Optimal. Leaf size=247 \[ \frac {(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right ) (A b (a d (m+3)+b (c-c m))+a B (b c (m+1)-a d (m+5)))}{2 a^2 b^3 e (m+1)}-\frac {d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+3))-a B (2 b c (m+3)-a d (m+5)))}{2 a b^3 e (m+1)}-\frac {d^2 (e x)^{m+3} (A b (m+3)-a B (m+5))}{2 a b^2 e^3 (m+3)}+\frac {\left (c+d x^2\right )^2 (e x)^{m+1} (A b-a B)}{2 a b e \left (a+b x^2\right )} \]
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Rubi [A] time = 0.44, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {577, 570, 364} \[ \frac {(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right ) (A b (a d (m+3)+b (c-c m))+a B (b c (m+1)-a d (m+5)))}{2 a^2 b^3 e (m+1)}-\frac {d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+3))-a B (2 b c (m+3)-a d (m+5)))}{2 a b^3 e (m+1)}-\frac {d^2 (e x)^{m+3} (A b (m+3)-a B (m+5))}{2 a b^2 e^3 (m+3)}+\frac {\left (c+d x^2\right )^2 (e x)^{m+1} (A b-a B)}{2 a b e \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 570
Rule 577
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{2 a b e \left (a+b x^2\right )}-\frac {\int \frac {(e x)^m \left (c+d x^2\right ) \left (-c (A b (1-m)+a B (1+m))+d (A b (3+m)-a B (5+m)) x^2\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{2 a b e \left (a+b x^2\right )}-\frac {\int \left (\frac {d (A b (2 b c (1+m)-a d (3+m))-a B (2 b c (3+m)-a d (5+m))) (e x)^m}{b^2}+\frac {d^2 (A b (3+m)-a B (5+m)) (e x)^{2+m}}{b e^2}+\frac {\left (-A b^3 c^2-a b^2 B c^2-2 a A b^2 c d+6 a^2 b B c d+3 a^2 A b d^2-5 a^3 B d^2+A b^3 c^2 m-a b^2 B c^2 m-2 a A b^2 c d m+2 a^2 b B c d m+a^2 A b d^2 m-a^3 B d^2 m\right ) (e x)^m}{b^2 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=-\frac {d (A b (2 b c (1+m)-a d (3+m))-a B (2 b c (3+m)-a d (5+m))) (e x)^{1+m}}{2 a b^3 e (1+m)}-\frac {d^2 (A b (3+m)-a B (5+m)) (e x)^{3+m}}{2 a b^2 e^3 (3+m)}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{2 a b e \left (a+b x^2\right )}-\frac {\left (-A b^3 c^2-a b^2 B c^2-2 a A b^2 c d+6 a^2 b B c d+3 a^2 A b d^2-5 a^3 B d^2+A b^3 c^2 m-a b^2 B c^2 m-2 a A b^2 c d m+2 a^2 b B c d m+a^2 A b d^2 m-a^3 B d^2 m\right ) \int \frac {(e x)^m}{a+b x^2} \, dx}{2 a b^3}\\ &=-\frac {d (A b (2 b c (1+m)-a d (3+m))-a B (2 b c (3+m)-a d (5+m))) (e x)^{1+m}}{2 a b^3 e (1+m)}-\frac {d^2 (A b (3+m)-a B (5+m)) (e x)^{3+m}}{2 a b^2 e^3 (3+m)}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{2 a b e \left (a+b x^2\right )}+\frac {(b c-a d) (A b (b c (1-m)+a d (3+m))+a B (b c (1+m)-a d (5+m))) (e 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^3 e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 156, normalized size = 0.63 \[ \frac {x (e x)^m \left (\frac {(A b-a B) (b c-a d)^2 \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a^2 (m+1)}+\frac {(b c-a d) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right ) (-3 a B d+2 A b d+b B c)}{a (m+1)}+\frac {d (-2 a B d+A b d+2 b B c)}{m+1}+\frac {b B d^2 x^2}{m+3}\right )}{b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B d^{2} x^{6} + {\left (2 \, B c d + A d^{2}\right )} x^{4} + A c^{2} + {\left (B c^{2} + 2 \, A c d\right )} x^{2}\right )} \left (e x\right )^{m}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{2}+A \right ) \left (d \,x^{2}+c \right )^{2} \left (e x \right )^{m}}{\left (b \,x^{2}+a \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^2} \,d x \]
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 \[ \int \frac {\left (e x\right )^{m} \left (A + B x^{2}\right ) \left (c + d x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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