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

Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} \sqrt{b c-a d}} \]

[Out]

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

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Rubi [A]  time = 0.057907, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.1516, size = 42, normalized size = 0.86 \[ \frac{\operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\sqrt{a} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0375766, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.006, size = 306, normalized size = 6.2 \[ -{\frac{1}{2}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{1}{2}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(-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/(-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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302111, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} + 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, \sqrt{-a b c + a^{2} d}}, \frac{\arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{2 \, \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.247413, size = 95, normalized size = 1.94 \[ -\frac{\sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(d)*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)