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

Optimal. Leaf size=100 \[ \frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)}-\frac{1}{3 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.455182, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)}-\frac{1}{3 a c x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 86.3471, size = 85, normalized size = 0.85 \[ \frac{d^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{c^{\frac{5}{2}} \left (a d - b c\right )} - \frac{1}{3 a c x^{3}} + \frac{a d + b c}{a^{2} c^{2} x} - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**(5/2)*atan(sqrt(d)*x/sqrt(c))/(c**(5/2)*(a*d - b*c)) - 1/(3*a*c*x**3) + (a*d
+ b*c)/(a**2*c**2*x) - b**(5/2)*atan(sqrt(b)*x/sqrt(a))/(a**(5/2)*(a*d - b*c))

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Mathematica [A]  time = 0.252064, size = 101, normalized size = 1.01 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (a d-b c)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)}-\frac{1}{3 a c x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 98, normalized size = 1. \[ -{\frac{1}{3\,ac{x}^{3}}}+{\frac{d}{ax{c}^{2}}}+{\frac{b}{{a}^{2}cx}}+{\frac{{d}^{3}}{{c}^{2} \left ( ad-bc \right ) }\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{b}^{3}}{{a}^{2} \left ( ad-bc \right ) }\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)*(d*x^2 + c)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278437, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \, b^{2} c^{2} x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 3 \, a^{2} d^{2} x^{3} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) + 2 \, a b c^{2} - 2 \, a^{2} c d - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{6 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}}, -\frac{6 \, a^{2} d^{2} x^{3} \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right ) + 3 \, b^{2} c^{2} x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 2 \, a b c^{2} - 2 \, a^{2} c d - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{6 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}}, \frac{6 \, b^{2} c^{2} x^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) - 3 \, a^{2} d^{2} x^{3} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) - 2 \, a b c^{2} + 2 \, a^{2} c d + 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{6 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}}, \frac{3 \, b^{2} c^{2} x^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) - 3 \, a^{2} d^{2} x^{3} \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right ) - a b c^{2} + a^{2} c d + 3 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2}}{3 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 30.8518, size = 1353, normalized size = 13.53 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-b**5/a**5)*log(x + (-a**10*c**5*d**5*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 +
2*a**9*b*c**6*d**4*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 - a**8*b**2*c**7*d**3*(-b*
*5/a**5)**(3/2)/(a*d - b*c)**3 - a**8*d**8*sqrt(-b**5/a**5)/(a*d - b*c) - a**7*b
**3*c**8*d**2*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 + 2*a**6*b**4*c**9*d*(-b**5/a**
5)**(3/2)/(a*d - b*c)**3 - a**5*b**5*c**10*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 -
b**8*c**8*sqrt(-b**5/a**5)/(a*d - b*c))/(a**4*b**3*d**7 + a**3*b**4*c*d**6 + a**
2*b**5*c**2*d**5 + a*b**6*c**3*d**4 + b**7*c**4*d**3))/(2*(a*d - b*c)) - sqrt(-b
**5/a**5)*log(x + (a**10*c**5*d**5*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 - 2*a**9*b
*c**6*d**4*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 + a**8*b**2*c**7*d**3*(-b**5/a**5)
**(3/2)/(a*d - b*c)**3 + a**8*d**8*sqrt(-b**5/a**5)/(a*d - b*c) + a**7*b**3*c**8
*d**2*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 - 2*a**6*b**4*c**9*d*(-b**5/a**5)**(3/2
)/(a*d - b*c)**3 + a**5*b**5*c**10*(-b**5/a**5)**(3/2)/(a*d - b*c)**3 + b**8*c**
8*sqrt(-b**5/a**5)/(a*d - b*c))/(a**4*b**3*d**7 + a**3*b**4*c*d**6 + a**2*b**5*c
**2*d**5 + a*b**6*c**3*d**4 + b**7*c**4*d**3))/(2*(a*d - b*c)) + sqrt(-d**5/c**5
)*log(x + (-a**10*c**5*d**5*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 + 2*a**9*b*c**6*d
**4*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 - a**8*b**2*c**7*d**3*(-d**5/c**5)**(3/2)
/(a*d - b*c)**3 - a**8*d**8*sqrt(-d**5/c**5)/(a*d - b*c) - a**7*b**3*c**8*d**2*(
-d**5/c**5)**(3/2)/(a*d - b*c)**3 + 2*a**6*b**4*c**9*d*(-d**5/c**5)**(3/2)/(a*d
- b*c)**3 - a**5*b**5*c**10*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 - b**8*c**8*sqrt(
-d**5/c**5)/(a*d - b*c))/(a**4*b**3*d**7 + a**3*b**4*c*d**6 + a**2*b**5*c**2*d**
5 + a*b**6*c**3*d**4 + b**7*c**4*d**3))/(2*(a*d - b*c)) - sqrt(-d**5/c**5)*log(x
+ (a**10*c**5*d**5*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 - 2*a**9*b*c**6*d**4*(-d*
*5/c**5)**(3/2)/(a*d - b*c)**3 + a**8*b**2*c**7*d**3*(-d**5/c**5)**(3/2)/(a*d -
b*c)**3 + a**8*d**8*sqrt(-d**5/c**5)/(a*d - b*c) + a**7*b**3*c**8*d**2*(-d**5/c*
*5)**(3/2)/(a*d - b*c)**3 - 2*a**6*b**4*c**9*d*(-d**5/c**5)**(3/2)/(a*d - b*c)**
3 + a**5*b**5*c**10*(-d**5/c**5)**(3/2)/(a*d - b*c)**3 + b**8*c**8*sqrt(-d**5/c*
*5)/(a*d - b*c))/(a**4*b**3*d**7 + a**3*b**4*c*d**6 + a**2*b**5*c**2*d**5 + a*b*
*6*c**3*d**4 + b**7*c**4*d**3))/(2*(a*d - b*c)) + (-a*c + x**2*(3*a*d + 3*b*c))/
(3*a**2*c**2*x**3)

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GIAC/XCAS [A]  time = 0.29586, size = 728, normalized size = 7.28 \[ -\frac{{\left (\sqrt{c d} a^{2} b^{3} c^{4}{\left | d \right |} + \sqrt{c d} a^{4} b c^{2} d^{2}{\left | d \right |} - \sqrt{c d} b^{2} c{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |}{\left | d \right |} - \sqrt{c d} a b d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |}{\left | d \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a^{2} b c^{3} + a^{3} c^{2} d + \sqrt{-4 \, a^{5} b c^{5} d +{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )}^{2}}}{a^{2} b c^{2} d}}}\right )}{a^{2} b c^{3} d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} + a^{3} c^{2} d^{2}{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} +{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )}^{2} d} + \frac{{\left (\sqrt{a b} a^{2} b^{2} c^{4} d{\left | b \right |} + \sqrt{a b} a^{4} c^{2} d^{3}{\left | b \right |} + \sqrt{a b} b c d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |}{\left | b \right |} + \sqrt{a b} a d^{2}{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |}{\left | b \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a^{2} b c^{3} + a^{3} c^{2} d - \sqrt{-4 \, a^{5} b c^{5} d +{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )}^{2}}}{a^{2} b c^{2} d}}}\right )}{a^{2} b^{2} c^{3}{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} + a^{3} b c^{2} d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} -{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )}^{2} b} + \frac{3 \, b c x^{2} + 3 \, a d x^{2} - a c}{3 \, a^{2} c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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