Optimal. Leaf size=86 \[ -\frac{\sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{x \sqrt{a+b x^3} \sqrt{c+d x^3}} \]
[Out]
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Rubi [A] time = 0.378873, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{x \sqrt{a+b x^3} \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]
[Out]
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Rubi in Sympy [A] time = 28.0139, size = 75, normalized size = 0.87 \[ - \frac{\sqrt{a + b x^{3}} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{2}{3},- \frac{b x^{3}}{a},- \frac{d x^{3}}{c} \right )}}{a c x \sqrt{1 + \frac{b x^{3}}{a}} \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)
[Out]
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Mathematica [B] time = 0.566252, size = 357, normalized size = 4.15 \[ \frac{-\frac{25 x^3 (a d+b c) F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{3 x^3 \left (a d F_1\left (\frac{5}{3};\frac{1}{2},\frac{3}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b c F_1\left (\frac{5}{3};\frac{3}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-10 a c F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}-\frac{64 b d x^6 F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{3 x^3 \left (a d F_1\left (\frac{8}{3};\frac{1}{2},\frac{3}{2};\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b c F_1\left (\frac{8}{3};\frac{3}{2},\frac{1}{2};\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-16 a c F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}-\frac{10 \left (a+b x^3\right ) \left (c+d x^3\right )}{a c}}{10 x \sqrt{a+b x^3} \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]
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Maple [F] time = 0.085, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^2),x, algorithm="giac")
[Out]