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

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

[Out]

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

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Rubi [A]  time = 0.182265, 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 \[ \frac{x}{2 \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^2}+\frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.8828, size = 88, normalized size = 0.85 \[ - \frac{\sqrt{a} \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\left (a d - b c\right )^{2}} - \frac{x}{2 \left (c + d x^{2}\right ) \left (a d - b c\right )} + \frac{\left (a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 \sqrt{c} \sqrt{d} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.213511, size = 90, normalized size = 0.87 \[ \frac{\frac{x (b c-a d)}{c+d x^2}+\frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} \sqrt{d}}-2 \sqrt{a} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280536, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (d x^{2} + c\right )} \sqrt{-a b} \sqrt{-c d} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 2 \,{\left (b c - a d\right )} \sqrt{-c d} x +{\left (b c^{2} + a c d +{\left (b c d + a d^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right )}{4 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \sqrt{-c d}}, \frac{{\left (d x^{2} + c\right )} \sqrt{-a b} \sqrt{c d} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) +{\left (b c - a d\right )} \sqrt{c d} x +{\left (b c^{2} + a c d +{\left (b c d + a d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{2 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \sqrt{c d}}, -\frac{4 \,{\left (d x^{2} + c\right )} \sqrt{a b} \sqrt{-c d} \arctan \left (\frac{b x}{\sqrt{a b}}\right ) - 2 \,{\left (b c - a d\right )} \sqrt{-c d} x -{\left (b c^{2} + a c d +{\left (b c d + a d^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right )}{4 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \sqrt{-c d}}, -\frac{2 \,{\left (d x^{2} + c\right )} \sqrt{a b} \sqrt{c d} \arctan \left (\frac{b x}{\sqrt{a b}}\right ) -{\left (b c - a d\right )} \sqrt{c d} x -{\left (b c^{2} + a c d +{\left (b c d + a d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{2 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \sqrt{c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 23.4487, 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)/(d*x**2+c)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.270926, size = 149, normalized size = 1.43 \[ -\frac{a b \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b}} + \frac{{\left (b c + a d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{c d}} + \frac{x}{2 \,{\left (d x^{2} + c\right )}{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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