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

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

[Out]

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

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Rubi [A]  time = 0.339085, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac{d x}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{b} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^3}-\frac{\sqrt{d} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 68.3915, size = 126, normalized size = 0.86 \[ - \frac{d x}{\left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} + \frac{x}{2 \left (a + b x^{2}\right ) \left (c + d x^{2}\right ) \left (a d - b c\right )} + \frac{\sqrt{d} \left (a d + 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 \sqrt{c} \left (a d - b c\right )^{3}} - \frac{\sqrt{b} \left (3 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.2994, size = 137, normalized size = 0.93 \[ \frac{1}{2} \left (-\frac{b x}{\left (a+b x^2\right ) (b c-a d)^2}-\frac{d x}{\left (c+d x^2\right ) (b c-a d)^2}-\frac{\sqrt{b} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (a d-b c)^3}-\frac{\sqrt{d} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 222, normalized size = 1.5 \[ -{\frac{{d}^{2}xa}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{dxbc}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{a{d}^{2}}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,bcd}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{xabd}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{x{b}^{2}c}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,abd}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{2}c}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*d^2/(a*d-b*c)^3*x/(d*x^2+c)*a+1/2*d/(a*d-b*c)^3*x/(d*x^2+c)*b*c+1/2*d^2/(a*
d-b*c)^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a+3/2*d/(a*d-b*c)^3/(c*d)^(1/2)*arc
tan(x*d/(c*d)^(1/2))*b*c-1/2*b/(a*d-b*c)^3*x/(b*x^2+a)*a*d+1/2*b^2/(a*d-b*c)^3*x
/(b*x^2+a)*c-3/2*b/(a*d-b*c)^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*a*d-1/2*b^2/(
a*d-b*c)^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.523413, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 97.8929, size = 2399, normalized size = 16.32 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-b/a)*(3*a*d + b*c)*log(x + (-a**9*c*d**8*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a
*d - b*c)**9 + 20*a**7*b**2*c**3*d**6*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)
**9 - 64*a**6*b**3*c**4*d**5*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 90*
a**5*b**4*c**5*d**4*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - a**5*d**5*sq
rt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 - 64*a**4*b**5*c**6*d**3*(-b/a)**(3/2)*(3*
a*d + b*c)**3/(a*d - b*c)**9 - 9*a**4*b*c*d**4*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b
*c)**3 + 20*a**3*b**6*c**7*d**2*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 -
54*a**3*b**2*c**2*d**3*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 - 54*a**2*b**3*c*
*3*d**2*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 - a*b**8*c**9*(-b/a)**(3/2)*(3*a
*d + b*c)**3/(a*d - b*c)**9 - 9*a*b**4*c**4*d*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*
c)**3 - b**5*c**5*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3)/(3*a**2*b*d**3 + 10*a
*b**2*c*d**2 + 3*b**3*c**2*d))/(4*(a*d - b*c)**3) - sqrt(-b/a)*(3*a*d + b*c)*log
(x + (a**9*c*d**8*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 20*a**7*b**2*c
**3*d**6*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 64*a**6*b**3*c**4*d**5*
(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 90*a**5*b**4*c**5*d**4*(-b/a)**(
3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + a**5*d**5*sqrt(-b/a)*(3*a*d + b*c)/(a*d -
b*c)**3 + 64*a**4*b**5*c**6*d**3*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9
+ 9*a**4*b*c*d**4*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 - 20*a**3*b**6*c**7*d*
*2*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 54*a**3*b**2*c**2*d**3*sqrt(-
b/a)*(3*a*d + b*c)/(a*d - b*c)**3 + 54*a**2*b**3*c**3*d**2*sqrt(-b/a)*(3*a*d + b
*c)/(a*d - b*c)**3 + a*b**8*c**9*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 +
9*a*b**4*c**4*d*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 + b**5*c**5*sqrt(-b/a)*
(3*a*d + b*c)/(a*d - b*c)**3)/(3*a**2*b*d**3 + 10*a*b**2*c*d**2 + 3*b**3*c**2*d)
)/(4*(a*d - b*c)**3) + sqrt(-d/c)*(a*d + 3*b*c)*log(x + (-a**9*c*d**8*(-d/c)**(3
/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 20*a**7*b**2*c**3*d**6*(-d/c)**(3/2)*(a*d
+ 3*b*c)**3/(a*d - b*c)**9 - 64*a**6*b**3*c**4*d**5*(-d/c)**(3/2)*(a*d + 3*b*c)*
*3/(a*d - b*c)**9 + 90*a**5*b**4*c**5*d**4*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d -
b*c)**9 - a**5*d**5*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3 - 64*a**4*b**5*c**6
*d**3*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 9*a**4*b*c*d**4*sqrt(-d/c)
*(a*d + 3*b*c)/(a*d - b*c)**3 + 20*a**3*b**6*c**7*d**2*(-d/c)**(3/2)*(a*d + 3*b*
c)**3/(a*d - b*c)**9 - 54*a**3*b**2*c**2*d**3*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*
c)**3 - 54*a**2*b**3*c**3*d**2*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3 - a*b**8*
c**9*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 9*a*b**4*c**4*d*sqrt(-d/c)*
(a*d + 3*b*c)/(a*d - b*c)**3 - b**5*c**5*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3
)/(3*a**2*b*d**3 + 10*a*b**2*c*d**2 + 3*b**3*c**2*d))/(4*(a*d - b*c)**3) - sqrt(
-d/c)*(a*d + 3*b*c)*log(x + (a**9*c*d**8*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b
*c)**9 - 20*a**7*b**2*c**3*d**6*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 +
64*a**6*b**3*c**4*d**5*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 90*a**5*b
**4*c**5*d**4*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + a**5*d**5*sqrt(-d/
c)*(a*d + 3*b*c)/(a*d - b*c)**3 + 64*a**4*b**5*c**6*d**3*(-d/c)**(3/2)*(a*d + 3*
b*c)**3/(a*d - b*c)**9 + 9*a**4*b*c*d**4*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3
- 20*a**3*b**6*c**7*d**2*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 54*a**
3*b**2*c**2*d**3*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3 + 54*a**2*b**3*c**3*d**
2*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3 + a*b**8*c**9*(-d/c)**(3/2)*(a*d + 3*b
*c)**3/(a*d - b*c)**9 + 9*a*b**4*c**4*d*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3
+ b**5*c**5*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3)/(3*a**2*b*d**3 + 10*a*b**2*
c*d**2 + 3*b**3*c**2*d))/(4*(a*d - b*c)**3) - (2*b*d*x**3 + x*(a*d + b*c))/(2*a*
*3*c*d**2 - 4*a**2*b*c**2*d + 2*a*b**2*c**3 + x**4*(2*a**2*b*d**3 - 4*a*b**2*c*d
**2 + 2*b**3*c**2*d) + x**2*(2*a**3*d**3 - 2*a**2*b*c*d**2 - 2*a*b**2*c**2*d + 2
*b**3*c**3))

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GIAC/XCAS [A]  time = 0.307961, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done