Optimal. Leaf size=58 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{36 c^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{12 c^{3/2}} \]
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Rubi [A] time = 0.1725, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{36 c^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{12 c^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]
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Rubi in Sympy [A] time = 18.1933, size = 48, normalized size = 0.83 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{36 c^{\frac{3}{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{12 c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(-d*x**3+8*c)/(d*x**3+c)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0801071, size = 161, normalized size = 2.78 \[ \frac{10 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{9 \left (d x^3-8 c\right ) \sqrt{c+d x^3} \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]
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Maple [C] time = 0.013, size = 433, normalized size = 7.5 \[ -{\frac{1}{12}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}-{\frac{{\frac{i}{216}}\sqrt{2}}{{c}^{2}{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x),x, algorithm="maxima")
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Fricas [A] time = 0.252032, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 3 \, \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right )}{72 \, c^{\frac{3}{2}}}, -\frac{\arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 3 \, \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right )}{36 \, \sqrt{-c} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(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. \[ - \int \frac{1}{- 8 c x \sqrt{c + d x^{3}} + d x^{4} \sqrt{c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(-d*x**3+8*c)/(d*x**3+c)**(1/2),x)
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GIAC/XCAS [A] time = 0.218373, size = 73, normalized size = 1.26 \[ \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{12 \, \sqrt{-c} c} - \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{36 \, \sqrt{-c} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x),x, algorithm="giac")
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