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

Optimal. Leaf size=202 \[ \frac{b x \left (-3 a^2 d^2-16 a b c d+4 b^2 c^2\right )}{6 a^2 c \sqrt{a+b x^2} (b c-a d)^3}+\frac{d^2 (6 b c-a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{7/2}}-\frac{d x}{2 c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (b c-a d)}+\frac{b x (3 a d+2 b c)}{6 a c \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.619936, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{b x \left (-3 a^2 d^2-16 a b c d+4 b^2 c^2\right )}{6 a^2 c \sqrt{a+b x^2} (b c-a d)^3}+\frac{d^2 (6 b c-a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{7/2}}-\frac{d x}{2 c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (b c-a d)}+\frac{b x (3 a d+2 b c)}{6 a c \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 116.464, size = 178, normalized size = 0.88 \[ \frac{d x}{2 c \left (a + b x^{2}\right )^{\frac{3}{2}} \left (c + d x^{2}\right ) \left (a d - b c\right )} + \frac{d^{2} \left (a d - 6 b c\right ) \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{2 c^{\frac{3}{2}} \left (a d - b c\right )^{\frac{7}{2}}} + \frac{b x \left (3 a d + 2 b c\right )}{6 a c \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{b x \left (3 a^{2} d^{2} + 16 a b c d - 4 b^{2} c^{2}\right )}{6 a^{2} c \sqrt{a + b x^{2}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/(2*c*(a + b*x**2)**(3/2)*(c + d*x**2)*(a*d - b*c)) + d**2*(a*d - 6*b*c)*atan
(x*sqrt(a*d - b*c)/(sqrt(c)*sqrt(a + b*x**2)))/(2*c**(3/2)*(a*d - b*c)**(7/2)) +
b*x*(3*a*d + 2*b*c)/(6*a*c*(a + b*x**2)**(3/2)*(a*d - b*c)**2) + b*x*(3*a**2*d*
*2 + 16*a*b*c*d - 4*b**2*c**2)/(6*a**2*c*sqrt(a + b*x**2)*(a*d - b*c)**3)

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Mathematica [A]  time = 1.02774, size = 170, normalized size = 0.84 \[ \frac{1}{6} \left (x \sqrt{a+b x^2} \left (\frac{4 b^2 (4 a d-b c)}{a^2 \left (a+b x^2\right ) (a d-b c)^3}+\frac{2 b^2}{a \left (a+b x^2\right )^2 (b c-a d)^2}-\frac{3 d^3}{c \left (c+d x^2\right ) (b c-a d)^3}\right )+\frac{3 d^2 (a d-6 b c) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{c^{3/2} (a d-b c)^{7/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.036, size = 2371, normalized size = 11.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.22512, 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)^(5/2)*(d*x^2 + c)^2),x, algorithm="fricas")

[Out]

[1/24*(4*((4*b^4*c^2*d - 16*a*b^3*c*d^2 - 3*a^2*b^2*d^3)*x^5 + 2*(2*b^4*c^3 - 5*
a*b^3*c^2*d - 9*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*x^3 + 3*(2*a*b^3*c^3 - 6*a^2*b^2*c^
2*d - a^4*d^3)*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a) + 3*(6*a^4*b*c^2*d^2 - a^5
*c*d^3 + (6*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^6 + (6*a^2*b^3*c^2*d^2 + 11*a^3*b^2*c
*d^3 - 2*a^4*b*d^4)*x^4 + (12*a^3*b^2*c^2*d^2 + 4*a^4*b*c*d^3 - a^5*d^4)*x^2)*lo
g((((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*
x^2)*sqrt(b*c^2 - a*c*d) + 4*((2*b^2*c^3 - 3*a*b*c^2*d + a^2*c*d^2)*x^3 + (a*b*c
^3 - a^2*c^2*d)*x)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)))/((a^4*b^3*c^5
- 3*a^5*b^2*c^4*d + 3*a^6*b*c^3*d^2 - a^7*c^2*d^3 + (a^2*b^5*c^4*d - 3*a^3*b^4*c
^3*d^2 + 3*a^4*b^3*c^2*d^3 - a^5*b^2*c*d^4)*x^6 + (a^2*b^5*c^5 - a^3*b^4*c^4*d -
3*a^4*b^3*c^3*d^2 + 5*a^5*b^2*c^2*d^3 - 2*a^6*b*c*d^4)*x^4 + (2*a^3*b^4*c^5 - 5
*a^4*b^3*c^4*d + 3*a^5*b^2*c^3*d^2 + a^6*b*c^2*d^3 - a^7*c*d^4)*x^2)*sqrt(b*c^2
- a*c*d)), 1/12*(2*((4*b^4*c^2*d - 16*a*b^3*c*d^2 - 3*a^2*b^2*d^3)*x^5 + 2*(2*b^
4*c^3 - 5*a*b^3*c^2*d - 9*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*x^3 + 3*(2*a*b^3*c^3 - 6*
a^2*b^2*c^2*d - a^4*d^3)*x)*sqrt(-b*c^2 + a*c*d)*sqrt(b*x^2 + a) + 3*(6*a^4*b*c^
2*d^2 - a^5*c*d^3 + (6*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^6 + (6*a^2*b^3*c^2*d^2 + 1
1*a^3*b^2*c*d^3 - 2*a^4*b*d^4)*x^4 + (12*a^3*b^2*c^2*d^2 + 4*a^4*b*c*d^3 - a^5*d
^4)*x^2)*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 + a*c)/((b*c^2 - a*c
*d)*sqrt(b*x^2 + a)*x)))/((a^4*b^3*c^5 - 3*a^5*b^2*c^4*d + 3*a^6*b*c^3*d^2 - a^7
*c^2*d^3 + (a^2*b^5*c^4*d - 3*a^3*b^4*c^3*d^2 + 3*a^4*b^3*c^2*d^3 - a^5*b^2*c*d^
4)*x^6 + (a^2*b^5*c^5 - a^3*b^4*c^4*d - 3*a^4*b^3*c^3*d^2 + 5*a^5*b^2*c^2*d^3 -
2*a^6*b*c*d^4)*x^4 + (2*a^3*b^4*c^5 - 5*a^4*b^3*c^4*d + 3*a^5*b^2*c^3*d^2 + a^6*
b*c^2*d^3 - a^7*c*d^4)*x^2)*sqrt(-b*c^2 + a*c*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/(b*x**2+a)**(5/2)/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 4.94363, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x