Optimal. Leaf size=246 \[ -\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (A d (m+3)-B c (m+5)))}{2 c^2 d^3 e (m+1)}-\frac{b (e x)^{m+1} (2 a d (A d (m+1)-B c (m+3))-b c (A d (m+3)-B c (m+5)))}{2 c d^3 e (m+1)}-\frac{\left (a+b x^2\right )^2 (e x)^{m+1} (B c-A d)}{2 c d e \left (c+d x^2\right )}-\frac{b^2 (e x)^{m+3} (A d (m+3)-B c (m+5))}{2 c d^2 e^3 (m+3)} \]
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Rubi [A] time = 0.404804, antiderivative size = 246, normalized size of antiderivative = 1., 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} \[ -\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (A d (m+3)-B c (m+5)))}{2 c^2 d^3 e (m+1)}-\frac{b (e x)^{m+1} (2 a d (A d (m+1)-B c (m+3))-b c (A d (m+3)-B c (m+5)))}{2 c d^3 e (m+1)}-\frac{\left (a+b x^2\right )^2 (e x)^{m+1} (B c-A d)}{2 c d e \left (c+d x^2\right )}-\frac{b^2 (e x)^{m+3} (A d (m+3)-B c (m+5))}{2 c d^2 e^3 (m+3)} \]
Antiderivative was successfully verified.
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Rule 577
Rule 570
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right )}{\left (c+d x^2\right )^2} \, dx &=-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^2}{2 c d e \left (c+d x^2\right )}-\frac{\int \frac{(e x)^m \left (a+b x^2\right ) \left (-a (A d (1-m)+B c (1+m))+b (A d (3+m)-B c (5+m)) x^2\right )}{c+d x^2} \, dx}{2 c d}\\ &=-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^2}{2 c d e \left (c+d x^2\right )}-\frac{\int \left (\frac{b (2 a d (A d (1+m)-B c (3+m))-b c (A d (3+m)-B c (5+m))) (e x)^m}{d^2}+\frac{b^2 (A d (3+m)-B c (5+m)) (e x)^{2+m}}{d e^2}+\frac{\left (-5 b^2 B c^3+3 A b^2 c^2 d+6 a b B c^2 d-2 a A b c d^2-a^2 B c d^2-a^2 A d^3-b^2 B c^3 m+A b^2 c^2 d m+2 a b B c^2 d m-2 a A b c d^2 m-a^2 B c d^2 m+a^2 A d^3 m\right ) (e x)^m}{d^2 \left (c+d x^2\right )}\right ) \, dx}{2 c d}\\ &=-\frac{b (2 a d (A d (1+m)-B c (3+m))-b c (A d (3+m)-B c (5+m))) (e x)^{1+m}}{2 c d^3 e (1+m)}-\frac{b^2 (A d (3+m)-B c (5+m)) (e x)^{3+m}}{2 c d^2 e^3 (3+m)}-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^2}{2 c d e \left (c+d x^2\right )}-\frac{((b c-a d) (a d (A d (1-m)+B c (1+m))+b c (A d (3+m)-B c (5+m)))) \int \frac{(e x)^m}{c+d x^2} \, dx}{2 c d^3}\\ &=-\frac{b (2 a d (A d (1+m)-B c (3+m))-b c (A d (3+m)-B c (5+m))) (e x)^{1+m}}{2 c d^3 e (1+m)}-\frac{b^2 (A d (3+m)-B c (5+m)) (e x)^{3+m}}{2 c d^2 e^3 (3+m)}-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^2}{2 c d e \left (c+d x^2\right )}-\frac{(b c-a d) (a d (A d (1-m)+B c (1+m))+b c (A d (3+m)-B c (5+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^3 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.223013, size = 158, normalized size = 0.64 \[ \frac{x (e x)^m \left (-\frac{(b c-a d)^2 (B c-A d) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c^2 (m+1)}+\frac{(b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (-a B d-2 A b d+3 b B c)}{c (m+1)}+\frac{b (2 a B d+A b d-2 b B c)}{m+1}+\frac{b^2 B d x^2}{m+3}\right )}{d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( B{x}^{2}+A \right ) \left ( b{x}^{2}+a \right ) ^{2} \left ( ex \right ) ^{m}}{ \left ( d{x}^{2}+c \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )}{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B b^{2} x^{6} +{\left (2 \, B a b + A b^{2}\right )} x^{4} + A a^{2} +{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )} \left (e x\right )^{m}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m} \left (A + B x^{2}\right ) \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )}{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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