3.400 \(\int \frac{x^5 \sqrt{c+d x^3}}{\left (8 c-d x^3\right )^2} \, dx\)

Optimal. Leaf size=82 \[ \frac{8 \left (c+d x^3\right )^{3/2}}{27 d^2 \left (8 c-d x^3\right )}+\frac{26 \sqrt{c+d x^3}}{27 d^2}-\frac{26 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^2} \]

[Out]

(26*Sqrt[c + d*x^3])/(27*d^2) + (8*(c + d*x^3)^(3/2))/(27*d^2*(8*c - d*x^3)) - (
26*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d^2)

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Rubi [A]  time = 0.18679, antiderivative size = 82, 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 \[ \frac{8 \left (c+d x^3\right )^{3/2}}{27 d^2 \left (8 c-d x^3\right )}+\frac{26 \sqrt{c+d x^3}}{27 d^2}-\frac{26 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]

[Out]

(26*Sqrt[c + d*x^3])/(27*d^2) + (8*(c + d*x^3)^(3/2))/(27*d^2*(8*c - d*x^3)) - (
26*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d^2)

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Rubi in Sympy [A]  time = 21.3165, size = 71, normalized size = 0.87 \[ - \frac{26 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{9 d^{2}} + \frac{8 \left (c + d x^{3}\right )^{\frac{3}{2}}}{27 d^{2} \left (8 c - d x^{3}\right )} + \frac{26 \sqrt{c + d x^{3}}}{27 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(d*x**3+c)**(1/2)/(-d*x**3+8*c)**2,x)

[Out]

-26*sqrt(c)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(9*d**2) + 8*(c + d*x**3)**(3/2)
/(27*d**2*(8*c - d*x**3)) + 26*sqrt(c + d*x**3)/(27*d**2)

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Mathematica [A]  time = 0.140842, size = 66, normalized size = 0.8 \[ \frac{2 \left (3 \sqrt{c+d x^3} \left (\frac{4 c}{8 c-d x^3}+1\right )-13 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{9 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]

[Out]

(2*(3*Sqrt[c + d*x^3]*(1 + (4*c)/(8*c - d*x^3)) - 13*Sqrt[c]*ArcTanh[Sqrt[c + d*
x^3]/(3*Sqrt[c])]))/(9*d^2)

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Maple [C]  time = 0.016, size = 874, normalized size = 10.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^2,x)

[Out]

1/d*(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)))+
8*c/d*(-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=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(sqrt(d*x^3 + c)*x^5/(d*x^3 - 8*c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222271, size = 1, normalized size = 0.01 \[ \left [\frac{13 \,{\left (d x^{3} - 8 \, c\right )} \sqrt{c} \log \left (\frac{d x^{3} - 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 6 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 12 \, c\right )}}{9 \,{\left (d^{3} x^{3} - 8 \, c d^{2}\right )}}, -\frac{2 \,{\left (13 \,{\left (d x^{3} - 8 \, c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) - 3 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 12 \, c\right )}\right )}}{9 \,{\left (d^{3} x^{3} - 8 \, c d^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)*x^5/(d*x^3 - 8*c)^2,x, algorithm="fricas")

[Out]

[1/9*(13*(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)) + 6*sqrt(d*x^3 + c)*(d*x^3 - 12*c))/(d^3*x^3 - 8*c*d^2), -2/9*(13*(
d*x^3 - 8*c)*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) - 3*sqrt(d*x^3 + c)*(
d*x^3 - 12*c))/(d^3*x^3 - 8*c*d^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5} \sqrt{c + d x^{3}}}{\left (- 8 c + d x^{3}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(d*x**3+c)**(1/2)/(-d*x**3+8*c)**2,x)

[Out]

Integral(x**5*sqrt(c + d*x**3)/(-8*c + d*x**3)**2, x)

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GIAC/XCAS [A]  time = 0.216678, size = 100, normalized size = 1.22 \[ \frac{2 \,{\left (\frac{13 \, c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d} + \frac{3 \, \sqrt{d x^{3} + c}}{d} - \frac{12 \, \sqrt{d x^{3} + c} c}{{\left (d x^{3} - 8 \, c\right )} d}\right )}}{9 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)*x^5/(d*x^3 - 8*c)^2,x, algorithm="giac")

[Out]

2/9*(13*c*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d) + 3*sqrt(d*x^3 + c)/
d - 12*sqrt(d*x^3 + c)*c/((d*x^3 - 8*c)*d))/d