3.236 \(\int \frac{x^8}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=120 \[ \frac{2 x^3 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3 c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.196579, antiderivative size = 120, 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 \[ \frac{2 x^3 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.6022, size = 107, normalized size = 0.89 \[ - \frac{2 b \sqrt{a + b x^{3} + c x^{6}}}{3 c \left (- 4 a c + b^{2}\right )} + \frac{2 x^{3} \left (2 a + b x^{3}\right )}{3 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}} + \frac{\operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{3 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.179836, size = 95, normalized size = 0.79 \[ \frac{2 \left (a b-2 a c x^3+b^2 x^3\right )}{3 c \left (4 a c-b^2\right ) \sqrt{a+b x^3+c x^6}}+\frac{\log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{3 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*b + b^2*x^3 - 2*a*c*x^3))/(3*c*(-b^2 + 4*a*c)*Sqrt[a + b*x^3 + c*x^6]) + L
og[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]]/(3*c^(3/2))

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \[ \int{{x}^{8} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^8/(c*x^6+b*x^3+a)^(3/2),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/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29223, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left ({\left (b^{2} - 2 \, a c\right )} x^{3} + a b\right )} \sqrt{c} -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} +{\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )} \log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{6 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} + a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3}\right )} \sqrt{c}}, -\frac{2 \, \sqrt{c x^{6} + b x^{3} + a}{\left ({\left (b^{2} - 2 \, a c\right )} x^{3} + a b\right )} \sqrt{-c} -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} +{\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{3 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} + a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**8/(a + b*x**3 + c*x**6)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^8/(c*x^6 + b*x^3 + a)^(3/2), x)