3.96 \(\int \frac{1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=313 \[ \frac{b x \left (-3 a^2 d^2-40 a b c d+8 b^2 c^2\right )}{12 a^2 c \sqrt{a+b x^2} \left (c+d x^2\right ) (b c-a d)^3}+\frac{d^2 \left (3 a^2 d^2-16 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{9/2}}+\frac{d x \sqrt{a+b x^2} \left (9 a^3 d^3-42 a^2 b c d^2-88 a b^2 c^2 d+16 b^3 c^3\right )}{24 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}-\frac{d x}{4 c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (b c-a d)}+\frac{b x (3 a d+4 b c)}{12 a c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

-(d*x)/(4*c*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^2) + (b*(4*b*c + 3*a*d)*x)
/(12*a*c*(b*c - a*d)^2*(a + b*x^2)^(3/2)*(c + d*x^2)) + (b*(8*b^2*c^2 - 40*a*b*c
*d - 3*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*Sqrt[a + b*x^2]*(c + d*x^2)) + (d*(16
*b^3*c^3 - 88*a*b^2*c^2*d - 42*a^2*b*c*d^2 + 9*a^3*d^3)*x*Sqrt[a + b*x^2])/(24*a
^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (d^2*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*A
rcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(b*c - a*d)^(9
/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.08367, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{b x \left (-3 a^2 d^2-40 a b c d+8 b^2 c^2\right )}{12 a^2 c \sqrt{a+b x^2} \left (c+d x^2\right ) (b c-a d)^3}+\frac{d^2 \left (3 a^2 d^2-16 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{9/2}}+\frac{d x \sqrt{a+b x^2} \left (9 a^3 d^3-42 a^2 b c d^2-88 a b^2 c^2 d+16 b^3 c^3\right )}{24 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}-\frac{d x}{4 c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (b c-a d)}+\frac{b x (3 a d+4 b c)}{12 a c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x]

[Out]

-(d*x)/(4*c*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^2) + (b*(4*b*c + 3*a*d)*x)
/(12*a*c*(b*c - a*d)^2*(a + b*x^2)^(3/2)*(c + d*x^2)) + (b*(8*b^2*c^2 - 40*a*b*c
*d - 3*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*Sqrt[a + b*x^2]*(c + d*x^2)) + (d*(16
*b^3*c^3 - 88*a*b^2*c^2*d - 42*a^2*b*c*d^2 + 9*a^3*d^3)*x*Sqrt[a + b*x^2])/(24*a
^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (d^2*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*A
rcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(b*c - a*d)^(9
/2))

_______________________________________________________________________________________

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/(b*x**2+a)**(5/2)/(d*x**2+c)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 1.22183, size = 221, normalized size = 0.71 \[ \frac{1}{24} \left (x \sqrt{a+b x^2} \left (\frac{8 b^3 (2 b c-11 a d)}{a^2 \left (a+b x^2\right ) (b c-a d)^4}-\frac{8 b^3}{a \left (a+b x^2\right )^2 (a d-b c)^3}+\frac{3 d^3 (3 a d-14 b c)}{c^2 \left (c+d x^2\right ) (b c-a d)^4}-\frac{6 d^3}{c \left (c+d x^2\right )^2 (b c-a d)^3}\right )+\frac{3 d^2 \left (3 a^2 d^2-16 a b c d+48 b^2 c^2\right ) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{c^{5/2} (a d-b c)^{9/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x]

[Out]

(x*Sqrt[a + b*x^2]*((-8*b^3)/(a*(-(b*c) + a*d)^3*(a + b*x^2)^2) + (8*b^3*(2*b*c
- 11*a*d))/(a^2*(b*c - a*d)^4*(a + b*x^2)) - (6*d^3)/(c*(b*c - a*d)^3*(c + d*x^2
)^2) + (3*d^3*(-14*b*c + 3*a*d))/(c^2*(b*c - a*d)^4*(c + d*x^2))) + (3*d^2*(48*b
^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a +
b*x^2])])/(c^(5/2)*(-(b*c) + a*d)^(9/2)))/24

_______________________________________________________________________________________

Maple [B]  time = 0.046, size = 4495, normalized size = 14.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3,x)

[Out]

-7/16/c*b/(a*d-b*c)^2/(x-(-c*d)^(1/2)/d)/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2
)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)+1/16/(-c*d)^(1/2)/c/(a*d-b*c)/(x-(-c*d
)^(1/2)/d)^2/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-
b*c)/d)^(3/2)-3/16/(-c*d)^(1/2)/c^2*d^2/(a*d-b*c)^2/((a*d-b*c)/d)^(1/2)*ln((2*(a
*d-b*c)/d+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)
^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d
)^(1/2)/d))-35/16*d*b^3/(a*d-b*c)^4/a/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d
*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x-35/16/(-c*d)^(1/2)*d^2*b^2/(a*d-b*c)^4/
((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*((
a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+
(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))+5/48/(-c*d)^(1/2)/c*d*b/(a*d-b*c)^2/((x-
(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)+5/1
6/(-c*d)^(1/2)/c*d^2*b/(a*d-b*c)^3/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x
-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-15/16/c^2*d*b*(-c*d)^(1/2)/(a*d-b*c)^3/((x+(
-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)+3/16
/c^2*b/(a*d-b*c)/a/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)
+(a*d-b*c)/d)^(3/2)*x+3/8/c^2*b/(a*d-b*c)/a^2/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)
^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x-3/8/c*b^2/(a*d-b*c)^2/a/((x+(-c
*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)*x-3/4/
c*b^2/(a*d-b*c)^2/a^2/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)
/d)+(a*d-b*c)/d)^(1/2)*x+3/16/(-c*d)^(1/2)/c^2*d^2/(a*d-b*c)^2/((a*d-b*c)/d)^(1/
2)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)
*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2
))/(x+(-c*d)^(1/2)/d))+3/16/c^2*b/(a*d-b*c)/a/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)
^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)*x+3/8/c^2*b/(a*d-b*c)/a^2/((x-(-c
*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x-3/8/
c*b^2/(a*d-b*c)^2/a/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d
)+(a*d-b*c)/d)^(3/2)*x-3/4/c*b^2/(a*d-b*c)^2/a^2/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c
*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x+15/16/c^2*d*b*(-c*d)^(1/2)/(
a*d-b*c)^3/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*
c)/d)^(1/2)-35/16*d*b^3/(a*d-b*c)^4/a/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d
*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x+35/16/(-c*d)^(1/2)*d^2*b^2/(a*d-b*c)^4/
((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((
a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+
(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))-5/48/(-c*d)^(1/2)/c*d*b/(a*d-b*c)^2/((x+
(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)-5/1
6/(-c*d)^(1/2)/c*d^2*b/(a*d-b*c)^3/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x
+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)+5/16/(-c*d)^(1/2)/c*d^2*b/(a*d-b*c)^3/((a*d-
b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((a*d-b*
c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b
*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))-3/16/c^2*d/(a*d-b*c)^2/a/((x-(-c*d)^(1/2)/d)^2
*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x*b-3/16/c^2*d/(a*d-
b*c)^2/a/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)
/d)^(1/2)*x*b+5/8/c*d*b^2/(a*d-b*c)^3/a/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)
/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x+15/16/c^2*d*b*(-c*d)^(1/2)/(a*d-b*c)^
3/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*
((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d
)+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))+5/8/c*d*b^2/(a*d-b*c)^3/a/((x-(-c*d)^(
1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x-15/16/c^2
*d*b*(-c*d)^(1/2)/(a*d-b*c)^3/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2*b*(-c*d)^(
1/2)/d*(x-(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*
d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))-5/16/(-c*d
)^(1/2)/c*d^2*b/(a*d-b*c)^3/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2*b*(-c*d)^(1/
2)/d*(x-(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)
^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))+35/48/(-c*d)
^(1/2)*d*b^2/(a*d-b*c)^3/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1
/2)/d)+(a*d-b*c)/d)^(3/2)-35/48*b^3/(a*d-b*c)^3/a/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-
c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)*x-35/24*b^3/(a*d-b*c)^3/a^2/(
(x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*
x+35/16/(-c*d)^(1/2)*d^2*b^2/(a*d-b*c)^4/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2
)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-1/16/(-c*d)^(1/2)/c/(a*d-b*c)/(x+(-c*d
)^(1/2)/d)^2/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-
b*c)/d)^(3/2)-7/16/c*b/(a*d-b*c)^2/(x+(-c*d)^(1/2)/d)/((x+(-c*d)^(1/2)/d)^2*b-2*
b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)-35/48/(-c*d)^(1/2)*d*b^2/
(a*d-b*c)^3/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b
*c)/d)^(3/2)-35/48*b^3/(a*d-b*c)^3/a/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*
(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)*x-35/24*b^3/(a*d-b*c)^3/a^2/((x+(-c*d)^(1/
2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x-35/16/(-c*d
)^(1/2)*d^2*b^2/(a*d-b*c)^4/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)
^(1/2)/d)+(a*d-b*c)/d)^(1/2)+3/16/(-c*d)^(1/2)/c^2*d^2/(a*d-b*c)^2/((x-(-c*d)^(1
/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-1/16/(-c*d)^
(1/2)/c^2/(a*d-b*c)*d/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)
/d)+(a*d-b*c)/d)^(3/2)-3/16/(-c*d)^(1/2)/c^2*d^2/(a*d-b*c)^2/((x+(-c*d)^(1/2)/d)
^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-5/16/c^2*b*(-c*d)^
(1/2)/(a*d-b*c)^2/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+
(a*d-b*c)/d)^(3/2)+1/16/(-c*d)^(1/2)/c^2/(a*d-b*c)*d/((x-(-c*d)^(1/2)/d)^2*b+2*b
*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)+5/16/c^2*b*(-c*d)^(1/2)/(a
*d-b*c)^2/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c
)/d)^(3/2)+3/16/c^2/(a*d-b*c)/(x+(-c*d)^(1/2)/d)/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c
*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)+3/16/c^2/(a*d-b*c)/(x-(-c*d)^(
1/2)/d)/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/
d)^(3/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}{\left (d x^{2} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^3),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^3), x)

_______________________________________________________________________________________

Fricas [A]  time = 3.90419, 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)^(5/2)*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

[1/96*(4*((16*b^5*c^3*d^2 - 88*a*b^4*c^2*d^3 - 42*a^2*b^3*c*d^4 + 9*a^3*b^2*d^5)
*x^7 + (32*b^5*c^4*d - 152*a*b^4*c^3*d^2 - 144*a^2*b^3*c^2*d^3 - 69*a^3*b^2*c*d^
4 + 18*a^4*b*d^5)*x^5 + (16*b^5*c^5 - 40*a*b^4*c^4*d - 192*a^2*b^3*c^3*d^2 - 96*
a^3*b^2*c^2*d^3 - 12*a^4*b*c*d^4 + 9*a^5*d^5)*x^3 + 3*(8*a*b^4*c^5 - 32*a^2*b^3*
c^4*d - 16*a^4*b*c^2*d^3 + 5*a^5*c*d^4)*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a) +
3*(48*a^4*b^2*c^4*d^2 - 16*a^5*b*c^3*d^3 + 3*a^6*c^2*d^4 + (48*a^2*b^4*c^2*d^4
- 16*a^3*b^3*c*d^5 + 3*a^4*b^2*d^6)*x^8 + 2*(48*a^2*b^4*c^3*d^3 + 32*a^3*b^3*c^2
*d^4 - 13*a^4*b^2*c*d^5 + 3*a^5*b*d^6)*x^6 + (48*a^2*b^4*c^4*d^2 + 176*a^3*b^3*c
^3*d^3 - 13*a^4*b^2*c^2*d^4 - 4*a^5*b*c*d^5 + 3*a^6*d^6)*x^4 + 2*(48*a^3*b^3*c^4
*d^2 + 32*a^4*b^2*c^3*d^3 - 13*a^5*b*c^2*d^4 + 3*a^6*c*d^5)*x^2)*log((((8*b^2*c^
2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2)*sqrt(b*c
^2 - a*c*d) + 4*((2*b^2*c^3 - 3*a*b*c^2*d + a^2*c*d^2)*x^3 + (a*b*c^3 - a^2*c^2*
d)*x)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)))/((a^4*b^4*c^8 - 4*a^5*b^3*c
^7*d + 6*a^6*b^2*c^6*d^2 - 4*a^7*b*c^5*d^3 + a^8*c^4*d^4 + (a^2*b^6*c^6*d^2 - 4*
a^3*b^5*c^5*d^3 + 6*a^4*b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*b^2*c^2*d^6)*x^8 +
2*(a^2*b^6*c^7*d - 3*a^3*b^5*c^6*d^2 + 2*a^4*b^4*c^5*d^3 + 2*a^5*b^3*c^4*d^4 -
3*a^6*b^2*c^3*d^5 + a^7*b*c^2*d^6)*x^6 + (a^2*b^6*c^8 - 9*a^4*b^4*c^6*d^2 + 16*a
^5*b^3*c^5*d^3 - 9*a^6*b^2*c^4*d^4 + a^8*c^2*d^6)*x^4 + 2*(a^3*b^5*c^8 - 3*a^4*b
^4*c^7*d + 2*a^5*b^3*c^6*d^2 + 2*a^6*b^2*c^5*d^3 - 3*a^7*b*c^4*d^4 + a^8*c^3*d^5
)*x^2)*sqrt(b*c^2 - a*c*d)), 1/48*(2*((16*b^5*c^3*d^2 - 88*a*b^4*c^2*d^3 - 42*a^
2*b^3*c*d^4 + 9*a^3*b^2*d^5)*x^7 + (32*b^5*c^4*d - 152*a*b^4*c^3*d^2 - 144*a^2*b
^3*c^2*d^3 - 69*a^3*b^2*c*d^4 + 18*a^4*b*d^5)*x^5 + (16*b^5*c^5 - 40*a*b^4*c^4*d
- 192*a^2*b^3*c^3*d^2 - 96*a^3*b^2*c^2*d^3 - 12*a^4*b*c*d^4 + 9*a^5*d^5)*x^3 +
3*(8*a*b^4*c^5 - 32*a^2*b^3*c^4*d - 16*a^4*b*c^2*d^3 + 5*a^5*c*d^4)*x)*sqrt(-b*c
^2 + a*c*d)*sqrt(b*x^2 + a) + 3*(48*a^4*b^2*c^4*d^2 - 16*a^5*b*c^3*d^3 + 3*a^6*c
^2*d^4 + (48*a^2*b^4*c^2*d^4 - 16*a^3*b^3*c*d^5 + 3*a^4*b^2*d^6)*x^8 + 2*(48*a^2
*b^4*c^3*d^3 + 32*a^3*b^3*c^2*d^4 - 13*a^4*b^2*c*d^5 + 3*a^5*b*d^6)*x^6 + (48*a^
2*b^4*c^4*d^2 + 176*a^3*b^3*c^3*d^3 - 13*a^4*b^2*c^2*d^4 - 4*a^5*b*c*d^5 + 3*a^6
*d^6)*x^4 + 2*(48*a^3*b^3*c^4*d^2 + 32*a^4*b^2*c^3*d^3 - 13*a^5*b*c^2*d^4 + 3*a^
6*c*d^5)*x^2)*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 + a*c)/((b*c^2
- a*c*d)*sqrt(b*x^2 + a)*x)))/((a^4*b^4*c^8 - 4*a^5*b^3*c^7*d + 6*a^6*b^2*c^6*d^
2 - 4*a^7*b*c^5*d^3 + a^8*c^4*d^4 + (a^2*b^6*c^6*d^2 - 4*a^3*b^5*c^5*d^3 + 6*a^4
*b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*b^2*c^2*d^6)*x^8 + 2*(a^2*b^6*c^7*d - 3*a
^3*b^5*c^6*d^2 + 2*a^4*b^4*c^5*d^3 + 2*a^5*b^3*c^4*d^4 - 3*a^6*b^2*c^3*d^5 + a^7
*b*c^2*d^6)*x^6 + (a^2*b^6*c^8 - 9*a^4*b^4*c^6*d^2 + 16*a^5*b^3*c^5*d^3 - 9*a^6*
b^2*c^4*d^4 + a^8*c^2*d^6)*x^4 + 2*(a^3*b^5*c^8 - 3*a^4*b^4*c^7*d + 2*a^5*b^3*c^
6*d^2 + 2*a^6*b^2*c^5*d^3 - 3*a^7*b*c^4*d^4 + a^8*c^3*d^5)*x^2)*sqrt(-b*c^2 + a*
c*d))]

_______________________________________________________________________________________

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/(b*x**2+a)**(5/2)/(d*x**2+c)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 1.16482, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^3),x, algorithm="giac")

[Out]

sage0*x