Optimal. Leaf size=90 \[ \frac{1024 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^4}+\frac{2 c^2}{27 d^4 \sqrt{c+d x^3}}-\frac{4 c \sqrt{c+d x^3}}{d^4}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^4} \]
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Rubi [A] time = 0.266037, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{1024 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^4}+\frac{2 c^2}{27 d^4 \sqrt{c+d x^3}}-\frac{4 c \sqrt{c+d x^3}}{d^4}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^4} \]
Antiderivative was successfully verified.
[In] Int[x^11/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 31.0834, size = 83, normalized size = 0.92 \[ \frac{1024 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{81 d^{4}} + \frac{2 c^{2}}{27 d^{4} \sqrt{c + d x^{3}}} - \frac{4 c \sqrt{c + d x^{3}}}{d^{4}} - \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**11/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.181066, size = 70, normalized size = 0.78 \[ \frac{2 \left (512 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-\frac{3 \left (56 c^2+60 c d x^3+3 d^2 x^6\right )}{\sqrt{c+d x^3}}\right )}{81 d^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^11/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
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Maple [C] time = 0.084, size = 560, normalized size = 6.2 \[ -{\frac{1}{d} \left ( -{\frac{2\,{c}^{2}}{3\,{d}^{3}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{2\,{x}^{3}}{9\,{d}^{2}}\sqrt{d{x}^{3}+c}}-{\frac{10\,c}{9\,{d}^{3}}\sqrt{d{x}^{3}+c}} \right ) }-8\,{\frac{c}{{d}^{2}} \left ( 2/3\,{\frac{c}{{d}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+2/3\,{\frac{\sqrt{d{x}^{3}+c}}{{d}^{2}}} \right ) }+{\frac{128\,{c}^{2}}{3\,{d}^{4}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}-512\,{\frac{{c}^{3}}{{d}^{3}} \left ({\frac{2}{27\,cd}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{{\frac{i}{243}}\sqrt{2}}{{d}^{3}{c}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}-8\,c \right ) }{\frac{\sqrt [3]{-c{d}^{2}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{2/3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}}} \left ( x-{\frac{\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{id\sqrt{3}}{\sqrt [3]{-c{d}^{2}}} \left ( x+1/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) }},-1/18\,{\frac{2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^11/(-d*x^3+8*c)/(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^11/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)),x, algorithm="maxima")
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Fricas [A] time = 0.237121, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (9 \, d^{2} x^{6} + 180 \, c d x^{3} - 256 \, \sqrt{d x^{3} + c} c^{\frac{3}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 168 \, c^{2}\right )}}{81 \, \sqrt{d x^{3} + c} d^{4}}, -\frac{2 \,{\left (9 \, d^{2} x^{6} + 180 \, c d x^{3} - 512 \, \sqrt{d x^{3} + c} \sqrt{-c} c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) + 168 \, c^{2}\right )}}{81 \, \sqrt{d x^{3} + c} d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^11/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)),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**11/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.219514, size = 111, normalized size = 1.23 \[ -\frac{1024 \, c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{81 \, \sqrt{-c} d^{4}} + \frac{2 \, c^{2}}{27 \, \sqrt{d x^{3} + c} d^{4}} - \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{8} + 18 \, \sqrt{d x^{3} + c} c d^{8}\right )}}{9 \, d^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^11/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)),x, algorithm="giac")
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