Optimal. Leaf size=495 \[ \frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b^3 e (m+2 p+3) (m+2 p+5) (m+2 p+7)}-\frac {(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {b x^2}{a}\right ) (b c (m+2 p+3) ((m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 b c (p+2) (a B (m+1)-A b (m+2 p+7)))-a (m+1) (2 b c d (p+2) (a B (m+1)-A b (m+2 p+7))+d (m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 (b c-a d) (a B d (m+5)-b (A d (m+2 p+7)+4 B c))))}{b^3 e (m+1) (m+2 p+3) (m+2 p+5) (m+2 p+7)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (a B d (m+5)-b (A d (m+2 p+7)+4 B c))}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)} \]
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Rubi [A] time = 0.75, antiderivative size = 464, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {581, 459, 365, 364} \[ -\frac {(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {b x^2}{a}\right ) \left (\frac {a \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b (m+2 p+3)}+c \left ((b c-a d) (a B (m+5)-A b (m+2 p+7))+\frac {2 b c (p+2) (a B (m+1)-A b (m+2 p+7))}{m+1}\right )\right )}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b^3 e (m+2 p+3) (m+2 p+5) (m+2 p+7)}+\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 459
Rule 581
Rubi steps
\begin {align*} \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx &=\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}+\frac {\int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right ) \left (-c (a B (1+m)-A b (7+m+2 p))+(4 b B c-a B d (5+m)+A b d (7+m+2 p)) x^2\right ) \, dx}{b (7+m+2 p)}\\ &=\frac {(4 b B c-a B d (5+m)+A b d (7+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^2 e (5+m+2 p) (7+m+2 p)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}+\frac {\int (e x)^m \left (a+b x^2\right )^p \left (-c (2 b c (2+p) (a B (1+m)-A b (7+m+2 p))+(b c-a d) (1+m) (a B (5+m)-A b (7+m+2 p)))-(2 b c d (2+p) (a B (1+m)-A b (7+m+2 p))+d (b c-a d) (1+m) (a B (5+m)-A b (7+m+2 p))-2 (b c-a d) (4 b B c-a B d (5+m)+A b d (7+m+2 p))) x^2\right ) \, dx}{b^2 (5+m+2 p) (7+m+2 p)}\\ &=\frac {\left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^3 e (3+m+2 p) (5+m+2 p) (7+m+2 p)}+\frac {(4 b B c-a B d (5+m)+A b d (7+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^2 e (5+m+2 p) (7+m+2 p)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}-\frac {\left (c (2 b c (2+p) (a B (1+m)-A b (7+m+2 p))+(b c-a d) (1+m) (a B (5+m)-A b (7+m+2 p)))+\frac {a (1+m) \left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\right )\right )}{b (3+m+2 p)}\right ) \int (e x)^m \left (a+b x^2\right )^p \, dx}{b^2 (5+m+2 p) (7+m+2 p)}\\ &=\frac {\left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^3 e (3+m+2 p) (5+m+2 p) (7+m+2 p)}+\frac {(4 b B c-a B d (5+m)+A b d (7+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^2 e (5+m+2 p) (7+m+2 p)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}-\frac {\left (\left (c (2 b c (2+p) (a B (1+m)-A b (7+m+2 p))+(b c-a d) (1+m) (a B (5+m)-A b (7+m+2 p)))+\frac {a (1+m) \left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\right )\right )}{b (3+m+2 p)}\right ) \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}{b^2 (5+m+2 p) (7+m+2 p)}\\ &=\frac {\left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^3 e (3+m+2 p) (5+m+2 p) (7+m+2 p)}+\frac {(4 b B c-a B d (5+m)+A b d (7+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^2 e (5+m+2 p) (7+m+2 p)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}-\frac {\left (c (2 b c (2+p) (a B (1+m)-A b (7+m+2 p))+(b c-a d) (1+m) (a B (5+m)-A b (7+m+2 p)))+\frac {a (1+m) \left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\right )\right )}{b (3+m+2 p)}\right ) (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 )}{b^2 e (1+m) (5+m+2 p) (7+m+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 198, normalized size = 0.40 \[ x (e x)^m \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (\frac {c x^2 (2 A d+B c) \, _2F_1\left (\frac {m+3}{2},-p;\frac {m+5}{2};-\frac {b x^2}{a}\right )}{m+3}+d x^4 \left (\frac {(A d+2 B c) \, _2F_1\left (\frac {m+5}{2},-p;\frac {m+7}{2};-\frac {b x^2}{a}\right )}{m+5}+\frac {B d x^2 \, _2F_1\left (\frac {m+7}{2},-p;\frac {m+9}{2};-\frac {b x^2}{a}\right )}{m+7}\right )+\frac {A c^2 \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {b x^2}{a}\right )}{m+1}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B d^{2} x^{6} + {\left (2 \, B c d + A d^{2}\right )} x^{4} + A c^{2} + {\left (B c^{2} + 2 \, A c d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}, x\right ) \]
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 \[ \int {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \]
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 \[ \int \left (B \,x^{2}+A \right ) \left (d \,x^{2}+c \right )^{2} \left (e x \right )^{m} \left (b \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \]
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 \[ \int \left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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