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

Optimal. Leaf size=104 \[ -\frac{x}{2 \left (a+b x^2\right ) (b c-a d)}+\frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} (b c-a d)^2}-\frac{\sqrt{c} \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{(b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.175464, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{x}{2 \left (a+b x^2\right ) (b c-a d)}+\frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} (b c-a d)^2}-\frac{\sqrt{c} \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{(b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.173, size = 88, normalized size = 0.85 \[ - \frac{\sqrt{c} \sqrt{d} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{\left (a d - b c\right )^{2}} + \frac{x}{2 \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{\left (a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{b} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c)*sqrt(d)*atan(sqrt(d)*x/sqrt(c))/(a*d - b*c)**2 + x/(2*(a + b*x**2)*(a*d
- b*c)) + (a*d + b*c)*atan(sqrt(b)*x/sqrt(a))/(2*sqrt(a)*sqrt(b)*(a*d - b*c)**2
)

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Mathematica [A]  time = 0.245064, size = 104, normalized size = 1. \[ \frac{x}{2 \left (a+b x^2\right ) (a d-b c)}+\frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} (a d-b c)^2}-\frac{\sqrt{c} \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{(b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 134, normalized size = 1.3 \[ -{\frac{cd}{ \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{axd}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{bxc}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{ad}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{bc}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300203, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b x^{2} + a\right )} \sqrt{-a b} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) - 2 \, \sqrt{-a b}{\left (b c - a d\right )} x +{\left (a b c + a^{2} d +{\left (b^{2} c + a b d\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right )}{4 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2}\right )} \sqrt{-a b}}, \frac{{\left (b x^{2} + a\right )} \sqrt{a b} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) - \sqrt{a b}{\left (b c - a d\right )} x +{\left (a b c + a^{2} d +{\left (b^{2} c + a b d\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2}\right )} \sqrt{a b}}, -\frac{4 \,{\left (b x^{2} + a\right )} \sqrt{-a b} \sqrt{c d} \arctan \left (\frac{d x}{\sqrt{c d}}\right ) + 2 \, \sqrt{-a b}{\left (b c - a d\right )} x -{\left (a b c + a^{2} d +{\left (b^{2} c + a b d\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right )}{4 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2}\right )} \sqrt{-a b}}, -\frac{2 \,{\left (b x^{2} + a\right )} \sqrt{a b} \sqrt{c d} \arctan \left (\frac{d x}{\sqrt{c d}}\right ) + \sqrt{a b}{\left (b c - a d\right )} x -{\left (a b c + a^{2} d +{\left (b^{2} c + a b d\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*(b*x^2 + a)*sqrt(-a*b)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^
2 + c)) - 2*sqrt(-a*b)*(b*c - a*d)*x + (a*b*c + a^2*d + (b^2*c + a*b*d)*x^2)*log
((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)))/((a*b^2*c^2 - 2*a^2*b*c*d + a^
3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^2)*sqrt(-a*b)), 1/2*((b*x^2 + a)*s
qrt(a*b)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) - sqrt(a*b)*(b
*c - a*d)*x + (a*b*c + a^2*d + (b^2*c + a*b*d)*x^2)*arctan(sqrt(a*b)*x/a))/((a*b
^2*c^2 - 2*a^2*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^2)*sqrt(a
*b)), -1/4*(4*(b*x^2 + a)*sqrt(-a*b)*sqrt(c*d)*arctan(d*x/sqrt(c*d)) + 2*sqrt(-a
*b)*(b*c - a*d)*x - (a*b*c + a^2*d + (b^2*c + a*b*d)*x^2)*log((2*a*b*x + (b*x^2
- a)*sqrt(-a*b))/(b*x^2 + a)))/((a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2 + (b^3*c^2 -
2*a*b^2*c*d + a^2*b*d^2)*x^2)*sqrt(-a*b)), -1/2*(2*(b*x^2 + a)*sqrt(a*b)*sqrt(c*
d)*arctan(d*x/sqrt(c*d)) + sqrt(a*b)*(b*c - a*d)*x - (a*b*c + a^2*d + (b^2*c + a
*b*d)*x^2)*arctan(sqrt(a*b)*x/a))/((a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2 + (b^3*c^2
- 2*a*b^2*c*d + a^2*b*d^2)*x^2)*sqrt(a*b))]

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Sympy [A]  time = 23.5807, size = 1530, normalized size = 14.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*a**2*d - 2*a*b*c + x**2*(2*a*b*d - 2*b**2*c)) + sqrt(-1/(a*b))*(a*d + b*c)*
log(x + (-3*a**6*b*d**5*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(2*(a*d - b*c)**6) + 11
*a**5*b**2*c*d**4*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(2*(a*d - b*c)**6) - 7*a**4*b
**3*c**2*d**3*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(a*d - b*c)**6 + 3*a**3*b**4*c**3
*d**2*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(a*d - b*c)**6 - a**3*d**3*sqrt(-1/(a*b))
*(a*d + b*c)/(2*(a*d - b*c)**2) + a**2*b**5*c**4*d*(-1/(a*b))**(3/2)*(a*d + b*c)
**3/(2*(a*d - b*c)**6) - 11*a**2*b*c*d**2*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d - b
*c)**2) - a*b**6*c**5*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(2*(a*d - b*c)**6) - 3*a*
b**2*c**2*d*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d - b*c)**2) - b**3*c**3*sqrt(-1/(a
*b))*(a*d + b*c)/(2*(a*d - b*c)**2))/(a*d**2 + b*c*d))/(4*(a*d - b*c)**2) - sqrt
(-1/(a*b))*(a*d + b*c)*log(x + (3*a**6*b*d**5*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(
2*(a*d - b*c)**6) - 11*a**5*b**2*c*d**4*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(2*(a*d
- b*c)**6) + 7*a**4*b**3*c**2*d**3*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(a*d - b*c)
**6 - 3*a**3*b**4*c**3*d**2*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(a*d - b*c)**6 + a*
*3*d**3*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d - b*c)**2) - a**2*b**5*c**4*d*(-1/(a*
b))**(3/2)*(a*d + b*c)**3/(2*(a*d - b*c)**6) + 11*a**2*b*c*d**2*sqrt(-1/(a*b))*(
a*d + b*c)/(2*(a*d - b*c)**2) + a*b**6*c**5*(-1/(a*b))**(3/2)*(a*d + b*c)**3/(2*
(a*d - b*c)**6) + 3*a*b**2*c**2*d*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d - b*c)**2)
+ b**3*c**3*sqrt(-1/(a*b))*(a*d + b*c)/(2*(a*d - b*c)**2))/(a*d**2 + b*c*d))/(4*
(a*d - b*c)**2) + sqrt(-c*d)*log(x + (-12*a**6*b*d**5*(-c*d)**(3/2)/(a*d - b*c)*
*6 + 44*a**5*b**2*c*d**4*(-c*d)**(3/2)/(a*d - b*c)**6 - 56*a**4*b**3*c**2*d**3*(
-c*d)**(3/2)/(a*d - b*c)**6 + 24*a**3*b**4*c**3*d**2*(-c*d)**(3/2)/(a*d - b*c)**
6 - a**3*d**3*sqrt(-c*d)/(a*d - b*c)**2 + 4*a**2*b**5*c**4*d*(-c*d)**(3/2)/(a*d
- b*c)**6 - 11*a**2*b*c*d**2*sqrt(-c*d)/(a*d - b*c)**2 - 4*a*b**6*c**5*(-c*d)**(
3/2)/(a*d - b*c)**6 - 3*a*b**2*c**2*d*sqrt(-c*d)/(a*d - b*c)**2 - b**3*c**3*sqrt
(-c*d)/(a*d - b*c)**2)/(a*d**2 + b*c*d))/(2*(a*d - b*c)**2) - sqrt(-c*d)*log(x +
(12*a**6*b*d**5*(-c*d)**(3/2)/(a*d - b*c)**6 - 44*a**5*b**2*c*d**4*(-c*d)**(3/2
)/(a*d - b*c)**6 + 56*a**4*b**3*c**2*d**3*(-c*d)**(3/2)/(a*d - b*c)**6 - 24*a**3
*b**4*c**3*d**2*(-c*d)**(3/2)/(a*d - b*c)**6 + a**3*d**3*sqrt(-c*d)/(a*d - b*c)*
*2 - 4*a**2*b**5*c**4*d*(-c*d)**(3/2)/(a*d - b*c)**6 + 11*a**2*b*c*d**2*sqrt(-c*
d)/(a*d - b*c)**2 + 4*a*b**6*c**5*(-c*d)**(3/2)/(a*d - b*c)**6 + 3*a*b**2*c**2*d
*sqrt(-c*d)/(a*d - b*c)**2 + b**3*c**3*sqrt(-c*d)/(a*d - b*c)**2)/(a*d**2 + b*c*
d))/(2*(a*d - b*c)**2)

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GIAC/XCAS [A]  time = 0.23621, size = 149, normalized size = 1.43 \[ -\frac{c d \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{c d}} + \frac{{\left (b c + a d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b}} - \frac{x}{2 \,{\left (b x^{2} + a\right )}{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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