3.79 \(\int \frac{d+e x^2}{x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{d \left (a+b x^2\right )}{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a+b x^2\right ) (b d-a e) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.194222, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{d \left (a+b x^2\right )}{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a+b x^2\right ) (b d-a e) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2)/(x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.0583863, size = 72, normalized size = 0.71 \[ \frac{\left (a+b x^2\right ) \left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) (a e x-b d x)-\sqrt{a} \sqrt{b} d\right )}{a^{3/2} \sqrt{b} x \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2)/(x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 67, normalized size = 0.7 \[ -{\frac{b{x}^{2}+a}{ax} \left ( -\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) xae+\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) xbd+d\sqrt{ab} \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*x^2+a)*(-arctan(x*b/(a*b)^(1/2))*x*a*e+arctan(x*b/(a*b)^(1/2))*x*b*d+d*(a*b)
^(1/2))/((b*x^2+a)^2)^(1/2)/a/(a*b)^(1/2)/x

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.284924, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b d - a e\right )} x \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \, \sqrt{-a b} d}{2 \, \sqrt{-a b} a x}, -\frac{{\left (b d - a e\right )} x \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + \sqrt{a b} d}{\sqrt{a b} a x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((b*d - a*e)*x*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*sqr
t(-a*b)*d)/(sqrt(-a*b)*a*x), -((b*d - a*e)*x*arctan(sqrt(a*b)*x/a) + sqrt(a*b)*d
)/(sqrt(a*b)*a*x)]

_______________________________________________________________________________________

Sympy [A]  time = 1.8109, size = 82, normalized size = 0.81 \[ - \frac{\sqrt{- \frac{1}{a^{3} b}} \left (a e - b d\right ) \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a^{3} b}} \left (a e - b d\right ) \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{2} - \frac{d}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**3*b))*(a*e - b*d)*log(-a**2*sqrt(-1/(a**3*b)) + x)/2 + sqrt(-1/(a**
3*b))*(a*e - b*d)*log(a**2*sqrt(-1/(a**3*b)) + x)/2 - d/(a*x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.265662, size = 84, normalized size = 0.83 \[ -\frac{{\left (b d{\rm sign}\left (b x^{2} + a\right ) - a e{\rm sign}\left (b x^{2} + a\right )\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{d{\rm sign}\left (b x^{2} + a\right )}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*d*sign(b*x^2 + a) - a*e*sign(b*x^2 + a))*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a)
- d*sign(b*x^2 + a)/(a*x)