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

Optimal. Leaf size=100 \[ \frac{(b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{3/2}}+\frac{b x \sqrt{c+d x^2}}{2 a \left (a+b x^2\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.14441, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{(b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{3/2}}+\frac{b x \sqrt{c+d x^2}}{2 a \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.8175, size = 83, normalized size = 0.83 \[ - \frac{b x \sqrt{c + d x^{2}}}{2 a \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{\left (2 a d - b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*x*sqrt(c + d*x**2)/(2*a*(a + b*x**2)*(a*d - b*c)) + (2*a*d - b*c)*atanh(x*sqr
t(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/(2*a**(3/2)*(a*d - b*c)**(3/2))

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Mathematica [A]  time = 0.176456, size = 100, normalized size = 1. \[ \frac{\frac{\sqrt{a} b x \sqrt{c+d x^2}}{\left (a+b x^2\right ) (b c-a d)}+\frac{(b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{3/2}}}{2 a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.041, size = 823, normalized size = 8.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.355736, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} b x +{\left (a b c - 2 \, a^{2} d +{\left (b^{2} c - 2 \, a b d\right )} x^{2}\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} + 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \,{\left (a^{2} b c - a^{3} d +{\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d}}, \frac{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} b x +{\left (a b c - 2 \, a^{2} d +{\left (b^{2} c - 2 \, a b d\right )} x^{2}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{4 \,{\left (a^{2} b c - a^{3} d +{\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.232772, size = 304, normalized size = 3.04 \[ -\frac{1}{2} \, d^{\frac{3}{2}}{\left (\frac{{\left (b c - 2 \, a d\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{{\left (a b c d - a^{2} d^{2}\right )}^{\frac{3}{2}}} + \frac{2 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d - b c^{2}\right )}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}{\left (a b c d - a^{2} d^{2}\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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