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

Optimal. Leaf size=67 \[ \frac{18 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d}-\frac{6 c \sqrt{c+d x^3}}{d}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d} \]

[Out]

(-6*c*Sqrt[c + d*x^3])/d - (2*(c + d*x^3)^(3/2))/(9*d) + (18*c^(3/2)*ArcTanh[Sqr
t[c + d*x^3]/(3*Sqrt[c])])/d

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Rubi [A]  time = 0.181104, antiderivative size = 67, 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 \[ \frac{18 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d}-\frac{6 c \sqrt{c+d x^3}}{d}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d} \]

Antiderivative was successfully verified.

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

[Out]

(-6*c*Sqrt[c + d*x^3])/d - (2*(c + d*x^3)^(3/2))/(9*d) + (18*c^(3/2)*ArcTanh[Sqr
t[c + d*x^3]/(3*Sqrt[c])])/d

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Rubi in Sympy [A]  time = 19.901, size = 56, normalized size = 0.84 \[ \frac{18 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{d} - \frac{6 c \sqrt{c + d x^{3}}}{d} - \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18*c**(3/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/d - 6*c*sqrt(c + d*x**3)/d - 2*(
c + d*x**3)**(3/2)/(9*d)

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Mathematica [A]  time = 0.0787161, size = 58, normalized size = 0.87 \[ \frac{162 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-2 \sqrt{c+d x^3} \left (28 c+d x^3\right )}{9 d} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c + d*x^3]*(28*c + d*x^3) + 162*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt
[c])])/(9*d)

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Maple [C]  time = 0.011, size = 441, normalized size = 6.6 \[ -{\frac{2\,{x}^{3}}{9}\sqrt{d{x}^{3}+c}}-{\frac{56\,c}{9\,d}\sqrt{d{x}^{3}+c}}-{\frac{3\,ic\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245254, size = 1, normalized size = 0.01 \[ \left [\frac{81 \, c^{\frac{3}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - 2 \,{\left (d x^{3} + 28 \, c\right )} \sqrt{d x^{3} + c}}{9 \, d}, \frac{2 \,{\left (81 \, \sqrt{-c} c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) -{\left (d x^{3} + 28 \, c\right )} \sqrt{d x^{3} + c}\right )}}{9 \, d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/9*(81*c^(3/2)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) -
2*(d*x^3 + 28*c)*sqrt(d*x^3 + c))/d, 2/9*(81*sqrt(-c)*c*arctan(1/3*sqrt(d*x^3 +
c)/sqrt(-c)) - (d*x^3 + 28*c)*sqrt(d*x^3 + c))/d]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217737, size = 88, normalized size = 1.31 \[ -\frac{18 \, c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d} - \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{2} + 27 \, \sqrt{d x^{3} + c} c d^{2}\right )}}{9 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-18*c^2*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d) - 2/9*((d*x^3 + c)^(3/
2)*d^2 + 27*sqrt(d*x^3 + c)*c*d^2)/d^3