3.285 \(\int \frac{x^2 \sqrt{c+d x^3}}{8 c-d x^3} \, dx\)

Optimal. Leaf size=50 \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d}-\frac{2 \sqrt{c+d x^3}}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.147602, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d}-\frac{2 \sqrt{c+d x^3}}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.2959, size = 41, normalized size = 0.82 \[ \frac{2 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{d} - \frac{2 \sqrt{c + d x^{3}}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0399521, size = 47, normalized size = 0.94 \[ -\frac{2 \left (\sqrt{c+d x^3}-3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.011, size = 425, normalized size = 8.5 \[ -{\frac{2}{3\,d}\sqrt{d{x}^{3}+c}}-{\frac{{\frac{i}{3}}\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244359, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{c} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - 2 \, \sqrt{d x^{3} + c}}{3 \, d}, \frac{2 \,{\left (3 \, \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) - \sqrt{d x^{3} + c}\right )}}{3 \, d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(3*sqrt(c)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) -
2*sqrt(d*x^3 + c))/d, 2/3*(3*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) - sqr
t(d*x^3 + c))/d]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213088, size = 58, normalized size = 1.16 \[ -\frac{2 \, c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d} - \frac{2 \, \sqrt{d x^{3} + c}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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