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

Optimal. Leaf size=149 \[ -\frac{b (3 b c-4 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{5/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} (3 b c-2 a d)}{4 a^2 c x^2 (b c-a d)}+\frac{b \sqrt{c+d x^4}}{4 a x^2 \left (a+b x^4\right ) (b c-a d)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.515258, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b (3 b c-4 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{5/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} (3 b c-2 a d)}{4 a^2 c x^2 (b c-a d)}+\frac{b \sqrt{c+d x^4}}{4 a x^2 \left (a+b x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 66.9363, size = 129, normalized size = 0.87 \[ - \frac{b \sqrt{c + d x^{4}}}{4 a x^{2} \left (a + b x^{4}\right ) \left (a d - b c\right )} - \frac{\sqrt{c + d x^{4}} \left (2 a d - 3 b c\right )}{4 a^{2} c x^{2} \left (a d - b c\right )} - \frac{b \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{4 a^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 2.24231, size = 172, normalized size = 1.15 \[ \frac{1}{4} \sqrt{c+d x^4} \left (\frac{b x^6 (4 a d-3 b c) \sin ^{-1}\left (\frac{\sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^4}{a}+1}}\right )}{a^4 c^2 \sqrt{\frac{b x^4}{a}+1} \left (\frac{x^4 (b c-a d)}{a c}\right )^{3/2} \sqrt{\frac{a \left (c+d x^4\right )}{c \left (a+b x^4\right )}}}+\frac{\frac{b^2 x^4}{\left (a+b x^4\right ) (a d-b c)}-\frac{2}{c}}{a^2 x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^4]*((-2/c + (b^2*x^4)/((-(b*c) + a*d)*(a + b*x^4)))/(a^2*x^2) + (b
*(-3*b*c + 4*a*d)*x^6*ArcSin[Sqrt[(b/a - d/c)*x^4]/Sqrt[1 + (b*x^4)/a]])/(a^4*c^
2*(((b*c - a*d)*x^4)/(a*c))^(3/2)*Sqrt[1 + (b*x^4)/a]*Sqrt[(a*(c + d*x^4))/(c*(a
+ b*x^4))])))/4

_______________________________________________________________________________________

Maple [B]  time = 0.021, size = 885, normalized size = 5.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/a^2/c/x^2*(d*x^4+c)^(1/2)+1/8*b/a^2/(a*d-b*c)/(x^2-1/b*(-a*b)^(1/2))*((x^2-
1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x^2-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/
2)-1/8/a^2*d*(-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^2-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2-1/b*(-a*b)^(1
/2))^2*d+2*d*(-a*b)^(1/2)/b*(x^2-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2-1/b*
(-a*b)^(1/2)))+1/8*b/a^2/(a*d-b*c)/(x^2+1/b*(-a*b)^(1/2))*((x^2+1/b*(-a*b)^(1/2)
)^2*d-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/8/a^2*d*(-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^2+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2+1/b*(-a*b)^(1/2))^2*d-2*d*(-a
*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2+1/b*(-a*b)^(1/2)))+3
/8*b/a^2/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b
*(x^2-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2-1/b*(-a*b)^(1/2))^2*d+2*d*(
-a*b)^(1/2)/b*(x^2-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2-1/b*(-a*b)^(1/2)))
-3/8*b/a^2/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)
/b*(x^2+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2+1/b*(-a*b)^(1/2))^2*d-2*d
*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2+1/b*(-a*b)^(1/2)
))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.404334, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{4} + 2 \, a b c - 2 \, a^{2} d\right )} \sqrt{d x^{4} + c} \sqrt{-a b c + a^{2} d} -{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{6} +{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \log \left (-\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{6} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2}\right )} \sqrt{d x^{4} + c} -{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{16 \,{\left ({\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{6} +{\left (a^{3} b c^{2} - a^{4} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \,{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{4} + 2 \, a b c - 2 \, a^{2} d\right )} \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} +{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{6} +{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} x^{2}}\right )}{8 \,{\left ({\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{6} +{\left (a^{3} b c^{2} - a^{4} c d\right )} x^{2}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(4*((3*b^2*c - 2*a*b*d)*x^4 + 2*a*b*c - 2*a^2*d)*sqrt(d*x^4 + c)*sqrt(-a*
b*c + a^2*d) - ((3*b^3*c^2 - 4*a*b^2*c*d)*x^6 + (3*a*b^2*c^2 - 4*a^2*b*c*d)*x^2)
*log(-(4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^6 - (a^2*b*c^2 - a^3*c*d)*x^2)
*sqrt(d*x^4 + c) - ((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2
*c*d)*x^4 + a^2*c^2)*sqrt(-a*b*c + a^2*d))/(b^2*x^8 + 2*a*b*x^4 + a^2)))/(((a^2*
b^2*c^2 - a^3*b*c*d)*x^6 + (a^3*b*c^2 - a^4*c*d)*x^2)*sqrt(-a*b*c + a^2*d)), -1/
8*(2*((3*b^2*c - 2*a*b*d)*x^4 + 2*a*b*c - 2*a^2*d)*sqrt(d*x^4 + c)*sqrt(a*b*c -
a^2*d) + ((3*b^3*c^2 - 4*a*b^2*c*d)*x^6 + (3*a*b^2*c^2 - 4*a^2*b*c*d)*x^2)*arcta
n(1/2*((b*c - 2*a*d)*x^4 - a*c)/(sqrt(d*x^4 + c)*sqrt(a*b*c - a^2*d)*x^2)))/(((a
^2*b^2*c^2 - a^3*b*c*d)*x^6 + (a^3*b*c^2 - a^4*c*d)*x^2)*sqrt(a*b*c - a^2*d))]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225331, size = 181, normalized size = 1.21 \[ -\frac{b^{2} c \sqrt{d + \frac{c}{x^{4}}}}{4 \,{\left (a^{2} b c - a^{3} d\right )}{\left (b c + a{\left (d + \frac{c}{x^{4}}\right )} - a d\right )}} + \frac{{\left (3 \, b^{2} c - 4 \, a b d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{4 \,{\left (a^{2} b c - a^{3} d\right )} \sqrt{a b c - a^{2} d}} - \frac{\sqrt{d + \frac{c}{x^{4}}}}{2 \, a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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