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3.254 \(\int \frac{x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=155 \[ \frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d} (b c-a d)^3}-\frac{\sqrt{a} b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{x (a d+3 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^2}+\frac{x}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

x/(4*(b*c - a*d)*(c + d*x^2)^2) + ((3*b*c + a*d)*x)/(8*c*(b*c - a*d)^2*(c + d*x^
2)) - (Sqrt[a]*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((3*b^2*c^2
+ 6*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*Sqrt[d]*(b*c - a*
d)^3)

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Rubi [A]  time = 0.360506, antiderivative size = 155, 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 \[ \frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d} (b c-a d)^3}-\frac{\sqrt{a} b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{x (a d+3 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^2}+\frac{x}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

x/(4*(b*c - a*d)*(c + d*x^2)^2) + ((3*b*c + a*d)*x)/(8*c*(b*c - a*d)^2*(c + d*x^
2)) - (Sqrt[a]*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((3*b^2*c^2
+ 6*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*Sqrt[d]*(b*c - a*
d)^3)

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Rubi in Sympy [A]  time = 70.5863, size = 136, normalized size = 0.88 \[ \frac{\sqrt{a} b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\left (a d - b c\right )^{3}} - \frac{x}{4 \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} + \frac{x \left (a d + 3 b c\right )}{8 c \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} + \frac{\left (a^{2} d^{2} - 6 a b c d - 3 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{3}{2}} \sqrt{d} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

sqrt(a)*b**(3/2)*atan(sqrt(b)*x/sqrt(a))/(a*d - b*c)**3 - x/(4*(c + d*x**2)**2*(
a*d - b*c)) + x*(a*d + 3*b*c)/(8*c*(c + d*x**2)*(a*d - b*c)**2) + (a**2*d**2 - 6
*a*b*c*d - 3*b**2*c**2)*atan(sqrt(d)*x/sqrt(c))/(8*c**(3/2)*sqrt(d)*(a*d - b*c)*
*3)

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Mathematica [A]  time = 0.397621, size = 151, normalized size = 0.97 \[ \frac{1}{8} \left (\frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} \sqrt{d} (b c-a d)^3}+\frac{8 \sqrt{a} b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(a d-b c)^3}+\frac{x (a d+3 b c)}{c \left (c+d x^2\right ) (b c-a d)^2}+\frac{2 x}{\left (c+d x^2\right )^2 (b c-a d)}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

((2*x)/((b*c - a*d)*(c + d*x^2)^2) + ((3*b*c + a*d)*x)/(c*(b*c - a*d)^2*(c + d*x
^2)) + (8*Sqrt[a]*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(-(b*c) + a*d)^3 + ((3*b^
2*c^2 + 6*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*Sqrt[d]*(b*c
- a*d)^3))/8

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Maple [B]  time = 0.017, size = 298, normalized size = 1.9 \[{\frac{{x}^{3}{a}^{2}{d}^{3}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}c}}+{\frac{{x}^{3}ab{d}^{2}}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,{x}^{3}{b}^{2}cd}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,abcdx}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{5\,{b}^{2}{c}^{2}x}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{a}^{2}{d}^{2}x}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{a}^{2}{d}^{2}}{8\, \left ( ad-bc \right ) ^{3}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{3\,abd}{4\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{3\,{b}^{2}c}{8\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{a{b}^{2}}{ \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^2+a)/(d*x^2+c)^3,x)

[Out]

1/8/(a*d-b*c)^3/(d*x^2+c)^2*d^3/c*x^3*a^2+1/4/(a*d-b*c)^3/(d*x^2+c)^2*x^3*a*b*d^
2-3/8/(a*d-b*c)^3/(d*x^2+c)^2*x^3*b^2*c*d+3/4/(a*d-b*c)^3/(d*x^2+c)^2*a*b*c*d*x-
5/8/(a*d-b*c)^3/(d*x^2+c)^2*b^2*c^2*x-1/8/(a*d-b*c)^3/(d*x^2+c)^2*a^2*d^2*x+1/8/
(a*d-b*c)^3/c/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2*d^2-3/4/(a*d-b*c)^3/(c*d)^
(1/2)*arctan(x*d/(c*d)^(1/2))*a*b*d-3/8/(a*d-b*c)^3*c/(c*d)^(1/2)*arctan(x*d/(c*
d)^(1/2))*b^2+a*b^2/(a*d-b*c)^3/(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)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.616961, 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)*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

[-1/16*(8*(b*c*d^2*x^4 + 2*b*c^2*d*x^2 + b*c^3)*sqrt(-a*b)*sqrt(-c*d)*log((b*x^2
 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - (3*b^2*c^4 + 6*a*b*c^3*d - a^2*c^2*d^2 + (
3*b^2*c^2*d^2 + 6*a*b*c*d^3 - a^2*d^4)*x^4 + 2*(3*b^2*c^3*d + 6*a*b*c^2*d^2 - a^
2*c*d^3)*x^2)*log((2*c*d*x + (d*x^2 - c)*sqrt(-c*d))/(d*x^2 + c)) - 2*((3*b^2*c^
2*d - 2*a*b*c*d^2 - a^2*d^3)*x^3 + (5*b^2*c^3 - 6*a*b*c^2*d + a^2*c*d^2)*x)*sqrt
(-c*d))/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3 + (b^3*c^4*d^2
 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)*x^4 + 2*(b^3*c^5*d - 3*a*b^2*c
^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*d^4)*x^2)*sqrt(-c*d)), -1/8*(4*(b*c*d^2*x^4 +
 2*b*c^2*d*x^2 + b*c^3)*sqrt(-a*b)*sqrt(c*d)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b
*x^2 + a)) - (3*b^2*c^4 + 6*a*b*c^3*d - a^2*c^2*d^2 + (3*b^2*c^2*d^2 + 6*a*b*c*d
^3 - a^2*d^4)*x^4 + 2*(3*b^2*c^3*d + 6*a*b*c^2*d^2 - a^2*c*d^3)*x^2)*arctan(sqrt
(c*d)*x/c) - ((3*b^2*c^2*d - 2*a*b*c*d^2 - a^2*d^3)*x^3 + (5*b^2*c^3 - 6*a*b*c^2
*d + a^2*c*d^2)*x)*sqrt(c*d))/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*
c^3*d^3 + (b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)*x^4 + 2*
(b^3*c^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*d^4)*x^2)*sqrt(c*d)), -
1/16*(16*(b*c*d^2*x^4 + 2*b*c^2*d*x^2 + b*c^3)*sqrt(a*b)*sqrt(-c*d)*arctan(b*x/s
qrt(a*b)) - (3*b^2*c^4 + 6*a*b*c^3*d - a^2*c^2*d^2 + (3*b^2*c^2*d^2 + 6*a*b*c*d^
3 - a^2*d^4)*x^4 + 2*(3*b^2*c^3*d + 6*a*b*c^2*d^2 - a^2*c*d^3)*x^2)*log((2*c*d*x
 + (d*x^2 - c)*sqrt(-c*d))/(d*x^2 + c)) - 2*((3*b^2*c^2*d - 2*a*b*c*d^2 - a^2*d^
3)*x^3 + (5*b^2*c^3 - 6*a*b*c^2*d + a^2*c*d^2)*x)*sqrt(-c*d))/((b^3*c^6 - 3*a*b^
2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3 + (b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2
*b*c^2*d^4 - a^3*c*d^5)*x^4 + 2*(b^3*c^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 -
 a^3*c^2*d^4)*x^2)*sqrt(-c*d)), -1/8*(8*(b*c*d^2*x^4 + 2*b*c^2*d*x^2 + b*c^3)*sq
rt(a*b)*sqrt(c*d)*arctan(b*x/sqrt(a*b)) - (3*b^2*c^4 + 6*a*b*c^3*d - a^2*c^2*d^2
 + (3*b^2*c^2*d^2 + 6*a*b*c*d^3 - a^2*d^4)*x^4 + 2*(3*b^2*c^3*d + 6*a*b*c^2*d^2
- a^2*c*d^3)*x^2)*arctan(sqrt(c*d)*x/c) - ((3*b^2*c^2*d - 2*a*b*c*d^2 - a^2*d^3)
*x^3 + (5*b^2*c^3 - 6*a*b*c^2*d + a^2*c*d^2)*x)*sqrt(c*d))/((b^3*c^6 - 3*a*b^2*c
^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3 + (b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*
c^2*d^4 - a^3*c*d^5)*x^4 + 2*(b^3*c^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 - a^
3*c^2*d^4)*x^2)*sqrt(c*d))]

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Sympy [A]  time = 164.665, size = 3386, normalized size = 21.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

-sqrt(-a*b**3)*log(x + (-64*a**8*c**3*d**9*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 768
*a**7*b*c**4*d**8*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 2560*a**6*b**2*c**5*d**7*(-a
*b**3)**(3/2)/(a*d - b*c)**9 - a**6*d**6*sqrt(-a*b**3)/(a*d - b*c)**3 + 2816*a**
5*b**3*c**6*d**6*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 18*a**5*b*c*d**5*sqrt(-a*b**3
)/(a*d - b*c)**3 + 1920*a**4*b**4*c**7*d**5*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 99
*a**4*b**2*c**2*d**4*sqrt(-a*b**3)/(a*d - b*c)**3 - 7936*a**3*b**5*c**8*d**4*(-a
*b**3)**(3/2)/(a*d - b*c)**9 + 108*a**3*b**3*c**3*d**3*sqrt(-a*b**3)/(a*d - b*c)
**3 + 8192*a**2*b**6*c**9*d**3*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 297*a**2*b**4*c
**4*d**2*sqrt(-a*b**3)/(a*d - b*c)**3 - 3840*a*b**7*c**10*d**2*(-a*b**3)**(3/2)/
(a*d - b*c)**9 + 674*a*b**5*c**5*d*sqrt(-a*b**3)/(a*d - b*c)**3 + 704*b**8*c**11
*d*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 27*b**6*c**6*sqrt(-a*b**3)/(a*d - b*c)**3)/
(a**3*b**2*d**3 - 15*a**2*b**3*c*d**2 + 51*a*b**4*c**2*d + 27*b**5*c**3))/(2*(a*
d - b*c)**3) + sqrt(-a*b**3)*log(x + (64*a**8*c**3*d**9*(-a*b**3)**(3/2)/(a*d -
b*c)**9 - 768*a**7*b*c**4*d**8*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 2560*a**6*b**2*
c**5*d**7*(-a*b**3)**(3/2)/(a*d - b*c)**9 + a**6*d**6*sqrt(-a*b**3)/(a*d - b*c)*
*3 - 2816*a**5*b**3*c**6*d**6*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 18*a**5*b*c*d**5
*sqrt(-a*b**3)/(a*d - b*c)**3 - 1920*a**4*b**4*c**7*d**5*(-a*b**3)**(3/2)/(a*d -
 b*c)**9 + 99*a**4*b**2*c**2*d**4*sqrt(-a*b**3)/(a*d - b*c)**3 + 7936*a**3*b**5*
c**8*d**4*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 108*a**3*b**3*c**3*d**3*sqrt(-a*b**3
)/(a*d - b*c)**3 - 8192*a**2*b**6*c**9*d**3*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 29
7*a**2*b**4*c**4*d**2*sqrt(-a*b**3)/(a*d - b*c)**3 + 3840*a*b**7*c**10*d**2*(-a*
b**3)**(3/2)/(a*d - b*c)**9 - 674*a*b**5*c**5*d*sqrt(-a*b**3)/(a*d - b*c)**3 - 7
04*b**8*c**11*d*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 27*b**6*c**6*sqrt(-a*b**3)/(a*
d - b*c)**3)/(a**3*b**2*d**3 - 15*a**2*b**3*c*d**2 + 51*a*b**4*c**2*d + 27*b**5*
c**3))/(2*(a*d - b*c)**3) - sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c*
*2)*log(x + (-a**8*c**3*d**9*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**
2*c**2)**3/(8*(a*d - b*c)**9) + 3*a**7*b*c**4*d**8*(-1/(c**3*d))**(3/2)*(a**2*d*
*2 - 6*a*b*c*d - 3*b**2*c**2)**3/(2*(a*d - b*c)**9) - 5*a**6*b**2*c**5*d**7*(-1/
(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(a*d - b*c)**9 - a**6*
d**6*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3)
+ 11*a**5*b**3*c**6*d**6*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c*
*2)**3/(2*(a*d - b*c)**9) + 9*a**5*b*c*d**5*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b
*c*d - 3*b**2*c**2)/(4*(a*d - b*c)**3) + 15*a**4*b**4*c**7*d**5*(-1/(c**3*d))**(
3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(4*(a*d - b*c)**9) - 99*a**4*b**2*
c**2*d**4*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)
**3) - 31*a**3*b**5*c**8*d**4*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b*
*2*c**2)**3/(2*(a*d - b*c)**9) + 27*a**3*b**3*c**3*d**3*sqrt(-1/(c**3*d))*(a**2*
d**2 - 6*a*b*c*d - 3*b**2*c**2)/(2*(a*d - b*c)**3) + 16*a**2*b**6*c**9*d**3*(-1/
(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(a*d - b*c)**9 + 297*a
**2*b**4*c**4*d**2*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a
*d - b*c)**3) - 15*a*b**7*c**10*d**2*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d
 - 3*b**2*c**2)**3/(2*(a*d - b*c)**9) + 337*a*b**5*c**5*d*sqrt(-1/(c**3*d))*(a**
2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(4*(a*d - b*c)**3) + 11*b**8*c**11*d*(-1/(c**3
*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(8*(a*d - b*c)**9) + 27*b**
6*c**6*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3
))/(a**3*b**2*d**3 - 15*a**2*b**3*c*d**2 + 51*a*b**4*c**2*d + 27*b**5*c**3))/(16
*(a*d - b*c)**3) + sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)*log(x
 + (a**8*c**3*d**9*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3
/(8*(a*d - b*c)**9) - 3*a**7*b*c**4*d**8*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b
*c*d - 3*b**2*c**2)**3/(2*(a*d - b*c)**9) + 5*a**6*b**2*c**5*d**7*(-1/(c**3*d))*
*(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(a*d - b*c)**9 + a**6*d**6*sqrt(
-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3) - 11*a**5*
b**3*c**6*d**6*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(2*
(a*d - b*c)**9) - 9*a**5*b*c*d**5*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b
**2*c**2)/(4*(a*d - b*c)**3) - 15*a**4*b**4*c**7*d**5*(-1/(c**3*d))**(3/2)*(a**2
*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(4*(a*d - b*c)**9) + 99*a**4*b**2*c**2*d**4*
sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3) + 31*
a**3*b**5*c**8*d**4*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**
3/(2*(a*d - b*c)**9) - 27*a**3*b**3*c**3*d**3*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a
*b*c*d - 3*b**2*c**2)/(2*(a*d - b*c)**3) - 16*a**2*b**6*c**9*d**3*(-1/(c**3*d))*
*(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(a*d - b*c)**9 - 297*a**2*b**4*c
**4*d**2*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)*
*3) + 15*a*b**7*c**10*d**2*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*
c**2)**3/(2*(a*d - b*c)**9) - 337*a*b**5*c**5*d*sqrt(-1/(c**3*d))*(a**2*d**2 - 6
*a*b*c*d - 3*b**2*c**2)/(4*(a*d - b*c)**3) - 11*b**8*c**11*d*(-1/(c**3*d))**(3/2
)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(8*(a*d - b*c)**9) - 27*b**6*c**6*sqr
t(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3))/(a**3*b
**2*d**3 - 15*a**2*b**3*c*d**2 + 51*a*b**4*c**2*d + 27*b**5*c**3))/(16*(a*d - b*
c)**3) + (x**3*(a*d**2 + 3*b*c*d) + x*(-a*c*d + 5*b*c**2))/(8*a**2*c**3*d**2 - 1
6*a*b*c**4*d + 8*b**2*c**5 + x**4*(8*a**2*c*d**4 - 16*a*b*c**2*d**3 + 8*b**2*c**
3*d**2) + x**2*(16*a**2*c**2*d**3 - 32*a*b*c**3*d**2 + 16*b**2*c**4*d))

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GIAC/XCAS [A]  time = 0.289198, size = 278, normalized size = 1.79 \[ -\frac{a b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{a b}} + \frac{{\left (3 \, b^{2} c^{2} + 6 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \sqrt{c d}} + \frac{3 \, b c d x^{3} + a d^{2} x^{3} + 5 \, b c^{2} x - a c d x}{8 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}{\left (d x^{2} + c\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="giac")

[Out]

-a*b^2*arctan(b*x/sqrt(a*b))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3
)*sqrt(a*b)) + 1/8*(3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*arctan(d*x/sqrt(c*d))/((b^3
*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*sqrt(c*d)) + 1/8*(3*b*c*d*x^
3 + a*d^2*x^3 + 5*b*c^2*x - a*c*d*x)/((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*(d*x^2
 + c)^2)

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