Optimal. Leaf size=58 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{4 \sqrt{c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{12 \sqrt{c}} \]
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Rubi [A] time = 0.178895, antiderivative size = 58, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{4 \sqrt{c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{12 \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^3]/(x*(8*c - d*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 18.1555, size = 48, normalized size = 0.83 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{4 \sqrt{c}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{12 \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c)**(1/2)/x/(-d*x**3+8*c),x)
[Out]
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Mathematica [C] time = 0.314318, size = 158, normalized size = 2.72 \[ \frac{2 d x^3 \sqrt{c+d x^3} F_1\left (\frac{1}{2};-\frac{1}{2},1;\frac{3}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (d x^3-8 c\right ) \left (3 d x^3 F_1\left (\frac{1}{2};-\frac{1}{2},1;\frac{3}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+c \left (16 F_1\left (\frac{3}{2};-\frac{1}{2},2;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[c + d*x^3]/(x*(8*c - d*x^3)),x]
[Out]
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Maple [C] time = 0.016, size = 468, normalized size = 8.1 \[{\frac{1}{8\,c} \left ({\frac{2}{3}\sqrt{d{x}^{3}+c}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ) \sqrt{c}} \right ) }-{\frac{d}{8\,c} \left ({\frac{2}{3\,d}\sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{3}}\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^(1/2)/x/(-d*x^3+8*c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(d*x^3 + c)/((d*x^3 - 8*c)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.262771, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right )}{24 \, \sqrt{c}}, -\frac{3 \, \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right )}{12 \, \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(d*x^3 + c)/((d*x^3 - 8*c)*x),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{c + d x^{3}}}{- 8 c x + d x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**3+c)**(1/2)/x/(-d*x**3+8*c),x)
[Out]
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GIAC/XCAS [A] time = 0.215631, size = 65, normalized size = 1.12 \[ \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{12 \, \sqrt{-c}} - \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{4 \, \sqrt{-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(d*x^3 + c)/((d*x^3 - 8*c)*x),x, algorithm="giac")
[Out]