3.692 \(\int \frac{1}{x^4 \left (a+b x^6\right ) \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{b \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^6}}{3 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.245905, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{b \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^6}}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

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Rubi in Sympy [A]  time = 30.7548, size = 68, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{6}}}{3 a c x^{3}} - \frac{b \operatorname{atanh}{\left (\frac{x^{3} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{6}}} \right )}}{3 a^{\frac{3}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

-sqrt(c + d*x**6)/(3*a*c*x**3) - b*atanh(x**3*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c +
d*x**6)))/(3*a**(3/2)*sqrt(a*d - b*c))

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Mathematica [A]  time = 1.29092, size = 129, normalized size = 1.61 \[ \frac{\sqrt{c+d x^6} \left (-\frac{b x^6 \sin ^{-1}\left (\frac{\sqrt{x^6 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^6}{a}+1}}\right )}{\sqrt{\frac{b x^6}{a}+1} \sqrt{x^6 \left (\frac{b}{a}-\frac{d}{c}\right )} \sqrt{\frac{a \left (c+d x^6\right )}{c \left (a+b x^6\right )}}}-a\right )}{3 a^2 c x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4} \left ( b{x}^{6}+a \right ) }{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^6+a)/(d*x^6+c)^(1/2),x)

[Out]

int(1/x^4/(b*x^6+a)/(d*x^6+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{6} + a\right )} \sqrt{d x^{6} + c} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^4), x)

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Fricas [A]  time = 0.288729, size = 1, normalized size = 0.01 \[ \left [\frac{b c x^{3} \log \left (-\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{9} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{3}\right )} \sqrt{d x^{6} + c} -{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) - 4 \, \sqrt{d x^{6} + c} \sqrt{-a b c + a^{2} d}}{12 \, \sqrt{-a b c + a^{2} d} a c x^{3}}, -\frac{b c x^{3} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{6} - a c}{2 \, \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d} x^{3}}\right ) + 2 \, \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d}}{6 \, \sqrt{a b c - a^{2} d} a c x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^4),x, algorithm="fricas")

[Out]

[1/12*(b*c*x^3*log(-(4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^9 - (a^2*b*c^2 -
a^3*c*d)*x^3)*sqrt(d*x^6 + c) - ((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^12 - 2*(3*
a*b*c^2 - 4*a^2*c*d)*x^6 + a^2*c^2)*sqrt(-a*b*c + a^2*d))/(b^2*x^12 + 2*a*b*x^6
+ a^2)) - 4*sqrt(d*x^6 + c)*sqrt(-a*b*c + a^2*d))/(sqrt(-a*b*c + a^2*d)*a*c*x^3)
, -1/6*(b*c*x^3*arctan(1/2*((b*c - 2*a*d)*x^6 - a*c)/(sqrt(d*x^6 + c)*sqrt(a*b*c
- a^2*d)*x^3)) + 2*sqrt(d*x^6 + c)*sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*a*
c*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (a + b x^{6}\right ) \sqrt{c + d x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

Integral(1/(x**4*(a + b*x**6)*sqrt(c + d*x**6)), x)

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GIAC/XCAS [A]  time = 0.254104, size = 185, normalized size = 2.31 \[ \frac{\frac{b c \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{6}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d} a{\rm sign}\left (x\right )} - \frac{\sqrt{d + \frac{c}{x^{6}}}}{a{\rm sign}\left (x\right )}}{3 \, c} - \frac{{\left (b c \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) - \sqrt{a b c - a^{2} d} \sqrt{d}\right )}{\rm sign}\left (x\right )}{3 \, \sqrt{a b c - a^{2} d} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^4),x, algorithm="giac")

[Out]

1/3*(b*c*arctan(a*sqrt(d + c/x^6)/sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*a*si
gn(x)) - sqrt(d + c/x^6)/(a*sign(x)))/c - 1/3*(b*c*arctan(a*sqrt(d)/sqrt(a*b*c -
a^2*d)) - sqrt(a*b*c - a^2*d)*sqrt(d))*sign(x)/(sqrt(a*b*c - a^2*d)*a*c)