Optimal. Leaf size=103 \[ \frac {(e x)^{m+1} (A d (1-m)+B c (m+1)) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{2 c^2 d e (m+1)}-\frac {(e x)^{m+1} (B c-A d)}{2 c d e \left (c+d x^2\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 364} \[ \frac {(e x)^{m+1} (A d (1-m)+B c (m+1)) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{2 c^2 d e (m+1)}-\frac {(e x)^{m+1} (B c-A d)}{2 c d e \left (c+d x^2\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 457
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^2\right )}{\left (c+d x^2\right )^2} \, dx &=-\frac {(B c-A d) (e x)^{1+m}}{2 c d e \left (c+d x^2\right )}+\frac {(-A d (-1+m)+B c (1+m)) \int \frac {(e x)^m}{c+d x^2} \, dx}{2 c d}\\ &=-\frac {(B c-A d) (e x)^{1+m}}{2 c d e \left (c+d x^2\right )}+\frac {(A d (1-m)+B c (1+m)) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{2 c^2 d e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 81, normalized size = 0.79 \[ \frac {x (e x)^m \left ((A d-B c) \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )+B c \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )\right )}{c^2 d (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{2}+A \right ) \left (e x \right )^{m}}{\left (d \,x^{2}+c \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m}{{\left (d\,x^2+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 42.98, size = 954, normalized size = 9.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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