Optimal. Leaf size=258 \[ \frac {d (e x)^{m+3} \left (a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right )}{b^3 e^3 (m+3)}-\frac {(e x)^{m+1} \left (a^3 B d^3-a^2 b d^2 (A d+3 B c)+3 a b^2 c d (A d+B c)+b^3 \left (-c^2\right ) (3 A d+B c)\right )}{b^4 e (m+1)}+\frac {(e x)^{m+1} (A b-a B) (b c-a d)^3 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a b^4 e (m+1)}+\frac {d^2 (e x)^{m+5} (-a B d+A b d+3 b B c)}{b^2 e^5 (m+5)}+\frac {B d^3 (e x)^{m+7}}{b e^7 (m+7)} \]
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Rubi [A] time = 0.28, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {570, 364} \[ -\frac {(e x)^{m+1} \left (-a^2 b d^2 (A d+3 B c)+a^3 B d^3+3 a b^2 c d (A d+B c)+b^3 \left (-c^2\right ) (3 A d+B c)\right )}{b^4 e (m+1)}+\frac {d (e x)^{m+3} \left (a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right )}{b^3 e^3 (m+3)}+\frac {d^2 (e x)^{m+5} (-a B d+A b d+3 b B c)}{b^2 e^5 (m+5)}+\frac {(e x)^{m+1} (A b-a B) (b c-a d)^3 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a b^4 e (m+1)}+\frac {B d^3 (e x)^{m+7}}{b e^7 (m+7)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 570
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\int \left (-\frac {\left (a^3 B d^3+3 a b^2 c d (B c+A d)-a^2 b d^2 (3 B c+A d)-b^3 c^2 (B c+3 A d)\right ) (e x)^m}{b^4}+\frac {d \left (a^2 B d^2+3 b^2 c (B c+A d)-a b d (3 B c+A d)\right ) (e x)^{2+m}}{b^3 e^2}+\frac {d^2 (3 b B c+A b d-a B d) (e x)^{4+m}}{b^2 e^4}+\frac {B d^3 (e x)^{6+m}}{b e^6}+\frac {\left (A b^4 c^3-a b^3 B c^3-3 a A b^3 c^2 d+3 a^2 b^2 B c^2 d+3 a^2 A b^2 c d^2-3 a^3 b B c d^2-a^3 A b d^3+a^4 B d^3\right ) (e x)^m}{b^4 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {\left (a^3 B d^3+3 a b^2 c d (B c+A d)-a^2 b d^2 (3 B c+A d)-b^3 c^2 (B c+3 A d)\right ) (e x)^{1+m}}{b^4 e (1+m)}+\frac {d \left (a^2 B d^2+3 b^2 c (B c+A d)-a b d (3 B c+A d)\right ) (e x)^{3+m}}{b^3 e^3 (3+m)}+\frac {d^2 (3 b B c+A b d-a B d) (e x)^{5+m}}{b^2 e^5 (5+m)}+\frac {B d^3 (e x)^{7+m}}{b e^7 (7+m)}+\frac {\left ((A b-a B) (b c-a d)^3\right ) \int \frac {(e x)^m}{a+b x^2} \, dx}{b^4}\\ &=-\frac {\left (a^3 B d^3+3 a b^2 c d (B c+A d)-a^2 b d^2 (3 B c+A d)-b^3 c^2 (B c+3 A d)\right ) (e x)^{1+m}}{b^4 e (1+m)}+\frac {d \left (a^2 B d^2+3 b^2 c (B c+A d)-a b d (3 B c+A d)\right ) (e x)^{3+m}}{b^3 e^3 (3+m)}+\frac {d^2 (3 b B c+A b d-a B d) (e x)^{5+m}}{b^2 e^5 (5+m)}+\frac {B d^3 (e x)^{7+m}}{b e^7 (7+m)}+\frac {(A b-a B) (b c-a d)^3 (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a b^4 e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 217, normalized size = 0.84 \[ \frac {x (e x)^m \left (\frac {b d x^2 \left (a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right )}{m+3}+\frac {-a^3 B d^3+a^2 b d^2 (A d+3 B c)-3 a b^2 c d (A d+B c)+b^3 c^2 (3 A d+B c)}{m+1}+\frac {b^2 d^2 x^4 (-a B d+A b d+3 b B c)}{m+5}+\frac {(a B-A b) (a d-b c)^3 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a (m+1)}+\frac {b^3 B d^3 x^6}{m+7}\right )}{b^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B d^{3} x^{8} + {\left (3 \, B c d^{2} + A d^{3}\right )} x^{6} + 3 \, {\left (B c^{2} d + A c d^{2}\right )} x^{4} + A c^{3} + {\left (B c^{3} + 3 \, A c^{2} d\right )} x^{2}\right )} \left (e x\right )^{m}}{b x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} \left (e x\right )^{m}}{b x^{2} + a}\,{d x} \]
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 \[ \int \frac {\left (B \,x^{2}+A \right ) \left (d \,x^{2}+c \right )^{3} \left (e x \right )^{m}}{b \,x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} \left (e x\right )^{m}}{b x^{2} + a}\,{d x} \]
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 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,{\left (d\,x^2+c\right )}^3}{b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 29.05, size = 911, normalized size = 3.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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