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

Optimal. Leaf size=88 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{243 c^{5/2} d}-\frac{1}{81 c^2 d \sqrt{c+d x^3}}+\frac{1}{27 c d \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]

[Out]

-1/(81*c^2*d*Sqrt[c + d*x^3]) + 1/(27*c*d*(8*c - d*x^3)*Sqrt[c + d*x^3]) + ArcTa
nh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(243*c^(5/2)*d)

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Rubi [A]  time = 0.187863, antiderivative size = 88, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{243 c^{5/2} d}-\frac{1}{81 c^2 d \sqrt{c+d x^3}}+\frac{1}{27 c d \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(81*c^2*d*Sqrt[c + d*x^3]) + 1/(27*c*d*(8*c - d*x^3)*Sqrt[c + d*x^3]) + ArcTa
nh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(243*c^(5/2)*d)

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Rubi in Sympy [A]  time = 23.7357, size = 76, normalized size = 0.86 \[ - \frac{2}{27 c d \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} + \frac{\sqrt{c + d x^{3}}}{81 c^{2} d \left (8 c - d x^{3}\right )} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{243 c^{\frac{5}{2}} d} \]

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)**(3/2),x)

[Out]

-2/(27*c*d*sqrt(c + d*x**3)*(8*c - d*x**3)) + sqrt(c + d*x**3)/(81*c**2*d*(8*c -
d*x**3)) + atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(243*c**(5/2)*d)

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Mathematica [A]  time = 0.195383, size = 72, normalized size = 0.82 \[ \frac{\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{c^{5/2}}+\frac{3 d x^3-15 c}{c^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}}{243 d} \]

Antiderivative was successfully verified.

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

[Out]

((-15*c + 3*d*x^3)/(c^2*(8*c - d*x^3)*Sqrt[c + d*x^3]) + ArcTanh[Sqrt[c + d*x^3]
/(3*Sqrt[c])]/c^(5/2))/(243*d)

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Maple [C]  time = 0.01, size = 463, normalized size = 5.3 \[ -{\frac{1}{243\,d{c}^{2} \left ( d{x}^{3}-8\,c \right ) }\sqrt{d{x}^{3}+c}}-{\frac{2}{243\,d{c}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}-{\frac{{\frac{i}{1458}}\sqrt{2}}{{d}^{3}{c}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/243/d/c^2*(d*x^3+c)^(1/2)/(d*x^3-8*c)-2/243/d/c^2/((x^3+c/d)*d)^(1/2)-1/1458*
I/d^3/c^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)*_al
pha*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(x^2/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221395, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 6 \,{\left (d x^{3} - 5 \, c\right )} \sqrt{c}}{486 \,{\left (c^{2} d^{2} x^{3} - 8 \, c^{3} d\right )} \sqrt{d x^{3} + c} \sqrt{c}}, -\frac{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 3 \,{\left (d x^{3} - 5 \, c\right )} \sqrt{-c}}{243 \,{\left (c^{2} d^{2} x^{3} - 8 \, c^{3} d\right )} \sqrt{d x^{3} + c} \sqrt{-c}}\right ] \]

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)^2),x, algorithm="fricas")

[Out]

[1/486*(sqrt(d*x^3 + c)*(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*(d*x^3 - 5*c)*sqrt(c))/((c^2*d^2*x^3 - 8*c^3*d)*sqrt
(d*x^3 + c)*sqrt(c)), -1/243*(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*arctan(3*c/(sqrt(d*x
^3 + c)*sqrt(-c))) + 3*(d*x^3 - 5*c)*sqrt(-c))/((c^2*d^2*x^3 - 8*c^3*d)*sqrt(d*x
^3 + c)*sqrt(-c))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219872, size = 97, normalized size = 1.1 \[ -\frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{243 \, \sqrt{-c} c^{2} d} - \frac{d x^{3} - 5 \, c}{81 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )} c^{2} d} \]

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)^2),x, algorithm="giac")

[Out]

-1/243*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^2*d) - 1/81*(d*x^3 - 5*c
)/(((d*x^3 + c)^(3/2) - 9*sqrt(d*x^3 + c)*c)*c^2*d)