Optimal. Leaf size=93 \[ \frac{x^{m+1} (a B (m+1)+A b (5-m)) \, _2F_1\left (2,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{6 a^3 b (m+1)}+\frac{x^{m+1} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
[Out]
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Rubi [A] time = 0.139544, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x^{m+1} (a B (m+1)+A b (5-m)) \, _2F_1\left (2,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{6 a^3 b (m+1)}+\frac{x^{m+1} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(x^m*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Rubi in Sympy [A] time = 11.9777, size = 73, normalized size = 0.78 \[ \frac{x^{m + 1} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x^{m + 1} \left (A b \left (- m + 5\right ) + B a \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{6 a^{3} b \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.0894682, size = 80, normalized size = 0.86 \[ \frac{x^{m+1} \left ((A b-a B) \, _2F_1\left (3,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )+a B \, _2F_1\left (2,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )\right )}{a^3 b (m+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(x^m*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Maple [F] time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( B{x}^{3}+A \right ) }{ \left ( b{x}^{3}+a \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(B*x^3+A)/(b*x^3+a)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )} x^{m}}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^m/(b*x^3 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{3} + A\right )} x^{m}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^m/(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**m*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )} x^{m}}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^m/(b*x^3 + a)^3,x, algorithm="giac")
[Out]