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

Optimal. Leaf size=153 \[ -\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{384 c^{7/2}}+\frac{\left (5 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{192 c^3}-\frac{5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{3/2}}{12 c} \]

[Out]

((5*b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(192*c^3) - (5*b*(a + b*
x^3 + c*x^6)^(3/2))/(72*c^2) + (x^3*(a + b*x^3 + c*x^6)^(3/2))/(12*c) - ((b^2 -
4*a*c)*(5*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])
])/(384*c^(7/2))

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Rubi [A]  time = 0.27897, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{384 c^{7/2}}+\frac{\left (5 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{192 c^3}-\frac{5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{3/2}}{12 c} \]

Antiderivative was successfully verified.

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

[Out]

((5*b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(192*c^3) - (5*b*(a + b*
x^3 + c*x^6)^(3/2))/(72*c^2) + (x^3*(a + b*x^3 + c*x^6)^(3/2))/(12*c) - ((b^2 -
4*a*c)*(5*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])
])/(384*c^(7/2))

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Rubi in Sympy [A]  time = 29.9161, size = 141, normalized size = 0.92 \[ - \frac{5 b \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{72 c^{2}} + \frac{x^{3} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{12 c} + \frac{\left (b + 2 c x^{3}\right ) \left (- 4 a c + 5 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{192 c^{3}} - \frac{\left (- 4 a c + b^{2}\right ) \left (- 4 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{384 c^{\frac{7}{2}}} \]

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]

-5*b*(a + b*x**3 + c*x**6)**(3/2)/(72*c**2) + x**3*(a + b*x**3 + c*x**6)**(3/2)/
(12*c) + (b + 2*c*x**3)*(-4*a*c + 5*b**2)*sqrt(a + b*x**3 + c*x**6)/(192*c**3) -
(-4*a*c + b**2)*(-4*a*c + 5*b**2)*atanh((b + 2*c*x**3)/(2*sqrt(c)*sqrt(a + b*x*
*3 + c*x**6)))/(384*c**(7/2))

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Mathematica [A]  time = 0.143219, size = 134, normalized size = 0.88 \[ \frac{2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (b \left (8 c^2 x^6-52 a c\right )+24 c^2 x^3 \left (a+2 c x^6\right )+15 b^3-10 b^2 c x^3\right )-3 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{1152 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(15*b^3 - 10*b^2*c*x^3 + 24*c^2*x^3*(a + 2*c*
x^6) + b*(-52*a*c + 8*c^2*x^6)) - 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*Log[b + 2*
c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/(1152*c^(7/2))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{x}^{8}\sqrt{c{x}^{6}+b{x}^{3}+a}\, dx \]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283242, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (48 \, c^{3} x^{9} + 8 \, b c^{2} x^{6} - 2 \,{\left (5 \, b^{2} c - 12 \, a c^{2}\right )} x^{3} + 15 \, b^{3} - 52 \, a b c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} + 3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left (4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{2304 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (48 \, c^{3} x^{9} + 8 \, b c^{2} x^{6} - 2 \,{\left (5 \, b^{2} c - 12 \, a c^{2}\right )} x^{3} + 15 \, b^{3} - 52 \, a b c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} - 3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{1152 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2304*(4*(48*c^3*x^9 + 8*b*c^2*x^6 - 2*(5*b^2*c - 12*a*c^2)*x^3 + 15*b^3 - 52*
a*b*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(c) + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*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^(7/2), 1/1152*(2*(48*c^3*x^9 + 8*b*c^2*x^6 - 2*(5*b^2*c - 12*a*
c^2)*x^3 + 15*b^3 - 52*a*b*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c) - 3*(5*b^4 - 24*a
*b^2*c + 16*a^2*c^2)*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)/(sqrt(c*x^6 + b*x^3 + a)*
c)))/(sqrt(-c)*c^3)]

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

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. \[ \int \sqrt{c x^{6} + b x^{3} + a} x^{8}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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