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

Optimal. Leaf size=122 \[ \frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d) \sqrt{d e-c f}} \]

[Out]

(b*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])])/(Sqrt[a]*(b*c - a*d)*S
qrt[b*e - a*f]) - (d*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqr
t[c]*(b*c - a*d)*Sqrt[d*e - c*f])

_______________________________________________________________________________________

Rubi [A]  time = 0.344605, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d) \sqrt{d e-c f}} \]

Antiderivative was successfully verified.

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

[Out]

(b*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])])/(Sqrt[a]*(b*c - a*d)*S
qrt[b*e - a*f]) - (d*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqr
t[c]*(b*c - a*d)*Sqrt[d*e - c*f])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 41.8518, size = 104, normalized size = 0.85 \[ \frac{d \operatorname{atanh}{\left (\frac{x \sqrt{c f - d e}}{\sqrt{c} \sqrt{e + f x^{2}}} \right )}}{\sqrt{c} \left (a d - b c\right ) \sqrt{c f - d e}} - \frac{b \operatorname{atanh}{\left (\frac{x \sqrt{a f - b e}}{\sqrt{a} \sqrt{e + f x^{2}}} \right )}}{\sqrt{a} \left (a d - b c\right ) \sqrt{a f - b e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.276808, size = 113, normalized size = 0.93 \[ \frac{\frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}}}{b c-a d} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.056, size = 782, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 92.5344, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(sqrt(-c*d*e + c^2*f)*b*log((((b^2*e^2 - 8*a*b*e*f + 8*a^2*f^2)*x^4 + a^2*
e^2 - 2*(3*a*b*e^2 - 4*a^2*e*f)*x^2)*sqrt(-a*b*e + a^2*f) - 4*((a*b^2*e^2 - 3*a^
2*b*e*f + 2*a^3*f^2)*x^3 - (a^2*b*e^2 - a^3*e*f)*x)*sqrt(f*x^2 + e))/(b^2*x^4 +
2*a*b*x^2 + a^2)) + sqrt(-a*b*e + a^2*f)*d*log((((d^2*e^2 - 8*c*d*e*f + 8*c^2*f^
2)*x^4 + c^2*e^2 - 2*(3*c*d*e^2 - 4*c^2*e*f)*x^2)*sqrt(-c*d*e + c^2*f) + 4*((c*d
^2*e^2 - 3*c^2*d*e*f + 2*c^3*f^2)*x^3 - (c^2*d*e^2 - c^3*e*f)*x)*sqrt(f*x^2 + e)
)/(d^2*x^4 + 2*c*d*x^2 + c^2)))/(sqrt(-a*b*e + a^2*f)*sqrt(-c*d*e + c^2*f)*(b*c
- a*d)), 1/4*(2*sqrt(-c*d*e + c^2*f)*b*arctan(1/2*((b*e - 2*a*f)*x^2 - a*e)/(sqr
t(a*b*e - a^2*f)*sqrt(f*x^2 + e)*x)) - sqrt(a*b*e - a^2*f)*d*log((((d^2*e^2 - 8*
c*d*e*f + 8*c^2*f^2)*x^4 + c^2*e^2 - 2*(3*c*d*e^2 - 4*c^2*e*f)*x^2)*sqrt(-c*d*e
+ c^2*f) + 4*((c*d^2*e^2 - 3*c^2*d*e*f + 2*c^3*f^2)*x^3 - (c^2*d*e^2 - c^3*e*f)*
x)*sqrt(f*x^2 + e))/(d^2*x^4 + 2*c*d*x^2 + c^2)))/(sqrt(a*b*e - a^2*f)*sqrt(-c*d
*e + c^2*f)*(b*c - a*d)), -1/4*(2*sqrt(-a*b*e + a^2*f)*d*arctan(1/2*((d*e - 2*c*
f)*x^2 - c*e)/(sqrt(c*d*e - c^2*f)*sqrt(f*x^2 + e)*x)) + sqrt(c*d*e - c^2*f)*b*l
og((((b^2*e^2 - 8*a*b*e*f + 8*a^2*f^2)*x^4 + a^2*e^2 - 2*(3*a*b*e^2 - 4*a^2*e*f)
*x^2)*sqrt(-a*b*e + a^2*f) - 4*((a*b^2*e^2 - 3*a^2*b*e*f + 2*a^3*f^2)*x^3 - (a^2
*b*e^2 - a^3*e*f)*x)*sqrt(f*x^2 + e))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/(sqrt(-a*b*e
+ a^2*f)*sqrt(c*d*e - c^2*f)*(b*c - a*d)), 1/2*(sqrt(c*d*e - c^2*f)*b*arctan(1/
2*((b*e - 2*a*f)*x^2 - a*e)/(sqrt(a*b*e - a^2*f)*sqrt(f*x^2 + e)*x)) - sqrt(a*b*
e - a^2*f)*d*arctan(1/2*((d*e - 2*c*f)*x^2 - c*e)/(sqrt(c*d*e - c^2*f)*sqrt(f*x^
2 + e)*x)))/(sqrt(a*b*e - a^2*f)*sqrt(c*d*e - c^2*f)*(b*c - a*d))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.267234, size = 234, normalized size = 1.92 \[ -f^{\frac{3}{2}}{\left (\frac{b \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} b + 2 \, a f - b e}{2 \, \sqrt{-a^{2} f^{2} + a b f e}}\right )}{\sqrt{-a^{2} f^{2} + a b f e}{\left (b c f - a d f\right )}} - \frac{d \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt{-c^{2} f^{2} + c d f e}}\right )}{\sqrt{-c^{2} f^{2} + c d f e}{\left (b c f - a d f\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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