Optimal. Leaf size=483 \[ \frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (a^2 d^2 (1-m) (A d (3-m)+B c (m+1))-2 a b c d (A d (1-m) (-m-2 p+3)+B c (m+1) (-m-2 p+1))+b^2 c^2 (-m-2 p+1) (A d (-m-2 p+3)+B c (m+2 p+1))\right ) F_1\left (\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{8 c^3 d e (m+1) (b c-a d)^2}-\frac{b (m+2 p+1) (e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right ) (a d (A d (3-m)+B c (m+1))+b c (B c (-m-2 p+1)-A d (-m-2 p+5)))}{8 c^2 d e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} \left (a+b x^2\right )^{p+1} (a d (A d (3-m)+B c (m+1))+b c (B c (-m-2 p+1)-A d (-m-2 p+5)))}{8 c^2 e \left (c+d x^2\right ) (b c-a d)^2}+\frac{(e x)^{m+1} \left (a+b x^2\right )^{p+1} (B c-A d)}{4 c e \left (c+d x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 1.03914, antiderivative size = 483, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {579, 584, 365, 364, 511, 510} \[ \frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (a^2 d^2 (1-m) (A d (3-m)+B c (m+1))-2 a b c d (A d (1-m) (-m-2 p+3)+B c (m+1) (-m-2 p+1))+b^2 c^2 (-m-2 p+1) (A d (-m-2 p+3)+B c (m+2 p+1))\right ) F_1\left (\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{8 c^3 d e (m+1) (b c-a d)^2}-\frac{b (m+2 p+1) (e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right ) (a d (A d (3-m)+B c (m+1))+b c (B c (-m-2 p+1)-A d (-m-2 p+5)))}{8 c^2 d e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} \left (a+b x^2\right )^{p+1} (a d (A d (3-m)+B c (m+1))+b c (B c (-m-2 p+1)-A d (-m-2 p+5)))}{8 c^2 e \left (c+d x^2\right ) (b c-a d)^2}+\frac{(e x)^{m+1} \left (a+b x^2\right )^{p+1} (B c-A d)}{4 c e \left (c+d x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 579
Rule 584
Rule 365
Rule 364
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right )}{\left (c+d x^2\right )^3} \, dx &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{4 c (b c-a d) e \left (c+d x^2\right )^2}+\frac{\int \frac{(e x)^m \left (a+b x^2\right )^p \left (4 A b c-a A d (3-m)-a B c (1+m)+b (B c-A d) (1-m-2 p) x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)}\\ &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{4 c (b c-a d) e \left (c+d x^2\right )^2}+\frac{(a d (A d (3-m)+B c (1+m))+b c (B c (1-m-2 p)-A d (5-m-2 p))) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{8 c^2 (b c-a d)^2 e \left (c+d x^2\right )}+\frac{\int \frac{(e x)^m \left (a+b x^2\right )^p \left (a B c (1+m) (a d (1-m)-b c (3-m-2 p))+A \left (8 b^2 c^2+a^2 d^2 \left (3-4 m+m^2\right )-a b c d \left (9+m^2-2 m (3-p)+2 p\right )\right )+b (d (4 A b c-a A d (3-m)-a B c (1+m))-b c (B c-A d) (1-m-2 p)) (1+m+2 p) x^2\right )}{c+d x^2} \, dx}{8 c^2 (b c-a d)^2}\\ &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{4 c (b c-a d) e \left (c+d x^2\right )^2}+\frac{(a d (A d (3-m)+B c (1+m))+b c (B c (1-m-2 p)-A d (5-m-2 p))) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{8 c^2 (b c-a d)^2 e \left (c+d x^2\right )}+\frac{\int \left (\frac{b (d (4 A b c-a A d (3-m)-a B c (1+m))-b c (B c-A d) (1-m-2 p)) (1+m+2 p) (e x)^m \left (a+b x^2\right )^p}{d}+\frac{\left (-b c (d (4 A b c-a A d (3-m)-a B c (1+m))-b c (B c-A d) (1-m-2 p)) (1+m+2 p)+d \left (a B c (1+m) (a d (1-m)-b c (3-m-2 p))+A \left (8 b^2 c^2+a^2 d^2 \left (3-4 m+m^2\right )-a b c d \left (9+m^2-2 m (3-p)+2 p\right )\right )\right )\right ) (e x)^m \left (a+b x^2\right )^p}{d \left (c+d x^2\right )}\right ) \, dx}{8 c^2 (b c-a d)^2}\\ &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{4 c (b c-a d) e \left (c+d x^2\right )^2}+\frac{(a d (A d (3-m)+B c (1+m))+b c (B c (1-m-2 p)-A d (5-m-2 p))) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{8 c^2 (b c-a d)^2 e \left (c+d x^2\right )}-\frac{(b (a d (A d (3-m)+B c (1+m))+b c (B c (1-m-2 p)-A d (5-m-2 p))) (1+m+2 p)) \int (e x)^m \left (a+b x^2\right )^p \, dx}{8 c^2 d (b c-a d)^2}+\frac{\left (a^2 d^2 (1-m) (A d (3-m)+B c (1+m))-2 a b c d (B c (1+m) (1-m-2 p)+A d (1-m) (3-m-2 p))+b^2 c^2 (1-m-2 p) (A d (3-m-2 p)+B c (1+m+2 p))\right ) \int \frac{(e x)^m \left (a+b x^2\right )^p}{c+d x^2} \, dx}{8 c^2 d (b c-a d)^2}\\ &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{4 c (b c-a d) e \left (c+d x^2\right )^2}+\frac{(a d (A d (3-m)+B c (1+m))+b c (B c (1-m-2 p)-A d (5-m-2 p))) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{8 c^2 (b c-a d)^2 e \left (c+d x^2\right )}-\frac{\left (b (a d (A d (3-m)+B c (1+m))+b c (B c (1-m-2 p)-A d (5-m-2 p))) (1+m+2 p) \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac{b x^2}{a}\right )^p \, dx}{8 c^2 d (b c-a d)^2}+\frac{\left (\left (a^2 d^2 (1-m) (A d (3-m)+B c (1+m))-2 a b c d (B c (1+m) (1-m-2 p)+A d (1-m) (3-m-2 p))+b^2 c^2 (1-m-2 p) (A d (3-m-2 p)+B c (1+m+2 p))\right ) \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int \frac{(e x)^m \left (1+\frac{b x^2}{a}\right )^p}{c+d x^2} \, dx}{8 c^2 d (b c-a d)^2}\\ &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{4 c (b c-a d) e \left (c+d x^2\right )^2}+\frac{(a d (A d (3-m)+B c (1+m))+b c (B c (1-m-2 p)-A d (5-m-2 p))) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{8 c^2 (b c-a d)^2 e \left (c+d x^2\right )}+\frac{\left (a^2 d^2 (1-m) (A d (3-m)+B c (1+m))-2 a b c d (B c (1+m) (1-m-2 p)+A d (1-m) (3-m-2 p))+b^2 c^2 (1-m-2 p) (A d (3-m-2 p)+B c (1+m+2 p))\right ) (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} F_1\left (\frac{1+m}{2};-p,1;\frac{3+m}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{8 c^3 d (b c-a d)^2 e (1+m)}-\frac{b (a d (A d (3-m)+B c (1+m))+b c (B c (1-m-2 p)-A d (5-m-2 p))) (1+m+2 p) (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \, _2F_1\left (\frac{1+m}{2},-p;\frac{3+m}{2};-\frac{b x^2}{a}\right )}{8 c^2 d (b c-a d)^2 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.383708, size = 128, normalized size = 0.27 \[ \frac{x (e x)^m \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left ((A d-B c) F_1\left (\frac{m+1}{2};-p,3;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+B c F_1\left (\frac{m+1}{2};-p,2;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{c^3 d (m+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.081, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) ^{p} \left ( B{x}^{2}+A \right ) }{ \left ( d{x}^{2}+c \right ) ^{3}}}\, 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 )}^{p} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{3}}\,{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 x^{2} + A\right )}{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}}{d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{p} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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