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

Optimal. Leaf size=147 \[ -\frac{b (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2} (3 b c-2 a d)}{2 a^2 c x (b c-a d)}+\frac{b \sqrt{c+d x^2}}{2 a x \left (a+b x^2\right ) (b c-a d)} \]

[Out]

-((3*b*c - 2*a*d)*Sqrt[c + d*x^2])/(2*a^2*c*(b*c - a*d)*x) + (b*Sqrt[c + d*x^2])
/(2*a*(b*c - a*d)*x*(a + b*x^2)) - (b*(3*b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)
/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.412581, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{b (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2} (3 b c-2 a d)}{2 a^2 c x (b c-a d)}+\frac{b \sqrt{c+d x^2}}{2 a x \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-((3*b*c - 2*a*d)*Sqrt[c + d*x^2])/(2*a^2*c*(b*c - a*d)*x) + (b*Sqrt[c + d*x^2])
/(2*a*(b*c - a*d)*x*(a + b*x^2)) - (b*(3*b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)
/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 61.0116, size = 124, normalized size = 0.84 \[ - \frac{b \sqrt{c + d x^{2}}}{2 a x \left (a + b x^{2}\right ) \left (a d - b c\right )} - \frac{\sqrt{c + d x^{2}} \left (2 a d - 3 b c\right )}{2 a^{2} c x \left (a d - b c\right )} - \frac{b \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*sqrt(c + d*x**2)/(2*a*x*(a + b*x**2)*(a*d - b*c)) - sqrt(c + d*x**2)*(2*a*d -
3*b*c)/(2*a**2*c*x*(a*d - b*c)) - b*(4*a*d - 3*b*c)*atanh(x*sqrt(a*d - b*c)/(sq
rt(a)*sqrt(c + d*x**2)))/(2*a**(5/2)*(a*d - b*c)**(3/2))

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Mathematica [A]  time = 0.267638, size = 116, normalized size = 0.79 \[ \frac{\sqrt{c+d x^2} \left (\frac{b^2 x^2}{\left (a+b x^2\right ) (a d-b c)}-\frac{2}{c}\right )}{2 a^2 x}-\frac{b (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^2]*(-2/c + (b^2*x^2)/((-(b*c) + a*d)*(a + b*x^2))))/(2*a^2*x) - (b
*(3*b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/
2)*(b*c - a*d)^(3/2))

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Maple [B]  time = 0.024, size = 841, normalized size = 5.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a^2/c/x*(d*x^2+c)^(1/2)+1/4/a^2/(a*d-b*c)*b/(x-1/b*(-a*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)-1/4/a^2
*d*(-a*b)^(1/2)/(a*d-b*c)/(-(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)))+1/4
/a^2/(a*d-b*c)*b/(x+1/b*(-a*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)+1/4/a^2*d*(-a*b)^(1/2)/(a*d-b*c)/(-(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)))+3/4*b/a^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)))-3/4*b/a^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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.404023, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (2 \, a b c - 2 \, a^{2} d +{\left (3 \, b^{2} c - 2 \, a b d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} -{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{3} +{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} - 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \,{\left ({\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{3} +{\left (a^{3} b c^{2} - a^{4} c d\right )} x\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \,{\left (2 \, a b c - 2 \, a^{2} d +{\left (3 \, b^{2} c - 2 \, a b d\right )} x^{2}\right )} \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} +{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{3} +{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{4 \,{\left ({\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{3} +{\left (a^{3} b c^{2} - a^{4} c d\right )} x\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 2.1349, size = 535, normalized size = 3.64 \[ \frac{1}{2} \, d^{\frac{5}{2}}{\left (\frac{{\left (3 \, b^{2} c - 4 \, a b d\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{a b c d - a^{2} d^{2}}} + \frac{2 \,{\left (3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{2} c - 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b d - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c^{2} + 14 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c d - 8 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} d^{2} + 3 \, b^{2} c^{3} - 2 \, a b c^{2} d\right )}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b - 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a d + 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c^{2} - 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a c d - b c^{3}\right )}{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^(5/2)*((3*b^2*c - 4*a*b*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))/((a^2*b*c*d^2 - a^3*d^3)*sqrt(a*b*c*d - a
^2*d^2)) + 2*(3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c - 4*(sqrt(d)*x - sqrt(d*x^
2 + c))^4*a*b*d - 6*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^2 + 14*(sqrt(d)*x - sq
rt(d*x^2 + c))^2*a*b*c*d - 8*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*d^2 + 3*b^2*c^3
- 2*a*b*c^2*d)/(((sqrt(d)*x - sqrt(d*x^2 + c))^6*b - 3*(sqrt(d)*x - sqrt(d*x^2
+ c))^4*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*d + 3*(sqrt(d)*x - sqrt(d*x^2
+ c))^2*b*c^2 - 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*c*d - b*c^3)*(a^2*b*c*d^2 -
a^3*d^3)))