Optimal. Leaf size=104 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{12 a^{3/2} b^{5/2}}-\frac{x^{3/2} (3 a B+A b)}{12 a b^2 \left (a+b x^3\right )}+\frac{x^{9/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.173112, antiderivative size = 104, 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 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{12 a^{3/2} b^{5/2}}-\frac{x^{3/2} (3 a B+A b)}{12 a b^2 \left (a+b x^3\right )}+\frac{x^{9/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(x^(7/2)*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Rubi in Sympy [A] time = 19.3873, size = 88, normalized size = 0.85 \[ \frac{x^{\frac{9}{2}} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} - \frac{x^{\frac{3}{2}} \left (A b + 3 B a\right )}{12 a b^{2} \left (a + b x^{3}\right )} + \frac{\left (A b + 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{12 a^{\frac{3}{2}} b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(7/2)*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.33952, size = 179, normalized size = 1.72 \[ \frac{-\frac{(3 a B+A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{a^{3/2}}+\frac{(3 a B+A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{a^{3/2}}-\frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{a^{3/2}}+\frac{\sqrt{b} x^{3/2} (A b-5 a B)}{a \left (a+b x^3\right )}-\frac{2 \sqrt{b} x^{3/2} (A b-a B)}{\left (a+b x^3\right )^2}}{12 b^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(7/2)*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Maple [A] time = 0.025, size = 96, normalized size = 0.9 \[{\frac{2}{3\, \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{Ab-5\,Ba}{8\,ab}{x}^{{\frac{9}{2}}}}-{\frac{Ab+3\,Ba}{8\,{b}^{2}}{x}^{{\frac{3}{2}}}} \right ) }+{\frac{A}{12\,ab}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{4\,{b}^{2}}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(7/2)*(B*x^3+A)/(b*x^3+a)^3,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)*x^(7/2)/(b*x^3 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254935, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left ({\left (5 \, B a b - A b^{2}\right )} x^{4} +{\left (3 \, B a^{2} + A a b\right )} x\right )} \sqrt{-a b} \sqrt{x} -{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 3 \, B a^{3} + A a^{2} b + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, a b x^{\frac{3}{2}} +{\left (b x^{3} - a\right )} \sqrt{-a b}}{b x^{3} + a}\right )}{24 \,{\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )} \sqrt{-a b}}, -\frac{{\left ({\left (5 \, B a b - A b^{2}\right )} x^{4} +{\left (3 \, B a^{2} + A a b\right )} x\right )} \sqrt{a b} \sqrt{x} -{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 3 \, B a^{3} + A a^{2} b + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right )}{12 \,{\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^(7/2)/(b*x^3 + a)^3,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)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.222604, size = 113, normalized size = 1.09 \[ \frac{{\left (3 \, B a + A b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{12 \, \sqrt{a b} a b^{2}} - \frac{5 \, B a b x^{\frac{9}{2}} - A b^{2} x^{\frac{9}{2}} + 3 \, B a^{2} x^{\frac{3}{2}} + A a b x^{\frac{3}{2}}}{12 \,{\left (b x^{3} + a\right )}^{2} a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^(7/2)/(b*x^3 + a)^3,x, algorithm="giac")
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