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

Optimal. Leaf size=71 \[ -\frac{2 c}{27 d^3 \sqrt{c+d x^3}}-\frac{2 \sqrt{c+d x^3}}{3 d^3}+\frac{128 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^3} \]

[Out]

(-2*c)/(27*d^3*Sqrt[c + d*x^3]) - (2*Sqrt[c + d*x^3])/(3*d^3) + (128*Sqrt[c]*Arc
Tanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*d^3)

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Rubi [A]  time = 0.212102, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{2 c}{27 d^3 \sqrt{c+d x^3}}-\frac{2 \sqrt{c+d x^3}}{3 d^3}+\frac{128 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c)/(27*d^3*Sqrt[c + d*x^3]) - (2*Sqrt[c + d*x^3])/(3*d^3) + (128*Sqrt[c]*Arc
Tanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*d^3)

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Rubi in Sympy [A]  time = 26.4641, size = 65, normalized size = 0.92 \[ \frac{128 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{81 d^{3}} - \frac{2 c}{27 d^{3} \sqrt{c + d x^{3}}} - \frac{2 \sqrt{c + d x^{3}}}{3 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

128*sqrt(c)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(81*d**3) - 2*c/(27*d**3*sqrt(c
+ d*x**3)) - 2*sqrt(c + d*x**3)/(3*d**3)

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Mathematica [A]  time = 0.107375, size = 67, normalized size = 0.94 \[ \frac{2 \left (64 \sqrt{c} \sqrt{c+d x^3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-30 c-27 d x^3\right )}{81 d^3 \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-30*c - 27*d*x^3 + 64*Sqrt[c]*Sqrt[c + d*x^3]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqr
t[c])]))/(81*d^3*Sqrt[c + d*x^3])

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Maple [C]  time = 0.016, size = 501, normalized size = 7.1 \[ -{\frac{1}{{d}^{2}} \left ( d \left ({\frac{2\,c}{3\,{d}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{2}{3\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) -{\frac{16\,c}{3\,d}{\frac{1}{\sqrt{d{x}^{3}+c}}}} \right ) }-64\,{\frac{{c}^{2}}{{d}^{2}} \left ({\frac{2}{27\,cd}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{{\frac{i}{243}}\sqrt{2}}{{d}^{3}{c}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{\frac{\sqrt [3]{-c{d}^{2}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{2/3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}}} \left ( x-{\frac{\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{i\sqrt{3}d}{\sqrt [3]{-c{d}^{2}}} \left ( x+1/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) }},-1/18\,{\frac{2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/d^2*(d*(2/3/d^2*c/((x^3+c/d)*d)^(1/2)+2/3*(d*x^3+c)^(1/2)/d^2)-16/3*c/d/(d*x^
3+c)^(1/2))-64*c^2/d^2*(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)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232643, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (27 \, d x^{3} - 32 \, \sqrt{d x^{3} + c} \sqrt{c} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 30 \, c\right )}}{81 \, \sqrt{d x^{3} + c} d^{3}}, -\frac{2 \,{\left (27 \, d x^{3} - 64 \, \sqrt{d x^{3} + c} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) + 30 \, c\right )}}{81 \, \sqrt{d x^{3} + c} d^{3}}\right ] \]

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)),x, algorithm="fricas")

[Out]

[-2/81*(27*d*x^3 - 32*sqrt(d*x^3 + c)*sqrt(c)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqr
t(c) + 10*c)/(d*x^3 - 8*c)) + 30*c)/(sqrt(d*x^3 + c)*d^3), -2/81*(27*d*x^3 - 64*
sqrt(d*x^3 + c)*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) + 30*c)/(sqrt(d*x^
3 + 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+8*c)/(d*x**3+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219454, size = 78, normalized size = 1.1 \[ -\frac{128 \, c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{81 \, \sqrt{-c} d^{3}} - \frac{2 \, \sqrt{d x^{3} + c}}{3 \, d^{3}} - \frac{2 \, c}{27 \, \sqrt{d x^{3} + c} d^{3}} \]

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)),x, algorithm="giac")

[Out]

-128/81*c*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^3) - 2/3*sqrt(d*x^3 +
c)/d^3 - 2/27*c/(sqrt(d*x^3 + c)*d^3)