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

Optimal. Leaf size=92 \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)}-\frac{1}{2 a c x^2} \]

[Out]

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

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Rubi [A]  time = 0.311112, antiderivative size = 92, 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 \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)}-\frac{1}{2 a c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 51.5054, size = 76, normalized size = 0.83 \[ - \frac{d^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c}} \right )}}{2 c^{\frac{3}{2}} \left (a d - b c\right )} - \frac{1}{2 a c x^{2}} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.382706, size = 169, normalized size = 1.84 \[ \frac{-\frac{b^{3/2} x^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/2}}-\frac{b^{3/2} x^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{3/2}}+\frac{b}{a}+\frac{d^{3/2} x^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{3/2}}+\frac{d^{3/2} x^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{3/2}}-\frac{d}{c}}{2 x^2 (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 81, normalized size = 0.9 \[ -{\frac{{d}^{2}}{2\,c \left ( ad-bc \right ) }\arctan \left ({d{x}^{2}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}}{2\,a \left ( ad-bc \right ) }\arctan \left ({b{x}^{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{2\,ac{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.611711, size = 1, normalized size = 0.01 \[ \left [-\frac{b c x^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{4} + 2 \, a x^{2} \sqrt{-\frac{b}{a}} - a}{b x^{4} + a}\right ) + a d x^{2} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{4} - 2 \, c x^{2} \sqrt{-\frac{d}{c}} - c}{d x^{4} + c}\right ) + 2 \, b c - 2 \, a d}{4 \,{\left (a b c^{2} - a^{2} c d\right )} x^{2}}, -\frac{2 \, a d x^{2} \sqrt{\frac{d}{c}} \arctan \left (\frac{c \sqrt{\frac{d}{c}}}{d x^{2}}\right ) + b c x^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{4} + 2 \, a x^{2} \sqrt{-\frac{b}{a}} - a}{b x^{4} + a}\right ) + 2 \, b c - 2 \, a d}{4 \,{\left (a b c^{2} - a^{2} c d\right )} x^{2}}, \frac{2 \, b c x^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{2}}\right ) - a d x^{2} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{4} - 2 \, c x^{2} \sqrt{-\frac{d}{c}} - c}{d x^{4} + c}\right ) - 2 \, b c + 2 \, a d}{4 \,{\left (a b c^{2} - a^{2} c d\right )} x^{2}}, \frac{b c x^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{2}}\right ) - a d x^{2} \sqrt{\frac{d}{c}} \arctan \left (\frac{c \sqrt{\frac{d}{c}}}{d x^{2}}\right ) - b c + a d}{2 \,{\left (a b c^{2} - a^{2} c d\right )} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(b*c*x^2*sqrt(-b/a)*log((b*x^4 + 2*a*x^2*sqrt(-b/a) - a)/(b*x^4 + a)) + a*
d*x^2*sqrt(-d/c)*log((d*x^4 - 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)) + 2*b*c - 2*a
*d)/((a*b*c^2 - a^2*c*d)*x^2), -1/4*(2*a*d*x^2*sqrt(d/c)*arctan(c*sqrt(d/c)/(d*x
^2)) + b*c*x^2*sqrt(-b/a)*log((b*x^4 + 2*a*x^2*sqrt(-b/a) - a)/(b*x^4 + a)) + 2*
b*c - 2*a*d)/((a*b*c^2 - a^2*c*d)*x^2), 1/4*(2*b*c*x^2*sqrt(b/a)*arctan(a*sqrt(b
/a)/(b*x^2)) - a*d*x^2*sqrt(-d/c)*log((d*x^4 - 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 +
c)) - 2*b*c + 2*a*d)/((a*b*c^2 - a^2*c*d)*x^2), 1/2*(b*c*x^2*sqrt(b/a)*arctan(a*
sqrt(b/a)/(b*x^2)) - a*d*x^2*sqrt(d/c)*arctan(c*sqrt(d/c)/(d*x^2)) - b*c + a*d)/
((a*b*c^2 - a^2*c*d)*x^2)]

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Sympy [A]  time = 20.0406, size = 1103, normalized size = 11.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-b**3/a**3)*log(x**2 + (-a**7*c**3*d**4*(-b**3/a**3)**(3/2)/(a*d - b*c)**3
+ 2*a**6*b*c**4*d**3*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - 2*a**5*b**2*c**5*d**2
*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - a**5*d**5*sqrt(-b**3/a**3)/(a*d - b*c) + 2
*a**4*b**3*c**6*d*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - a**3*b**4*c**7*(-b**3/a**
3)**(3/2)/(a*d - b*c)**3 - b**5*c**5*sqrt(-b**3/a**3)/(a*d - b*c))/(a**2*b**2*d*
*4 + a*b**3*c*d**3 + b**4*c**2*d**2))/(4*(a*d - b*c)) + sqrt(-b**3/a**3)*log(x**
2 + (a**7*c**3*d**4*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - 2*a**6*b*c**4*d**3*(-b*
*3/a**3)**(3/2)/(a*d - b*c)**3 + 2*a**5*b**2*c**5*d**2*(-b**3/a**3)**(3/2)/(a*d
- b*c)**3 + a**5*d**5*sqrt(-b**3/a**3)/(a*d - b*c) - 2*a**4*b**3*c**6*d*(-b**3/a
**3)**(3/2)/(a*d - b*c)**3 + a**3*b**4*c**7*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 +
b**5*c**5*sqrt(-b**3/a**3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b**3*c*d**3 + b**4*
c**2*d**2))/(4*(a*d - b*c)) - sqrt(-d**3/c**3)*log(x**2 + (-a**7*c**3*d**4*(-d**
3/c**3)**(3/2)/(a*d - b*c)**3 + 2*a**6*b*c**4*d**3*(-d**3/c**3)**(3/2)/(a*d - b*
c)**3 - 2*a**5*b**2*c**5*d**2*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - a**5*d**5*sqr
t(-d**3/c**3)/(a*d - b*c) + 2*a**4*b**3*c**6*d*(-d**3/c**3)**(3/2)/(a*d - b*c)**
3 - a**3*b**4*c**7*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - b**5*c**5*sqrt(-d**3/c**
3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b**3*c*d**3 + b**4*c**2*d**2))/(4*(a*d - b*c
)) + sqrt(-d**3/c**3)*log(x**2 + (a**7*c**3*d**4*(-d**3/c**3)**(3/2)/(a*d - b*c)
**3 - 2*a**6*b*c**4*d**3*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 + 2*a**5*b**2*c**5*d
**2*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 + a**5*d**5*sqrt(-d**3/c**3)/(a*d - b*c)
- 2*a**4*b**3*c**6*d*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 + a**3*b**4*c**7*(-d**3/
c**3)**(3/2)/(a*d - b*c)**3 + b**5*c**5*sqrt(-d**3/c**3)/(a*d - b*c))/(a**2*b**2
*d**4 + a*b**3*c*d**3 + b**4*c**2*d**2))/(4*(a*d - b*c)) - 1/(2*a*c*x**2)

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GIAC/XCAS [A]  time = 0.247899, size = 508, normalized size = 5.52 \[ \frac{{\left (\sqrt{c d} a b^{2} c d x^{4}{\left | d \right |} + \sqrt{c d} a b^{2} c^{2}{\left | d \right |} + \sqrt{c d} a^{2} b c d{\left | d \right |}\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, a b c^{2} + 2 \, a^{2} c d + \sqrt{-16 \, a^{3} b c^{3} d + 4 \,{\left (a b c^{2} + a^{2} c d\right )}^{2}}}{a b c d}}}\right )}{a b c^{2} d{\left | a b c^{2} - a^{2} c d \right |} + a^{2} c d^{2}{\left | a b c^{2} - a^{2} c d \right |} +{\left (a b c^{2} - a^{2} c d\right )}^{2} d} - \frac{{\left (\sqrt{a b} a b c d^{2} x^{4}{\left | b \right |} + \sqrt{a b} a b c^{2} d{\left | b \right |} + \sqrt{a b} a^{2} c d^{2}{\left | b \right |}\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, a b c^{2} + 2 \, a^{2} c d - \sqrt{-16 \, a^{3} b c^{3} d + 4 \,{\left (a b c^{2} + a^{2} c d\right )}^{2}}}{a b c d}}}\right )}{a b^{2} c^{2}{\left | a b c^{2} - a^{2} c d \right |} + a^{2} b c d{\left | a b c^{2} - a^{2} c d \right |} -{\left (a b c^{2} - a^{2} c d\right )}^{2} b} - \frac{1}{2 \, a c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(sqrt(c*d)*a*b^2*c*d*x^4*abs(d) + sqrt(c*d)*a*b^2*c^2*abs(d) + sqrt(c*d)*a^2*b*c
*d*abs(d))*arctan(2*x^2/sqrt((2*a*b*c^2 + 2*a^2*c*d + sqrt(-16*a^3*b*c^3*d + 4*(
a*b*c^2 + a^2*c*d)^2))/(a*b*c*d)))/(a*b*c^2*d*abs(a*b*c^2 - a^2*c*d) + a^2*c*d^2
*abs(a*b*c^2 - a^2*c*d) + (a*b*c^2 - a^2*c*d)^2*d) - (sqrt(a*b)*a*b*c*d^2*x^4*ab
s(b) + sqrt(a*b)*a*b*c^2*d*abs(b) + sqrt(a*b)*a^2*c*d^2*abs(b))*arctan(2*x^2/sqr
t((2*a*b*c^2 + 2*a^2*c*d - sqrt(-16*a^3*b*c^3*d + 4*(a*b*c^2 + a^2*c*d)^2))/(a*b
*c*d)))/(a*b^2*c^2*abs(a*b*c^2 - a^2*c*d) + a^2*b*c*d*abs(a*b*c^2 - a^2*c*d) - (
a*b*c^2 - a^2*c*d)^2*b) - 1/2/(a*c*x^2)