Optimal. Leaf size=88 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{24 \sqrt{3} c^{5/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{8 c^{5/2}}-\frac{\sqrt{c+d x^3}}{12 c^2 x^3} \]
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Rubi [A] time = 0.321398, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{24 \sqrt{3} c^{5/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{8 c^{5/2}}-\frac{\sqrt{c+d x^3}}{12 c^2 x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 30.3218, size = 80, normalized size = 0.91 \[ - \frac{\sqrt{c + d x^{3}}}{12 c^{2} x^{3}} + \frac{\sqrt{3} d \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{72 c^{\frac{5}{2}}} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{8 c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)
[Out]
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Mathematica [C] time = 0.381572, size = 324, normalized size = 3.68 \[ \frac{-\frac{4 c d^2 x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (8 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )}-\frac{10 c d^2 x^6 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )}{\left (4 c+d x^3\right ) \left (-5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )+8 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )+c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )\right )}-c-d x^3}{12 c^2 x^3 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^4*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]
[Out]
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Maple [C] time = 0.033, size = 477, normalized size = 5.4 \[{\frac{1}{4\,c} \left ( -{\frac{1}{3\,c{x}^{3}}\sqrt{d{x}^{3}+c}}+{\frac{d}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \right ) }+{\frac{d}{24}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{5}{2}}}}-{\frac{{\frac{i}{144}}\sqrt{2}}{d{c}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}+4\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{1}{6\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(d*x^3+4*c)/(d*x^3+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265511, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{3}{\left (3 \, \sqrt{3} \sqrt{c} d x^{3} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 2 \, \sqrt{c} d x^{3} \arctan \left (\frac{\sqrt{3} \sqrt{c}}{\sqrt{d x^{3} + c}}\right ) - 4 \, \sqrt{3} \sqrt{d x^{3} + c} c\right )}}{144 \, c^{3} x^{3}}, \frac{\sqrt{3}{\left (6 \, \sqrt{3} \sqrt{-c} d x^{3} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - \sqrt{-c} d x^{3} \log \left (\frac{\sqrt{3}{\left (d x^{3} - 2 \, c\right )} \sqrt{-c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} + 4 \, c}\right ) - 4 \, \sqrt{3} \sqrt{d x^{3} + c} c\right )}}{144 \, c^{3} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{c + d x^{3}} \left (4 c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216284, size = 101, normalized size = 1.15 \[ \frac{1}{72} \, d{\left (\frac{\sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right )}{c^{\frac{5}{2}}} - \frac{9 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{6 \, \sqrt{d x^{3} + c}}{c^{2} d x^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^4),x, algorithm="giac")
[Out]