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

Optimal. Leaf size=74 \[ -\frac{b \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2}}{a c x} \]

[Out]

-(Sqrt[c + d*x^2]/(a*c*x)) - (b*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x
^2])])/(a^(3/2)*Sqrt[b*c - a*d])

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Rubi [A]  time = 0.154837, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{b \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2}}{a c x} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d*x^2]/(a*c*x)) - (b*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x
^2])])/(a^(3/2)*Sqrt[b*c - a*d])

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Rubi in Sympy [A]  time = 25.1114, size = 61, normalized size = 0.82 \[ - \frac{\sqrt{c + d x^{2}}}{a c x} - \frac{b \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{a^{\frac{3}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**2)/(a*c*x) - b*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2))
)/(a**(3/2)*sqrt(a*d - b*c))

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Mathematica [A]  time = 0.0800713, size = 74, normalized size = 1. \[ -\frac{b \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2}}{a c x} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d*x^2]/(a*c*x)) - (b*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x
^2])])/(a^(3/2)*Sqrt[b*c - a*d])

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Maple [B]  time = 0.019, size = 334, normalized size = 4.5 \[ -{\frac{1}{acx}\sqrt{d{x}^{2}+c}}+{\frac{b}{2\,a}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{b}{2\,a}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d*x^2+c)^(1/2)/a/c/x+1/2*b/a/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c
)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b
)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*
(-a*b)^(1/2)))-1/2*b/a/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*
(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))
^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(
1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.324314, size = 1, normalized size = 0.01 \[ \left [\frac{b c x \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} - 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c}}{4 \, \sqrt{-a b c + a^{2} d} a c x}, -\frac{b c x \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right ) + 2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c}}{2 \, \sqrt{a b c - a^{2} d} a c x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(b*c*x*log((((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2
- 4*a^2*c*d)*x^2)*sqrt(-a*b*c + a^2*d) - 4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^
2)*x^3 - (a^2*b*c^2 - a^3*c*d)*x)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2))
- 4*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(sqrt(-a*b*c + a^2*d)*a*c*x), -1/2*(b*
c*x*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)
) + 2*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c))/(sqrt(a*b*c - a^2*d)*a*c*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (a + b x^{2}\right ) \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.242706, size = 150, normalized size = 2.03 \[ d^{\frac{3}{2}}{\left (\frac{b \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} a d} + \frac{2}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )} a d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^(3/2)*(b*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b
*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a*d) + 2/(((sqrt(d)*x - sqrt(d*x^2 + c
))^2 - c)*a*d))