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

Optimal. Leaf size=206 \[ \frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2} (5 b c-2 a d)}{6 a^2 c x^3 (b c-a d)}+\frac{\sqrt{c+d x^2} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{6 a^3 c^2 x (b c-a d)}+\frac{b \sqrt{c+d x^2}}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.741834, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2} (5 b c-2 a d)}{6 a^2 c x^3 (b c-a d)}+\frac{\sqrt{c+d x^2} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{6 a^3 c^2 x (b c-a d)}+\frac{b \sqrt{c+d x^2}}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 128.861, size = 180, normalized size = 0.87 \[ - \frac{b \sqrt{c + d x^{2}}}{2 a x^{3} \left (a + b x^{2}\right ) \left (a d - b c\right )} - \frac{\sqrt{c + d x^{2}} \left (2 a d - 5 b c\right )}{6 a^{2} c x^{3} \left (a d - b c\right )} + \frac{\sqrt{c + d x^{2}} \left (4 a^{2} d^{2} + 8 a b c d - 15 b^{2} c^{2}\right )}{6 a^{3} c^{2} x \left (a d - b c\right )} + \frac{b^{2} \left (6 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{7}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

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)**(1/2),x)

[Out]

-b*sqrt(c + d*x**2)/(2*a*x**3*(a + b*x**2)*(a*d - b*c)) - sqrt(c + d*x**2)*(2*a*
d - 5*b*c)/(6*a**2*c*x**3*(a*d - b*c)) + sqrt(c + d*x**2)*(4*a**2*d**2 + 8*a*b*c
*d - 15*b**2*c**2)/(6*a**3*c**2*x*(a*d - b*c)) + b**2*(6*a*d - 5*b*c)*atanh(x*sq
rt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/(2*a**(7/2)*(a*d - b*c)**(3/2))

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Mathematica [A]  time = 0.410323, size = 136, normalized size = 0.66 \[ \frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^2} \left (\frac{3 b^3 x^4}{\left (a+b x^2\right ) (b c-a d)}+\frac{4 x^2 (a d+3 b c)}{c^2}-\frac{2 a}{c}\right )}{6 a^3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 893, normalized size = 4.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.558448, size = 1, normalized size = 0. \[ \left [-\frac{4 \,{\left (2 \, a^{2} b c^{2} - 2 \, a^{3} c d -{\left (15 \, b^{3} c^{2} - 8 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{4} - 2 \,{\left (5 \, a b^{2} c^{2} - 3 \, a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} - 3 \,{\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{5} +{\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{3}\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} + 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{24 \,{\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{5} +{\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{3}\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \,{\left (2 \, a^{2} b c^{2} - 2 \, a^{3} c d -{\left (15 \, b^{3} c^{2} - 8 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{4} - 2 \,{\left (5 \, a b^{2} c^{2} - 3 \, a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{2}\right )} \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} - 3 \,{\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{5} +{\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{3}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{12 \,{\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{5} +{\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{3}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/24*(4*(2*a^2*b*c^2 - 2*a^3*c*d - (15*b^3*c^2 - 8*a*b^2*c*d - 4*a^2*b*d^2)*x^
4 - 2*(5*a*b^2*c^2 - 3*a^2*b*c*d - 2*a^3*d^2)*x^2)*sqrt(-a*b*c + a^2*d)*sqrt(d*x
^2 + c) - 3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^5 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*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^2*c^3 - a^4*b*c^2*d)*x^5 + (a^4*b*c^3 - a^5*c^2*d)*x^3)*sqrt(-a*b*c + a^2*d)),
-1/12*(2*(2*a^2*b*c^2 - 2*a^3*c*d - (15*b^3*c^2 - 8*a*b^2*c*d - 4*a^2*b*d^2)*x^
4 - 2*(5*a*b^2*c^2 - 3*a^2*b*c*d - 2*a^3*d^2)*x^2)*sqrt(a*b*c - a^2*d)*sqrt(d*x^
2 + c) - 3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^5 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*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^2*c^3 - a^4*b*c^2*d)*x^5 + (a^4*b*c^3 - a^5*c^2*d)*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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 4.63135, size = 506, normalized size = 2.46 \[ -\frac{1}{6} \, d^{\frac{7}{2}}{\left (\frac{3 \,{\left (5 \, b^{3} c - 6 \, a b^{2} d\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt{a b c d - a^{2} d^{2}}} + \frac{6 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{3} c - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b^{2} d - b^{3} c^{2}\right )}}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )}{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}} + \frac{8 \,{\left (3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c - 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + 3 \, b c^{2} + a c d\right )}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3} d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*d^(7/2)*(3*(5*b^3*c - 6*a*b^2*d)*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^
2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/((a^3*b*c*d^3 - a^4*d^4)*sqrt(a*b*c*
d - a^2*d^2)) + 6*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b^3*c - 2*(sqrt(d)*x - sqrt(d
*x^2 + c))^2*a*b^2*d - b^3*c^2)/((a^3*b*c*d^3 - a^4*d^4)*((sqrt(d)*x - sqrt(d*x^
2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2
+ c))^2*a*d + b*c^2)) + 8*(3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 6*(sqrt(d)*x -
sqrt(d*x^2 + c))^2*b*c - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + 3*b*c^2 + a*c*d
)/(((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^3*a^3*d^3))