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

Optimal. Leaf size=225 \[ -\frac{3 d \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{7/2}}+\frac{d x \sqrt{a+b x^2} (4 b c-a d) (3 a d+2 b c)}{8 a c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{b x (a d+4 b c)}{4 a c \sqrt{a+b x^2} \left (c+d x^2\right ) (b c-a d)^2}-\frac{d x}{4 c \sqrt{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(d*x)/(4*c*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^2) + (b*(4*b*c + a*d)*x)/(4*
a*c*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)) + (d*(4*b*c - a*d)*(2*b*c + 3*a*d
)*x*Sqrt[a + b*x^2])/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*d*(8*b^2*c^2 - 4*a
*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(
5/2)*(b*c - a*d)^(7/2))

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Rubi [A]  time = 0.678708, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{3 d \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{7/2}}+\frac{d x \sqrt{a+b x^2} (4 b c-a d) (3 a d+2 b c)}{8 a c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{b x (a d+4 b c)}{4 a c \sqrt{a+b x^2} \left (c+d x^2\right ) (b c-a d)^2}-\frac{d x}{4 c \sqrt{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(d*x)/(4*c*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^2) + (b*(4*b*c + a*d)*x)/(4*
a*c*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)) + (d*(4*b*c - a*d)*(2*b*c + 3*a*d
)*x*Sqrt[a + b*x^2])/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*d*(8*b^2*c^2 - 4*a
*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(
5/2)*(b*c - a*d)^(7/2))

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Rubi in Sympy [A]  time = 115.598, size = 196, normalized size = 0.87 \[ \frac{d x}{4 c \sqrt{a + b x^{2}} \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} + \frac{d x \left (3 a d - 8 b c\right )}{8 c^{2} \sqrt{a + b x^{2}} \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} + \frac{3 d \left (a^{2} d^{2} - 4 a b c d + 8 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{8 c^{\frac{5}{2}} \left (a d - b c\right )^{\frac{7}{2}}} + \frac{b x \left (a d - 4 b c\right ) \left (3 a d + 2 b c\right )}{8 a c^{2} \sqrt{a + b x^{2}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/(4*c*sqrt(a + b*x**2)*(c + d*x**2)**2*(a*d - b*c)) + d*x*(3*a*d - 8*b*c)/(8*
c**2*sqrt(a + b*x**2)*(c + d*x**2)*(a*d - b*c)**2) + 3*d*(a**2*d**2 - 4*a*b*c*d
+ 8*b**2*c**2)*atan(x*sqrt(a*d - b*c)/(sqrt(c)*sqrt(a + b*x**2)))/(8*c**(5/2)*(a
*d - b*c)**(7/2)) + b*x*(a*d - 4*b*c)*(3*a*d + 2*b*c)/(8*a*c**2*sqrt(a + b*x**2)
*(a*d - b*c)**3)

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Mathematica [A]  time = 0.794328, size = 181, normalized size = 0.8 \[ \frac{1}{8} \left (\frac{3 d \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{c^{5/2} (a d-b c)^{7/2}}+x \sqrt{a+b x^2} \left (-\frac{8 b^3}{a \left (a+b x^2\right ) (a d-b c)^3}+\frac{d^2 (10 b c-3 a d)}{c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{2 d^2}{c \left (c+d x^2\right )^2 (b c-a d)^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^2)^(3/2)*(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*d^2)/(c*(b*c
- a*d)^2*(c + d*x^2)^2) + (d^2*(10*b*c - 3*a*d))/(c^2*(b*c - a*d)^3*(c + d*x^2))
) + (3*d*(8*b^2*c^2 - 4*a*b*c*d + 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)^(7/2)))/8

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Maple [B]  time = 0.04, size = 2919, normalized size = 13. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/16/(-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)^(1/2)-15/16*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)^(1/2)*x+9/16/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)^(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)^(1/2)-5/16/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)^(1/2)-15/16/(-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)^(1/2)-15/16*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)^(1/2)*x-3/16/(-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)^(1/2)-9/16/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)^(1/2)+3/16/(-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)^(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)^(1/2)+15/16/(-c*d)^(1/2)*d*b^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))-3/16/(-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)^(1/2)-15/16/(-c*d)^(1/2)*d*b^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))+3/16/(-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)
^(1/2)-1/4/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)^(1/2)*x-9/16/c^2*b*(-c*d)^(1/2)/(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/(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)^(1/2)*x*b-1/4/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)^(1/2)
*x+9/16/c^2*b*(-c*d)^(1/2)/(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/(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)^(1/2)*x*b-3/16/(-c*d)^(1/2)/c^2/(a*d-b*c)*d/((a*d-b*c)/d)^(1/2)*l
n((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*d)^(1/2)/c^2/(a*d-b*c)*d/((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*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)^(1/2)-3/16/(-c*d)^(1/2)/c*d*b/(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/(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)^(1/2)+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)^(1/2)+3/16/(-c*d)^(1/2)/c*d*b/(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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.30652, 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)^(3/2)*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

[1/32*(4*((8*b^3*c^2*d^2 + 10*a*b^2*c*d^3 - 3*a^2*b*d^4)*x^5 + (16*b^3*c^3*d + 1
2*a*b^2*c^2*d^2 + 5*a^2*b*c*d^3 - 3*a^3*d^4)*x^3 + (8*b^3*c^4 + 12*a^2*b*c^2*d^2
- 5*a^3*c*d^3)*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a) - 3*(8*a^2*b^2*c^4*d - 4*
a^3*b*c^3*d^2 + a^4*c^2*d^3 + (8*a*b^3*c^2*d^3 - 4*a^2*b^2*c*d^4 + a^3*b*d^5)*x^
6 + (16*a*b^3*c^3*d^2 - 2*a^3*b*c*d^4 + a^4*d^5)*x^4 + (8*a*b^3*c^4*d + 12*a^2*b
^2*c^3*d^2 - 7*a^3*b*c^2*d^3 + 2*a^4*c*d^4)*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^2*b^3*c^7 - 3*a^3*b^2*c^6*d + 3*a^4*b*c
^5*d^2 - a^5*c^4*d^3 + (a*b^4*c^5*d^2 - 3*a^2*b^3*c^4*d^3 + 3*a^3*b^2*c^3*d^4 -
a^4*b*c^2*d^5)*x^6 + (2*a*b^4*c^6*d - 5*a^2*b^3*c^5*d^2 + 3*a^3*b^2*c^4*d^3 + a^
4*b*c^3*d^4 - a^5*c^2*d^5)*x^4 + (a*b^4*c^7 - a^2*b^3*c^6*d - 3*a^3*b^2*c^5*d^2
+ 5*a^4*b*c^4*d^3 - 2*a^5*c^3*d^4)*x^2)*sqrt(b*c^2 - a*c*d)), 1/16*(2*((8*b^3*c^
2*d^2 + 10*a*b^2*c*d^3 - 3*a^2*b*d^4)*x^5 + (16*b^3*c^3*d + 12*a*b^2*c^2*d^2 + 5
*a^2*b*c*d^3 - 3*a^3*d^4)*x^3 + (8*b^3*c^4 + 12*a^2*b*c^2*d^2 - 5*a^3*c*d^3)*x)*
sqrt(-b*c^2 + a*c*d)*sqrt(b*x^2 + a) - 3*(8*a^2*b^2*c^4*d - 4*a^3*b*c^3*d^2 + a^
4*c^2*d^3 + (8*a*b^3*c^2*d^3 - 4*a^2*b^2*c*d^4 + a^3*b*d^5)*x^6 + (16*a*b^3*c^3*
d^2 - 2*a^3*b*c*d^4 + a^4*d^5)*x^4 + (8*a*b^3*c^4*d + 12*a^2*b^2*c^3*d^2 - 7*a^3
*b*c^2*d^3 + 2*a^4*c*d^4)*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^2*b^3*c^7 - 3*a^3*b^2*c^6*d +
3*a^4*b*c^5*d^2 - a^5*c^4*d^3 + (a*b^4*c^5*d^2 - 3*a^2*b^3*c^4*d^3 + 3*a^3*b^2*
c^3*d^4 - a^4*b*c^2*d^5)*x^6 + (2*a*b^4*c^6*d - 5*a^2*b^3*c^5*d^2 + 3*a^3*b^2*c^
4*d^3 + a^4*b*c^3*d^4 - a^5*c^2*d^5)*x^4 + (a*b^4*c^7 - a^2*b^3*c^6*d - 3*a^3*b^
2*c^5*d^2 + 5*a^4*b*c^4*d^3 - 2*a^5*c^3*d^4)*x^2)*sqrt(-b*c^2 + a*c*d))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 12.073, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x