Integrand size = 29, antiderivative size = 816 \[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\frac {(f x)^{1+m} \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) f n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {(f x)^{1+m} \left (\left (b^2-2 a c\right ) \left (a b e (1+m)+2 a c d (1+m-4 n)-b^2 d (1+m-2 n)\right )+a b c (b d-2 a e) (1+m-3 n)+c \left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 f n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {c \left (\left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n)+\frac {a b^3 e (1+m) (1+m-n)-4 a^2 b c e \left (1+m^2+m (2-n)-n-3 n^2\right )-b^4 d \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )+6 a b^2 c d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )-8 a^2 c^2 d \left (1+m^2+m (2-6 n)-6 n+8 n^2\right )}{\sqrt {b^2-4 a c}}\right ) (f x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) f (1+m) n^2}-\frac {c \left (\left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n)-\frac {a b^3 e (1+m) (1+m-n)-4 a^2 b c e \left (1+m^2+m (2-n)-n-3 n^2\right )-b^4 d \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )+6 a b^2 c d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )-8 a^2 c^2 d \left (1+m^2+m (2-6 n)-6 n+8 n^2\right )}{\sqrt {b^2-4 a c}}\right ) (f x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b+\sqrt {b^2-4 a c}\right ) f (1+m) n^2} \]
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Time = 3.53 (sec) , antiderivative size = 816, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1572, 1574, 371} \[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=-\frac {c \left (\left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1)+\frac {-d \left (m^2+(2-3 n) m+2 n^2-3 n+1\right ) b^4+a e (m+1) (m-n+1) b^3+6 a c d \left (m^2+(2-4 n) m+3 n^2-4 n+1\right ) b^2-4 a^2 c e \left (m^2+(2-n) m-3 n^2-n+1\right ) b-8 a^2 c^2 d \left (m^2+(2-6 n) m+8 n^2-6 n+1\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right ) (f x)^{m+1}}{2 a^2 \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) f (m+1) n^2}-\frac {c \left (\left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1)-\frac {-d \left (m^2+(2-3 n) m+2 n^2-3 n+1\right ) b^4+a e (m+1) (m-n+1) b^3+6 a c d \left (m^2+(2-4 n) m+3 n^2-4 n+1\right ) b^2-4 a^2 c e \left (m^2+(2-n) m-3 n^2-n+1\right ) b-8 a^2 c^2 d \left (m^2+(2-6 n) m+8 n^2-6 n+1\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right ) (f x)^{m+1}}{2 a^2 \left (b^2-4 a c\right )^2 \left (b+\sqrt {b^2-4 a c}\right ) f (m+1) n^2}+\frac {\left (c \left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) x^n+\left (b^2-2 a c\right ) \left (-d (m-2 n+1) b^2+a e (m+1) b+2 a c d (m-4 n+1)\right )+a b c (b d-2 a e) (m-3 n+1)\right ) (f x)^{m+1}}{2 a^2 \left (b^2-4 a c\right )^2 f n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac {\left (c (b d-2 a e) x^n+b^2 d-2 a c d-a b e\right ) (f x)^{m+1}}{2 a \left (b^2-4 a c\right ) f n \left (b x^n+c x^{2 n}+a\right )^2} \]
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Rule 371
Rule 1572
Rule 1574
Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) f n \left (a+b x^n+c x^{2 n}\right )^2}-\frac {\int \frac {(f x)^m \left (-a b e (1+m)-2 a c d (1+m-4 n)+b^2 d (1+m-2 n)+c (b d-2 a e) (1+m-3 n) x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{2 a \left (b^2-4 a c\right ) n} \\ & = \frac {(f x)^{1+m} \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) f n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {(f x)^{1+m} \left (\left (b^2-2 a c\right ) \left (a b e (1+m)+2 a c d (1+m-4 n)-b^2 d (1+m-2 n)\right )+a b c (b d-2 a e) (1+m-3 n)+c \left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 f n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {\int \frac {(f x)^m \left (\left (a b e (1+m)+2 a c d (1+m-4 n)-b^2 d (1+m-2 n)\right ) \left (2 a c (1+m-2 n)-b^2 (1+m-n)\right )-a b c (b d-2 a e) (1+m) (1+m-3 n)-c \left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n) x^n\right )}{a+b x^n+c x^{2 n}} \, dx}{2 a^2 \left (b^2-4 a c\right )^2 n^2} \\ & = \frac {(f x)^{1+m} \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) f n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {(f x)^{1+m} \left (\left (b^2-2 a c\right ) \left (a b e (1+m)+2 a c d (1+m-4 n)-b^2 d (1+m-2 n)\right )+a b c (b d-2 a e) (1+m-3 n)+c \left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 f n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {\int \left (\frac {\left (-c \left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n)+\frac {c \left (b^4 d-6 a b^2 c d+8 a^2 c^2 d-a b^3 e+4 a^2 b c e+2 b^4 d m-12 a b^2 c d m+16 a^2 c^2 d m-2 a b^3 e m+8 a^2 b c e m+b^4 d m^2-6 a b^2 c d m^2+8 a^2 c^2 d m^2-a b^3 e m^2+4 a^2 b c e m^2-3 b^4 d n+24 a b^2 c d n-48 a^2 c^2 d n+a b^3 e n-4 a^2 b c e n-3 b^4 d m n+24 a b^2 c d m n-48 a^2 c^2 d m n+a b^3 e m n-4 a^2 b c e m n+2 b^4 d n^2-18 a b^2 c d n^2+64 a^2 c^2 d n^2-12 a^2 b c e n^2\right )}{\sqrt {b^2-4 a c}}\right ) (f x)^m}{b-\sqrt {b^2-4 a c}+2 c x^n}+\frac {\left (-c \left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n)-\frac {c \left (b^4 d-6 a b^2 c d+8 a^2 c^2 d-a b^3 e+4 a^2 b c e+2 b^4 d m-12 a b^2 c d m+16 a^2 c^2 d m-2 a b^3 e m+8 a^2 b c e m+b^4 d m^2-6 a b^2 c d m^2+8 a^2 c^2 d m^2-a b^3 e m^2+4 a^2 b c e m^2-3 b^4 d n+24 a b^2 c d n-48 a^2 c^2 d n+a b^3 e n-4 a^2 b c e n-3 b^4 d m n+24 a b^2 c d m n-48 a^2 c^2 d m n+a b^3 e m n-4 a^2 b c e m n+2 b^4 d n^2-18 a b^2 c d n^2+64 a^2 c^2 d n^2-12 a^2 b c e n^2\right )}{\sqrt {b^2-4 a c}}\right ) (f x)^m}{b+\sqrt {b^2-4 a c}+2 c x^n}\right ) \, dx}{2 a^2 \left (b^2-4 a c\right )^2 n^2} \\ & = \frac {(f x)^{1+m} \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) f n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {(f x)^{1+m} \left (\left (b^2-2 a c\right ) \left (a b e (1+m)+2 a c d (1+m-4 n)-b^2 d (1+m-2 n)\right )+a b c (b d-2 a e) (1+m-3 n)+c \left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 f n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {\left (c \left (\left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n)-\frac {a b^3 e (1+m) (1+m-n)-4 a^2 b c e \left (1+m^2+m (2-n)-n-3 n^2\right )-b^4 d \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )+6 a b^2 c d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )-8 a^2 c^2 d \left (1+m^2+m (2-6 n)-6 n+8 n^2\right )}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {(f x)^m}{b+\sqrt {b^2-4 a c}+2 c x^n} \, dx}{2 a^2 \left (b^2-4 a c\right )^2 n^2}-\frac {\left (c \left (\left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n)+\frac {a b^3 e (1+m) (1+m-n)-4 a^2 b c e \left (1+m^2+m (2-n)-n-3 n^2\right )-b^4 d \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )+6 a b^2 c d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )-8 a^2 c^2 d \left (1+m^2+m (2-6 n)-6 n+8 n^2\right )}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {(f x)^m}{b-\sqrt {b^2-4 a c}+2 c x^n} \, dx}{2 a^2 \left (b^2-4 a c\right )^2 n^2} \\ & = \frac {(f x)^{1+m} \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) f n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {(f x)^{1+m} \left (\left (b^2-2 a c\right ) \left (a b e (1+m)+2 a c d (1+m-4 n)-b^2 d (1+m-2 n)\right )+a b c (b d-2 a e) (1+m-3 n)+c \left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 f n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {c \left (\left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n)+\frac {a b^3 e (1+m) (1+m-n)-4 a^2 b c e \left (1+m^2+m (2-n)-n-3 n^2\right )-b^4 d \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )+6 a b^2 c d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )-8 a^2 c^2 d \left (1+m^2+m (2-6 n)-6 n+8 n^2\right )}{\sqrt {b^2-4 a c}}\right ) (f x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) f (1+m) n^2}-\frac {c \left (\left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n)-\frac {a b^3 e (1+m) (1+m-n)-4 a^2 b c e \left (1+m^2+m (2-n)-n-3 n^2\right )-b^4 d \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )+6 a b^2 c d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )-8 a^2 c^2 d \left (1+m^2+m (2-6 n)-6 n+8 n^2\right )}{\sqrt {b^2-4 a c}}\right ) (f x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b+\sqrt {b^2-4 a c}\right ) f (1+m) n^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(20515\) vs. \(2(816)=1632\).
Time = 8.44 (sec) , antiderivative size = 20515, normalized size of antiderivative = 25.14 \[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (f x \right )^{m} \left (d +e \,x^{n}\right )}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{3}}d x\]
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\[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )} \left (f x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )} \left (f x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \]
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\[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )} \left (f x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (d+e\,x^n\right )}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^3} \,d x \]
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