Optimal. Leaf size=64 \[ -\frac{\sqrt{c+d x^3} F_1\left (-\frac{2}{3};2,-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a^2 x^2 \sqrt{\frac{d x^3}{c}+1}} \]
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Rubi [A] time = 0.189473, antiderivative size = 64, 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 \[ -\frac{\sqrt{c+d x^3} F_1\left (-\frac{2}{3};2,-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a^2 x^2 \sqrt{\frac{d x^3}{c}+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^3]/(x^3*(a + b*x^3)^2),x]
[Out]
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Rubi in Sympy [A] time = 22.4174, size = 56, normalized size = 0.88 \[ - \frac{\sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},- \frac{1}{2},2,\frac{1}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{2 a^{2} x^{2} \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c)**(1/2)/x**3/(b*x**3+a)**2,x)
[Out]
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Mathematica [B] time = 0.721181, size = 347, normalized size = 5.42 \[ \frac{\frac{16 a c x^3 (9 a d-20 b c) F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )-3 x^3 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}+\frac{35 a b c d x^6 F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-14 a c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}-4 \left (3 a+5 b x^3\right ) \left (c+d x^3\right )}{24 a^2 x^2 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[c + d*x^3]/(x^3*(a + b*x^3)^2),x]
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Maple [C] time = 0.018, size = 1768, normalized size = 27.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^(1/2)/x^3/(b*x^3+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{3} + c}}{{\left (b x^{3} + a\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)/((b*x^3 + a)^2*x^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(sqrt(d*x^3 + c)/((b*x^3 + a)^2*x^3),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**3+c)**(1/2)/x**3/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{3} + c}}{{\left (b x^{3} + a\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)/((b*x^3 + a)^2*x^3),x, algorithm="giac")
[Out]