3.318 \(\int \frac{1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=377 \[ \frac{b^{9/2} (5 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^4}+\frac{d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{8 a c^2 x^3 \left (c+d x^2\right ) (b c-a d)^3}+\frac{d^{7/2} \left (35 a^2 d^2-110 a b c d+99 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} (b c-a d)^4}-\frac{-35 a^3 d^3+75 a^2 b c d^2-24 a b^2 c^2 d+20 b^3 c^3}{24 a^2 c^3 x^3 (b c-a d)^3}+\frac{-35 a^4 d^4+75 a^3 b c d^3-24 a^2 b^2 c^2 d^2-24 a b^3 c^3 d+20 b^4 c^4}{8 a^3 c^4 x (b c-a d)^3}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{d (a d+2 b c)}{4 a c x^3 \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

-(20*b^3*c^3 - 24*a*b^2*c^2*d + 75*a^2*b*c*d^2 - 35*a^3*d^3)/(24*a^2*c^3*(b*c -
a*d)^3*x^3) + (20*b^4*c^4 - 24*a*b^3*c^3*d - 24*a^2*b^2*c^2*d^2 + 75*a^3*b*c*d^3
- 35*a^4*d^4)/(8*a^3*c^4*(b*c - a*d)^3*x) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d
)^2*x^3*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*(c + d*x^2)^2) + (d*
(4*b^2*c^2 + 15*a*b*c*d - 7*a^2*d^2))/(8*a*c^2*(b*c - a*d)^3*x^3*(c + d*x^2)) +
(b^(9/2)*(5*b*c - 11*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(b*c - a*d)^4)
+ (d^(7/2)*(99*b^2*c^2 - 110*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])
/(8*c^(9/2)*(b*c - a*d)^4)

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Rubi [A]  time = 1.78343, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{b^{9/2} (5 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^4}+\frac{d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{8 a c^2 x^3 \left (c+d x^2\right ) (b c-a d)^3}+\frac{d^{7/2} \left (35 a^2 d^2-110 a b c d+99 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} (b c-a d)^4}-\frac{-35 a^3 d^3+75 a^2 b c d^2-24 a b^2 c^2 d+20 b^3 c^3}{24 a^2 c^3 x^3 (b c-a d)^3}+\frac{-35 a^4 d^4+75 a^3 b c d^3-24 a^2 b^2 c^2 d^2-24 a b^3 c^3 d+20 b^4 c^4}{8 a^3 c^4 x (b c-a d)^3}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{d (a d+2 b c)}{4 a c x^3 \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

-(20*b^3*c^3 - 24*a*b^2*c^2*d + 75*a^2*b*c*d^2 - 35*a^3*d^3)/(24*a^2*c^3*(b*c -
a*d)^3*x^3) + (20*b^4*c^4 - 24*a*b^3*c^3*d - 24*a^2*b^2*c^2*d^2 + 75*a^3*b*c*d^3
- 35*a^4*d^4)/(8*a^3*c^4*(b*c - a*d)^3*x) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d
)^2*x^3*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*(c + d*x^2)^2) + (d*
(4*b^2*c^2 + 15*a*b*c*d - 7*a^2*d^2))/(8*a*c^2*(b*c - a*d)^3*x^3*(c + d*x^2)) +
(b^(9/2)*(5*b*c - 11*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(b*c - a*d)^4)
+ (d^(7/2)*(99*b^2*c^2 - 110*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])
/(8*c^(9/2)*(b*c - a*d)^4)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 1.05648, size = 230, normalized size = 0.61 \[ \frac{1}{24} \left (\frac{12 b^{9/2} (5 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2} (b c-a d)^4}-\frac{12 b^5 x}{a^3 \left (a+b x^2\right ) (a d-b c)^3}+\frac{72 a d+48 b c}{a^3 c^4 x}+\frac{3 d^{7/2} \left (35 a^2 d^2-110 a b c d+99 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{9/2} (b c-a d)^4}-\frac{8}{a^2 c^3 x^3}+\frac{3 d^4 x (19 b c-11 a d)}{c^4 \left (c+d x^2\right ) (b c-a d)^3}+\frac{6 d^4 x}{c^3 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(-8/(a^2*c^3*x^3) + (48*b*c + 72*a*d)/(a^3*c^4*x) - (12*b^5*x)/(a^3*(-(b*c) + a*
d)^3*(a + b*x^2)) + (6*d^4*x)/(c^3*(b*c - a*d)^2*(c + d*x^2)^2) + (3*d^4*(19*b*c
- 11*a*d)*x)/(c^4*(b*c - a*d)^3*(c + d*x^2)) + (12*b^(9/2)*(5*b*c - 11*a*d)*Arc
Tan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(b*c - a*d)^4) + (3*d^(7/2)*(99*b^2*c^2 - 110
*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(9/2)*(b*c - a*d)^4))/24

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Maple [A]  time = 0.035, size = 455, normalized size = 1.2 \[ -{\frac{1}{3\,{a}^{2}{c}^{3}{x}^{3}}}+3\,{\frac{d}{{a}^{2}x{c}^{4}}}+2\,{\frac{b}{x{a}^{3}{c}^{3}}}+{\frac{11\,{d}^{7}{x}^{3}{a}^{2}}{8\,{c}^{4} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{15\,{d}^{6}{x}^{3}ab}{4\,{c}^{3} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{19\,{d}^{5}{x}^{3}{b}^{2}}{8\,{c}^{2} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{13\,{d}^{6}x{a}^{2}}{8\,{c}^{3} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{17\,{d}^{5}xab}{4\,{c}^{2} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{21\,{d}^{4}x{b}^{2}}{8\,c \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{35\,{d}^{6}{a}^{2}}{8\,{c}^{4} \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{55\,{d}^{5}ab}{4\,{c}^{3} \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{99\,{d}^{4}{b}^{2}}{8\,{c}^{2} \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{b}^{5}xd}{2\,{a}^{2} \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{6}xc}{2\,{a}^{3} \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{11\,{b}^{5}d}{2\,{a}^{2} \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{6}c}{2\,{a}^{3} \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

-1/3/a^2/c^3/x^3+3/x/a^2/c^4*d+2/x/a^3/c^3*b+11/8*d^7/c^4/(a*d-b*c)^4/(d*x^2+c)^
2*x^3*a^2-15/4*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x^3*a*b+19/8*d^5/c^2/(a*d-b*c)^4/
(d*x^2+c)^2*x^3*b^2+13/8*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x*a^2-17/4*d^5/c^2/(a*d
-b*c)^4/(d*x^2+c)^2*x*a*b+21/8*d^4/c/(a*d-b*c)^4/(d*x^2+c)^2*x*b^2+35/8*d^6/c^4/
(a*d-b*c)^4/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2-55/4*d^5/c^3/(a*d-b*c)^4/(c*
d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b+99/8*d^4/c^2/(a*d-b*c)^4/(c*d)^(1/2)*arctan
(x*d/(c*d)^(1/2))*b^2-1/2*b^5/a^2/(a*d-b*c)^4*x/(b*x^2+a)*d+1/2*b^6/a^3/(a*d-b*c
)^4*x/(b*x^2+a)*c-11/2*b^5/a^2/(a*d-b*c)^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d
+5/2*b^6/a^3/(a*d-b*c)^4/(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(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 20.7274, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^4),x, algorithm="fricas")

[Out]

[-1/48*(16*a^2*b^4*c^7 - 64*a^3*b^3*c^6*d + 96*a^4*b^2*c^5*d^2 - 64*a^5*b*c^4*d^
3 + 16*a^6*c^3*d^4 - 6*(20*b^6*c^5*d^2 - 44*a*b^5*c^4*d^3 + 99*a^3*b^3*c^2*d^5 -
110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^8 - 2*(120*b^6*c^6*d - 224*a*b^5*c^5*d^2 -
88*a^2*b^4*c^4*d^3 + 495*a^3*b^3*c^3*d^4 - 253*a^4*b^2*c^2*d^5 - 155*a^5*b*c*d^6
+ 105*a^6*d^7)*x^6 - 2*(60*b^6*c^7 - 52*a*b^5*c^6*d - 184*a^2*b^4*c^5*d^2 + 176
*a^3*b^3*c^4*d^3 + 319*a^4*b^2*c^3*d^4 - 494*a^5*b*c^2*d^5 + 175*a^6*c*d^6)*x^4
- 16*(5*a*b^5*c^7 - 13*a^2*b^4*c^6*d + 2*a^3*b^3*c^5*d^2 + 22*a^4*b^2*c^4*d^3 -
23*a^5*b*c^3*d^4 + 7*a^6*c^2*d^5)*x^2 + 12*((5*b^6*c^5*d^2 - 11*a*b^5*c^4*d^3)*x
^9 + (10*b^6*c^6*d - 17*a*b^5*c^5*d^2 - 11*a^2*b^4*c^4*d^3)*x^7 + (5*b^6*c^7 - a
*b^5*c^6*d - 22*a^2*b^4*c^5*d^2)*x^5 + (5*a*b^5*c^7 - 11*a^2*b^4*c^6*d)*x^3)*sqr
t(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 3*((99*a^3*b^3*c^2*d^5
- 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^9 + (198*a^3*b^3*c^3*d^4 - 121*a^4*b^2*c^
2*d^5 - 40*a^5*b*c*d^6 + 35*a^6*d^7)*x^7 + (99*a^3*b^3*c^4*d^3 + 88*a^4*b^2*c^3*
d^4 - 185*a^5*b*c^2*d^5 + 70*a^6*c*d^6)*x^5 + (99*a^4*b^2*c^4*d^3 - 110*a^5*b*c^
3*d^4 + 35*a^6*c^2*d^5)*x^3)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^
2 + c)))/((a^3*b^5*c^8*d^2 - 4*a^4*b^4*c^7*d^3 + 6*a^5*b^3*c^6*d^4 - 4*a^6*b^2*c
^5*d^5 + a^7*b*c^4*d^6)*x^9 + (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d^2 + 8*a^5*b^3*c
^7*d^3 - 2*a^6*b^2*c^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x^7 + (a^3*b^5*c^10
- 2*a^4*b^4*c^9*d - 2*a^5*b^3*c^8*d^2 + 8*a^6*b^2*c^7*d^3 - 7*a^7*b*c^6*d^4 + 2*
a^8*c^5*d^5)*x^5 + (a^4*b^4*c^10 - 4*a^5*b^3*c^9*d + 6*a^6*b^2*c^8*d^2 - 4*a^7*b
*c^7*d^3 + a^8*c^6*d^4)*x^3), -1/24*(8*a^2*b^4*c^7 - 32*a^3*b^3*c^6*d + 48*a^4*b
^2*c^5*d^2 - 32*a^5*b*c^4*d^3 + 8*a^6*c^3*d^4 - 3*(20*b^6*c^5*d^2 - 44*a*b^5*c^4
*d^3 + 99*a^3*b^3*c^2*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^8 - (120*b^6*c^6
*d - 224*a*b^5*c^5*d^2 - 88*a^2*b^4*c^4*d^3 + 495*a^3*b^3*c^3*d^4 - 253*a^4*b^2*
c^2*d^5 - 155*a^5*b*c*d^6 + 105*a^6*d^7)*x^6 - (60*b^6*c^7 - 52*a*b^5*c^6*d - 18
4*a^2*b^4*c^5*d^2 + 176*a^3*b^3*c^4*d^3 + 319*a^4*b^2*c^3*d^4 - 494*a^5*b*c^2*d^
5 + 175*a^6*c*d^6)*x^4 - 8*(5*a*b^5*c^7 - 13*a^2*b^4*c^6*d + 2*a^3*b^3*c^5*d^2 +
22*a^4*b^2*c^4*d^3 - 23*a^5*b*c^3*d^4 + 7*a^6*c^2*d^5)*x^2 - 3*((99*a^3*b^3*c^2
*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^9 + (198*a^3*b^3*c^3*d^4 - 121*a^4*b^
2*c^2*d^5 - 40*a^5*b*c*d^6 + 35*a^6*d^7)*x^7 + (99*a^3*b^3*c^4*d^3 + 88*a^4*b^2*
c^3*d^4 - 185*a^5*b*c^2*d^5 + 70*a^6*c*d^6)*x^5 + (99*a^4*b^2*c^4*d^3 - 110*a^5*
b*c^3*d^4 + 35*a^6*c^2*d^5)*x^3)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))) + 6*((5*b^6
*c^5*d^2 - 11*a*b^5*c^4*d^3)*x^9 + (10*b^6*c^6*d - 17*a*b^5*c^5*d^2 - 11*a^2*b^4
*c^4*d^3)*x^7 + (5*b^6*c^7 - a*b^5*c^6*d - 22*a^2*b^4*c^5*d^2)*x^5 + (5*a*b^5*c^
7 - 11*a^2*b^4*c^6*d)*x^3)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2
+ a)))/((a^3*b^5*c^8*d^2 - 4*a^4*b^4*c^7*d^3 + 6*a^5*b^3*c^6*d^4 - 4*a^6*b^2*c^5
*d^5 + a^7*b*c^4*d^6)*x^9 + (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d^2 + 8*a^5*b^3*c^7
*d^3 - 2*a^6*b^2*c^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x^7 + (a^3*b^5*c^10 -
2*a^4*b^4*c^9*d - 2*a^5*b^3*c^8*d^2 + 8*a^6*b^2*c^7*d^3 - 7*a^7*b*c^6*d^4 + 2*a^
8*c^5*d^5)*x^5 + (a^4*b^4*c^10 - 4*a^5*b^3*c^9*d + 6*a^6*b^2*c^8*d^2 - 4*a^7*b*c
^7*d^3 + a^8*c^6*d^4)*x^3), -1/48*(16*a^2*b^4*c^7 - 64*a^3*b^3*c^6*d + 96*a^4*b^
2*c^5*d^2 - 64*a^5*b*c^4*d^3 + 16*a^6*c^3*d^4 - 6*(20*b^6*c^5*d^2 - 44*a*b^5*c^4
*d^3 + 99*a^3*b^3*c^2*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^8 - 2*(120*b^6*c
^6*d - 224*a*b^5*c^5*d^2 - 88*a^2*b^4*c^4*d^3 + 495*a^3*b^3*c^3*d^4 - 253*a^4*b^
2*c^2*d^5 - 155*a^5*b*c*d^6 + 105*a^6*d^7)*x^6 - 2*(60*b^6*c^7 - 52*a*b^5*c^6*d
- 184*a^2*b^4*c^5*d^2 + 176*a^3*b^3*c^4*d^3 + 319*a^4*b^2*c^3*d^4 - 494*a^5*b*c^
2*d^5 + 175*a^6*c*d^6)*x^4 - 16*(5*a*b^5*c^7 - 13*a^2*b^4*c^6*d + 2*a^3*b^3*c^5*
d^2 + 22*a^4*b^2*c^4*d^3 - 23*a^5*b*c^3*d^4 + 7*a^6*c^2*d^5)*x^2 - 24*((5*b^6*c^
5*d^2 - 11*a*b^5*c^4*d^3)*x^9 + (10*b^6*c^6*d - 17*a*b^5*c^5*d^2 - 11*a^2*b^4*c^
4*d^3)*x^7 + (5*b^6*c^7 - a*b^5*c^6*d - 22*a^2*b^4*c^5*d^2)*x^5 + (5*a*b^5*c^7 -
11*a^2*b^4*c^6*d)*x^3)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - 3*((99*a^3*b^3*c^2
*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^9 + (198*a^3*b^3*c^3*d^4 - 121*a^4*b^
2*c^2*d^5 - 40*a^5*b*c*d^6 + 35*a^6*d^7)*x^7 + (99*a^3*b^3*c^4*d^3 + 88*a^4*b^2*
c^3*d^4 - 185*a^5*b*c^2*d^5 + 70*a^6*c*d^6)*x^5 + (99*a^4*b^2*c^4*d^3 - 110*a^5*
b*c^3*d^4 + 35*a^6*c^2*d^5)*x^3)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(
d*x^2 + c)))/((a^3*b^5*c^8*d^2 - 4*a^4*b^4*c^7*d^3 + 6*a^5*b^3*c^6*d^4 - 4*a^6*b
^2*c^5*d^5 + a^7*b*c^4*d^6)*x^9 + (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d^2 + 8*a^5*b
^3*c^7*d^3 - 2*a^6*b^2*c^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x^7 + (a^3*b^5*c
^10 - 2*a^4*b^4*c^9*d - 2*a^5*b^3*c^8*d^2 + 8*a^6*b^2*c^7*d^3 - 7*a^7*b*c^6*d^4
+ 2*a^8*c^5*d^5)*x^5 + (a^4*b^4*c^10 - 4*a^5*b^3*c^9*d + 6*a^6*b^2*c^8*d^2 - 4*a
^7*b*c^7*d^3 + a^8*c^6*d^4)*x^3), -1/24*(8*a^2*b^4*c^7 - 32*a^3*b^3*c^6*d + 48*a
^4*b^2*c^5*d^2 - 32*a^5*b*c^4*d^3 + 8*a^6*c^3*d^4 - 3*(20*b^6*c^5*d^2 - 44*a*b^5
*c^4*d^3 + 99*a^3*b^3*c^2*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^8 - (120*b^6
*c^6*d - 224*a*b^5*c^5*d^2 - 88*a^2*b^4*c^4*d^3 + 495*a^3*b^3*c^3*d^4 - 253*a^4*
b^2*c^2*d^5 - 155*a^5*b*c*d^6 + 105*a^6*d^7)*x^6 - (60*b^6*c^7 - 52*a*b^5*c^6*d
- 184*a^2*b^4*c^5*d^2 + 176*a^3*b^3*c^4*d^3 + 319*a^4*b^2*c^3*d^4 - 494*a^5*b*c^
2*d^5 + 175*a^6*c*d^6)*x^4 - 8*(5*a*b^5*c^7 - 13*a^2*b^4*c^6*d + 2*a^3*b^3*c^5*d
^2 + 22*a^4*b^2*c^4*d^3 - 23*a^5*b*c^3*d^4 + 7*a^6*c^2*d^5)*x^2 - 12*((5*b^6*c^5
*d^2 - 11*a*b^5*c^4*d^3)*x^9 + (10*b^6*c^6*d - 17*a*b^5*c^5*d^2 - 11*a^2*b^4*c^4
*d^3)*x^7 + (5*b^6*c^7 - a*b^5*c^6*d - 22*a^2*b^4*c^5*d^2)*x^5 + (5*a*b^5*c^7 -
11*a^2*b^4*c^6*d)*x^3)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - 3*((99*a^3*b^3*c^2*
d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^9 + (198*a^3*b^3*c^3*d^4 - 121*a^4*b^2
*c^2*d^5 - 40*a^5*b*c*d^6 + 35*a^6*d^7)*x^7 + (99*a^3*b^3*c^4*d^3 + 88*a^4*b^2*c
^3*d^4 - 185*a^5*b*c^2*d^5 + 70*a^6*c*d^6)*x^5 + (99*a^4*b^2*c^4*d^3 - 110*a^5*b
*c^3*d^4 + 35*a^6*c^2*d^5)*x^3)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))))/((a^3*b^5*c
^8*d^2 - 4*a^4*b^4*c^7*d^3 + 6*a^5*b^3*c^6*d^4 - 4*a^6*b^2*c^5*d^5 + a^7*b*c^4*d
^6)*x^9 + (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d^2 + 8*a^5*b^3*c^7*d^3 - 2*a^6*b^2*c
^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x^7 + (a^3*b^5*c^10 - 2*a^4*b^4*c^9*d -
2*a^5*b^3*c^8*d^2 + 8*a^6*b^2*c^7*d^3 - 7*a^7*b*c^6*d^4 + 2*a^8*c^5*d^5)*x^5 + (
a^4*b^4*c^10 - 4*a^5*b^3*c^9*d + 6*a^6*b^2*c^8*d^2 - 4*a^7*b*c^7*d^3 + a^8*c^6*d
^4)*x^3)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.247669, size = 495, normalized size = 1.31 \[ \frac{b^{5} x}{2 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )}{\left (b x^{2} + a\right )}} + \frac{{\left (5 \, b^{6} c - 11 \, a b^{5} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} \sqrt{a b}} + \frac{{\left (99 \, b^{2} c^{2} d^{4} - 110 \, a b c d^{5} + 35 \, a^{2} d^{6}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} \sqrt{c d}} + \frac{19 \, b c d^{5} x^{3} - 11 \, a d^{6} x^{3} + 21 \, b c^{2} d^{4} x - 13 \, a c d^{5} x}{8 \,{\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )}{\left (d x^{2} + c\right )}^{2}} + \frac{6 \, b c x^{2} + 9 \, a d x^{2} - a c}{3 \, a^{3} c^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^4),x, algorithm="giac")

[Out]

1/2*b^5*x/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*(b*x^2 + a)
) + 1/2*(5*b^6*c - 11*a*b^5*d)*arctan(b*x/sqrt(a*b))/((a^3*b^4*c^4 - 4*a^4*b^3*c
^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*sqrt(a*b)) + 1/8*(99*b^2*c^2
*d^4 - 110*a*b*c*d^5 + 35*a^2*d^6)*arctan(d*x/sqrt(c*d))/((b^4*c^8 - 4*a*b^3*c^7
*d + 6*a^2*b^2*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4)*sqrt(c*d)) + 1/8*(19*b*c
*d^5*x^3 - 11*a*d^6*x^3 + 21*b*c^2*d^4*x - 13*a*c*d^5*x)/((b^3*c^7 - 3*a*b^2*c^6
*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*(d*x^2 + c)^2) + 1/3*(6*b*c*x^2 + 9*a*d*x^2
- a*c)/(a^3*c^4*x^3)