Optimal. Leaf size=90 \[ \frac{128 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^3}-\frac{128 c^2 \sqrt{c+d x^3}}{3 d^3}-\frac{14 c \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^3} \]
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Rubi [A] time = 0.257817, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{128 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^3}-\frac{128 c^2 \sqrt{c+d x^3}}{3 d^3}-\frac{14 c \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^3} \]
Antiderivative was successfully verified.
[In] Int[(x^8*Sqrt[c + d*x^3])/(8*c - d*x^3),x]
[Out]
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Rubi in Sympy [A] time = 27.7941, size = 83, normalized size = 0.92 \[ \frac{128 c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{d^{3}} - \frac{128 c^{2} \sqrt{c + d x^{3}}}{3 d^{3}} - \frac{14 c \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{3}} - \frac{2 \left (c + d x^{3}\right )^{\frac{5}{2}}}{15 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(d*x**3+c)**(1/2)/(-d*x**3+8*c),x)
[Out]
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Mathematica [A] time = 0.0975081, size = 70, normalized size = 0.78 \[ \frac{5760 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-2 \sqrt{c+d x^3} \left (998 c^2+41 c d x^3+3 d^2 x^6\right )}{45 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*Sqrt[c + d*x^3])/(8*c - d*x^3),x]
[Out]
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Maple [C] time = 0.016, size = 507, normalized size = 5.6 \[ -{\frac{1}{{d}^{2}} \left ( d \left ({\frac{2\,{x}^{6}}{15}\sqrt{d{x}^{3}+c}}+{\frac{2\,c{x}^{3}}{45\,d}\sqrt{d{x}^{3}+c}}-{\frac{4\,{c}^{2}}{45\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) +{\frac{16\,c}{9\,d} \left ( d{x}^{3}+c \right ) ^{{\frac{3}{2}}}} \right ) }-64\,{\frac{{c}^{2}}{{d}^{2}} \left ( 2/3\,{\frac{\sqrt{d{x}^{3}+c}}{d}}+{\frac{i/3\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{\frac{\sqrt [3]{-c{d}^{2}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{2/3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}}} \left ( x-{\frac{\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{i\sqrt{3}d}{\sqrt [3]{-c{d}^{2}}} \left ( x+1/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) }},-1/18\,{\frac{2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(d*x^3+c)^(1/2)/(-d*x^3+8*c),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(-sqrt(d*x^3 + c)*x^8/(d*x^3 - 8*c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.252609, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (1440 \, c^{\frac{5}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) -{\left (3 \, d^{2} x^{6} + 41 \, c d x^{3} + 998 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{45 \, d^{3}}, \frac{2 \,{\left (2880 \, \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) -{\left (3 \, d^{2} x^{6} + 41 \, c d x^{3} + 998 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{45 \, d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(d*x^3 + c)*x^8/(d*x^3 - 8*c),x, algorithm="fricas")
[Out]
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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**8*(d*x**3+c)**(1/2)/(-d*x**3+8*c),x)
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GIAC/XCAS [A] time = 0.218015, size = 112, normalized size = 1.24 \[ -\frac{128 \, c^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d^{3}} - \frac{2 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} d^{12} + 35 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c d^{12} + 960 \, \sqrt{d x^{3} + c} c^{2} d^{12}\right )}}{45 \, d^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(d*x^3 + c)*x^8/(d*x^3 - 8*c),x, algorithm="giac")
[Out]