Optimal. Leaf size=120 \[ \frac{e^{7/2} (2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{5/2}}-\frac{e^2 (e x)^{3/2} (2 A b-3 a B)}{3 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.261849, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{e^{7/2} (2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{5/2}}-\frac{e^2 (e x)^{3/2} (2 A b-3 a B)}{3 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(7/2)*(A + B*x^3))/(a + b*x^3)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 24.4732, size = 112, normalized size = 0.93 \[ \frac{B \left (e x\right )^{\frac{9}{2}}}{3 b e \sqrt{a + b x^{3}}} - \frac{2 e^{2} \left (e x\right )^{\frac{3}{2}} \left (A b - \frac{3 B a}{2}\right )}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 e^{\frac{7}{2}} \left (A b - \frac{3 B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{3 b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(7/2)*(B*x**3+A)/(b*x**3+a)**(3/2),x)
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Mathematica [A] time = 0.228545, size = 91, normalized size = 0.76 \[ \frac{e^2 (e x)^{3/2} \left (\sqrt{b} \left (3 a B-2 A b+b B x^3\right )+\sqrt{\frac{a}{x^3}+b} (2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )\right )}{3 b^{5/2} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(7/2)*(A + B*x^3))/(a + b*x^3)^(3/2),x]
[Out]
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Maple [C] time = 0.079, size = 7016, normalized size = 58.5 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(7/2)*(B*x^3+A)/(b*x^3+a)^(3/2),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)*(e*x)^(7/2)/(b*x^3 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.652507, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (3 \, B a b - 2 \, A b^{2}\right )} e^{3} x^{3} +{\left (3 \, B a^{2} - 2 \, A a b\right )} e^{3}\right )} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (B b e^{3} x^{4} +{\left (3 \, B a - 2 \, A b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{12 \,{\left (b^{3} x^{3} + a b^{2}\right )}}, -\frac{{\left ({\left (3 \, B a b - 2 \, A b^{2}\right )} e^{3} x^{3} +{\left (3 \, B a^{2} - 2 \, A a b\right )} e^{3}\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) - 2 \,{\left (B b e^{3} x^{4} +{\left (3 \, B a - 2 \, A b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{6 \,{\left (b^{3} x^{3} + a b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*(e*x)^(7/2)/(b*x^3 + a)^(3/2),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((e*x)**(7/2)*(B*x**3+A)/(b*x**3+a)**(3/2),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)*(e*x)^(7/2)/(b*x^3 + a)^(3/2),x, algorithm="giac")
[Out]