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

Optimal. Leaf size=91 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{3 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 b \sqrt{b c-a d}} \]

[Out]

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

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Rubi [A]  time = 0.255327, antiderivative size = 91, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{3 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 b \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.1934, size = 76, normalized size = 0.84 \[ - \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{x^{3} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{6}}} \right )}}{3 b \sqrt{a d - b c}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{d} x^{3}}{\sqrt{c + d x^{6}}} \right )}}{3 b \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0857718, size = 90, normalized size = 0.99 \[ \frac{\frac{\log \left (\sqrt{d} \sqrt{c+d x^6}+d x^3\right )}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{\sqrt{b c-a d}}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{{x}^{8}}{b{x}^{6}+a}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304182, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{d} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{9} -{\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt{d x^{6} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) + 2 \, \log \left (-2 \, \sqrt{d x^{6} + c} d x^{3} -{\left (2 \, d x^{6} + c\right )} \sqrt{d}\right )}{12 \, b \sqrt{d}}, \frac{\sqrt{-d} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{9} -{\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt{d x^{6} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) + 4 \, \arctan \left (\frac{\sqrt{-d} x^{3}}{\sqrt{d x^{6} + c}}\right )}{12 \, b \sqrt{-d}}, -\frac{\sqrt{d} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{6} - a c}{2 \, \sqrt{d x^{6} + c}{\left (b c - a d\right )} x^{3} \sqrt{\frac{a}{b c - a d}}}\right ) - \log \left (-2 \, \sqrt{d x^{6} + c} d x^{3} -{\left (2 \, d x^{6} + c\right )} \sqrt{d}\right )}{6 \, b \sqrt{d}}, -\frac{\sqrt{-d} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{6} - a c}{2 \, \sqrt{d x^{6} + c}{\left (b c - a d\right )} x^{3} \sqrt{\frac{a}{b c - a d}}}\right ) - 2 \, \arctan \left (\frac{\sqrt{-d} x^{3}}{\sqrt{d x^{6} + c}}\right )}{6 \, b \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**8/((a + b*x**6)*sqrt(c + d*x**6)), x)

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GIAC/XCAS [A]  time = 0.232113, size = 223, normalized size = 2.45 \[ \frac{1}{3} \, c{\left (\frac{a \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{6}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d} b c{\rm sign}\left (x\right )} - \frac{\arctan \left (\frac{\sqrt{d + \frac{c}{x^{6}}}}{\sqrt{-d}}\right )}{b c \sqrt{-d}{\rm sign}\left (x\right )}\right )} - \frac{{\left (a \sqrt{-d} \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) - \sqrt{a b c - a^{2} d} \arctan \left (\frac{\sqrt{d}}{\sqrt{-d}}\right )\right )}{\rm sign}\left (x\right )}{3 \, \sqrt{a b c - a^{2} d} b \sqrt{-d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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