Optimal. Leaf size=64 \[ \frac{x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a^2 \sqrt{c+d x^3}} \]
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Rubi [A] time = 0.163049, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a^2 \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]
[Out]
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Rubi in Sympy [A] time = 20.5183, size = 53, normalized size = 0.83 \[ \frac{x^{2} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},2,\frac{5}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{2 a^{2} c \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**3+a)**2/(d*x**3+c)**(1/2),x)
[Out]
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Mathematica [B] time = 0.519086, size = 342, normalized size = 5.34 \[ \frac{x^2 \left (-\frac{8 b c d x^3 F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-16 a c F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{25 c (b c-3 a d) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-10 a c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}-\frac{5 b \left (c+d x^3\right )}{a}\right )}{15 \left (a+b x^3\right ) \sqrt{c+d x^3} (a d-b c)} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]
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Maple [C] time = 0.054, size = 923, normalized size = 14.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^3+a)^2/(d*x^3+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{3} + a\right )}^{2} \sqrt{d x^{3} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^3 + a)^2*sqrt(d*x^3 + c)),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(x/((b*x^3 + a)^2*sqrt(d*x^3 + c)),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(x/(b*x**3+a)**2/(d*x**3+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{3} + a\right )}^{2} \sqrt{d x^{3} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^3 + a)^2*sqrt(d*x^3 + c)),x, algorithm="giac")
[Out]