Optimal. Leaf size=66 \[ \frac{x^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a b (m+1)}+\frac{B x^{m+1}}{b (m+1)} \]
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Rubi [A] time = 0.108708, antiderivative size = 66, 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 \[ \frac{x^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a b (m+1)}+\frac{B x^{m+1}}{b (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(x^m*(A + B*x^3))/(a + b*x^3),x]
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Rubi in Sympy [A] time = 10.4017, size = 49, normalized size = 0.74 \[ \frac{B x^{m + 1}}{b \left (m + 1\right )} + \frac{x^{m + 1} \left (A b - B a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{a b \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(B*x**3+A)/(b*x**3+a),x)
[Out]
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Mathematica [A] time = 0.0907811, size = 55, normalized size = 0.83 \[ \frac{x^{m+1} \left ((A b-a B) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )+a B\right )}{a b (m+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(x^m*(A + B*x^3))/(a + b*x^3),x]
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Maple [F] time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( B{x}^{3}+A \right ) }{b{x}^{3}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(B*x^3+A)/(b*x^3+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )} x^{m}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^m/(b*x^3 + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{3} + A\right )} x^{m}}{b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^m/(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 93.2513, size = 190, normalized size = 2.88 \[ \frac{A m x x^{m} \Phi \left (\frac{b x^{3} e^{i \pi }}{a}, 1, \frac{m}{3} + \frac{1}{3}\right ) \Gamma \left (\frac{m}{3} + \frac{1}{3}\right )}{9 a \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} + \frac{A x x^{m} \Phi \left (\frac{b x^{3} e^{i \pi }}{a}, 1, \frac{m}{3} + \frac{1}{3}\right ) \Gamma \left (\frac{m}{3} + \frac{1}{3}\right )}{9 a \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} + \frac{B m x^{4} x^{m} \Phi \left (\frac{b x^{3} e^{i \pi }}{a}, 1, \frac{m}{3} + \frac{4}{3}\right ) \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )}{9 a \Gamma \left (\frac{m}{3} + \frac{7}{3}\right )} + \frac{4 B x^{4} x^{m} \Phi \left (\frac{b x^{3} e^{i \pi }}{a}, 1, \frac{m}{3} + \frac{4}{3}\right ) \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )}{9 a \Gamma \left (\frac{m}{3} + \frac{7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(B*x**3+A)/(b*x**3+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )} x^{m}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^m/(b*x^3 + a),x, algorithm="giac")
[Out]