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

Optimal. Leaf size=102 \[ -\frac{352 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^3}+\frac{64 c \left (c+d x^3\right )^{3/2}}{27 d^3 \left (8 c-d x^3\right )}+\frac{352 c \sqrt{c+d x^3}}{27 d^3}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3} \]

[Out]

(352*c*Sqrt[c + d*x^3])/(27*d^3) + (2*(c + d*x^3)^(3/2))/(9*d^3) + (64*c*(c + d*
x^3)^(3/2))/(27*d^3*(8*c - d*x^3)) - (352*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqr
t[c])])/(9*d^3)

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Rubi [A]  time = 0.256414, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{352 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^3}+\frac{64 c \left (c+d x^3\right )^{3/2}}{27 d^3 \left (8 c-d x^3\right )}+\frac{352 c \sqrt{c+d x^3}}{27 d^3}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(352*c*Sqrt[c + d*x^3])/(27*d^3) + (2*(c + d*x^3)^(3/2))/(9*d^3) + (64*c*(c + d*
x^3)^(3/2))/(27*d^3*(8*c - d*x^3)) - (352*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqr
t[c])])/(9*d^3)

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Rubi in Sympy [A]  time = 30.5617, size = 92, normalized size = 0.9 \[ - \frac{352 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{9 d^{3}} + \frac{64 c \left (c + d x^{3}\right )^{\frac{3}{2}}}{27 d^{3} \left (8 c - d x^{3}\right )} + \frac{352 c \sqrt{c + d x^{3}}}{27 d^{3}} + \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-352*c**(3/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(9*d**3) + 64*c*(c + d*x**3)**
(3/2)/(27*d**3*(8*c - d*x**3)) + 352*c*sqrt(c + d*x**3)/(27*d**3) + 2*(c + d*x**
3)**(3/2)/(9*d**3)

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Mathematica [A]  time = 0.154123, size = 79, normalized size = 0.77 \[ \frac{2 \left (\frac{\sqrt{c+d x^3} \left (-488 c^2+41 c d x^3+d^2 x^6\right )}{d x^3-8 c}-176 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{9 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[c + d*x^3]*(-488*c^2 + 41*c*d*x^3 + d^2*x^6))/(-8*c + d*x^3) - 176*c^(
3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]))/(9*d^3)

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Maple [C]  time = 0.019, size = 892, normalized size = 8.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(d*x^3+c)^(3/2)/d^3+64*c^2/d^2*(-1/3/d*(d*x^3+c)^(1/2)/(d*x^3-8*c)+1/54*I/d^
3/c*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)))+16*c/d^2*(2/3*(d*x^3+c)^(1/2)/d+1
/3*I/d^3*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)*_alp
ha*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(sqrt(d*x^3 + c)*x^8/(d*x^3 - 8*c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221799, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (88 \,{\left (c d x^{3} - 8 \, c^{2}\right )} \sqrt{c} \log \left (\frac{d x^{3} - 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) +{\left (d^{2} x^{6} + 41 \, c d x^{3} - 488 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{9 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}, -\frac{2 \,{\left (176 \,{\left (c d x^{3} - 8 \, c^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) -{\left (d^{2} x^{6} + 41 \, c d x^{3} - 488 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{9 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2/9*(88*(c*d*x^3 - 8*c^2)*sqrt(c)*log((d*x^3 - 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c
)/(d*x^3 - 8*c)) + (d^2*x^6 + 41*c*d*x^3 - 488*c^2)*sqrt(d*x^3 + c))/(d^4*x^3 -
8*c*d^3), -2/9*(176*(c*d*x^3 - 8*c^2)*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-
c)) - (d^2*x^6 + 41*c*d*x^3 - 488*c^2)*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+c)**(1/2)/(-d*x**3+8*c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216427, size = 126, normalized size = 1.24 \[ \frac{352 \, c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{9 \, \sqrt{-c} d^{3}} - \frac{64 \, \sqrt{d x^{3} + c} c^{2}}{3 \,{\left (d x^{3} - 8 \, c\right )} d^{3}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{6} + 48 \, \sqrt{d x^{3} + c} c d^{6}\right )}}{9 \, d^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

352/9*c^2*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^3) - 64/3*sqrt(d*x^3
+ c)*c^2/((d*x^3 - 8*c)*d^3) + 2/9*((d*x^3 + c)^(3/2)*d^6 + 48*sqrt(d*x^3 + c)*c
*d^6)/d^9