3.710 \(\int \frac{1}{x^{10} \left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=208 \[ \frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^6} (5 b c-2 a d)}{18 a^2 c x^9 (b c-a d)}+\frac{\sqrt{c+d x^6} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{18 a^3 c^2 x^3 (b c-a d)}+\frac{b \sqrt{c+d x^6}}{6 a x^9 \left (a+b x^6\right ) (b c-a d)} \]

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^6])/(18*a^2*c*(b*c - a*d)*x^9) + ((15*b^2*c^2 - 8
*a*b*c*d - 4*a^2*d^2)*Sqrt[c + d*x^6])/(18*a^3*c^2*(b*c - a*d)*x^3) + (b*Sqrt[c
+ d*x^6])/(6*a*(b*c - a*d)*x^9*(a + b*x^6)) + (b^2*(5*b*c - 6*a*d)*ArcTan[(Sqrt[
b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(6*a^(7/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.865718, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^6} (5 b c-2 a d)}{18 a^2 c x^9 (b c-a d)}+\frac{\sqrt{c+d x^6} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{18 a^3 c^2 x^3 (b c-a d)}+\frac{b \sqrt{c+d x^6}}{6 a x^9 \left (a+b x^6\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^6])/(18*a^2*c*(b*c - a*d)*x^9) + ((15*b^2*c^2 - 8
*a*b*c*d - 4*a^2*d^2)*Sqrt[c + d*x^6])/(18*a^3*c^2*(b*c - a*d)*x^3) + (b*Sqrt[c
+ d*x^6])/(6*a*(b*c - a*d)*x^9*(a + b*x^6)) + (b^2*(5*b*c - 6*a*d)*ArcTan[(Sqrt[
b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(6*a^(7/2)*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 136.273, size = 184, normalized size = 0.88 \[ - \frac{b \sqrt{c + d x^{6}}}{6 a x^{9} \left (a + b x^{6}\right ) \left (a d - b c\right )} - \frac{\sqrt{c + d x^{6}} \left (2 a d - 5 b c\right )}{18 a^{2} c x^{9} \left (a d - b c\right )} + \frac{\sqrt{c + d x^{6}} \left (4 a^{2} d^{2} + 8 a b c d - 15 b^{2} c^{2}\right )}{18 a^{3} c^{2} x^{3} \left (a d - b c\right )} + \frac{b^{2} \left (6 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{x^{3} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{6}}} \right )}}{6 a^{\frac{7}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*sqrt(c + d*x**6)/(6*a*x**9*(a + b*x**6)*(a*d - b*c)) - sqrt(c + d*x**6)*(2*a*
d - 5*b*c)/(18*a**2*c*x**9*(a*d - b*c)) + sqrt(c + d*x**6)*(4*a**2*d**2 + 8*a*b*
c*d - 15*b**2*c**2)/(18*a**3*c**2*x**3*(a*d - b*c)) + b**2*(6*a*d - 5*b*c)*atanh
(x**3*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**6)))/(6*a**(7/2)*(a*d - b*c)**(3/2)
)

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Mathematica [A]  time = 1.93734, size = 195, normalized size = 0.94 \[ \frac{\sqrt{c+d x^6} \left (-\frac{2 a^2}{c}+\frac{3 a b^3 x^{12}}{\left (a+b x^6\right ) (b c-a d)}+\frac{3 b^2 x^{18} (5 b c-6 a d) \sin ^{-1}\left (\frac{\sqrt{x^6 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^6}{a}+1}}\right )}{a c^2 \sqrt{\frac{b x^6}{a}+1} \left (\frac{x^6 (b c-a d)}{a c}\right )^{3/2} \sqrt{\frac{a \left (c+d x^6\right )}{c \left (a+b x^6\right )}}}+\frac{4 a x^6 (a d+3 b c)}{c^2}\right )}{18 a^4 x^9} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^6]*((-2*a^2)/c + (4*a*(3*b*c + a*d)*x^6)/c^2 + (3*a*b^3*x^12)/((b*
c - a*d)*(a + b*x^6)) + (3*b^2*(5*b*c - 6*a*d)*x^18*ArcSin[Sqrt[(b/a - d/c)*x^6]
/Sqrt[1 + (b*x^6)/a]])/(a*c^2*(((b*c - a*d)*x^6)/(a*c))^(3/2)*Sqrt[1 + (b*x^6)/a
]*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))])))/(18*a^4*x^9)

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Maple [F]  time = 0.116, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{10} \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^10),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.562031, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left ({\left (15 \, b^{3} c^{2} - 8 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{12} + 2 \,{\left (5 \, a b^{2} c^{2} - 3 \, a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{6} - 2 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} \sqrt{d x^{6} + c} \sqrt{-a b c + a^{2} d} + 3 \,{\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{15} +{\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{9}\right )} \log \left (\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{9} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{3}\right )} \sqrt{d x^{6} + c} +{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right )}{72 \,{\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{15} +{\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{9}\right )} \sqrt{-a b c + a^{2} d}}, \frac{2 \,{\left ({\left (15 \, b^{3} c^{2} - 8 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{12} + 2 \,{\left (5 \, a b^{2} c^{2} - 3 \, a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{6} - 2 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d} + 3 \,{\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{15} +{\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{9}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{6} - a c}{2 \, \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d} x^{3}}\right )}{36 \,{\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{15} +{\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{9}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^10),x, algorithm="fricas")

[Out]

[1/72*(4*((15*b^3*c^2 - 8*a*b^2*c*d - 4*a^2*b*d^2)*x^12 + 2*(5*a*b^2*c^2 - 3*a^2
*b*c*d - 2*a^3*d^2)*x^6 - 2*a^2*b*c^2 + 2*a^3*c*d)*sqrt(d*x^6 + c)*sqrt(-a*b*c +
a^2*d) + 3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^15 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*
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)))/((
(a^3*b^2*c^3 - a^4*b*c^2*d)*x^15 + (a^4*b*c^3 - a^5*c^2*d)*x^9)*sqrt(-a*b*c + a^
2*d)), 1/36*(2*((15*b^3*c^2 - 8*a*b^2*c*d - 4*a^2*b*d^2)*x^12 + 2*(5*a*b^2*c^2 -
3*a^2*b*c*d - 2*a^3*d^2)*x^6 - 2*a^2*b*c^2 + 2*a^3*c*d)*sqrt(d*x^6 + c)*sqrt(a*
b*c - a^2*d) + 3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^15 + (5*a*b^3*c^3 - 6*a^2*b^2*c^
2*d)*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)))/(((a^3*b^2*c^3 - a^4*b*c^2*d)*x^15 + (a^4*b*c^3 - a^5*c^2*d)*x^9)*sqr
t(a*b*c - a^2*d))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.307738, size = 494, normalized size = 2.38 \[ \frac{b^{3} c \sqrt{d + \frac{c}{x^{6}}}}{6 \,{\left (a^{3} b c{\rm sign}\left (x\right ) - a^{4} d{\rm sign}\left (x\right )\right )}{\left (b c + a{\left (d + \frac{c}{x^{6}}\right )} - a d\right )}} + \frac{{\left (15 \, b^{3} c^{3} \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) - 18 \, a b^{2} c^{2} d \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) - 15 \, \sqrt{a b c - a^{2} d} b^{2} c^{2} \sqrt{d} + 8 \, \sqrt{a b c - a^{2} d} a b c d^{\frac{3}{2}} + 4 \, \sqrt{a b c - a^{2} d} a^{2} d^{\frac{5}{2}}\right )}{\rm sign}\left (x\right )}{18 \,{\left (\sqrt{a b c - a^{2} d} a^{3} b c^{3} - \sqrt{a b c - a^{2} d} a^{4} c^{2} d\right )}} - \frac{{\left (5 \, b^{3} c - 6 \, a b^{2} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{6}}}}{\sqrt{a b c - a^{2} d}}\right )}{6 \,{\left (a^{3} b c{\rm sign}\left (x\right ) - a^{4} d{\rm sign}\left (x\right )\right )} \sqrt{a b c - a^{2} d}} + \frac{6 \, a^{3} b c^{5} \sqrt{d + \frac{c}{x^{6}}} - a^{4} c^{4}{\left (d + \frac{c}{x^{6}}\right )}^{\frac{3}{2}} + 3 \, a^{4} c^{4} \sqrt{d + \frac{c}{x^{6}}} d}{9 \, a^{6} c^{6}{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^10),x, algorithm="giac")

[Out]

1/6*b^3*c*sqrt(d + c/x^6)/((a^3*b*c*sign(x) - a^4*d*sign(x))*(b*c + a*(d + c/x^6
) - a*d)) + 1/18*(15*b^3*c^3*arctan(a*sqrt(d)/sqrt(a*b*c - a^2*d)) - 18*a*b^2*c^
2*d*arctan(a*sqrt(d)/sqrt(a*b*c - a^2*d)) - 15*sqrt(a*b*c - a^2*d)*b^2*c^2*sqrt(
d) + 8*sqrt(a*b*c - a^2*d)*a*b*c*d^(3/2) + 4*sqrt(a*b*c - a^2*d)*a^2*d^(5/2))*si
gn(x)/(sqrt(a*b*c - a^2*d)*a^3*b*c^3 - sqrt(a*b*c - a^2*d)*a^4*c^2*d) - 1/6*(5*b
^3*c - 6*a*b^2*d)*arctan(a*sqrt(d + c/x^6)/sqrt(a*b*c - a^2*d))/((a^3*b*c*sign(x
) - a^4*d*sign(x))*sqrt(a*b*c - a^2*d)) + 1/9*(6*a^3*b*c^5*sqrt(d + c/x^6) - a^4
*c^4*(d + c/x^6)^(3/2) + 3*a^4*c^4*sqrt(d + c/x^6)*d)/(a^6*c^6*sign(x))