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

Optimal. Leaf size=119 \[ -\frac{480 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^3}+\frac{160 c^2 \sqrt{c+d x^3}}{d^3}+\frac{64 c \left (c+d x^3\right )^{5/2}}{27 d^3 \left (8 c-d x^3\right )}+\frac{160 c \left (c+d x^3\right )^{3/2}}{27 d^3}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^3} \]

[Out]

(160*c^2*Sqrt[c + d*x^3])/d^3 + (160*c*(c + d*x^3)^(3/2))/(27*d^3) + (2*(c + d*x
^3)^(5/2))/(15*d^3) + (64*c*(c + d*x^3)^(5/2))/(27*d^3*(8*c - d*x^3)) - (480*c^(
5/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^3

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Rubi [A]  time = 0.290337, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{480 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^3}+\frac{160 c^2 \sqrt{c+d x^3}}{d^3}+\frac{64 c \left (c+d x^3\right )^{5/2}}{27 d^3 \left (8 c-d x^3\right )}+\frac{160 c \left (c+d x^3\right )^{3/2}}{27 d^3}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(160*c^2*Sqrt[c + d*x^3])/d^3 + (160*c*(c + d*x^3)^(3/2))/(27*d^3) + (2*(c + d*x
^3)^(5/2))/(15*d^3) + (64*c*(c + d*x^3)^(5/2))/(27*d^3*(8*c - d*x^3)) - (480*c^(
5/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^3

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Rubi in Sympy [A]  time = 37.1209, size = 109, normalized size = 0.92 \[ - \frac{480 c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{d^{3}} + \frac{160 c^{2} \sqrt{c + d x^{3}}}{d^{3}} + \frac{64 c \left (c + d x^{3}\right )^{\frac{5}{2}}}{27 d^{3} \left (8 c - d x^{3}\right )} + \frac{160 c \left (c + d x^{3}\right )^{\frac{3}{2}}}{27 d^{3}} + \frac{2 \left (c + d x^{3}\right )^{\frac{5}{2}}}{15 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-480*c**(5/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/d**3 + 160*c**2*sqrt(c + d*x**
3)/d**3 + 64*c*(c + d*x**3)**(5/2)/(27*d**3*(8*c - d*x**3)) + 160*c*(c + d*x**3)
**(3/2)/(27*d**3) + 2*(c + d*x**3)**(5/2)/(15*d**3)

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Mathematica [A]  time = 0.254843, size = 91, normalized size = 0.76 \[ \frac{2 \left (\frac{\sqrt{c+d x^3} \left (-29944 c^3+2515 c^2 d x^3+62 c d^2 x^6+3 d^3 x^9\right )}{d x^3-8 c}-10800 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{45 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[c + d*x^3]*(-29944*c^3 + 2515*c^2*d*x^3 + 62*c*d^2*x^6 + 3*d^3*x^9))/(
-8*c + d*x^3) - 10800*c^(5/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]))/(45*d^3)

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Maple [C]  time = 0.019, size = 920, normalized size = 7.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(d*x^3+c)^(5/2)/d^3+64*c^2/d^2*(-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*_alph
a^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*_al
pha*(-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*(2/9
*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))*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=Ro
otOf(_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^8/(d*x^3 - 8*c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229451, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (5400 \,{\left (c^{2} d x^{3} - 8 \, c^{3}\right )} \sqrt{c} \log \left (\frac{d x^{3} - 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) +{\left (3 \, d^{3} x^{9} + 62 \, c d^{2} x^{6} + 2515 \, c^{2} d x^{3} - 29944 \, c^{3}\right )} \sqrt{d x^{3} + c}\right )}}{45 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}, -\frac{2 \,{\left (10800 \,{\left (c^{2} d x^{3} - 8 \, c^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) -{\left (3 \, d^{3} x^{9} + 62 \, c d^{2} x^{6} + 2515 \, c^{2} d x^{3} - 29944 \, c^{3}\right )} \sqrt{d x^{3} + c}\right )}}{45 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2/45*(5400*(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 + 62*c*d^2*x^6 + 2515*c^2*d*x^3 - 29944*c^3)*
sqrt(d*x^3 + c))/(d^4*x^3 - 8*c*d^3), -2/45*(10800*(c^2*d*x^3 - 8*c^3)*sqrt(-c)*
arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) - (3*d^3*x^9 + 62*c*d^2*x^6 + 2515*c^2*d*x^
3 - 29944*c^3)*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)**(3/2)/(-d*x**3+8*c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218764, size = 150, normalized size = 1.26 \[ \frac{480 \, c^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d^{3}} - \frac{192 \, \sqrt{d x^{3} + c} c^{3}}{{\left (d x^{3} - 8 \, c\right )} d^{3}} + \frac{2 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} d^{12} + 80 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c d^{12} + 3120 \, \sqrt{d x^{3} + c} c^{2} d^{12}\right )}}{45 \, d^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

480*c^3*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^3) - 192*sqrt(d*x^3 + c
)*c^3/((d*x^3 - 8*c)*d^3) + 2/45*(3*(d*x^3 + c)^(5/2)*d^12 + 80*(d*x^3 + c)^(3/2
)*c*d^12 + 3120*sqrt(d*x^3 + c)*c^2*d^12)/d^15