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