3.773 \(\int \frac{1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=205 \[ -\frac{3 b^2 (b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x^2} \left (4 a^2 d^2-4 a b c d+3 b^2 c^2\right )}{2 a^2 c^2 x (b c-a d)^2}+\frac{b}{2 a x \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c x \sqrt{c+d x^2} (b c-a d)^2} \]
[Out]
(d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*x*Sqrt[c + d*x^2]) + b/(2*a*(b*c - a*d)*x
*(a + b*x^2)*Sqrt[c + d*x^2]) - ((3*b^2*c^2 - 4*a*b*c*d + 4*a^2*d^2)*Sqrt[c + d*
x^2])/(2*a^2*c^2*(b*c - a*d)^2*x) - (3*b^2*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]
*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(5/2))
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Rubi [A] time = 0.776525, antiderivative size = 205, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25
\[ -\frac{3 b^2 (b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x^2} \left (4 a^2 d^2-4 a b c d+3 b^2 c^2\right )}{2 a^2 c^2 x (b c-a d)^2}+\frac{b}{2 a x \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c x \sqrt{c+d x^2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]
[Out]
(d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*x*Sqrt[c + d*x^2]) + b/(2*a*(b*c - a*d)*x
*(a + b*x^2)*Sqrt[c + d*x^2]) - ((3*b^2*c^2 - 4*a*b*c*d + 4*a^2*d^2)*Sqrt[c + d*
x^2])/(2*a^2*c^2*(b*c - a*d)^2*x) - (3*b^2*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]
*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(5/2))
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Rubi in Sympy [A] time = 135.396, size = 178, normalized size = 0.87 \[ - \frac{b}{2 a x \left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{d \left (2 a d + b c\right )}{2 a c x \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} - \frac{\sqrt{c + d x^{2}} \left (4 a^{2} d^{2} - 4 a b c d + 3 b^{2} c^{2}\right )}{2 a^{2} c^{2} x \left (a d - b c\right )^{2}} + \frac{3 b^{2} \left (2 a d - b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{5}{2}} \left (a d - b c\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**2+a)**2/(d*x**2+c)**(3/2),x)
[Out]
-b/(2*a*x*(a + b*x**2)*sqrt(c + d*x**2)*(a*d - b*c)) + d*(2*a*d + b*c)/(2*a*c*x*
sqrt(c + d*x**2)*(a*d - b*c)**2) - sqrt(c + d*x**2)*(4*a**2*d**2 - 4*a*b*c*d + 3
*b**2*c**2)/(2*a**2*c**2*x*(a*d - b*c)**2) + 3*b**2*(2*a*d - b*c)*atanh(x*sqrt(a
*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/(2*a**(5/2)*(a*d - b*c)**(5/2))
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Mathematica [A] time = 0.611781, size = 145, normalized size = 0.71 \[ \sqrt{c+d x^2} \left (-\frac{\frac{b^3 x}{2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{1}{c^2 x}}{a^2}-\frac{d^3 x}{c^2 \left (c+d x^2\right ) (b c-a d)^2}\right )-\frac{3 b^2 (b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]
[Out]
Sqrt[c + d*x^2]*(-((d^3*x)/(c^2*(b*c - a*d)^2*(c + d*x^2))) - (1/(c^2*x) + (b^3*
x)/(2*(b*c - a*d)^2*(a + b*x^2)))/a^2) - (3*b^2*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c -
a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(5/2))
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Maple [B] time = 0.026, size = 1524, normalized size = 7.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^2+a)^2/(d*x^2+c)^(3/2),x)
[Out]
-1/a^2/c/x/(d*x^2+c)^(1/2)-2/a^2*d/c^2*x/(d*x^2+c)^(1/2)+1/4/a^2/(a*d-b*c)*b/(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^2*d*(-a*b)^(1/2)/(a*d-b*c)^2*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)-3/4/a*d^2*b/(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^2*d*(-a*b)^(1/2)/(a*d-b*c)^2*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^2/(a*d-b*c)*b/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^2/(a*d-b*c)*b/(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^2*d*(-a*b)^(1/2)/(a*d-b*c)^2*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)-3/4/a*d^2*b/(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^2*d*(-a*b)^(1/2)/(a*d-b*c)^2*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^2/(a*d-b*c)*b/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+3/4*b^2/a^2/(-a*b)^(1/2
)/(a*d-b*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)-3/4*b^2/a^2/(-a*b)^(1/2)/(a*d-b*c)/(-(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)))-3/4*b^2/a^2/(-a*b)^(1/2)/(a*d-b*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)+3/4*b^2/a^2/(-a*b
)^(1/2)/(a*d-b*c)/(-(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)))
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]
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^2),x, algorithm="maxima")
[Out]
integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(3/2)*x^2), x)
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Fricas [A] time = 0.771603, 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)^(3/2)*x^2),x, algorithm="fricas")
[Out]
[-1/8*(4*(2*a*b^2*c^3 - 4*a^2*b*c^2*d + 2*a^3*c*d^2 + (3*b^3*c^2*d - 4*a*b^2*c*d
^2 + 4*a^2*b*d^3)*x^4 + (3*b^3*c^3 - 2*a*b^2*c^2*d - 2*a^2*b*c*d^2 + 4*a^3*d^3)*
x^2)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c) + 3*((b^4*c^3*d - 2*a*b^3*c^2*d^2)*x^5
+ (b^4*c^4 - a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2)*x^3 + (a*b^3*c^4 - 2*a^2*b^2*c^3*
d)*x)*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^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^5 + (a^2*b^3*c^5 - a^3*b^2*c^
4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^3 + (a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*
d^2)*x)*sqrt(-a*b*c + a^2*d)), -1/4*(2*(2*a*b^2*c^3 - 4*a^2*b*c^2*d + 2*a^3*c*d^
2 + (3*b^3*c^2*d - 4*a*b^2*c*d^2 + 4*a^2*b*d^3)*x^4 + (3*b^3*c^3 - 2*a*b^2*c^2*d
- 2*a^2*b*c*d^2 + 4*a^3*d^3)*x^2)*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c) + 3*((b^4
*c^3*d - 2*a*b^3*c^2*d^2)*x^5 + (b^4*c^4 - a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2)*x^3
+ (a*b^3*c^4 - 2*a^2*b^2*c^3*d)*x)*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^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c
^2*d^3)*x^5 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^3 +
(a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*x)*sqrt(a*b*c - a^2*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/x**2/(b*x**2+a)**2/(d*x**2+c)**(3/2),x)
[Out]
Timed out
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
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^2),x, algorithm="giac")
[Out]
Exception raised: TypeError