3.221 \(\int \frac{x^8}{\sqrt{a+b x^3+c x^6}} \, dx\)

Optimal. Leaf size=104 \[ \frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{24 c^{5/2}}-\frac{b \sqrt{a+b x^3+c x^6}}{4 c^2}+\frac{x^3 \sqrt{a+b x^3+c x^6}}{6 c} \]

[Out]

-(b*Sqrt[a + b*x^3 + c*x^6])/(4*c^2) + (x^3*Sqrt[a + b*x^3 + c*x^6])/(6*c) + ((3
*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(24*c^
(5/2))

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Rubi [A]  time = 0.195274, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{24 c^{5/2}}-\frac{b \sqrt{a+b x^3+c x^6}}{4 c^2}+\frac{x^3 \sqrt{a+b x^3+c x^6}}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[x^8/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

-(b*Sqrt[a + b*x^3 + c*x^6])/(4*c^2) + (x^3*Sqrt[a + b*x^3 + c*x^6])/(6*c) + ((3
*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(24*c^
(5/2))

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Rubi in Sympy [A]  time = 24.3176, size = 92, normalized size = 0.88 \[ - \frac{b \sqrt{a + b x^{3} + c x^{6}}}{4 c^{2}} + \frac{x^{3} \sqrt{a + b x^{3} + c x^{6}}}{6 c} + \frac{\left (- 4 a c + 3 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{24 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

-b*sqrt(a + b*x**3 + c*x**6)/(4*c**2) + x**3*sqrt(a + b*x**3 + c*x**6)/(6*c) + (
-4*a*c + 3*b**2)*atanh((b + 2*c*x**3)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x**6)))/(24
*c**(5/2))

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Mathematica [A]  time = 0.0686677, size = 86, normalized size = 0.83 \[ \frac{\left (3 b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )+2 \sqrt{c} \left (2 c x^3-3 b\right ) \sqrt{a+b x^3+c x^6}}{24 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

(2*Sqrt[c]*(-3*b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6] + (3*b^2 - 4*a*c)*Log[b + 2*
c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/(24*c^(5/2))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{{x}^{8}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(x^8/(c*x^6+b*x^3+a)^(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/sqrt(c*x^6 + b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277303, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} - 3 \, b\right )} \sqrt{c} -{\left (3 \, b^{2} - 4 \, a c\right )} \log \left (4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{48 \, c^{\frac{5}{2}}}, \frac{2 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} - 3 \, b\right )} \sqrt{-c} +{\left (3 \, b^{2} - 4 \, a c\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{24 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(c*x^6 + b*x^3 + a),x, algorithm="fricas")

[Out]

[1/48*(4*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 - 3*b)*sqrt(c) - (3*b^2 - 4*a*c)*log(4
*sqrt(c*x^6 + b*x^3 + a)*(2*c^2*x^3 + b*c) - (8*c^2*x^6 + 8*b*c*x^3 + b^2 + 4*a*
c)*sqrt(c)))/c^(5/2), 1/24*(2*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 - 3*b)*sqrt(-c) +
(3*b^2 - 4*a*c)*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)/(sqrt(c*x^6 + b*x^3 + a)*c)))
/(sqrt(-c)*c^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{a + b x^{3} + c x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(x**8/sqrt(a + b*x**3 + c*x**6), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{c x^{6} + b x^{3} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(c*x^6 + b*x^3 + a),x, algorithm="giac")

[Out]

integrate(x^8/sqrt(c*x^6 + b*x^3 + a), x)