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

Optimal. Leaf size=82 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} d} \]

[Out]

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

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Rubi [A]  time = 0.133035, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.4124, size = 70, normalized size = 0.85 \[ \frac{\sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{d} + \frac{\sqrt{a d - b c} \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{\sqrt{c} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0689451, size = 84, normalized size = 1.02 \[ \frac{\sqrt{a d-b c} \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} d}+\frac{\sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.055, size = 932, normalized size = 11.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250814, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + \sqrt{\frac{b c - a d}{c}} \log \left (\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \,{\left (a c^{2} x +{\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt{b x^{2} + a} \sqrt{\frac{b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d}, \frac{4 \, \sqrt{-b} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) + \sqrt{\frac{b c - a d}{c}} \log \left (\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \,{\left (a c^{2} x +{\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt{b x^{2} + a} \sqrt{\frac{b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d}, \frac{\sqrt{-\frac{b c - a d}{c}} \arctan \left (-\frac{{\left (2 \, b c - a d\right )} x^{2} + a c}{2 \, \sqrt{b x^{2} + a} c x \sqrt{-\frac{b c - a d}{c}}}\right ) + \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right )}{2 \, d}, \frac{2 \, \sqrt{-b} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) + \sqrt{-\frac{b c - a d}{c}} \arctan \left (-\frac{{\left (2 \, b c - a d\right )} x^{2} + a c}{2 \, \sqrt{b x^{2} + a} c x \sqrt{-\frac{b c - a d}{c}}}\right )}{2 \, d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260211, size = 150, normalized size = 1.83 \[ \frac{{\left (b^{\frac{3}{2}} c - a \sqrt{b} d\right )} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{\sqrt{-b^{2} c^{2} + a b c d} d} - \frac{\sqrt{b}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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