3.978 \(\int \frac{1}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=153 \[ \frac{d x \sqrt{a+b x^2}}{a c \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{a c x}-\frac{\sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

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

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Rubi [A]  time = 0.286172, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{d x \sqrt{a+b x^2}}{a c \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{a c x}-\frac{\sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

(d*x*Sqrt[a + b*x^2])/(a*c*Sqrt[c + d*x^2]) - (Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/
(a*c*x) - (Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b
*c)/(a*d)])/(a*Sqrt[c]*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 = 39.3731, size = 126, normalized size = 0.82 \[ \frac{b x \sqrt{c + d x^{2}}}{a c \sqrt{a + b x^{2}}} - \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{a c x} - \frac{\sqrt{b} \sqrt{c + d x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{\sqrt{a} c \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.528746, size = 146, normalized size = 0.95 \[ \frac{-\frac{\left (a+b x^2\right ) \left (c+d x^2\right )}{c x}-i a \sqrt{\frac{b}{a}} \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )\right )}{a \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.031, size = 224, normalized size = 1.5 \[{\frac{1}{cxa \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) } \left ( -\sqrt{-{\frac{b}{a}}}{x}^{4}bd-bc\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}x{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) +bc\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}x{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) -\sqrt{-{\frac{b}{a}}}{x}^{2}ad-\sqrt{-{\frac{b}{a}}}{x}^{2}bc-\sqrt{-{\frac{b}{a}}}ac \right ) \sqrt{d{x}^{2}+c}\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^2),x, algorithm="maxima")

[Out]

integrate(1/(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. \[{\rm integral}\left (\frac{1}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^2),x, algorithm="fricas")

[Out]

integral(1/(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. \[ \int \frac{1}{x^{2} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*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. \[ \int \frac{1}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^2), x)