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

Optimal. Leaf size=203 \[ \frac{b \left (4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right ) \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}} \]

[Out]

(b^2*x*Sqrt[e + f*x^2])/(2*a*(b*c - a*d)*(b*e - a*f)*(a + b*x^2)) + (b*(b^2*c*e
- 3*a*b*d*e - 2*a*b*c*f + 4*a^2*d*f)*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e
+ f*x^2])])/(2*a^(3/2)*(b*c - a*d)^2*(b*e - a*f)^(3/2)) + (d^2*ArcTan[(Sqrt[d*e
- c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*(b*c - a*d)^2*Sqrt[d*e - c*f])

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Rubi [A]  time = 0.7523, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{b \left (4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right ) \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}} \]

Antiderivative was successfully verified.

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

[Out]

(b^2*x*Sqrt[e + f*x^2])/(2*a*(b*c - a*d)*(b*e - a*f)*(a + b*x^2)) + (b*(b^2*c*e
- 3*a*b*d*e - 2*a*b*c*f + 4*a^2*d*f)*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e
+ f*x^2])])/(2*a^(3/2)*(b*c - a*d)^2*(b*e - a*f)^(3/2)) + (d^2*ArcTan[(Sqrt[d*e
- c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*(b*c - a*d)^2*Sqrt[d*e - c*f])

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Rubi in Sympy [A]  time = 97.7256, size = 182, normalized size = 0.9 \[ \frac{d^{2} \operatorname{atanh}{\left (\frac{x \sqrt{c f - d e}}{\sqrt{c} \sqrt{e + f x^{2}}} \right )}}{\sqrt{c} \left (a d - b c\right )^{2} \sqrt{c f - d e}} + \frac{b^{2} x \sqrt{e + f x^{2}}}{2 a \left (a + b x^{2}\right ) \left (a d - b c\right ) \left (a f - b e\right )} - \frac{b \left (4 a^{2} d f - 2 a b c f - 3 a b d e + b^{2} c e\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a f - b e}}{\sqrt{a} \sqrt{e + f x^{2}}} \right )}}{2 a^{\frac{3}{2}} \left (a d - b c\right )^{2} \left (a f - b e\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*atanh(x*sqrt(c*f - d*e)/(sqrt(c)*sqrt(e + f*x**2)))/(sqrt(c)*(a*d - b*c)**2
*sqrt(c*f - d*e)) + b**2*x*sqrt(e + f*x**2)/(2*a*(a + b*x**2)*(a*d - b*c)*(a*f -
b*e)) - b*(4*a**2*d*f - 2*a*b*c*f - 3*a*b*d*e + b**2*c*e)*atanh(x*sqrt(a*f - b*
e)/(sqrt(a)*sqrt(e + f*x**2)))/(2*a**(3/2)*(a*d - b*c)**2*(a*f - b*e)**(3/2))

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Mathematica [A]  time = 0.986749, size = 203, normalized size = 1. \[ \frac{1}{2} \left (\frac{b \left (4 a^2 d f-a b (2 c f+3 d e)+b^2 c e\right ) \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{a \left (a+b x^2\right ) (a d-b c) (a f-b e)}+\frac{2 d^2 \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((b^2*x*Sqrt[e + f*x^2])/(a*(-(b*c) + a*d)*(-(b*e) + a*f)*(a + b*x^2)) + (b*(b^2
*c*e + 4*a^2*d*f - a*b*(3*d*e + 2*c*f))*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt
[e + f*x^2])])/(a^(3/2)*(b*c - a*d)^2*(b*e - a*f)^(3/2)) + (2*d^2*ArcTan[(Sqrt[d
*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*(b*c - a*d)^2*Sqrt[d*e - c*f])
)/2

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Maple [B]  time = 0.082, size = 1865, normalized size = 9.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b^2*d^4/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)^2/(b*(-c*d)^(1/2)-(-a*b)^(1/2)*d)^2
/(-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^2*d^4/(b*(-c*d)^(
1/2)+(-a*b)^(1/2)*d)^2/(b*(-c*d)^(1/2)-(-a*b)^(1/2)*d)^2/(-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))+1/4*b^2*d/a/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)/(b*
(-c*d)^(1/2)-(-a*b)^(1/2)*d)/(a*f-b*e)/(x-1/b*(-a*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)-1/4*b*d/a/(b*(-
c*d)^(1/2)+(-a*b)^(1/2)*d)/(b*(-c*d)^(1/2)-(-a*b)^(1/2)*d)*f*(-a*b)^(1/2)/(a*f-b
*e)/(-(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)))+1/4*b^2*d/a/(b*(-c*d)^(1/
2)+(-a*b)^(1/2)*d)/(b*(-c*d)^(1/2)-(-a*b)^(1/2)*d)/(a*f-b*e)/(x+1/b*(-a*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)+1/4*b*d/a/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)/(b*(-c*d)^(1/2)-(-a*b)^(1/2)*d)
*f*(-a*b)^(1/2)/(a*f-b*e)/(-(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)))+3/4
*b^3*d^3/(-a*b)^(1/2)/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)^2/(b*(-c*d)^(1/2)-(-a*b)^(
1/2)*d)^2/(-(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)))-1/4*b^4*d^2/a/(-a*b
)^(1/2)/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)^2/(b*(-c*d)^(1/2)-(-a*b)^(1/2)*d)^2/(-(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)))*c-3/4*b^3*d^3/(-a*b)^(1/2)/(b*(-c
*d)^(1/2)+(-a*b)^(1/2)*d)^2/(b*(-c*d)^(1/2)-(-a*b)^(1/2)*d)^2/(-(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)))+1/4*b^4*d^2/a/(-a*b)^(1/2)/(b*(-c*d)^(1/2)+(-a*
b)^(1/2)*d)^2/(b*(-c*d)^(1/2)-(-a*b)^(1/2)*d)^2/(-(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)))*c

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 8.45473, size = 647, normalized size = 3.19 \[ -\frac{1}{2} \,{\left (\frac{2 \, d^{2} \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt{-c^{2} f^{2} + c d f e}}\right )}{{\left (b^{2} c^{2} f^{2} - 2 \, a b c d f^{2} + a^{2} d^{2} f^{2}\right )} \sqrt{-c^{2} f^{2} + c d f e}} + \frac{{\left (2 \, a b^{2} c f - 4 \, a^{2} b d f - b^{3} c e + 3 \, a b^{2} d e\right )} \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} b + 2 \, a f - b e}{2 \, \sqrt{-a^{2} f^{2} + a b f e}}\right )}{{\left (a^{2} b^{2} c^{2} f^{3} - 2 \, a^{3} b c d f^{3} + a^{4} d^{2} f^{3} - a b^{3} c^{2} f^{2} e + 2 \, a^{2} b^{2} c d f^{2} e - a^{3} b d^{2} f^{2} e\right )} \sqrt{-a^{2} f^{2} + a b f e}} + \frac{2 \,{\left (2 \,{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} a b f -{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} b^{2} e + b^{2} e^{2}\right )}}{{\left (a^{2} b c f^{3} - a^{3} d f^{3} - a b^{2} c f^{2} e + a^{2} b d f^{2} e\right )}{\left ({\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{4} b + 4 \,{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} a f - 2 \,{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} b e + b e^{2}\right )}}\right )} f^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*d^2*arctan(1/2*((sqrt(f)*x - sqrt(f*x^2 + e))^2*d + 2*c*f - d*e)/sqrt(-c
^2*f^2 + c*d*f*e))/((b^2*c^2*f^2 - 2*a*b*c*d*f^2 + a^2*d^2*f^2)*sqrt(-c^2*f^2 +
c*d*f*e)) + (2*a*b^2*c*f - 4*a^2*b*d*f - b^3*c*e + 3*a*b^2*d*e)*arctan(1/2*((sqr
t(f)*x - sqrt(f*x^2 + e))^2*b + 2*a*f - b*e)/sqrt(-a^2*f^2 + a*b*f*e))/((a^2*b^2
*c^2*f^3 - 2*a^3*b*c*d*f^3 + a^4*d^2*f^3 - a*b^3*c^2*f^2*e + 2*a^2*b^2*c*d*f^2*e
- a^3*b*d^2*f^2*e)*sqrt(-a^2*f^2 + a*b*f*e)) + 2*(2*(sqrt(f)*x - sqrt(f*x^2 + e
))^2*a*b*f - (sqrt(f)*x - sqrt(f*x^2 + e))^2*b^2*e + b^2*e^2)/((a^2*b*c*f^3 - a^
3*d*f^3 - a*b^2*c*f^2*e + a^2*b*d*f^2*e)*((sqrt(f)*x - sqrt(f*x^2 + e))^4*b + 4*
(sqrt(f)*x - sqrt(f*x^2 + e))^2*a*f - 2*(sqrt(f)*x - sqrt(f*x^2 + e))^2*b*e + b*
e^2)))*f^(5/2)