Optimal. Leaf size=66 \[ \frac{x^4 \sqrt{c+d x^3} F_1\left (\frac{4}{3};1,-\frac{1}{2};\frac{7}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{16 c \sqrt{\frac{d x^3}{c}+1}} \]
[Out]
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Rubi [A] time = 0.200824, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^4 \sqrt{c+d x^3} F_1\left (\frac{4}{3};1,-\frac{1}{2};\frac{7}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{16 c \sqrt{\frac{d x^3}{c}+1}} \]
Antiderivative was successfully verified.
[In] Int[(x^3*Sqrt[c + d*x^3])/(4*c + d*x^3),x]
[Out]
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Rubi in Sympy [A] time = 28.0432, size = 53, normalized size = 0.8 \[ \frac{x^{4} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{4}{3},- \frac{1}{2},1,\frac{7}{3},- \frac{d x^{3}}{c},- \frac{d x^{3}}{4 c} \right )}}{16 c \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(d*x**3+c)**(1/2)/(d*x**3+4*c),x)
[Out]
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Mathematica [B] time = 0.77796, size = 344, normalized size = 5.21 \[ \frac{x \left (\frac{128 c^3 F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{d \left (4 c+d x^3\right ) \left (3 d x^3 \left (F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )-16 c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )}-\frac{119 c^2 x^3 F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (28 c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-3 d x^3 \left (F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )}+2 \left (\frac{c}{d}+x^3\right )\right )}{5 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(x^3*Sqrt[c + d*x^3])/(4*c + d*x^3),x]
[Out]
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Maple [C] time = 0.055, size = 1003, normalized size = 15.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(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. \[ \int \frac{\sqrt{d x^{3} + c} x^{3}}{d x^{3} + 4 \, c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x^3/(d*x^3 + 4*c),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x^{3} + c} x^{3}}{d x^{3} + 4 \, c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x^3/(d*x^3 + 4*c),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \sqrt{c + d x^{3}}}{4 c + d x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(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. \[ \int \frac{\sqrt{d x^{3} + c} x^{3}}{d x^{3} + 4 \, c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x^3/(d*x^3 + 4*c),x, algorithm="giac")
[Out]