3.425 \(\int \frac{x^8}{\left (8 c-d x^3\right )^2 \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=83 \[ \frac{64 c \sqrt{c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}+\frac{2 \sqrt{c+d x^3}}{3 d^3}-\frac{224 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^3} \]

[Out]

(2*Sqrt[c + d*x^3])/(3*d^3) + (64*c*Sqrt[c + d*x^3])/(27*d^3*(8*c - d*x^3)) - (2
24*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*d^3)

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Rubi [A]  time = 0.228534, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{64 c \sqrt{c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}+\frac{2 \sqrt{c+d x^3}}{3 d^3}-\frac{224 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]

[Out]

(2*Sqrt[c + d*x^3])/(3*d^3) + (64*c*Sqrt[c + d*x^3])/(27*d^3*(8*c - d*x^3)) - (2
24*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*d^3)

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Rubi in Sympy [A]  time = 26.1207, size = 73, normalized size = 0.88 \[ - \frac{224 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{81 d^{3}} + \frac{64 c \sqrt{c + d x^{3}}}{27 d^{3} \left (8 c - d x^{3}\right )} + \frac{2 \sqrt{c + d x^{3}}}{3 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)

[Out]

-224*sqrt(c)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(81*d**3) + 64*c*sqrt(c + d*x**
3)/(27*d**3*(8*c - d*x**3)) + 2*sqrt(c + d*x**3)/(3*d**3)

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Mathematica [A]  time = 0.152742, size = 66, normalized size = 0.8 \[ \frac{2 \left (3 \sqrt{c+d x^3} \left (\frac{32 c}{8 c-d x^3}+9\right )-112 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{81 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]

[Out]

(2*(3*Sqrt[c + d*x^3]*(9 + (32*c)/(8*c - d*x^3)) - 112*Sqrt[c]*ArcTanh[Sqrt[c +
d*x^3]/(3*Sqrt[c])]))/(81*d^3)

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Maple [C]  time = 0.017, size = 874, normalized size = 10.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x)

[Out]

2/3*(d*x^3+c)^(1/2)/d^3+64*c^2/d^2*(-1/27/d/c*(d*x^3+c)^(1/2)/(d*x^3-8*c)-1/486*
I/d^3/c^2*2^(1/2)*sum((-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3
)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^
(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1
/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_al
pha*3^(1/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-
c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*
(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),-1/18/d*(2*I*_alpha^2*(-c*d^2)^(
1/3)*3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(
2/3)-3*c*d)/c,(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d
*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d-8*c)))+16/27*I/d^5*2^(1/2)*sum((-c
*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^
2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^
(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*
d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d+2*_alpha^2*
d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(
1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d
/(-c*d^2)^(1/3))^(1/2),-1/18/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)*d-I*_alpha*(
-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/c,(I*3^(1/2)/
d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),
_alpha=RootOf(_Z^3*d-8*c))

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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(d*x^3 + c)*(d*x^3 - 8*c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227659, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (56 \,{\left (d x^{3} - 8 \, c\right )} \sqrt{c} \log \left (\frac{d x^{3} - 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 3 \,{\left (9 \, d x^{3} - 104 \, c\right )} \sqrt{d x^{3} + c}\right )}}{81 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}, -\frac{2 \,{\left (112 \,{\left (d x^{3} - 8 \, c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) - 3 \,{\left (9 \, d x^{3} - 104 \, c\right )} \sqrt{d x^{3} + c}\right )}}{81 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2),x, algorithm="fricas")

[Out]

[2/81*(56*(d*x^3 - 8*c)*sqrt(c)*log((d*x^3 - 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(
d*x^3 - 8*c)) + 3*(9*d*x^3 - 104*c)*sqrt(d*x^3 + c))/(d^4*x^3 - 8*c*d^3), -2/81*
(112*(d*x^3 - 8*c)*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) - 3*(9*d*x^3 -
104*c)*sqrt(d*x^3 + c))/(d^4*x^3 - 8*c*d^3)]

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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/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215078, size = 93, normalized size = 1.12 \[ \frac{224 \, c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{81 \, \sqrt{-c} d^{3}} + \frac{2 \, \sqrt{d x^{3} + c}}{3 \, d^{3}} - \frac{64 \, \sqrt{d x^{3} + c} c}{27 \,{\left (d x^{3} - 8 \, c\right )} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2),x, algorithm="giac")

[Out]

224/81*c*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^3) + 2/3*sqrt(d*x^3 +
c)/d^3 - 64/27*sqrt(d*x^3 + c)*c/((d*x^3 - 8*c)*d^3)