{c} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

(d*x*Sqrt[a + b*x^2])/(c*Sqrt[c + d*x^2]) - (Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c
*x) - (Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/
(a*d)])/(Sqrt[c]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*Sqr
t[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a
*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 = 0.433286, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{d x \sqrt{a+b x^2}}{c \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{c x}+\frac{b \sqrt{c} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

(d*x*Sqrt[a + b*x^2])/(c*Sqrt[c + d*x^2]) - (Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c
*x) - (Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/
(a*d)])/(Sqrt[c]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*Sqr
t[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a
*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 = 57.4794, size = 197, normalized size = 0.85 \[ \frac{d x \sqrt{a + b x^{2}}}{c \sqrt{c + d x^{2}}} - \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{c x} - \frac{\sqrt{d} \sqrt{a + b x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{\sqrt{c} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{b \sqrt{c} \sqrt{a + b x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.477754, size = 111, normalized size = 0.48 \[ \frac{\frac{b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} E\left (\sin ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}}}-\left (a+b x^2\right ) \left (c+d x^2\right )}{c x \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-((a + b*x^2)*(c + d*x^2)) + (b*c*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ell
ipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/Sqrt[-(b/a)])/(c*x*Sqrt[a + b*x^2]*
Sqrt[c + d*x^2])

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Maple [A]  time = 0.038, size = 168, normalized size = 0.7 \[{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) cx}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( -\sqrt{-{\frac{b}{a}}}{x}^{4}bd+bc\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}x{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) -\sqrt{-{\frac{b}{a}}}{x}^{2}ad-\sqrt{-{\frac{b}{a}}}{x}^{2}bc-\sqrt{-{\frac{b}{a}}}ac \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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