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

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

[Out]

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

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Rubi [A]  time = 0.0580408, 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 \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.4196, size = 42, normalized size = 0.86 \[ \frac{\operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{\sqrt{c} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0444047, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} \sqrt{a d-b c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 300, normalized size = 6.1 \[ -{\frac{1}{2}\ln \left ({1 \left ( 2\,{\frac{ad-bc}{d}}+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{{\frac{ad-bc}{d}}}\sqrt{ \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) ^{2}b+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}} \right ) \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}+{\frac{1}{2}\ln \left ({1 \left ( 2\,{\frac{ad-bc}{d}}-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{{\frac{ad-bc}{d}}}\sqrt{ \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) ^{2}b-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}} \right ) \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260597, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{{\left ({\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{b c^{2} - a c d} + 4 \,{\left ({\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{3} +{\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt{b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, \sqrt{b c^{2} - a c d}}, \frac{\arctan \left (\frac{\sqrt{-b c^{2} + a c d}{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )}}{2 \,{\left (b c^{2} - a c d\right )} \sqrt{b x^{2} + a} x}\right )}{2 \, \sqrt{-b c^{2} + a c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241544, size = 95, normalized size = 1.94 \[ -\frac{\sqrt{b} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{\sqrt{-b^{2} c^{2} + a b c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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