Optimal. Leaf size=205 \[ \frac {(B c-A d) (e x)^{1+m}}{2 c (b c-a d) e \left (c+d x^2\right )}+\frac {b (A b-a B) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a (b c-a d)^2 e (1+m)}+\frac {(b c (B c (1-m)-A d (3-m))+a d (A d (1-m)+B c (1+m))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{2 c^2 (b c-a d)^2 e (1+m)} \]
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Rubi [A]
time = 0.26, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {593, 598, 371}
\begin {gather*} \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (B c (1-m)-A d (3-m)))}{2 c^2 e (m+1) (b c-a d)^2}+\frac {b (e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a e (m+1) (b c-a d)^2}+\frac {(e x)^{m+1} (B c-A d)}{2 c e \left (c+d x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 593
Rule 598
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^2\right )}{\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 (b c-a d) e \left (c+d x^2\right )}+\frac {\int \frac {(e x)^m \left (2 A b c-a A d (1-m)-a B c (1+m)+b (B c-A d) (1-m) x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=\frac {(B c-A d) (e x)^{1+m}}{2 c (b c-a d) e \left (c+d x^2\right )}+\frac {\int \left (\frac {2 b (A b-a B) c (e x)^m}{(b c-a d) \left (a+b x^2\right )}+\frac {(a d (A d (1-m)+B c (1+m))-b c (A d (3-m)-B (c-c m))) (e x)^m}{(b c-a d) \left (c+d x^2\right )}\right ) \, dx}{2 c (b c-a d)}\\ &=\frac {(B c-A d) (e x)^{1+m}}{2 c (b c-a d) e \left (c+d x^2\right )}+\frac {(b (A b-a B)) \int \frac {(e x)^m}{a+b x^2} \, dx}{(b c-a d)^2}+\frac {(a d (A d (1-m)+B c (1+m))-b c (A d (3-m)-B (c-c m))) \int \frac {(e x)^m}{c+d x^2} \, dx}{2 c (b c-a d)^2}\\ &=\frac {(B c-A d) (e x)^{1+m}}{2 c (b c-a d) e \left (c+d x^2\right )}+\frac {b (A b-a B) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a (b c-a d)^2 e (1+m)}+\frac {(b c (B c (1-m)-A d (3-m))+a d (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 (b c-a d)^2 e (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 147, normalized size = 0.72 \begin {gather*} \frac {x (e x)^m \left (b (A b-a B) c^2 \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )+a (-A b+a B) c d \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )+a (b c-a d) (B c-A d) \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )\right )}{a c^2 (b c-a d)^2 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (B \,x^{2}+A \right )}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x\right )^{m} \left (A + B x^{2}\right )}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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