3.284 \(\int \frac{x^5 \sqrt{c+d x^3}}{8 c-d x^3} \, dx\)

Optimal. Leaf size=69 \[ \frac{16 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^2}-\frac{16 c \sqrt{c+d x^3}}{3 d^2}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2} \]

[Out]

(-16*c*Sqrt[c + d*x^3])/(3*d^2) - (2*(c + d*x^3)^(3/2))/(9*d^2) + (16*c^(3/2)*Ar
cTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^2

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Rubi [A]  time = 0.193912, antiderivative size = 69, 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 \[ \frac{16 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^2}-\frac{16 c \sqrt{c+d x^3}}{3 d^2}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(-16*c*Sqrt[c + d*x^3])/(3*d^2) - (2*(c + d*x^3)^(3/2))/(9*d^2) + (16*c^(3/2)*Ar
cTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^2

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Rubi in Sympy [A]  time = 19.5435, size = 63, normalized size = 0.91 \[ \frac{16 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{d^{2}} - \frac{16 c \sqrt{c + d x^{3}}}{3 d^{2}} - \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{2}} \]

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),x)

[Out]

16*c**(3/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/d**2 - 16*c*sqrt(c + d*x**3)/(3*
d**2) - 2*(c + d*x**3)**(3/2)/(9*d**2)

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Mathematica [A]  time = 0.0721411, size = 58, normalized size = 0.84 \[ \frac{144 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-2 \sqrt{c+d x^3} \left (25 c+d x^3\right )}{9 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c + d*x^3]*(25*c + d*x^3) + 144*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt
[c])])/(9*d^2)

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Maple [C]  time = 0.015, size = 446, normalized size = 6.5 \[ -{\frac{2}{9\,{d}^{2}} \left ( d{x}^{3}+c \right ) ^{{\frac{3}{2}}}}-8\,{\frac{c}{d} \left ( 2/3\,{\frac{\sqrt{d{x}^{3}+c}}{d}}+{\frac{i/3\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{\frac{\sqrt [3]{-c{d}^{2}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{2/3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}}} \left ( x-{\frac{\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{i\sqrt{3}d}{\sqrt [3]{-c{d}^{2}}} \left ( x+1/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) }},-1/18\,{\frac{2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(d*x^3+c)^(3/2)/d^2-8*c/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)))

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239096, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (36 \, c^{\frac{3}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) -{\left (d x^{3} + 25 \, c\right )} \sqrt{d x^{3} + c}\right )}}{9 \, d^{2}}, \frac{2 \,{\left (72 \, \sqrt{-c} c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) -{\left (d x^{3} + 25 \, c\right )} \sqrt{d x^{3} + c}\right )}}{9 \, d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2/9*(36*c^(3/2)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) -
(d*x^3 + 25*c)*sqrt(d*x^3 + c))/d^2, 2/9*(72*sqrt(-c)*c*arctan(1/3*sqrt(d*x^3 +
c)/sqrt(-c)) - (d*x^3 + 25*c)*sqrt(d*x^3 + c))/d^2]

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

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213697, size = 93, normalized size = 1.35 \[ -\frac{2 \,{\left (\frac{72 \, c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d} + \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{2} + 24 \, \sqrt{d x^{3} + c} c d^{2}}{d^{3}}\right )}}{9 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(72*c^2*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d) + ((d*x^3 + c)^(3
/2)*d^2 + 24*sqrt(d*x^3 + c)*c*d^2)/d^3)/d