3.784 \(\int \frac{1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=362 \[ \frac{5 b^4 (b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{7/2}}+\frac{d \left (-4 a^2 d^2+8 a b c d+b^2 c^2\right )}{2 a c^2 x^3 \sqrt{c+d x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} \left (-16 a^3 d^3+32 a^2 b c d^2-6 a b^2 c^2 d+5 b^3 c^3\right )}{6 a^2 c^3 x^3 (b c-a d)^3}+\frac{\sqrt{c+d x^2} \left (-32 a^4 d^4+64 a^3 b c d^3-12 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+15 b^4 c^4\right )}{6 a^3 c^4 x (b c-a d)^3}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c x^3 \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*x^3*(c + d*x^2)^(3/2)) + b/(2*a*(b*c -
a*d)*x^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + (d*(b^2*c^2 + 8*a*b*c*d - 4*a^2*d^2))/
(2*a*c^2*(b*c - a*d)^3*x^3*Sqrt[c + d*x^2]) - ((5*b^3*c^3 - 6*a*b^2*c^2*d + 32*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^3) + ((15*b
^4*c^4 - 20*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 64*a^3*b*c*d^3 - 32*a^4*d^4)*Sqrt
[c + d*x^2])/(6*a^3*c^4*(b*c - a*d)^3*x) + (5*b^4*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c
- a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(b*c - a*d)^(7/2))

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Rubi [A]  time = 1.66449, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{5 b^4 (b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{7/2}}+\frac{d \left (-4 a^2 d^2+8 a b c d+b^2 c^2\right )}{2 a c^2 x^3 \sqrt{c+d x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} \left (-16 a^3 d^3+32 a^2 b c d^2-6 a b^2 c^2 d+5 b^3 c^3\right )}{6 a^2 c^3 x^3 (b c-a d)^3}+\frac{\sqrt{c+d x^2} \left (-32 a^4 d^4+64 a^3 b c d^3-12 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+15 b^4 c^4\right )}{6 a^3 c^4 x (b c-a d)^3}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c x^3 \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(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^3*(c + d*x^2)^(3/2)) + b/(2*a*(b*c -
a*d)*x^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + (d*(b^2*c^2 + 8*a*b*c*d - 4*a^2*d^2))/
(2*a*c^2*(b*c - a*d)^3*x^3*Sqrt[c + d*x^2]) - ((5*b^3*c^3 - 6*a*b^2*c^2*d + 32*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^3) + ((15*b
^4*c^4 - 20*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 64*a^3*b*c*d^3 - 32*a^4*d^4)*Sqrt
[c + d*x^2])/(6*a^3*c^4*(b*c - a*d)^3*x) + (5*b^4*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c
- a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(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**4/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 1.13437, size = 210, normalized size = 0.58 \[ \frac{5 b^4 (b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{7/2}}+\frac{\sqrt{c+d x^2} \left (-\frac{3 b^5 x^4}{a^3 \left (a+b x^2\right ) (a d-b c)^3}+\frac{4 x^2 (4 a d+3 b c)}{a^3 c^4}-\frac{2}{a^2 c^3}+\frac{4 d^4 x^4 (7 b c-4 a d)}{c^4 \left (c+d x^2\right ) (b c-a d)^3}+\frac{2 d^4 x^4}{c^3 \left (c+d x^2\right )^2 (b c-a d)^2}\right )}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]

[Out]

(Sqrt[c + d*x^2]*(-2/(a^2*c^3) + (4*(3*b*c + 4*a*d)*x^2)/(a^3*c^4) - (3*b^5*x^4)
/(a^3*(-(b*c) + a*d)^3*(a + b*x^2)) + (2*d^4*x^4)/(c^3*(b*c - a*d)^2*(c + d*x^2)
^2) + (4*d^4*(7*b*c - 4*a*d)*x^4)/(c^4*(b*c - a*d)^3*(c + d*x^2))))/(6*x^3) + (5
*b^4*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(
7/2)*(b*c - a*d)^(7/2))

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Maple [B]  time = 0.032, size = 2623, normalized size = 7.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^2+a)^2/(d*x^2+c)^(5/2),x)

[Out]

2/a^2*d/c^2/x/(d*x^2+c)^(3/2)+8/3/a^2*d^2/c^3*x/(d*x^2+c)^(3/2)+16/3/a^2*d^2/c^4
*x/(d*x^2+c)^(1/2)+2*b/a^3/c/x/(d*x^2+c)^(3/2)-5/12*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)^(3/2)+5/4*b^3/a^3*d*(-a*b)^(1/2)/(a*d-b*c)^3/((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)+8/3*b/a^3*d/c^2*x/(d*x^2
+c)^(3/2)+16/3*b/a^3*d/c^3*x/(d*x^2+c)^(1/2)-5/4*b^3/a^3*d*(-a*b)^(1/2)/(a*d-b*c
)^3/((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^4/a^3/(-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)))-5/4*b^4/a^3/(-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)))+5/12*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)^(3/2)-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)^(3/2)-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)^
(3/2)+1/6*b^2/a^3*d/(a*d-b*c)/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*b^3/a^3/(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-5/4*
b^3/a^3/(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+5/12*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)^(3/2)*x+5/6*b^2/a^2
*d^2/(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*b^3/a^2*d^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*b^3/a^3*d*(
-a*b)^(1/2)/(a*d-b*c)^3/(-(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*
b^2/a^3*d/(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)^(3/2)*x+5/12*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)^(3/2)-5/4*b^4/a^
3/(-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)-5/12*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)^(3/2)+5/4*b^4
/a^3/(-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)+5/12*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)^(3/2)*x+5/6*b^2
/a^2*d^2/(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*b^3/a^2*d^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*b^3/a^3
*d*(-a*b)^(1/2)/(a*d-b*c)^3/(-(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
/3/a^2/c/x^3/(d*x^2+c)^(3/2)+1/12*b^2/a^3*d/(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)^(3/2)*x+1/6*b^2/a^3*d/(a*
d-b*c)/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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{5}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^4), x)

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Fricas [A]  time = 3.27575, 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^4),x, algorithm="fricas")

[Out]

[-1/24*(4*(2*a^2*b^3*c^6 - 6*a^3*b^2*c^5*d + 6*a^4*b*c^4*d^2 - 2*a^5*c^3*d^3 - (
15*b^5*c^4*d^2 - 20*a*b^4*c^3*d^3 - 12*a^2*b^3*c^2*d^4 + 64*a^3*b^2*c*d^5 - 32*a
^4*b*d^6)*x^8 - 2*(15*b^5*c^5*d - 15*a*b^4*c^4*d^2 - 19*a^2*b^3*c^3*d^3 + 42*a^3
*b^2*c^2*d^4 + 8*a^4*b*c*d^5 - 16*a^5*d^6)*x^6 - 3*(5*b^5*c^6 - 14*a^2*b^3*c^4*d
^2 + 2*a^3*b^2*c^3*d^3 + 28*a^4*b*c^2*d^4 - 16*a^5*c*d^5)*x^4 - 2*(5*a*b^4*c^6 -
9*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 13*a^4*b*c^3*d^3 - 6*a^5*c^2*d^4)*x^2)*sq
rt(-a*b*c + a^2*d)*sqrt(d*x^2 + c) - 15*((b^6*c^5*d^2 - 2*a*b^5*c^4*d^3)*x^9 + (
2*b^6*c^6*d - 3*a*b^5*c^5*d^2 - 2*a^2*b^4*c^4*d^3)*x^7 + (b^6*c^7 - 4*a^2*b^4*c^
5*d^2)*x^5 + (a*b^5*c^7 - 2*a^2*b^4*c^6*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^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*
a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^9 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*
a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^7 + (a^3*b^4*c^9 - a^4*b^3*c^8*
d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^5 + (a^4*b^3*c^9 - 3*
a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^3)*sqrt(-a*b*c + a^2*d)), -1/12
*(2*(2*a^2*b^3*c^6 - 6*a^3*b^2*c^5*d + 6*a^4*b*c^4*d^2 - 2*a^5*c^3*d^3 - (15*b^5
*c^4*d^2 - 20*a*b^4*c^3*d^3 - 12*a^2*b^3*c^2*d^4 + 64*a^3*b^2*c*d^5 - 32*a^4*b*d
^6)*x^8 - 2*(15*b^5*c^5*d - 15*a*b^4*c^4*d^2 - 19*a^2*b^3*c^3*d^3 + 42*a^3*b^2*c
^2*d^4 + 8*a^4*b*c*d^5 - 16*a^5*d^6)*x^6 - 3*(5*b^5*c^6 - 14*a^2*b^3*c^4*d^2 + 2
*a^3*b^2*c^3*d^3 + 28*a^4*b*c^2*d^4 - 16*a^5*c*d^5)*x^4 - 2*(5*a*b^4*c^6 - 9*a^2
*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 13*a^4*b*c^3*d^3 - 6*a^5*c^2*d^4)*x^2)*sqrt(a*b
*c - a^2*d)*sqrt(d*x^2 + c) - 15*((b^6*c^5*d^2 - 2*a*b^5*c^4*d^3)*x^9 + (2*b^6*c
^6*d - 3*a*b^5*c^5*d^2 - 2*a^2*b^4*c^4*d^3)*x^7 + (b^6*c^7 - 4*a^2*b^4*c^5*d^2)*
x^5 + (a*b^5*c^7 - 2*a^2*b^4*c^6*d)*x^3)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)/(s
qrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)))/(((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 +
3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^9 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 +
3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^7 + (a^3*b^4*c^9 - a^4*b^3*c
^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^5 + (a^4*b^3*c^9 -
3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [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)^2*(d*x^2 + c)^(5/2)*x^4),x, algorithm="giac")

[Out]

Timed out