Optimal. Leaf size=199 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \]
[Out]
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Rubi [A] time = 0.263475, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(1/3)*Sqrt[c + d*x]*(4*c + d*x)),x]
[Out]
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Rubi in Sympy [A] time = 10.034, size = 49, normalized size = 0.25 \[ \frac{3 x^{\frac{2}{3}} \sqrt{c + d x} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},- \frac{d x}{c},- \frac{d x}{4 c} \right )}}{8 c^{2} \sqrt{1 + \frac{d x}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(1/3)/(d*x+4*c)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [C] time = 0.232704, size = 147, normalized size = 0.74 \[ \frac{30 c x^{2/3} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x}{c},-\frac{d x}{4 c}\right )}{\sqrt{c+d x} (4 c+d x) \left (20 c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x}{c},-\frac{d x}{4 c}\right )-3 d x \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x}{c},-\frac{d x}{4 c}\right )+2 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x}{c},-\frac{d x}{4 c}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^(1/3)*Sqrt[c + d*x]*(4*c + d*x)),x]
[Out]
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Maple [F] time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{1}{dx+4\,c}{\frac{1}{\sqrt [3]{x}}}{\frac{1}{\sqrt{dx+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(1/3)/(d*x+4*c)/(d*x+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x + 4 \, c\right )} \sqrt{d x + c} x^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x + 4*c)*sqrt(d*x + c)*x^(1/3)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x + 4*c)*sqrt(d*x + c)*x^(1/3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{x} \sqrt{c + d x} \left (4 c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(1/3)/(d*x+4*c)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x + 4 \, c\right )} \sqrt{d x + c} x^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x + 4*c)*sqrt(d*x + c)*x^(1/3)),x, algorithm="giac")
[Out]