3.721 \(\int \frac{1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=176 \[ \frac{b^3 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^2} (3 b c-4 a d) (2 a d+b c)}{3 a^2 c^3 x (b c-a d)}-\frac{\sqrt{c+d x^2} (b c-4 a d)}{3 a c^2 x^3 (b c-a d)}-\frac{d}{c x^3 \sqrt{c+d x^2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.685407, antiderivative size = 176, 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 \[ \frac{b^3 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^2} (3 b c-4 a d) (2 a d+b c)}{3 a^2 c^3 x (b c-a d)}-\frac{\sqrt{c+d x^2} (b c-4 a d)}{3 a c^2 x^3 (b c-a d)}-\frac{d}{c x^3 \sqrt{c+d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^2)*(c + d*x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 96.1181, size = 151, normalized size = 0.86 \[ \frac{d}{c x^{3} \sqrt{c + d x^{2}} \left (a d - b c\right )} - \frac{\sqrt{c + d x^{2}} \left (4 a d - b c\right )}{3 a c^{2} x^{3} \left (a d - b c\right )} + \frac{\sqrt{c + d x^{2}} \left (2 a d + b c\right ) \left (4 a d - 3 b c\right )}{3 a^{2} c^{3} x \left (a d - b c\right )} - \frac{b^{3} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{a^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**2+a)/(d*x**2+c)**(3/2),x)

[Out]

d/(c*x**3*sqrt(c + d*x**2)*(a*d - b*c)) - sqrt(c + d*x**2)*(4*a*d - b*c)/(3*a*c*
*2*x**3*(a*d - b*c)) + sqrt(c + d*x**2)*(2*a*d + b*c)*(4*a*d - 3*b*c)/(3*a**2*c*
*3*x*(a*d - b*c)) - b**3*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/(a*
*(5/2)*(a*d - b*c)**(3/2))

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Mathematica [A]  time = 0.36402, size = 124, normalized size = 0.7 \[ \frac{b^3 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^2} \left (\frac{x^2 (5 a d+3 b c)}{a^2}+\frac{3 d^3 x^4}{\left (c+d x^2\right ) (a d-b c)}-\frac{c}{a}\right )}{3 c^3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^2)*(c + d*x^2)^(3/2)),x]

[Out]

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

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Maple [B]  time = 0.025, size = 762, 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)/(d*x^2+c)^(3/2),x)

[Out]

-1/3/a/c/x^3/(d*x^2+c)^(1/2)+4/3/a*d/c^2/x/(d*x^2+c)^(1/2)+8/3/a*d^2/c^3*x/(d*x^
2+c)^(1/2)+b/a^2/c/x/(d*x^2+c)^(1/2)+2*b/a^2*d/c^2*x/(d*x^2+c)^(1/2)-1/2*b^3/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)^(1/2)+1/2*b^2/a^2/(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+1/2*b^3/a^2/(-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)))+1/2*b^3/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)^(1/2)+1/2*b^2/a^2/(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-1/2*b^3/a^2/(-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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(3/2)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.478106, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (a b c^{3} - a^{2} c^{2} d -{\left (3 \, b^{2} c^{2} d + 2 \, a b c d^{2} - 8 \, a^{2} d^{3}\right )} x^{4} -{\left (3 \, b^{2} c^{3} + a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} + 3 \,{\left (b^{3} c^{3} d x^{5} + b^{3} c^{4} x^{3}\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} - 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{12 \,{\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{5} +{\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{3}\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \,{\left (a b c^{3} - a^{2} c^{2} d -{\left (3 \, b^{2} c^{2} d + 2 \, a b c d^{2} - 8 \, a^{2} d^{3}\right )} x^{4} -{\left (3 \, b^{2} c^{3} + a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} - 3 \,{\left (b^{3} c^{3} d x^{5} + b^{3} c^{4} x^{3}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{6 \,{\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{5} +{\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{3}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(3/2)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**2+a)/(d*x**2+c)**(3/2),x)

[Out]

Integral(1/(x**4*(a + b*x**2)*(c + d*x**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 1.89573, size = 373, normalized size = 2.12 \[ -\frac{b^{3} \sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{{\left (a^{2} b c - a^{3} d\right )} \sqrt{a b c d - a^{2} d^{2}}} - \frac{d^{3} x}{{\left (b c^{4} - a c^{3} d\right )} \sqrt{d x^{2} + c}} - \frac{2 \,{\left (3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b c \sqrt{d} + 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a d^{\frac{3}{2}} - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c^{2} \sqrt{d} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a c d^{\frac{3}{2}} + 3 \, b c^{3} \sqrt{d} + 5 \, a c^{2} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(3/2)*x^4),x, algorithm="giac")

[Out]

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