3.442 \(\int \frac{x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt{c+d x^3}}-\frac{640 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{243 d^4}+\frac{8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]

[Out]

(8*x^6)/(27*d^2*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (2*(38*c + 39*d*x^3))/(81*d^4*S
qrt[c + d*x^3]) - (640*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(243*d^4)

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Rubi [A]  time = 0.270759, antiderivative size = 95, 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 \[ \frac{2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt{c+d x^3}}-\frac{640 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{243 d^4}+\frac{8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^11/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]

[Out]

(8*x^6)/(27*d^2*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (2*(38*c + 39*d*x^3))/(81*d^4*S
qrt[c + d*x^3]) - (640*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(243*d^4)

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Rubi in Sympy [A]  time = 33.2562, size = 83, normalized size = 0.87 \[ - \frac{640 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{243 d^{4}} + \frac{8 x^{6}}{27 d^{2} \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} + \frac{4 \left (57 c + \frac{117 d x^{3}}{2}\right )}{243 d^{4} \sqrt{c + d x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)

[Out]

-640*sqrt(c)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(243*d**4) + 8*x**6/(27*d**2*sq
rt(c + d*x**3)*(8*c - d*x**3)) + 4*(57*c + 117*d*x**3/2)/(243*d**4*sqrt(c + d*x*
*3))

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Mathematica [A]  time = 0.301883, size = 81, normalized size = 0.85 \[ \frac{2 \left (\frac{912 c^2+822 c d x^3-81 d^2 x^6}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}-320 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{243 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]

[Out]

(2*((912*c^2 + 822*c*d*x^3 - 81*d^2*x^6)/((8*c - d*x^3)*Sqrt[c + d*x^3]) - 320*S
qrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]))/(243*d^4)

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Maple [C]  time = 0.067, size = 970, normalized size = 10.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x)

[Out]

1/d^3*(d*(2/3/d^2*c/((x^3+c/d)*d)^(1/2)+2/3*(d*x^3+c)^(1/2)/d^2)-32/3*c/d/(d*x^3
+c)^(1/2))+192*c^2/d^3*(2/27/d/c/((x^3+c/d)*d)^(1/2)+1/243*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)*_alpha*3^(1/2)*d+2*_alph
a^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*Ellipti
cPi(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*_alp
ha*(-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)))+512*c^3/d^3*(-1/243/d/c^2*(d*x^3+c)^(1/2)/(d*x^3
-8*c)-2/243/d/c^2/((x^3+c/d)*d)^(1/2)-1/1458*I/d^3/c^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)*_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=Roo
tOf(_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^11/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223282, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (81 \, d^{2} x^{6} - 822 \, c d x^{3} + 160 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \sqrt{c} \log \left (\frac{d x^{3} - 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - 912 \, c^{2}\right )}}{243 \,{\left (d^{5} x^{3} - 8 \, c d^{4}\right )} \sqrt{d x^{3} + c}}, \frac{2 \,{\left (81 \, d^{2} x^{6} - 822 \, c d x^{3} - 320 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) - 912 \, c^{2}\right )}}{243 \,{\left (d^{5} x^{3} - 8 \, c d^{4}\right )} \sqrt{d x^{3} + c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2),x, algorithm="fricas")

[Out]

[2/243*(81*d^2*x^6 - 822*c*d*x^3 + 160*sqrt(d*x^3 + c)*(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)) - 912*c^2)/((d^5*x^3
- 8*c*d^4)*sqrt(d*x^3 + c)), 2/243*(81*d^2*x^6 - 822*c*d*x^3 - 320*sqrt(d*x^3 +
c)*(d*x^3 - 8*c)*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) - 912*c^2)/((d^5*
x^3 - 8*c*d^4)*sqrt(d*x^3 + c))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226763, size = 119, normalized size = 1.25 \[ \frac{640 \, c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{243 \, \sqrt{-c} d^{4}} + \frac{2 \, \sqrt{d x^{3} + c}}{3 \, d^{4}} - \frac{2 \,{\left (85 \,{\left (d x^{3} + c\right )} c + 3 \, c^{2}\right )}}{81 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2),x, algorithm="giac")

[Out]

640/243*c*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^4) + 2/3*sqrt(d*x^3 +
c)/d^4 - 2/81*(85*(d*x^3 + c)*c + 3*c^2)/(((d*x^3 + c)^(3/2) - 9*sqrt(d*x^3 + c
)*c)*d^4)