Optimal. Leaf size=93 \[ \frac {x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left (1,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{3 a^2 b (m+1)}+\frac {x^{m+1} (A b-a B)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 93, 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 = {457, 364} \[ \frac {x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left (1,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{3 a^2 b (m+1)}+\frac {x^{m+1} (A b-a B)}{3 a b \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 457
Rubi steps
\begin {align*} \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac {(A b-a B) x^{1+m}}{3 a b \left (a+b x^3\right )}+\frac {(-A b (-2+m)+a B (1+m)) \int \frac {x^m}{a+b x^3} \, dx}{3 a b}\\ &=\frac {(A b-a B) x^{1+m}}{3 a b \left (a+b x^3\right )}+\frac {(A b (2-m)+a B (1+m)) x^{1+m} \, _2F_1\left (1,\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{3 a^2 b (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 80, normalized size = 0.86 \[ \frac {x^{m+1} \left ((A b-a B) \, _2F_1\left (2,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )+a B \, _2F_1\left (1,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )\right )}{a^2 b (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} x^{m}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} x^{m}}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{3}+A \right ) x^{m}}{\left (b \,x^{3}+a \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} x^{m}}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,\left (B\,x^3+A\right )}{{\left (b\,x^3+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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