Optimal. Leaf size=73 \[ -\frac{2 \sqrt{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 b^{5/2}}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{9/2}}{9 b} \]
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Rubi [A] time = 0.148463, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 \sqrt{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 b^{5/2}}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{9/2}}{9 b} \]
Antiderivative was successfully verified.
[In] Int[(x^(7/2)*(A + B*x^3))/(a + b*x^3),x]
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Rubi in Sympy [A] time = 16.8753, size = 66, normalized size = 0.9 \[ \frac{2 B x^{\frac{9}{2}}}{9 b} - \frac{2 \sqrt{a} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 b^{\frac{5}{2}}} + \frac{2 x^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(7/2)*(B*x**3+A)/(b*x**3+a),x)
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Mathematica [B] time = 0.175819, size = 180, normalized size = 2.47 \[ \frac{2 \sqrt{a} (a B-A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}-\sqrt{3} \sqrt [6]{a}}{\sqrt [6]{a}}\right )}{3 b^{5/2}}+\frac{2 \sqrt{a} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 b^{5/2}}-\frac{2 \sqrt{a} (a B-A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 b^{5/2}}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{9/2}}{9 b} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(7/2)*(A + B*x^3))/(a + b*x^3),x]
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Maple [A] time = 0.013, size = 78, normalized size = 1.1 \[{\frac{2\,B}{9\,b}{x}^{{\frac{9}{2}}}}+{\frac{2\,A}{3\,b}{x}^{{\frac{3}{2}}}}-{\frac{2\,Ba}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}-{\frac{2\,Aa}{3\,b}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{2\,{a}^{2}B}{3\,{b}^{2}}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(7/2)*(B*x^3+A)/(b*x^3+a),x)
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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)*x^(7/2)/(b*x^3 + a),x, algorithm="maxima")
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Fricas [A] time = 0.244268, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B a - A b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{3} - 2 \, b x^{\frac{3}{2}} \sqrt{-\frac{a}{b}} - a}{b x^{3} + a}\right ) - 2 \,{\left (B b x^{4} - 3 \,{\left (B a - A b\right )} x\right )} \sqrt{x}}{9 \, b^{2}}, \frac{2 \,{\left (3 \,{\left (B a - A b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x^{\frac{3}{2}}}{\sqrt{\frac{a}{b}}}\right ) +{\left (B b x^{4} - 3 \,{\left (B a - A b\right )} x\right )} \sqrt{x}\right )}}{9 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^(7/2)/(b*x^3 + a),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**(7/2)*(B*x**3+A)/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.21628, size = 86, normalized size = 1.18 \[ \frac{2 \,{\left (B a^{2} - A a b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} b^{2}} + \frac{2 \,{\left (B b^{2} x^{\frac{9}{2}} - 3 \, B a b x^{\frac{3}{2}} + 3 \, A b^{2} x^{\frac{3}{2}}\right )}}{9 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^(7/2)/(b*x^3 + a),x, algorithm="giac")
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