Integrand size = 29, antiderivative size = 228 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^3} \, dx=-\frac {(A b (b c (1+m-2 n)-a d (1+m-n))-a B (b c (1+m)-a d (1+m+n))) (e x)^{1+m}}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2}-\frac {(b c (a B (1+m)-A b (1+m-2 n)) (1+m-n)+a d (1+m) (A b (1+m-n)-a B (1+m+n))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{2 a^3 b^2 e (1+m) n^2} \]
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Time = 0.17 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {608, 468, 371} \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^3} \, dx=-\frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right ) (b c (m-n+1) (a B (m+1)-A b (m-2 n+1))+a d (m+1) (A b (m-n+1)-a B (m+n+1)))}{2 a^3 b^2 e (m+1) n^2}+\frac {(e x)^{m+1} (b c (a B (m+1)-A b (m-2 n+1))+a d (A b (m-n+1)-a B (m+n+1)))}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2} \]
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Rule 371
Rule 468
Rule 608
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2}-\frac {\int \frac {(e x)^m \left (-c (a B (1+m)-A b (1+m-2 n))+d (A b (1+m-n)-a B (1+m+n)) x^n\right )}{\left (a+b x^n\right )^2} \, dx}{2 a b n} \\ & = \frac {(b c (a B (1+m)-A b (1+m-2 n))+a d (A b (1+m-n)-a B (1+m+n))) (e x)^{1+m}}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2}-\frac {(b c (a B (1+m)-A b (1+m-2 n)) (1+m-n)+a d (1+m) (A b (1+m-n)-a B (1+m+n))) \int \frac {(e x)^m}{a+b x^n} \, dx}{2 a^2 b^2 n^2} \\ & = \frac {(b c (a B (1+m)-A b (1+m-2 n))+a d (A b (1+m-n)-a B (1+m+n))) (e x)^{1+m}}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2}-\frac {(b c (a B (1+m)-A b (1+m-2 n)) (1+m-n)+a d (1+m) (A b (1+m-n)-a B (1+m+n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{2 a^3 b^2 e (1+m) n^2} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.60 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^3} \, dx=\frac {x (e x)^m \left (a^2 B d \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )+a (b B c+A b d-2 a B d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )+(A b-a B) (b c-a d) \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )\right )}{a^3 b^2 (1+m)} \]
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\[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )}{\left (a +b \,x^{n}\right )^{3}}d x\]
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\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \]
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\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,\left (c+d\,x^n\right )}{{\left (a+b\,x^n\right )}^3} \,d x \]
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