3.782 \(\int \frac{1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=279 \[ -\frac{b^3 (3 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{7/2}}+\frac{d \left (-8 a^2 d^2+20 a b c d+3 b^2 c^2\right )}{6 a c^2 x \sqrt{c+d x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} \left (-16 a^3 d^3+40 a^2 b c d^2-18 a b^2 c^2 d+9 b^3 c^3\right )}{6 a^2 c^3 x (b c-a d)^3}+\frac{b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c x \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]
[Out]
(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*x*(c + d*x^2)^(3/2)) + b/(2*a*(b*c - a*
d)*x*(a + b*x^2)*(c + d*x^2)^(3/2)) + (d*(3*b^2*c^2 + 20*a*b*c*d - 8*a^2*d^2))/(
6*a*c^2*(b*c - a*d)^3*x*Sqrt[c + d*x^2]) - ((9*b^3*c^3 - 18*a*b^2*c^2*d + 40*a^2
*b*c*d^2 - 16*a^3*d^3)*Sqrt[c + d*x^2])/(6*a^2*c^3*(b*c - a*d)^3*x) - (b^3*(3*b*
c - 8*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*
c - a*d)^(7/2))
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Rubi [A] time = 1.26694, antiderivative size = 279, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25
\[ -\frac{b^3 (3 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{7/2}}+\frac{d \left (-8 a^2 d^2+20 a b c d+3 b^2 c^2\right )}{6 a c^2 x \sqrt{c+d x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} \left (-16 a^3 d^3+40 a^2 b c d^2-18 a b^2 c^2 d+9 b^3 c^3\right )}{6 a^2 c^3 x (b c-a d)^3}+\frac{b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c x \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]
[Out]
(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*x*(c + d*x^2)^(3/2)) + b/(2*a*(b*c - a*
d)*x*(a + b*x^2)*(c + d*x^2)^(3/2)) + (d*(3*b^2*c^2 + 20*a*b*c*d - 8*a^2*d^2))/(
6*a*c^2*(b*c - a*d)^3*x*Sqrt[c + d*x^2]) - ((9*b^3*c^3 - 18*a*b^2*c^2*d + 40*a^2
*b*c*d^2 - 16*a^3*d^3)*Sqrt[c + d*x^2])/(6*a^2*c^3*(b*c - a*d)^3*x) - (b^3*(3*b*
c - 8*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*
c - a*d)^(7/2))
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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**2/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)
[Out]
Timed out
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Mathematica [A] time = 0.910351, size = 188, normalized size = 0.67 \[ \sqrt{c+d x^2} \left (\frac{b^4 x}{2 a^2 \left (a+b x^2\right ) (a d-b c)^3}-\frac{1}{a^2 c^3 x}+\frac{d^3 x (5 a d-11 b c)}{3 c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3 x}{3 c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right )-\frac{b^3 (3 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{5/2} (b c-a d)^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]
[Out]
Sqrt[c + d*x^2]*(-(1/(a^2*c^3*x)) + (b^4*x)/(2*a^2*(-(b*c) + a*d)^3*(a + b*x^2))
- (d^3*x)/(3*c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (d^3*(-11*b*c + 5*a*d)*x)/(3*c^
3*(b*c - a*d)^3*(c + d*x^2))) - (b^3*(3*b*c - 8*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/
(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(7/2))
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Maple [B] time = 0.029, size = 2513, normalized size = 9. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^2+a)^2/(d*x^2+c)^(5/2),x)
[Out]
-1/a^2/c/x/(d*x^2+c)^(3/2)-3/4*b^3/a^2/(-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)-1/4*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)^(3/2)-5/12/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)^(3/2)*x+1/6/a^2*d/(a*d-b*
c)*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)*x+3/4*b^2/a^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*d+3/4*b^2/a^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*d-5/6/a*d^2*b/(a*d-b*c)^2/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)*x+5/4/a*d^2*b^2/(a*d-b*c)^3/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-5/4/
a^2*d*(-a*b)^(1/2)/(a*d-b*c)^3*b^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)))+1/12/a^2*d/(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)^(3/2)*x-5/12/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)^(3/2)*x-5/6/a*d^
2*b/(a*d-b*c)^2/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)*x+5/4/a*d^2*b^2/(a*d-b*c)^3/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+5/4/a^2*d*(-a*b)^(
1/2)/(a*d-b*c)^3*b^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)))+1/12/a^2
*d/(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)^(3/2)*x+1/6/a^2*d/(a*d-b*c)*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)*x-5/12/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)^(3/2)+5/4/a^2*d*(-a*b)^(1/2)/(a*d-b*c)^3*b^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)+5/12/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)^(3/2)-5/4/a^2*d*(-a*b)^(1/2)/(a*d-b*c)^3*b^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*b^3/a
^2/(-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)))-3
/4*b^3/a^2/(-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)))+3/4*b^3/a^2/(-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)+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)^(3/2)+1/4*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)^(3/2)+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)^(3/2)-4/3/a^2*d/c^2*x/(d*x^2+c)^(3/2)-8/3/a^2*d/c^3*x/(d*x^2+c)^(
1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{5}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^2),x, algorithm="maxima")
[Out]
integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^2), x)
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Fricas [A] time = 1.76424, 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)^(5/2)*x^2),x, algorithm="fricas")
[Out]
[-1/24*(4*(6*a*b^3*c^5 - 18*a^2*b^2*c^4*d + 18*a^3*b*c^3*d^2 - 6*a^4*c^2*d^3 + (
9*b^4*c^3*d^2 - 18*a*b^3*c^2*d^3 + 40*a^2*b^2*c*d^4 - 16*a^3*b*d^5)*x^6 + 2*(9*b
^4*c^4*d - 15*a*b^3*c^3*d^2 + 21*a^2*b^2*c^2*d^3 + 8*a^3*b*c*d^4 - 8*a^4*d^5)*x^
4 + 3*(3*b^4*c^5 - 2*a*b^3*c^4*d - 6*a^2*b^2*c^3*d^2 + 18*a^3*b*c^2*d^3 - 8*a^4*
c*d^4)*x^2)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c) - 3*((3*b^5*c^4*d^2 - 8*a*b^4*c
^3*d^3)*x^7 + (6*b^5*c^5*d - 13*a*b^4*c^4*d^2 - 8*a^2*b^3*c^3*d^3)*x^5 + (3*b^5*
c^6 - 2*a*b^4*c^5*d - 16*a^2*b^3*c^4*d^2)*x^3 + (3*a*b^4*c^6 - 8*a^2*b^3*c^5*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
^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^7 + (2*a^2
*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5
)*x^5 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a
^6*c^4*d^4)*x^3 + (a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3
)*x)*sqrt(-a*b*c + a^2*d)), -1/12*(2*(6*a*b^3*c^5 - 18*a^2*b^2*c^4*d + 18*a^3*b*
c^3*d^2 - 6*a^4*c^2*d^3 + (9*b^4*c^3*d^2 - 18*a*b^3*c^2*d^3 + 40*a^2*b^2*c*d^4 -
16*a^3*b*d^5)*x^6 + 2*(9*b^4*c^4*d - 15*a*b^3*c^3*d^2 + 21*a^2*b^2*c^2*d^3 + 8*
a^3*b*c*d^4 - 8*a^4*d^5)*x^4 + 3*(3*b^4*c^5 - 2*a*b^3*c^4*d - 6*a^2*b^2*c^3*d^2
+ 18*a^3*b*c^2*d^3 - 8*a^4*c*d^4)*x^2)*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c) + 3*(
(3*b^5*c^4*d^2 - 8*a*b^4*c^3*d^3)*x^7 + (6*b^5*c^5*d - 13*a*b^4*c^4*d^2 - 8*a^2*
b^3*c^3*d^3)*x^5 + (3*b^5*c^6 - 2*a*b^4*c^5*d - 16*a^2*b^3*c^4*d^2)*x^3 + (3*a*b
^4*c^6 - 8*a^2*b^3*c^5*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^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c
^4*d^4 - a^5*b*c^3*d^5)*x^7 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c
^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)*x^5 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4
*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^3 + (a^3*b^3*c^8 - 3*a^4*b^2*c
^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3)*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)**(5/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 7.08523, 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)^2*(d*x^2 + c)^(5/2)*x^2),x, algorithm="giac")
[Out]
sage0*x