3.301 \(\int \frac{x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=162 \[ \frac{x (a d+b c)}{2 b \left (c+d x^2\right ) (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)^3}+\frac{\sqrt{c} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.411205, antiderivative size = 162, 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 \[ \frac{x (a d+b c)}{2 b \left (c+d x^2\right ) (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)^3}+\frac{\sqrt{c} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 77.951, size = 138, normalized size = 0.85 \[ \frac{\sqrt{a} \left (a d + 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b} \left (a d - b c\right )^{3}} - \frac{a x}{2 b \left (a + b x^{2}\right ) \left (c + d x^{2}\right ) \left (a d - b c\right )} - \frac{\sqrt{c} \left (3 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 \sqrt{d} \left (a d - b c\right )^{3}} + \frac{x \left (a d + b c\right )}{2 b \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.324419, size = 133, normalized size = 0.82 \[ \frac{1}{2} \left (\frac{a x}{\left (a+b x^2\right ) (b c-a d)^2}+\frac{c x}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{a} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (a d-b c)^3}+\frac{\sqrt{c} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 222, normalized size = 1.4 \[{\frac{acxd}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{{c}^{2}xb}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{3\,acd}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{b{c}^{2}}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{x{a}^{2}d}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{xabc}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{2}d}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,abc}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

1/2*c/(a*d-b*c)^3*x/(d*x^2+c)*a*d-1/2*c^2/(a*d-b*c)^3*x/(d*x^2+c)*b-3/2*c/(a*d-b
*c)^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*d-1/2*c^2/(a*d-b*c)^3/(c*d)^(1/2)*ar
ctan(x*d/(c*d)^(1/2))*b+1/2*a^2/(a*d-b*c)^3*x/(b*x^2+a)*d-1/2*a/(a*d-b*c)^3*x/(b
*x^2+a)*b*c+1/2*a^2/(a*d-b*c)^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d+3/2*a/(a*d
-b*c)^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*b*c

_______________________________________________________________________________________

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^4/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.399201, 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^4/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="fricas")

[Out]

[1/4*(2*(b^2*c^2 - a^2*d^2)*x^3 - ((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(-a/b)*log((b*x^2 + 2*b*x*sqrt(-
a/b) - a)/(b*x^2 + a)) - ((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(-c/d)*log((d*x^2 - 2*d*x*sqrt(-c/d) - c)
/(d*x^2 + c)) + 4*(a*b*c^2 - a^2*c*d)*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*(2*(b^2*c^
2 - a^2*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(a/b)*arctan(x/sqrt(a/b)) - ((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(-c
/d)*log((d*x^2 - 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) + 4*(a*b*c^2 - a^2*c*d)*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*(2*(b^2*c^2 - a^2*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)*sqrt(c/d)*a
rctan(x/sqrt(c/d)) - ((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(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*
x^2 + a)) + 4*(a*b*c^2 - a^2*c*d)*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*((b^2*c^2 - a^
2*d^2)*x^3 - ((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(a/b)*arctan(x/sqrt(a/b)) + ((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(c/d)*arcta
n(x/sqrt(c/d)) + 2*(a*b*c^2 - a^2*c*d)*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)]

_______________________________________________________________________________________

Sympy [A]  time = 83.168, size = 2378, normalized size = 14.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

sqrt(-a/b)*(a*d + 3*b*c)*log(x + (-4*a**7*b*d**8*(-a/b)**(3/2)*(a*d + 3*b*c)**3/
(a*d - b*c)**9 + 20*a**6*b**2*c*d**7*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)*
*9 - 36*a**5*b**3*c**2*d**6*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 20*a
**4*b**4*c**3*d**5*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - a**4*d**4*sqr
t(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3 + 20*a**3*b**5*c**4*d**4*(-a/b)**(3/2)*(a*d
+ 3*b*c)**3/(a*d - b*c)**9 - 36*a**3*b*c*d**3*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b
*c)**3 - 36*a**2*b**6*c**5*d**3*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 -
54*a**2*b**2*c**2*d**2*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3 + 20*a*b**7*c**6*
d**2*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 36*a*b**3*c**3*d*sqrt(-a/b)
*(a*d + 3*b*c)/(a*d - b*c)**3 - 4*b**8*c**7*d*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*
d - b*c)**9 - b**4*c**4*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3)/(3*a**2*d**2 +
10*a*b*c*d + 3*b**2*c**2))/(4*(a*d - b*c)**3) - sqrt(-a/b)*(a*d + 3*b*c)*log(x +
(4*a**7*b*d**8*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 20*a**6*b**2*c*d
**7*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 36*a**5*b**3*c**2*d**6*(-a/b
)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 20*a**4*b**4*c**3*d**5*(-a/b)**(3/2)*
(a*d + 3*b*c)**3/(a*d - b*c)**9 + a**4*d**4*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)
**3 - 20*a**3*b**5*c**4*d**4*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 36*
a**3*b*c*d**3*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3 + 36*a**2*b**6*c**5*d**3*(
-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 54*a**2*b**2*c**2*d**2*sqrt(-a/b)
*(a*d + 3*b*c)/(a*d - b*c)**3 - 20*a*b**7*c**6*d**2*(-a/b)**(3/2)*(a*d + 3*b*c)*
*3/(a*d - b*c)**9 + 36*a*b**3*c**3*d*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3 + 4
*b**8*c**7*d*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + b**4*c**4*sqrt(-a/b
)*(a*d + 3*b*c)/(a*d - b*c)**3)/(3*a**2*d**2 + 10*a*b*c*d + 3*b**2*c**2))/(4*(a*
d - b*c)**3) + sqrt(-c/d)*(3*a*d + b*c)*log(x + (-4*a**7*b*d**8*(-c/d)**(3/2)*(3
*a*d + b*c)**3/(a*d - b*c)**9 + 20*a**6*b**2*c*d**7*(-c/d)**(3/2)*(3*a*d + b*c)*
*3/(a*d - b*c)**9 - 36*a**5*b**3*c**2*d**6*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d -
b*c)**9 + 20*a**4*b**4*c**3*d**5*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9
- a**4*d**4*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3 + 20*a**3*b**5*c**4*d**4*(-c
/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 36*a**3*b*c*d**3*sqrt(-c/d)*(3*a*d
+ b*c)/(a*d - b*c)**3 - 36*a**2*b**6*c**5*d**3*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a
*d - b*c)**9 - 54*a**2*b**2*c**2*d**2*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3 +
20*a*b**7*c**6*d**2*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 36*a*b**3*c*
*3*d*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3 - 4*b**8*c**7*d*(-c/d)**(3/2)*(3*a*
d + b*c)**3/(a*d - b*c)**9 - b**4*c**4*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3)/
(3*a**2*d**2 + 10*a*b*c*d + 3*b**2*c**2))/(4*(a*d - b*c)**3) - sqrt(-c/d)*(3*a*d
+ b*c)*log(x + (4*a**7*b*d**8*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 2
0*a**6*b**2*c*d**7*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 36*a**5*b**3*
c**2*d**6*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 20*a**4*b**4*c**3*d**5
*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + a**4*d**4*sqrt(-c/d)*(3*a*d + b
*c)/(a*d - b*c)**3 - 20*a**3*b**5*c**4*d**4*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d
- b*c)**9 + 36*a**3*b*c*d**3*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3 + 36*a**2*b
**6*c**5*d**3*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 54*a**2*b**2*c**2*
d**2*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3 - 20*a*b**7*c**6*d**2*(-c/d)**(3/2)
*(3*a*d + b*c)**3/(a*d - b*c)**9 + 36*a*b**3*c**3*d*sqrt(-c/d)*(3*a*d + b*c)/(a*
d - b*c)**3 + 4*b**8*c**7*d*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + b**4
*c**4*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3)/(3*a**2*d**2 + 10*a*b*c*d + 3*b**
2*c**2))/(4*(a*d - b*c)**3) + (2*a*c*x + x**3*(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))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.340477, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="giac")

[Out]

Done