Optimal. Leaf size=48 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 \sqrt{b} \sqrt{d}} \]
[Out]
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Rubi [A] time = 0.157188, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 \sqrt{b} \sqrt{d}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]
[Out]
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Rubi in Sympy [A] time = 13.5054, size = 44, normalized size = 0.92 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{d} \sqrt{a + b x^{3}}} \right )}}{3 \sqrt{b} \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0431235, size = 63, normalized size = 1.31 \[ \frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^3} \sqrt{c+d x^3}+a d+b c+2 b d x^3\right )}{3 \sqrt{b} \sqrt{d}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]
[Out]
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Maple [F] time = 0.075, size = 0, normalized size = 0. \[ \int{{x}^{2}{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(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. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236023, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (4 \,{\left (2 \, b^{2} d^{2} x^{3} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} +{\left (8 \, b^{2} d^{2} x^{6} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{3}\right )} \sqrt{b d}\right )}{6 \, \sqrt{b d}}, \frac{\arctan \left (\frac{{\left (2 \, b d x^{3} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} b d}\right )}{3 \, \sqrt{-b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.230133, size = 73, normalized size = 1.52 \[ -\frac{2 \, b{\rm ln}\left ({\left | -\sqrt{b x^{3} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d} \right |}\right )}{3 \, \sqrt{b d}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="giac")
[Out]