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

Optimal. Leaf size=142 \[ \frac{b (b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{5/2}}+\frac{d x (2 a d+b c)}{2 a c \sqrt{c+d x^2} (b c-a d)^2}+\frac{b x}{2 a \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)} \]

[Out]

(d*(b*c + 2*a*d)*x)/(2*a*c*(b*c - a*d)^2*Sqrt[c + d*x^2]) + (b*x)/(2*a*(b*c - a*
d)*(a + b*x^2)*Sqrt[c + d*x^2]) + (b*(b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(S
qrt[a]*Sqrt[c + d*x^2])])/(2*a^(3/2)*(b*c - a*d)^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.28875, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{b (b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{5/2}}+\frac{d x (2 a d+b c)}{2 a c \sqrt{c+d x^2} (b c-a d)^2}+\frac{b x}{2 a \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(b*c + 2*a*d)*x)/(2*a*c*(b*c - a*d)^2*Sqrt[c + d*x^2]) + (b*x)/(2*a*(b*c - a*
d)*(a + b*x^2)*Sqrt[c + d*x^2]) + (b*(b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(S
qrt[a]*Sqrt[c + d*x^2])])/(2*a^(3/2)*(b*c - a*d)^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 55.4798, size = 121, normalized size = 0.85 \[ - \frac{b x}{2 a \left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{d x \left (2 a d + b c\right )}{2 a c \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} - \frac{b \left (4 a d - b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{3}{2}} \left (a d - b c\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.372684, size = 120, normalized size = 0.85 \[ \frac{1}{2} \left (\frac{b (b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}}+\frac{x \sqrt{c+d x^2} \left (\frac{b^2}{a^2+a b x^2}+\frac{2 d^2}{c^2+c d x^2}\right )}{(b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 1461, normalized size = 10.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/a/(a*d-b*c)/(x-1/b*(-a*b)^(1/2))/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)
/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+3/4/a*d*(-a*b)^(1/2)/(a*d-b*c)^2/((x-
1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)
+3/4*d^2/(a*d-b*c)^2/c/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b
)^(1/2))-(a*d-b*c)/b)^(1/2)*x-3/4/a*d*(-a*b)^(1/2)/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1
/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^
(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c
)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))-1/4/a/(a*d-b*c)/c/((x-1/b*(-a*b)^(1/2))^2*d+2*
d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d-1/4/a/(a*d-b*c)/(x+
1/b*(-a*b)^(1/2))/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/
2))-(a*d-b*c)/b)^(1/2)-3/4/a*d*(-a*b)^(1/2)/(a*d-b*c)^2/((x+1/b*(-a*b)^(1/2))^2*
d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+3/4*d^2/(a*d-b*c)^2
/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b
)^(1/2)*x+3/4/a*d*(-a*b)^(1/2)/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)
/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)
^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(
-a*b)^(1/2)))-1/4/a/(a*d-b*c)/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+
1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d-1/4/a/(-a*b)^(1/2)/(a*d-b*c)*b/((x-1/b*
(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/4
/a/(-a*b)^(1/2)/(a*d-b*c)*b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(
1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d
*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))+1
/4/a/(-a*b)^(1/2)/(a*d-b*c)*b/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/
b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1/4/a/(-a*b)^(1/2)/(a*d-b*c)*b/(-(a*d-b*c)/b)
^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/
b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-
b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.668301, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{3} +{\left (b^{2} c^{2} + 2 \, a^{2} d^{2}\right )} x\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} -{\left (a b^{2} c^{3} - 4 \, a^{2} b c^{2} d +{\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{4} +{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{2}\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} - 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \,{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} +{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{4} +{\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d}}, \frac{2 \,{\left ({\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{3} +{\left (b^{2} c^{2} + 2 \, a^{2} d^{2}\right )} x\right )} \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} +{\left (a b^{2} c^{3} - 4 \, a^{2} b c^{2} d +{\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{4} +{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{4 \,{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} +{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{4} +{\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{2}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(4*((b^2*c*d + 2*a*b*d^2)*x^3 + (b^2*c^2 + 2*a^2*d^2)*x)*sqrt(-a*b*c + a^2*
d)*sqrt(d*x^2 + c) - (a*b^2*c^3 - 4*a^2*b*c^2*d + (b^3*c^2*d - 4*a*b^2*c*d^2)*x^
4 + (b^3*c^3 - 3*a*b^2*c^2*d - 4*a^2*b*c*d^2)*x^2)*log((((b^2*c^2 - 8*a*b*c*d +
8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2)*sqrt(-a*b*c + a^2*d) -
4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^3 - (a^2*b*c^2 - a^3*c*d)*x)*sqrt(d*
x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*
d^2 + (a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x^4 + (a*b^3*c^4 - a^2*b^2
*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x^2)*sqrt(-a*b*c + a^2*d)), 1/4*(2*((b^2*c*d
+ 2*a*b*d^2)*x^3 + (b^2*c^2 + 2*a^2*d^2)*x)*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)
+ (a*b^2*c^3 - 4*a^2*b*c^2*d + (b^3*c^2*d - 4*a*b^2*c*d^2)*x^4 + (b^3*c^3 - 3*a
*b^2*c^2*d - 4*a^2*b*c*d^2)*x^2)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(a*b*
c - a^2*d)*sqrt(d*x^2 + c)*x)))/((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (a
*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x^4 + (a*b^3*c^4 - a^2*b^2*c^3*d -
a^3*b*c^2*d^2 + a^4*c*d^3)*x^2)*sqrt(a*b*c - a^2*d))]

_______________________________________________________________________________________

Sympy [F(-2)]  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)**(3/2),x)

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 3.29811, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x