Optimal. Leaf size=83 \[ \frac{\sqrt{e} (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{3/2}}+\frac{B (e x)^{3/2} \sqrt{a+b x^3}}{3 b e} \]
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Rubi [A] time = 0.202006, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{e} (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{3/2}}+\frac{B (e x)^{3/2} \sqrt{a+b x^3}}{3 b e} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[e*x]*(A + B*x^3))/Sqrt[a + b*x^3],x]
[Out]
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Rubi in Sympy [A] time = 19.0747, size = 73, normalized size = 0.88 \[ \frac{B \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}}}{3 b e} + \frac{2 \sqrt{e} \left (A b - \frac{B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{3 b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)*(e*x)**(1/2)/(b*x**3+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.170303, size = 82, normalized size = 0.99 \[ \frac{x \sqrt{e x} \left (\sqrt{\frac{a}{x^3}+b} (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )+\sqrt{b} B \left (a+b x^3\right )\right )}{3 b^{3/2} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[e*x]*(A + B*x^3))/Sqrt[a + b*x^3],x]
[Out]
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Maple [C] time = 0.04, size = 6424, normalized size = 77.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)*(e*x)^(1/2)/(b*x^3+a)^(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((B*x^3 + A)*sqrt(e*x)/sqrt(b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.634202, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{b x^{3} + a} \sqrt{e x} B x -{\left (B a - 2 \, A b\right )} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right )}{12 \, b}, \frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} B x -{\left (B a - 2 \, A b\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right )}{6 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(e*x)/sqrt(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.51888, size = 107, normalized size = 1.29 \[ \frac{2 A \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{3 \sqrt{b}} + \frac{B \sqrt{a} \left (e x\right )^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 b e} - \frac{B a \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{3 b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)*(e*x)**(1/2)/(b*x**3+a)**(1/2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(e*x)/sqrt(b*x^3 + a),x, algorithm="giac")
[Out]