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

Optimal. Leaf size=67 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 c^{3/2} d}+\frac{\sqrt{c+d x^3}}{27 c d \left (8 c-d x^3\right )} \]

[Out]

Sqrt[c + d*x^3]/(27*c*d*(8*c - d*x^3)) + ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(8
1*c^(3/2)*d)

_______________________________________________________________________________________

Rubi [A]  time = 0.156216, antiderivative size = 67, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 c^{3/2} d}+\frac{\sqrt{c+d x^3}}{27 c d \left (8 c-d x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x^3]/(27*c*d*(8*c - d*x^3)) + ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(8
1*c^(3/2)*d)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.4947, size = 49, normalized size = 0.73 \[ \frac{\sqrt{c + d x^{3}}}{27 c d \left (8 c - d x^{3}\right )} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{81 c^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c + d*x**3)/(27*c*d*(8*c - d*x**3)) + atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(
81*c**(3/2)*d)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0738962, size = 66, normalized size = 0.99 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 c^{3/2} d}-\frac{\sqrt{c+d x^3}}{27 c d \left (d x^3-8 c\right )} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[c + d*x^3]/(27*c*d*(-8*c + d*x^3)) + ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/
(81*c^(3/2)*d)

_______________________________________________________________________________________

Maple [C]  time = 0.01, size = 442, normalized size = 6.6 \[ -{\frac{1}{27\,cd \left ( d{x}^{3}-8\,c \right ) }\sqrt{d{x}^{3}+c}}-{\frac{{\frac{i}{486}}\sqrt{2}}{{d}^{3}{c}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27/d/c*(d*x^3+c)^(1/2)/(d*x^3-8*c)-1/486*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*_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=Root
Of(_Z^3*d-8*c))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.228882, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (d x^{3} - 8 \, c\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 6 \, \sqrt{d x^{3} + c} \sqrt{c}}{162 \,{\left (c d^{2} x^{3} - 8 \, c^{2} d\right )} \sqrt{c}}, -\frac{{\left (d x^{3} - 8 \, c\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 3 \, \sqrt{d x^{3} + c} \sqrt{-c}}{81 \,{\left (c d^{2} x^{3} - 8 \, c^{2} d\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/162*((d*x^3 - 8*c)*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c)*c)/(d*x^3
- 8*c)) - 6*sqrt(d*x^3 + c)*sqrt(c))/((c*d^2*x^3 - 8*c^2*d)*sqrt(c)), -1/81*((d*
x^3 - 8*c)*arctan(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) + 3*sqrt(d*x^3 + c)*sqrt(-c))/
((c*d^2*x^3 - 8*c^2*d)*sqrt(-c))]

_______________________________________________________________________________________

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**2/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214626, size = 80, normalized size = 1.19 \[ -\frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{81 \, \sqrt{-c} c d} - \frac{\sqrt{d x^{3} + c}}{27 \,{\left (d x^{3} - 8 \, c\right )} c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/81*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c*d) - 1/27*sqrt(d*x^3 + c)
/((d*x^3 - 8*c)*c*d)