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

Optimal. Leaf size=108 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)^2}-\frac{c x}{2 d \left (c+d x^2\right ) (b c-a d)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.243666, antiderivative size = 108, 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 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)^2}-\frac{c x}{2 d \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 38.5092, size = 94, normalized size = 0.87 \[ \frac{a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{b} \left (a d - b c\right )^{2}} - \frac{\sqrt{c} \left (3 a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 d^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{c x}{2 d \left (c + d x^{2}\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.218645, size = 108, normalized size = 1. \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (a d-b c)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)^2}+\frac{c x}{2 d \left (c+d x^2\right ) (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 144, normalized size = 1.3 \[{\frac{acx}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{c}^{2}xb}{2\, \left ( ad-bc \right ) ^{2}d \left ( d{x}^{2}+c \right ) }}-{\frac{3\,ac}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{c}^{2}b}{2\, \left ( ad-bc \right ) ^{2}d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{a}^{2}}{ \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.306612, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (a d^{2} x^{2} + a c d\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) -{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} - 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) - 2 \,{\left (b c^{2} - a c d\right )} x}{4 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac{4 \,{\left (a d^{2} x^{2} + a c d\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) -{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} - 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) - 2 \,{\left (b c^{2} - a c d\right )} x}{4 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac{{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{x}{\sqrt{\frac{c}{d}}}\right ) +{\left (a d^{2} x^{2} + a c d\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) -{\left (b c^{2} - a c d\right )} x}{2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac{2 \,{\left (a d^{2} x^{2} + a c d\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) +{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{x}{\sqrt{\frac{c}{d}}}\right ) -{\left (b c^{2} - a c d\right )} x}{2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 42.6211, size = 1850, normalized size = 17.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x/(2*a*c*d**2 - 2*b*c**2*d + x**2*(2*a*d**3 - 2*b*c*d**2)) + sqrt(-a**3/b)*log
(x + (-20*a**5*b*d**8*(-a**3/b)**(3/2)/(a*d - b*c)**6 + 84*a**4*b**2*c*d**7*(-a*
*3/b)**(3/2)/(a*d - b*c)**6 - 8*a**4*d**4*sqrt(-a**3/b)/(a*d - b*c)**2 - 136*a**
3*b**3*c**2*d**6*(-a**3/b)**(3/2)/(a*d - b*c)**6 - 27*a**3*b*c*d**3*sqrt(-a**3/b
)/(a*d - b*c)**2 + 104*a**2*b**4*c**3*d**5*(-a**3/b)**(3/2)/(a*d - b*c)**6 + 27*
a**2*b**2*c**2*d**2*sqrt(-a**3/b)/(a*d - b*c)**2 - 36*a*b**5*c**4*d**4*(-a**3/b)
**(3/2)/(a*d - b*c)**6 - 9*a*b**3*c**3*d*sqrt(-a**3/b)/(a*d - b*c)**2 + 4*b**6*c
**5*d**3*(-a**3/b)**(3/2)/(a*d - b*c)**6 + b**4*c**4*sqrt(-a**3/b)/(a*d - b*c)**
2)/(12*a**3*d**2 - 7*a**2*b*c*d + a*b**2*c**2))/(2*(a*d - b*c)**2) - sqrt(-a**3/
b)*log(x + (20*a**5*b*d**8*(-a**3/b)**(3/2)/(a*d - b*c)**6 - 84*a**4*b**2*c*d**7
*(-a**3/b)**(3/2)/(a*d - b*c)**6 + 8*a**4*d**4*sqrt(-a**3/b)/(a*d - b*c)**2 + 13
6*a**3*b**3*c**2*d**6*(-a**3/b)**(3/2)/(a*d - b*c)**6 + 27*a**3*b*c*d**3*sqrt(-a
**3/b)/(a*d - b*c)**2 - 104*a**2*b**4*c**3*d**5*(-a**3/b)**(3/2)/(a*d - b*c)**6
- 27*a**2*b**2*c**2*d**2*sqrt(-a**3/b)/(a*d - b*c)**2 + 36*a*b**5*c**4*d**4*(-a*
*3/b)**(3/2)/(a*d - b*c)**6 + 9*a*b**3*c**3*d*sqrt(-a**3/b)/(a*d - b*c)**2 - 4*b
**6*c**5*d**3*(-a**3/b)**(3/2)/(a*d - b*c)**6 - b**4*c**4*sqrt(-a**3/b)/(a*d - b
*c)**2)/(12*a**3*d**2 - 7*a**2*b*c*d + a*b**2*c**2))/(2*(a*d - b*c)**2) + sqrt(-
c/d**3)*(3*a*d - b*c)*log(x + (-5*a**5*b*d**8*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/
(2*(a*d - b*c)**6) + 21*a**4*b**2*c*d**7*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a
*d - b*c)**6) - 4*a**4*d**4*sqrt(-c/d**3)*(3*a*d - b*c)/(a*d - b*c)**2 - 17*a**3
*b**3*c**2*d**6*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(a*d - b*c)**6 - 27*a**3*b*c*d
**3*sqrt(-c/d**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) + 13*a**2*b**4*c**3*d**5*(-c/
d**3)**(3/2)*(3*a*d - b*c)**3/(a*d - b*c)**6 + 27*a**2*b**2*c**2*d**2*sqrt(-c/d*
*3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) - 9*a*b**5*c**4*d**4*(-c/d**3)**(3/2)*(3*a*
d - b*c)**3/(2*(a*d - b*c)**6) - 9*a*b**3*c**3*d*sqrt(-c/d**3)*(3*a*d - b*c)/(2*
(a*d - b*c)**2) + b**6*c**5*d**3*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c
)**6) + b**4*c**4*sqrt(-c/d**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2))/(12*a**3*d**2
- 7*a**2*b*c*d + a*b**2*c**2))/(4*(a*d - b*c)**2) - sqrt(-c/d**3)*(3*a*d - b*c)*
log(x + (5*a**5*b*d**8*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) - 21
*a**4*b**2*c*d**7*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) + 4*a**4*
d**4*sqrt(-c/d**3)*(3*a*d - b*c)/(a*d - b*c)**2 + 17*a**3*b**3*c**2*d**6*(-c/d**
3)**(3/2)*(3*a*d - b*c)**3/(a*d - b*c)**6 + 27*a**3*b*c*d**3*sqrt(-c/d**3)*(3*a*
d - b*c)/(2*(a*d - b*c)**2) - 13*a**2*b**4*c**3*d**5*(-c/d**3)**(3/2)*(3*a*d - b
*c)**3/(a*d - b*c)**6 - 27*a**2*b**2*c**2*d**2*sqrt(-c/d**3)*(3*a*d - b*c)/(2*(a
*d - b*c)**2) + 9*a*b**5*c**4*d**4*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b
*c)**6) + 9*a*b**3*c**3*d*sqrt(-c/d**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) - b**6*
c**5*d**3*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) - b**4*c**4*sqrt(
-c/d**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2))/(12*a**3*d**2 - 7*a**2*b*c*d + a*b**2
*c**2))/(4*(a*d - b*c)**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.288677, size = 163, normalized size = 1.51 \[ \frac{a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b}} + \frac{{\left (b c^{2} - 3 \, a c d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt{c d}} - \frac{c x}{2 \,{\left (b c d - a d^{2}\right )}{\left (d x^{2} + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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