Optimal. Leaf size=93 \[ \frac {x^{m+1} (a B (m+1)+A (b-b m)) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{2 a^2 b (m+1)}+\frac {x^{m+1} (A b-a B)}{2 a b \left (a+b x^2\right )} \]
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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} \[ \frac {x^{m+1} (a B (m+1)+A (b-b m)) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{2 a^2 b (m+1)}+\frac {x^{m+1} (A b-a B)}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 457
Rubi steps
\begin {align*} \int \frac {x^m \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {(A b-a B) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a B (1+m)+A (b-b m)) \int \frac {x^m}{a+b x^2} \, dx}{2 a b}\\ &=\frac {(A b-a B) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a B (1+m)+A (b-b m)) x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{2 a^2 b (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 80, normalized size = 0.86 \[ \frac {x^{m+1} \left ((A b-a B) \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )+a B \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )\right )}{a^2 b (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} x^{m}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{m}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{2}+A \right ) x^{m}}{\left (b \,x^{2}+a \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{m}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {x^m\,\left (B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 30.22, size = 906, normalized size = 9.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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