3.707 \(\int \frac{x^8}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=93 \[ \frac{c \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^3 \sqrt{c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.257695, antiderivative size = 93, 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 \[ \frac{c \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^3 \sqrt{c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^8/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

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

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Rubi in Sympy [A]  time = 29.5023, size = 76, normalized size = 0.82 \[ \frac{x^{3} \sqrt{c + d x^{6}}}{6 \left (a + b x^{6}\right ) \left (a d - b c\right )} - \frac{c \operatorname{atanh}{\left (\frac{x^{3} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{6}}} \right )}}{6 \sqrt{a} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.174096, size = 90, normalized size = 0.97 \[ \frac{\frac{x^3 \sqrt{c+d x^6}}{a+b x^6}-\frac{c \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{\sqrt{a} \sqrt{b c-a d}}}{6 a d-6 b c} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

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

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{{x}^{8}}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(x^8/(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. \[ \int \frac{x^{8}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="maxima")

[Out]

integrate(x^8/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)

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Fricas [A]  time = 0.321661, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{d x^{6} + c} \sqrt{-a b c + a^{2} d} x^{3} +{\left (b c x^{6} + a c\right )} \log \left (-\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{9} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{3}\right )} \sqrt{d x^{6} + c} -{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right )}{24 \,{\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \, \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d} x^{3} -{\left (b c x^{6} + a c\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{6} - a c}{2 \, \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d} x^{3}}\right )}{12 \,{\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="fricas")

[Out]

[-1/24*(4*sqrt(d*x^6 + c)*sqrt(-a*b*c + a^2*d)*x^3 + (b*c*x^6 + a*c)*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)))/(((b^2*c - a*b*d)
*x^6 + a*b*c - a^2*d)*sqrt(-a*b*c + a^2*d)), -1/12*(2*sqrt(d*x^6 + c)*sqrt(a*b*c
- a^2*d)*x^3 - (b*c*x^6 + a*c)*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)))/(((b^2*c - a*b*d)*x^6 + a*b*c - a^2*d)*sqrt(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(x**8/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.277513, size = 248, normalized size = 2.67 \[ -\frac{1}{6} \, c{\left (\frac{\arctan \left (\frac{a \sqrt{d + \frac{c}{x^{6}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d}{\left (b c{\rm sign}\left (x\right ) - a d{\rm sign}\left (x\right )\right )}} + \frac{\sqrt{d + \frac{c}{x^{6}}}}{{\left (b c{\rm sign}\left (x\right ) - a d{\rm sign}\left (x\right )\right )}{\left (b c + a{\left (d + \frac{c}{x^{6}}\right )} - a d\right )}}\right )} + \frac{{\left (b c \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) + \sqrt{a b c - a^{2} d} \sqrt{d}\right )}{\rm sign}\left (x\right )}{6 \,{\left (\sqrt{a b c - a^{2} d} b^{2} c - \sqrt{a b c - a^{2} d} a b d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="giac")

[Out]

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