3.299 \(\int \frac{1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=250 \[ -\frac{b^{7/2} (7 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2} (b c-a d)^2}-\frac{7 b c-2 a d}{10 a^2 c x^5 (b c-a d)}+\frac{-2 a^2 d^2-2 a b c d+7 b^2 c^2}{6 a^3 c^2 x^3 (b c-a d)}-\frac{-2 a^3 d^3-2 a^2 b c d^2-2 a b^2 c^2 d+7 b^3 c^3}{2 a^4 c^3 x (b c-a d)}-\frac{d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)^2}+\frac{b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)} \]

[Out]

-(7*b*c - 2*a*d)/(10*a^2*c*(b*c - a*d)*x^5) + (7*b^2*c^2 - 2*a*b*c*d - 2*a^2*d^2
)/(6*a^3*c^2*(b*c - a*d)*x^3) - (7*b^3*c^3 - 2*a*b^2*c^2*d - 2*a^2*b*c*d^2 - 2*a
^3*d^3)/(2*a^4*c^3*(b*c - a*d)*x) + b/(2*a*(b*c - a*d)*x^5*(a + b*x^2)) - (b^(7/
2)*(7*b*c - 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2)*(b*c - a*d)^2) - (d^(
9/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^2)

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Rubi [A]  time = 1.10313, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{b^{7/2} (7 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2} (b c-a d)^2}-\frac{7 b c-2 a d}{10 a^2 c x^5 (b c-a d)}+\frac{-2 a^2 d^2-2 a b c d+7 b^2 c^2}{6 a^3 c^2 x^3 (b c-a d)}-\frac{-2 a^3 d^3-2 a^2 b c d^2-2 a b^2 c^2 d+7 b^3 c^3}{2 a^4 c^3 x (b c-a d)}-\frac{d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)^2}+\frac{b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

-(7*b*c - 2*a*d)/(10*a^2*c*(b*c - a*d)*x^5) + (7*b^2*c^2 - 2*a*b*c*d - 2*a^2*d^2
)/(6*a^3*c^2*(b*c - a*d)*x^3) - (7*b^3*c^3 - 2*a*b^2*c^2*d - 2*a^2*b*c*d^2 - 2*a
^3*d^3)/(2*a^4*c^3*(b*c - a*d)*x) + b/(2*a*(b*c - a*d)*x^5*(a + b*x^2)) - (b^(7/
2)*(7*b*c - 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2)*(b*c - a*d)^2) - (d^(
9/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(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**6/(b*x**2+a)**2/(d*x**2+c),x)

[Out]

Timed out

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Mathematica [A]  time = 0.568445, size = 179, normalized size = 0.72 \[ \frac{b^{7/2} (9 a d-7 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2} (a d-b c)^2}+\frac{b^4 x}{2 a^4 \left (a+b x^2\right ) (a d-b c)}+\frac{a d+2 b c}{3 a^3 c^2 x^3}-\frac{1}{5 a^2 c x^5}+\frac{-a^2 d^2-2 a b c d-3 b^2 c^2}{a^4 c^3 x}-\frac{d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

-1/(5*a^2*c*x^5) + (2*b*c + a*d)/(3*a^3*c^2*x^3) + (-3*b^2*c^2 - 2*a*b*c*d - a^2
*d^2)/(a^4*c^3*x) + (b^4*x)/(2*a^4*(-(b*c) + a*d)*(a + b*x^2)) + (b^(7/2)*(-7*b*
c + 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2)*(-(b*c) + a*d)^2) - (d^(9/2)*
ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^2)

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Maple [A]  time = 0.024, size = 234, normalized size = 0.9 \[ -{\frac{1}{5\,{a}^{2}c{x}^{5}}}+{\frac{d}{3\,{a}^{2}{c}^{2}{x}^{3}}}+{\frac{2\,b}{3\,{x}^{3}{a}^{3}c}}-{\frac{{d}^{2}}{{a}^{2}{c}^{3}x}}-2\,{\frac{bd}{{a}^{3}{c}^{2}x}}-3\,{\frac{{b}^{2}}{{a}^{4}cx}}-{\frac{{d}^{5}}{{c}^{3} \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{4}xd}{2\,{a}^{3} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{5}xc}{2\,{a}^{4} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{9\,d{b}^{4}}{2\,{a}^{3} \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,{b}^{5}c}{2\,{a}^{4} \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^2+a)^2/(d*x^2+c),x)

[Out]

-1/5/a^2/c/x^5+1/3/x^3/a^2/c^2*d+2/3/x^3/a^3/c*b-1/a^2/c^3/x*d^2-2/a^3/c^2/x*b*d
-3/a^4/c/x*b^2-1/c^3*d^5/(a*d-b*c)^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))+1/2*b^4
/a^3/(a*d-b*c)^2*x/(b*x^2+a)*d-1/2*b^5/a^4/(a*d-b*c)^2*x/(b*x^2+a)*c+9/2*b^4/a^3
/(a*d-b*c)^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d-7/2*b^5/a^4/(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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.12234, 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)*x^6),x, algorithm="fricas")

[Out]

[-1/60*(12*a^3*b^2*c^4 - 24*a^4*b*c^3*d + 12*a^5*c^2*d^2 + 30*(7*b^5*c^4 - 9*a*b
^4*c^3*d + 2*a^4*b*d^4)*x^6 + 20*(7*a*b^4*c^4 - 9*a^2*b^3*c^3*d - a^4*b*c*d^3 +
3*a^5*d^4)*x^4 - 4*(7*a^2*b^3*c^4 - 9*a^3*b^2*c^3*d - 3*a^4*b*c^2*d^2 + 5*a^5*c*
d^3)*x^2 + 15*((7*b^5*c^4 - 9*a*b^4*c^3*d)*x^7 + (7*a*b^4*c^4 - 9*a^2*b^3*c^3*d)
*x^5)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 30*(a^4*b*d^4
*x^7 + a^5*d^4*x^5)*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/
((a^4*b^3*c^5 - 2*a^5*b^2*c^4*d + a^6*b*c^3*d^2)*x^7 + (a^5*b^2*c^5 - 2*a^6*b*c^
4*d + a^7*c^3*d^2)*x^5), -1/60*(12*a^3*b^2*c^4 - 24*a^4*b*c^3*d + 12*a^5*c^2*d^2
+ 30*(7*b^5*c^4 - 9*a*b^4*c^3*d + 2*a^4*b*d^4)*x^6 + 20*(7*a*b^4*c^4 - 9*a^2*b^
3*c^3*d - a^4*b*c*d^3 + 3*a^5*d^4)*x^4 - 4*(7*a^2*b^3*c^4 - 9*a^3*b^2*c^3*d - 3*
a^4*b*c^2*d^2 + 5*a^5*c*d^3)*x^2 + 60*(a^4*b*d^4*x^7 + a^5*d^4*x^5)*sqrt(d/c)*ar
ctan(d*x/(c*sqrt(d/c))) + 15*((7*b^5*c^4 - 9*a*b^4*c^3*d)*x^7 + (7*a*b^4*c^4 - 9
*a^2*b^3*c^3*d)*x^5)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))
/((a^4*b^3*c^5 - 2*a^5*b^2*c^4*d + a^6*b*c^3*d^2)*x^7 + (a^5*b^2*c^5 - 2*a^6*b*c
^4*d + a^7*c^3*d^2)*x^5), -1/30*(6*a^3*b^2*c^4 - 12*a^4*b*c^3*d + 6*a^5*c^2*d^2
+ 15*(7*b^5*c^4 - 9*a*b^4*c^3*d + 2*a^4*b*d^4)*x^6 + 10*(7*a*b^4*c^4 - 9*a^2*b^3
*c^3*d - a^4*b*c*d^3 + 3*a^5*d^4)*x^4 - 2*(7*a^2*b^3*c^4 - 9*a^3*b^2*c^3*d - 3*a
^4*b*c^2*d^2 + 5*a^5*c*d^3)*x^2 + 15*((7*b^5*c^4 - 9*a*b^4*c^3*d)*x^7 + (7*a*b^4
*c^4 - 9*a^2*b^3*c^3*d)*x^5)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - 15*(a^4*b*d^4
*x^7 + a^5*d^4*x^5)*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/
((a^4*b^3*c^5 - 2*a^5*b^2*c^4*d + a^6*b*c^3*d^2)*x^7 + (a^5*b^2*c^5 - 2*a^6*b*c^
4*d + a^7*c^3*d^2)*x^5), -1/30*(6*a^3*b^2*c^4 - 12*a^4*b*c^3*d + 6*a^5*c^2*d^2 +
15*(7*b^5*c^4 - 9*a*b^4*c^3*d + 2*a^4*b*d^4)*x^6 + 10*(7*a*b^4*c^4 - 9*a^2*b^3*
c^3*d - a^4*b*c*d^3 + 3*a^5*d^4)*x^4 - 2*(7*a^2*b^3*c^4 - 9*a^3*b^2*c^3*d - 3*a^
4*b*c^2*d^2 + 5*a^5*c*d^3)*x^2 + 15*((7*b^5*c^4 - 9*a*b^4*c^3*d)*x^7 + (7*a*b^4*
c^4 - 9*a^2*b^3*c^3*d)*x^5)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) + 30*(a^4*b*d^4*
x^7 + a^5*d^4*x^5)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))))/((a^4*b^3*c^5 - 2*a^5*b^
2*c^4*d + a^6*b*c^3*d^2)*x^7 + (a^5*b^2*c^5 - 2*a^6*b*c^4*d + a^7*c^3*d^2)*x^5)]

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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**6/(b*x**2+a)**2/(d*x**2+c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.238905, size = 279, normalized size = 1.12 \[ -\frac{d^{5} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} \sqrt{c d}} - \frac{b^{4} x}{2 \,{\left (a^{4} b c - a^{5} d\right )}{\left (b x^{2} + a\right )}} - \frac{{\left (7 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )} \sqrt{a b}} - \frac{45 \, b^{2} c^{2} x^{4} + 30 \, a b c d x^{4} + 15 \, a^{2} d^{2} x^{4} - 10 \, a b c^{2} x^{2} - 5 \, a^{2} c d x^{2} + 3 \, a^{2} c^{2}}{15 \, a^{4} c^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)*x^6),x, algorithm="giac")

[Out]

-d^5*arctan(d*x/sqrt(c*d))/((b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*sqrt(c*d)) - 1
/2*b^4*x/((a^4*b*c - a^5*d)*(b*x^2 + a)) - 1/2*(7*b^5*c - 9*a*b^4*d)*arctan(b*x/
sqrt(a*b))/((a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)*sqrt(a*b)) - 1/15*(45*b^2*c^2*
x^4 + 30*a*b*c*d*x^4 + 15*a^2*d^2*x^4 - 10*a*b*c^2*x^2 - 5*a^2*c*d*x^2 + 3*a^2*c
^2)/(a^4*c^3*x^5)