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

Optimal. Leaf size=117 \[ -\frac{3968 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^4}+\frac{2 c \sqrt{c+d x^3} \left (1146 c+47 d x^3\right )}{15 d^4}+\frac{7 x^6 \sqrt{c+d x^3}}{15 d^2}+\frac{x^9 \sqrt{c+d x^3}}{3 d \left (8 c-d x^3\right )} \]

[Out]

(7*x^6*Sqrt[c + d*x^3])/(15*d^2) + (x^9*Sqrt[c + d*x^3])/(3*d*(8*c - d*x^3)) + (
2*c*Sqrt[c + d*x^3]*(1146*c + 47*d*x^3))/(15*d^4) - (3968*c^(5/2)*ArcTanh[Sqrt[c
+ d*x^3]/(3*Sqrt[c])])/(9*d^4)

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Rubi [A]  time = 0.348105, antiderivative size = 117, 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 \[ -\frac{3968 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^4}+\frac{2 c \sqrt{c+d x^3} \left (1146 c+47 d x^3\right )}{15 d^4}+\frac{7 x^6 \sqrt{c+d x^3}}{15 d^2}+\frac{x^9 \sqrt{c+d x^3}}{3 d \left (8 c-d x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(7*x^6*Sqrt[c + d*x^3])/(15*d^2) + (x^9*Sqrt[c + d*x^3])/(3*d*(8*c - d*x^3)) + (
2*c*Sqrt[c + d*x^3]*(1146*c + 47*d*x^3))/(15*d^4) - (3968*c^(5/2)*ArcTanh[Sqrt[c
+ d*x^3]/(3*Sqrt[c])])/(9*d^4)

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Rubi in Sympy [A]  time = 43.6136, size = 107, normalized size = 0.91 \[ - \frac{3968 c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{9 d^{4}} + \frac{8 c \sqrt{c + d x^{3}} \left (\frac{1719 c}{2} + \frac{141 d x^{3}}{4}\right )}{45 d^{4}} + \frac{x^{9} \sqrt{c + d x^{3}}}{3 d \left (8 c - d x^{3}\right )} + \frac{7 x^{6} \sqrt{c + d x^{3}}}{15 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(d*x**3+c)**(1/2)/(-d*x**3+8*c)**2,x)

[Out]

-3968*c**(5/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(9*d**4) + 8*c*sqrt(c + d*x**
3)*(1719*c/2 + 141*d*x**3/4)/(45*d**4) + x**9*sqrt(c + d*x**3)/(3*d*(8*c - d*x**
3)) + 7*x**6*sqrt(c + d*x**3)/(15*d**2)

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Mathematica [A]  time = 0.149321, size = 100, normalized size = 0.85 \[ \frac{1}{3} \sqrt{c+d x^3} \left (-\frac{512 c^3}{d^4 \left (d x^3-8 c\right )}+\frac{1972 c^2}{5 d^4}+\frac{54 c x^3}{5 d^3}+\frac{2 x^6}{5 d^2}\right )-\frac{3968 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^3]*((1972*c^2)/(5*d^4) + (54*c*x^3)/(5*d^3) + (2*x^6)/(5*d^2) - (5
12*c^3)/(d^4*(-8*c + d*x^3))))/3 - (3968*c^(5/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt
[c])])/(9*d^4)

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Maple [C]  time = 0.058, size = 952, normalized size = 8.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^2,x)

[Out]

1/d^3*(d*(2/15*x^6*(d*x^3+c)^(1/2)+2/45*c/d*x^3*(d*x^3+c)^(1/2)-4/45*c^2*(d*x^3+
c)^(1/2)/d^2)+32/9*c/d*(d*x^3+c)^(3/2))+192*c^2/d^3*(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)*_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)))+512*c^3/d^3*(-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)))

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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^11/(d*x^3 - 8*c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221446, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (4960 \,{\left (c^{2} d x^{3} - 8 \, c^{3}\right )} \sqrt{c} \log \left (\frac{d x^{3} - 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 3 \,{\left (d^{3} x^{9} + 19 \, c d^{2} x^{6} + 770 \, c^{2} d x^{3} - 9168 \, c^{3}\right )} \sqrt{d x^{3} + c}\right )}}{45 \,{\left (d^{5} x^{3} - 8 \, c d^{4}\right )}}, -\frac{2 \,{\left (9920 \,{\left (c^{2} d x^{3} - 8 \, c^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) - 3 \,{\left (d^{3} x^{9} + 19 \, c d^{2} x^{6} + 770 \, c^{2} d x^{3} - 9168 \, c^{3}\right )} \sqrt{d x^{3} + c}\right )}}{45 \,{\left (d^{5} x^{3} - 8 \, c d^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2/45*(4960*(c^2*d*x^3 - 8*c^3)*sqrt(c)*log((d*x^3 - 6*sqrt(d*x^3 + c)*sqrt(c) +
10*c)/(d*x^3 - 8*c)) + 3*(d^3*x^9 + 19*c*d^2*x^6 + 770*c^2*d*x^3 - 9168*c^3)*sq
rt(d*x^3 + c))/(d^5*x^3 - 8*c*d^4), -2/45*(9920*(c^2*d*x^3 - 8*c^3)*sqrt(-c)*arc
tan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) - 3*(d^3*x^9 + 19*c*d^2*x^6 + 770*c^2*d*x^3 -
9168*c^3)*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)**(1/2)/(-d*x**3+8*c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216151, size = 149, normalized size = 1.27 \[ \frac{3968 \, c^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{9 \, \sqrt{-c} d^{4}} - \frac{512 \, \sqrt{d x^{3} + c} c^{3}}{3 \,{\left (d x^{3} - 8 \, c\right )} d^{4}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{5}{2}} d^{16} + 25 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c d^{16} + 960 \, \sqrt{d x^{3} + c} c^{2} d^{16}\right )}}{15 \, d^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3968/9*c^3*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^4) - 512/3*sqrt(d*x^
3 + c)*c^3/((d*x^3 - 8*c)*d^4) + 2/15*((d*x^3 + c)^(5/2)*d^16 + 25*(d*x^3 + c)^(
3/2)*c*d^16 + 960*sqrt(d*x^3 + c)*c^2*d^16)/d^20