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

Optimal. Leaf size=189 \[ \frac{b^{5/2} (5 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^2}-\frac{5 b c-2 a d}{6 a^2 c x^3 (b c-a d)}+\frac{-2 a^2 d^2-2 a b c d+5 b^2 c^2}{2 a^3 c^2 x (b c-a d)}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)^2}+\frac{b}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.762349, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{b^{5/2} (5 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^2}-\frac{5 b c-2 a d}{6 a^2 c x^3 (b c-a d)}+\frac{-2 a^2 d^2-2 a b c d+5 b^2 c^2}{2 a^3 c^2 x (b c-a d)}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)^2}+\frac{b}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.608628, size = 142, normalized size = 0.75 \[ -\frac{b^{5/2} (7 a d-5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (a d-b c)^2}-\frac{b^3 x}{2 a^3 \left (a+b x^2\right ) (a d-b c)}+\frac{a d+2 b c}{a^3 c^2 x}-\frac{1}{3 a^2 c x^3}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*a^2*c*x^3) + (2*b*c + a*d)/(a^3*c^2*x) - (b^3*x)/(2*a^3*(-(b*c) + a*d)*(a
+ b*x^2)) - (b^(5/2)*(-5*b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(-
(b*c) + a*d)^2) + (d^(7/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d)^2)

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Maple [A]  time = 0.023, size = 191, normalized size = 1. \[ -{\frac{1}{3\,{a}^{2}c{x}^{3}}}+{\frac{d}{{a}^{2}{c}^{2}x}}+2\,{\frac{b}{x{a}^{3}c}}+{\frac{{d}^{4}}{{c}^{2} \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{b}^{3}xd}{2\,{a}^{2} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{4}xc}{2\,{a}^{3} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,d{b}^{3}}{2\,{a}^{2} \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{4}c}{2\,{a}^{3} \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/a^2/c/x^3+1/x/a^2/c^2*d+2/x/a^3/c*b+1/c^2*d^4/(a*d-b*c)^2/(c*d)^(1/2)*arcta
n(x*d/(c*d)^(1/2))-1/2*b^3/a^2/(a*d-b*c)^2*x/(b*x^2+a)*d+1/2*b^4/a^3/(a*d-b*c)^2
*x/(b*x^2+a)*c-7/2*b^3/a^2/(a*d-b*c)^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d+5/2
*b^4/a^3/(a*d-b*c)^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.40961, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(4*a^2*b^2*c^3 - 8*a^3*b*c^2*d + 4*a^4*c*d^2 - 6*(5*b^4*c^3 - 7*a*b^3*c^2
*d + 2*a^3*b*d^3)*x^4 - 4*(5*a*b^3*c^3 - 7*a^2*b^2*c^2*d - a^3*b*c*d^2 + 3*a^4*d
^3)*x^2 + 3*((5*b^4*c^3 - 7*a*b^3*c^2*d)*x^5 + (5*a*b^3*c^3 - 7*a^2*b^2*c^2*d)*x
^3)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 6*(a^3*b*d^3*x^
5 + a^4*d^3*x^3)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a
^3*b^3*c^4 - 2*a^4*b^2*c^3*d + a^5*b*c^2*d^2)*x^5 + (a^4*b^2*c^4 - 2*a^5*b*c^3*d
+ a^6*c^2*d^2)*x^3), -1/12*(4*a^2*b^2*c^3 - 8*a^3*b*c^2*d + 4*a^4*c*d^2 - 6*(5*
b^4*c^3 - 7*a*b^3*c^2*d + 2*a^3*b*d^3)*x^4 - 4*(5*a*b^3*c^3 - 7*a^2*b^2*c^2*d -
a^3*b*c*d^2 + 3*a^4*d^3)*x^2 - 12*(a^3*b*d^3*x^5 + a^4*d^3*x^3)*sqrt(d/c)*arctan
(d*x/(c*sqrt(d/c))) + 3*((5*b^4*c^3 - 7*a*b^3*c^2*d)*x^5 + (5*a*b^3*c^3 - 7*a^2*
b^2*c^2*d)*x^3)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/((a^
3*b^3*c^4 - 2*a^4*b^2*c^3*d + a^5*b*c^2*d^2)*x^5 + (a^4*b^2*c^4 - 2*a^5*b*c^3*d
+ a^6*c^2*d^2)*x^3), -1/6*(2*a^2*b^2*c^3 - 4*a^3*b*c^2*d + 2*a^4*c*d^2 - 3*(5*b^
4*c^3 - 7*a*b^3*c^2*d + 2*a^3*b*d^3)*x^4 - 2*(5*a*b^3*c^3 - 7*a^2*b^2*c^2*d - a^
3*b*c*d^2 + 3*a^4*d^3)*x^2 - 3*((5*b^4*c^3 - 7*a*b^3*c^2*d)*x^5 + (5*a*b^3*c^3 -
7*a^2*b^2*c^2*d)*x^3)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - 3*(a^3*b*d^3*x^5 +
a^4*d^3*x^3)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^3*b
^3*c^4 - 2*a^4*b^2*c^3*d + a^5*b*c^2*d^2)*x^5 + (a^4*b^2*c^4 - 2*a^5*b*c^3*d + a
^6*c^2*d^2)*x^3), -1/6*(2*a^2*b^2*c^3 - 4*a^3*b*c^2*d + 2*a^4*c*d^2 - 3*(5*b^4*c
^3 - 7*a*b^3*c^2*d + 2*a^3*b*d^3)*x^4 - 2*(5*a*b^3*c^3 - 7*a^2*b^2*c^2*d - a^3*b
*c*d^2 + 3*a^4*d^3)*x^2 - 3*((5*b^4*c^3 - 7*a*b^3*c^2*d)*x^5 + (5*a*b^3*c^3 - 7*
a^2*b^2*c^2*d)*x^3)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - 6*(a^3*b*d^3*x^5 + a^4
*d^3*x^3)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))))/((a^3*b^3*c^4 - 2*a^4*b^2*c^3*d +
a^5*b*c^2*d^2)*x^5 + (a^4*b^2*c^4 - 2*a^5*b*c^3*d + a^6*c^2*d^2)*x^3)]

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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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283637, size = 223, normalized size = 1.18 \[ \frac{d^{4} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \sqrt{c d}} + \frac{b^{3} x}{2 \,{\left (a^{3} b c - a^{4} d\right )}{\left (b x^{2} + a\right )}} + \frac{{\left (5 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt{a b}} + \frac{6 \, b c x^{2} + 3 \, a d x^{2} - a c}{3 \, a^{3} c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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