3.693 \(\int \frac{1}{x^{10} \left (a+b x^6\right ) \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=115 \[ \frac{b^2 \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^6} (2 a d+3 b c)}{9 a^2 c^2 x^3}-\frac{\sqrt{c+d x^6}}{9 a c x^9} \]

[Out]

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

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Rubi [A]  time = 0.479913, antiderivative size = 115, 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 \[ \frac{b^2 \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^6} (2 a d+3 b c)}{9 a^2 c^2 x^3}-\frac{\sqrt{c+d x^6}}{9 a c x^9} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^10*(a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

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Rubi in Sympy [A]  time = 63.3972, size = 100, normalized size = 0.87 \[ - \frac{\sqrt{c + d x^{6}}}{9 a c x^{9}} + \frac{\sqrt{c + d x^{6}} \left (2 a d + 3 b c\right )}{9 a^{2} c^{2} x^{3}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{x^{3} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{6}}} \right )}}{3 a^{\frac{5}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

-sqrt(c + d*x**6)/(9*a*c*x**9) + sqrt(c + d*x**6)*(2*a*d + 3*b*c)/(9*a**2*c**2*x
**3) + b**2*atanh(x**3*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**6)))/(3*a**(5/2)*s
qrt(a*d - b*c))

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Mathematica [A]  time = 1.4021, size = 149, normalized size = 1.3 \[ \frac{\sqrt{c+d x^6} \left (-a^2 c+\frac{3 b^2 c x^{12} \sin ^{-1}\left (\frac{\sqrt{x^6 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^6}{a}+1}}\right )}{\sqrt{\frac{b x^6}{a}+1} \sqrt{x^6 \left (\frac{b}{a}-\frac{d}{c}\right )} \sqrt{\frac{a \left (c+d x^6\right )}{c \left (a+b x^6\right )}}}+a x^6 (2 a d+3 b c)\right )}{9 a^3 c^2 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^10*(a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

(Sqrt[c + d*x^6]*(-(a^2*c) + a*(3*b*c + 2*a*d)*x^6 + (3*b^2*c*x^12*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))])))/(9*a^3*c^2*x^9)

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Maple [F]  time = 0.099, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{10} \left ( b{x}^{6}+a \right ) }{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(b*x^6+a)/(d*x^6+c)^(1/2),x)

[Out]

int(1/x^10/(b*x^6+a)/(d*x^6+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{6} + a\right )} \sqrt{d x^{6} + c} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^10),x, algorithm="maxima")

[Out]

integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^10), x)

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Fricas [A]  time = 0.308903, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} c^{2} x^{9} \log \left (\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{9} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{3}\right )} \sqrt{d x^{6} + c} +{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) + 4 \,{\left ({\left (3 \, b c + 2 \, a d\right )} x^{6} - a c\right )} \sqrt{d x^{6} + c} \sqrt{-a b c + a^{2} d}}{36 \, \sqrt{-a b c + a^{2} d} a^{2} c^{2} x^{9}}, \frac{3 \, b^{2} c^{2} x^{9} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{6} - a c}{2 \, \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d} x^{3}}\right ) + 2 \,{\left ({\left (3 \, b c + 2 \, a d\right )} x^{6} - a c\right )} \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d}}{18 \, \sqrt{a b c - a^{2} d} a^{2} c^{2} x^{9}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^10),x, algorithm="fricas")

[Out]

[1/36*(3*b^2*c^2*x^9*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*((3*b*c + 2*a*d)*x^6 - a*c)*sqrt(d*x^6 + c)*sqrt(-a*b*c + a^2*d
))/(sqrt(-a*b*c + a^2*d)*a^2*c^2*x^9), 1/18*(3*b^2*c^2*x^9*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*((3*b*c + 2*a*d)*
x^6 - a*c)*sqrt(d*x^6 + c)*sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*a^2*c^2*x^9
)]

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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**10/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.2678, size = 277, normalized size = 2.41 \[ -\frac{b^{2} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{6}}}}{\sqrt{a b c - a^{2} d}}\right )}{3 \, \sqrt{a b c - a^{2} d} a^{2}{\rm sign}\left (x\right )} + \frac{{\left (3 \, b^{2} c^{2} \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) - 3 \, \sqrt{a b c - a^{2} d} b c \sqrt{d} - 2 \, \sqrt{a b c - a^{2} d} a d^{\frac{3}{2}}\right )}{\rm sign}\left (x\right )}{9 \, \sqrt{a b c - a^{2} d} a^{2} c^{2}} + \frac{3 \, a b c^{5} \sqrt{d + \frac{c}{x^{6}}} - a^{2} c^{4}{\left (d + \frac{c}{x^{6}}\right )}^{\frac{3}{2}} + 3 \, a^{2} c^{4} \sqrt{d + \frac{c}{x^{6}}} d}{9 \, a^{3} c^{6}{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^10),x, algorithm="giac")

[Out]

-1/3*b^2*arctan(a*sqrt(d + c/x^6)/sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*a^2*
sign(x)) + 1/9*(3*b^2*c^2*arctan(a*sqrt(d)/sqrt(a*b*c - a^2*d)) - 3*sqrt(a*b*c -
a^2*d)*b*c*sqrt(d) - 2*sqrt(a*b*c - a^2*d)*a*d^(3/2))*sign(x)/(sqrt(a*b*c - a^2
*d)*a^2*c^2) + 1/9*(3*a*b*c^5*sqrt(d + c/x^6) - a^2*c^4*(d + c/x^6)^(3/2) + 3*a^
2*c^4*sqrt(d + c/x^6)*d)/(a^3*c^6*sign(x))