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

Optimal. Leaf size=134 \[ -\frac{4992 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^4}+\frac{1664 c^3 \sqrt{c+d x^3}}{d^4}+\frac{2 c \left (c+d x^3\right )^{3/2} \left (694 c+51 d x^3\right )}{21 d^4}+\frac{3 x^6 \left (c+d x^3\right )^{3/2}}{7 d^2}+\frac{x^9 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )} \]

[Out]

(1664*c^3*Sqrt[c + d*x^3])/d^4 + (3*x^6*(c + d*x^3)^(3/2))/(7*d^2) + (x^9*(c + d
*x^3)^(3/2))/(3*d*(8*c - d*x^3)) + (2*c*(c + d*x^3)^(3/2)*(694*c + 51*d*x^3))/(2
1*d^4) - (4992*c^(7/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^4

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Rubi [A]  time = 0.374853, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{4992 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^4}+\frac{1664 c^3 \sqrt{c+d x^3}}{d^4}+\frac{2 c \left (c+d x^3\right )^{3/2} \left (694 c+51 d x^3\right )}{21 d^4}+\frac{3 x^6 \left (c+d x^3\right )^{3/2}}{7 d^2}+\frac{x^9 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(1664*c^3*Sqrt[c + d*x^3])/d^4 + (3*x^6*(c + d*x^3)^(3/2))/(7*d^2) + (x^9*(c + d
*x^3)^(3/2))/(3*d*(8*c - d*x^3)) + (2*c*(c + d*x^3)^(3/2)*(694*c + 51*d*x^3))/(2
1*d^4) - (4992*c^(7/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^4

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Rubi in Sympy [A]  time = 51.7636, size = 124, normalized size = 0.93 \[ - \frac{4992 c^{\frac{7}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{d^{4}} + \frac{1664 c^{3} \sqrt{c + d x^{3}}}{d^{4}} + \frac{8 c \left (c + d x^{3}\right )^{\frac{3}{2}} \left (\frac{5205 c}{2} + \frac{765 d x^{3}}{4}\right )}{315 d^{4}} + \frac{x^{9} \left (c + d x^{3}\right )^{\frac{3}{2}}}{3 d \left (8 c - d x^{3}\right )} + \frac{3 x^{6} \left (c + d x^{3}\right )^{\frac{3}{2}}}{7 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4992*c**(7/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/d**4 + 1664*c**3*sqrt(c + d*x
**3)/d**4 + 8*c*(c + d*x**3)**(3/2)*(5205*c/2 + 765*d*x**3/4)/(315*d**4) + x**9*
(c + d*x**3)**(3/2)/(3*d*(8*c - d*x**3)) + 3*x**6*(c + d*x**3)**(3/2)/(7*d**2)

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Mathematica [A]  time = 0.277794, size = 101, normalized size = 0.75 \[ \frac{2 \left (\frac{\sqrt{c+d x^3} \left (-145328 c^4+12206 c^3 d x^3+301 c^2 d^2 x^6+16 c d^3 x^9+d^4 x^{12}\right )}{d x^3-8 c}-52416 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{21 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[c + d*x^3]*(-145328*c^4 + 12206*c^3*d*x^3 + 301*c^2*d^2*x^6 + 16*c*d^3
*x^9 + d^4*x^12))/(-8*c + d*x^3) - 52416*c^(7/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt
[c])]))/(21*d^4)

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Maple [C]  time = 0.059, size = 998, normalized size = 7.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/d^3*(d*(2/21*d*x^9*(d*x^3+c)^(1/2)+16/105*c*x^6*(d*x^3+c)^(1/2)+2/105*c^2/d*x^
3*(d*x^3+c)^(1/2)-4/105*c^3/d^2*(d*x^3+c)^(1/2))+32/15*c/d*(d*x^3+c)^(5/2))+192*
c^2/d^3*(2/9*x^3*(d*x^3+c)^(1/2)+56/9*c*(d*x^3+c)^(1/2)/d+3*I*c/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)*_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))*Elliptic
Pi(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*_alph
a*(-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*(-3*c/d*(d*x^3+c)^(1/2)/(d*x^3-8*c)+2
/3*(d*x^3+c)^(1/2)/d+1/2*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)*_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((d*x^3 + c)^(3/2)*x^11/(d*x^3 - 8*c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227099, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (26208 \,{\left (c^{3} d x^{3} - 8 \, c^{4}\right )} \sqrt{c} \log \left (\frac{d x^{3} - 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) +{\left (d^{4} x^{12} + 16 \, c d^{3} x^{9} + 301 \, c^{2} d^{2} x^{6} + 12206 \, c^{3} d x^{3} - 145328 \, c^{4}\right )} \sqrt{d x^{3} + c}\right )}}{21 \,{\left (d^{5} x^{3} - 8 \, c d^{4}\right )}}, -\frac{2 \,{\left (52416 \,{\left (c^{3} d x^{3} - 8 \, c^{4}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) -{\left (d^{4} x^{12} + 16 \, c d^{3} x^{9} + 301 \, c^{2} d^{2} x^{6} + 12206 \, c^{3} d x^{3} - 145328 \, c^{4}\right )} \sqrt{d x^{3} + c}\right )}}{21 \,{\left (d^{5} x^{3} - 8 \, c d^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2/21*(26208*(c^3*d*x^3 - 8*c^4)*sqrt(c)*log((d*x^3 - 6*sqrt(d*x^3 + c)*sqrt(c)
+ 10*c)/(d*x^3 - 8*c)) + (d^4*x^12 + 16*c*d^3*x^9 + 301*c^2*d^2*x^6 + 12206*c^3*
d*x^3 - 145328*c^4)*sqrt(d*x^3 + c))/(d^5*x^3 - 8*c*d^4), -2/21*(52416*(c^3*d*x^
3 - 8*c^4)*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) - (d^4*x^12 + 16*c*d^3*
x^9 + 301*c^2*d^2*x^6 + 12206*c^3*d*x^3 - 145328*c^4)*sqrt(d*x^3 + c))/(d^5*x^3
- 8*c*d^4)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219363, size = 171, normalized size = 1.28 \[ \frac{4992 \, c^{4} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d^{4}} - \frac{1536 \, \sqrt{d x^{3} + c} c^{4}}{{\left (d x^{3} - 8 \, c\right )} d^{4}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{7}{2}} d^{24} + 21 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} c d^{24} + 448 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c^{2} d^{24} + 15680 \, \sqrt{d x^{3} + c} c^{3} d^{24}\right )}}{21 \, d^{28}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4992*c^4*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^4) - 1536*sqrt(d*x^3 +
c)*c^4/((d*x^3 - 8*c)*d^4) + 2/21*((d*x^3 + c)^(7/2)*d^24 + 21*(d*x^3 + c)^(5/2
)*c*d^24 + 448*(d*x^3 + c)^(3/2)*c^2*d^24 + 15680*sqrt(d*x^3 + c)*c^3*d^24)/d^28