3.775 \(\int \frac{1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=277 \[ \frac{b^3 (5 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x^2} \left (8 a^2 d^2-4 a b c d+5 b^2 c^2\right )}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac{\sqrt{c+d x^2} \left (16 a^3 d^3-8 a^2 b c d^2-14 a b^2 c^2 d+15 b^3 c^3\right )}{6 a^3 c^3 x (b c-a d)^2}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c x^3 \sqrt{c+d x^2} (b c-a d)^2} \]
[Out]
(d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*x^3*Sqrt[c + d*x^2]) + b/(2*a*(b*c - a*d)
*x^3*(a + b*x^2)*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*Sqrt[c
+ d*x^2])/(6*a^2*c^2*(b*c - a*d)^2*x^3) + ((15*b^3*c^3 - 14*a*b^2*c^2*d - 8*a^2*
b*c*d^2 + 16*a^3*d^3)*Sqrt[c + d*x^2])/(6*a^3*c^3*(b*c - a*d)^2*x) + (b^3*(5*b*c
- 8*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(b*c
- a*d)^(5/2))
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Rubi [A] time = 1.16252, antiderivative size = 277, 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
\[ \frac{b^3 (5 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x^2} \left (8 a^2 d^2-4 a b c d+5 b^2 c^2\right )}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac{\sqrt{c+d x^2} \left (16 a^3 d^3-8 a^2 b c d^2-14 a b^2 c^2 d+15 b^3 c^3\right )}{6 a^3 c^3 x (b c-a d)^2}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c x^3 \sqrt{c+d x^2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(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^3*Sqrt[c + d*x^2]) + b/(2*a*(b*c - a*d)
*x^3*(a + b*x^2)*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*Sqrt[c
+ d*x^2])/(6*a^2*c^2*(b*c - a*d)^2*x^3) + ((15*b^3*c^3 - 14*a*b^2*c^2*d - 8*a^2*
b*c*d^2 + 16*a^3*d^3)*Sqrt[c + d*x^2])/(6*a^3*c^3*(b*c - a*d)^2*x) + (b^3*(5*b*c
- 8*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(b*c
- a*d)^(5/2))
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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**4/(b*x**2+a)**2/(d*x**2+c)**(3/2),x)
[Out]
Timed out
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Mathematica [A] time = 0.745059, size = 167, normalized size = 0.6 \[ \frac{b^3 (5 b c-8 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{5/2}}+\sqrt{c+d x^2} \left (\frac{\frac{b^4 x}{2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{2 b}{c^2 x}}{a^3}-\frac{c-5 d x^2}{3 a^2 c^3 x^3}+\frac{d^4 x}{c^3 \left (c+d x^2\right ) (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]
[Out]
Sqrt[c + d*x^2]*(-(c - 5*d*x^2)/(3*a^2*c^3*x^3) + (d^4*x)/(c^3*(b*c - a*d)^2*(c
+ d*x^2)) + ((2*b)/(c^2*x) + (b^4*x)/(2*(b*c - a*d)^2*(a + b*x^2)))/a^3) + (b^3*
(5*b*c - 8*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2
)*(b*c - a*d)^(5/2))
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Maple [B] time = 0.028, size = 1608, normalized size = 5.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(b*x^2+a)^2/(d*x^2+c)^(3/2),x)
[Out]
-1/3/a^2/c/x^3/(d*x^2+c)^(1/2)+4/3/a^2*d/c^2/x/(d*x^2+c)^(1/2)+8/3/a^2*d^2/c^3*x
/(d*x^2+c)^(1/2)+2*b/a^3/c/x/(d*x^2+c)^(1/2)+4*b/a^3*d/c^2*x/(d*x^2+c)^(1/2)-1/4
*b^2/a^3/(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*b^2/a^3*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*b^2/a^2*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*b^2/a^3*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)))+3/4*b^2/a^3/(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*b^2/a^3/(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*b^2/a^3*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*b^2/a^2*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*b^2/a^3*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)))+3/4*b^2/a^3/
(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-5/4*b^3/a^3/(-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)+5/4*b^3/a^3/(-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)))+5/4*b^3/a^3/(
-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)-5/4*b^3/a^3/(-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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]
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^4),x, algorithm="maxima")
[Out]
integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(3/2)*x^4), x)
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Fricas [A] time = 1.25841, 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^4),x, algorithm="fricas")
[Out]
[-1/24*(4*(2*a^2*b^2*c^4 - 4*a^3*b*c^3*d + 2*a^4*c^2*d^2 - (15*b^4*c^3*d - 14*a*
b^3*c^2*d^2 - 8*a^2*b^2*c*d^3 + 16*a^3*b*d^4)*x^6 - (15*b^4*c^4 - 4*a*b^3*c^3*d
- 18*a^2*b^2*c^2*d^2 + 16*a^4*d^4)*x^4 - 2*(5*a*b^3*c^4 - 6*a^2*b^2*c^3*d - 3*a^
3*b*c^2*d^2 + 4*a^4*c*d^3)*x^2)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c) + 3*((5*b^5
*c^4*d - 8*a*b^4*c^3*d^2)*x^7 + (5*b^5*c^5 - 3*a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2)*
x^5 + (5*a*b^4*c^5 - 8*a^2*b^3*c^4*d)*x^3)*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^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3
*d^3)*x^7 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^5 + (a
^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x^3)*sqrt(-a*b*c + a^2*d)), -1/12*(2*(
2*a^2*b^2*c^4 - 4*a^3*b*c^3*d + 2*a^4*c^2*d^2 - (15*b^4*c^3*d - 14*a*b^3*c^2*d^2
- 8*a^2*b^2*c*d^3 + 16*a^3*b*d^4)*x^6 - (15*b^4*c^4 - 4*a*b^3*c^3*d - 18*a^2*b^
2*c^2*d^2 + 16*a^4*d^4)*x^4 - 2*(5*a*b^3*c^4 - 6*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2
+ 4*a^4*c*d^3)*x^2)*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c) - 3*((5*b^5*c^4*d - 8*a
*b^4*c^3*d^2)*x^7 + (5*b^5*c^5 - 3*a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2)*x^5 + (5*a*b
^4*c^5 - 8*a^2*b^3*c^4*d)*x^3)*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^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d
^3)*x^7 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^5 + (a^4
*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x^3)*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**4/(b*x**2+a)**2/(d*x**2+c)**(3/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 9.97954, 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)^(3/2)*x^4),x, algorithm="giac")
[Out]
sage0*x