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 = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {459, 364} \[ \frac {x^{m+1} (A b-a B) \, _2F_1\left (1,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{a b (m+1)}+\frac {B x^{m+1}}{b (m+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 459
Rubi steps
\begin {align*} \int \frac {x^m \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac {B x^{1+m}}{b (1+m)}-\frac {(-A b (1+m)+a B (1+m)) \int \frac {x^m}{a+b x^3} \, dx}{b (1+m)}\\ &=\frac {B x^{1+m}}{b (1+m)}+\frac {(A b-a B) x^{1+m} \, _2F_1\left (1,\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{a b (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.07, 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.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} x^{m}}{b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{3} + A\right )} x^{m}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{3}+A \right ) x^{m}}{b \,x^{3}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{3} + A\right )} x^{m}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m\,\left (B\,x^3+A\right )}{b\,x^3+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 22.06, 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.
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