Optimal. Leaf size=64 \[ \frac{x \sqrt{c+d x^3} F_1\left (\frac{1}{3};1,-\frac{1}{2};\frac{4}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{4 c \sqrt{\frac{d x^3}{c}+1}} \]
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Rubi [A] time = 0.100552, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{x \sqrt{c+d x^3} F_1\left (\frac{1}{3};1,-\frac{1}{2};\frac{4}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{4 c \sqrt{\frac{d x^3}{c}+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^3]/(4*c + d*x^3),x]
[Out]
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Rubi in Sympy [A] time = 26.1712, size = 51, normalized size = 0.8 \[ \frac{x \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{1}{3},- \frac{1}{2},1,\frac{4}{3},- \frac{d x^{3}}{c},- \frac{d x^{3}}{4 c} \right )}}{4 c \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c)**(1/2)/(d*x**3+4*c),x)
[Out]
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Mathematica [B] time = 0.257715, size = 165, normalized size = 2.58 \[ \frac{16 c x \sqrt{c+d x^3} F_1\left (\frac{1}{3};-\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (16 c F_1\left (\frac{1}{3};-\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-3 d x^3 \left (F_1\left (\frac{4}{3};-\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-2 F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[c + d*x^3]/(4*c + d*x^3),x]
[Out]
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Maple [C] time = 0.008, size = 696, normalized size = 10.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^(1/2)/(d*x^3+4*c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{3} + c}}{d x^{3} + 4 \, c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)/(d*x^3 + 4*c),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x^{3} + c}}{d x^{3} + 4 \, c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)/(d*x^3 + 4*c),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{3}}}{4 c + d x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**3+c)**(1/2)/(d*x**3+4*c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{3} + c}}{d x^{3} + 4 \, c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)/(d*x^3 + 4*c),x, algorithm="giac")
[Out]