Integrand size = 22, antiderivative size = 328 \[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\frac {(d x)^{1+m} \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) d n \left (a+b x^n+c x^{2 n}\right )}+\frac {c \left (\frac {4 a c (1+m-2 n)-b^2 (1+m-n)}{\sqrt {b^2-4 a c}}-b (1+m-n)\right ) (d x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{a \left (b^2-4 a c\right ) \left (b-\sqrt {b^2-4 a c}\right ) d (1+m) n}-\frac {c \left (4 a c (1+m-2 n)-b^2 (1+m-n)+b \sqrt {b^2-4 a c} (1+m-n)\right ) (d x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt {b^2-4 a c}\right ) d (1+m) n} \]
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Time = 0.61 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1398, 1574, 371} \[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\frac {c (d x)^{m+1} \left (\frac {4 a c (m-2 n+1)-b^2 (m-n+1)}{\sqrt {b^2-4 a c}}-b (m-n+1)\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{a d (m+1) n \left (b^2-4 a c\right ) \left (b-\sqrt {b^2-4 a c}\right )}-\frac {c (d x)^{m+1} \left (b (m-n+1) \sqrt {b^2-4 a c}+4 a c (m-2 n+1)-\left (b^2 (m-n+1)\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a d (m+1) n \left (b^2-4 a c\right )^{3/2} \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(d x)^{m+1} \left (-2 a c+b^2+b c x^n\right )}{a d n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]
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Rule 371
Rule 1398
Rule 1574
Rubi steps \begin{align*} \text {integral}& = \frac {(d x)^{1+m} \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) d n \left (a+b x^n+c x^{2 n}\right )}-\frac {\int \frac {(d x)^m \left (-2 a c (1+m-2 n)+b^2 (1+m-n)+b c (1+m-n) x^n\right )}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n} \\ & = \frac {(d x)^{1+m} \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) d n \left (a+b x^n+c x^{2 n}\right )}-\frac {\int \left (\frac {\left (b c (1+m-n)+\frac {c \left (b^2-4 a c+b^2 m-4 a c m-b^2 n+8 a c n\right )}{\sqrt {b^2-4 a c}}\right ) (d x)^m}{b-\sqrt {b^2-4 a c}+2 c x^n}+\frac {\left (b c (1+m-n)-\frac {c \left (b^2-4 a c+b^2 m-4 a c m-b^2 n+8 a c n\right )}{\sqrt {b^2-4 a c}}\right ) (d x)^m}{b+\sqrt {b^2-4 a c}+2 c x^n}\right ) \, dx}{a \left (b^2-4 a c\right ) n} \\ & = \frac {(d x)^{1+m} \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) d n \left (a+b x^n+c x^{2 n}\right )}+\frac {\left (c \left (4 a c (1+m-2 n)-b^2 (1+m-n)-b \sqrt {b^2-4 a c} (1+m-n)\right )\right ) \int \frac {(d x)^m}{b-\sqrt {b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}-\frac {\left (c \left (4 a c (1+m-2 n)-b^2 (1+m-n)+b \sqrt {b^2-4 a c} (1+m-n)\right )\right ) \int \frac {(d x)^m}{b+\sqrt {b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n} \\ & = \frac {(d x)^{1+m} \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) d n \left (a+b x^n+c x^{2 n}\right )}+\frac {c \left (4 a c (1+m-2 n)-b^2 (1+m-n)-b \sqrt {b^2-4 a c} (1+m-n)\right ) (d 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 )}{a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt {b^2-4 a c}\right ) d (1+m) n}-\frac {c \left (4 a c (1+m-2 n)-b^2 (1+m-n)+b \sqrt {b^2-4 a c} (1+m-n)\right ) (d 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 )}{a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt {b^2-4 a c}\right ) d (1+m) n} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1890\) vs. \(2(328)=656\).
Time = 4.41 (sec) , antiderivative size = 1890, normalized size of antiderivative = 5.76 \[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\frac {x (d x)^m \left (\frac {2 b^2 c}{a \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m)}-\frac {8 c^2}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m)}+\frac {2 b^2 c}{a \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) (1+m)}+\frac {8 c^2}{\left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right ) (1+m)}-\frac {2 b^2 c}{a \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m) n}+\frac {4 c^2}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m) n}+\frac {4 c^2}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) (1+m) n}+\frac {2 b^2 c}{a \left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right ) (1+m) n}-\frac {2 b^2 c m}{a \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m) n}+\frac {4 c^2 m}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m) n}+\frac {4 c^2 m}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) (1+m) n}+\frac {2 b^2 c m}{a \left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right ) (1+m) n}-\frac {b^2}{a n \left (a+x^n \left (b+c x^n\right )\right )}+\frac {2 c}{n \left (a+x^n \left (b+c x^n\right )\right )}-\frac {b c x^n}{a n \left (a+x^n \left (b+c x^n\right )\right )}+\frac {2^{-\frac {1+m}{n}} c \left (4 a c \sqrt {b^2-4 a c} (1+m-2 n)+4 a b c (1+m-n)-b^2 \sqrt {b^2-4 a c} (1+m-n)+b^3 (-1-m+n)\right ) \left (\frac {c x^n}{b-\sqrt {b^2-4 a c}+2 c x^n}\right )^{-\frac {1+m}{n}} \operatorname {Hypergeometric2F1}\left (-\frac {1+m}{n},-\frac {1+m}{n},1-\frac {1+m}{n},\frac {b-\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}+2 c x^n}\right )}{a \sqrt {b^2-4 a c} \left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right ) (1+m) n}+\frac {2^{-\frac {1+m}{n}} b c (-1-m+n) \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-\frac {1+m}{n}} \operatorname {Hypergeometric2F1}\left (-\frac {1+m}{n},-\frac {1+m}{n},1-\frac {1+m}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{a \sqrt {b^2-4 a c} (1+m) n}-\frac {2^{\frac {-1-m+n}{n}} b^2 c \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-\frac {1+m}{n}} \operatorname {Hypergeometric2F1}\left (-\frac {1+m}{n},-\frac {1+m}{n},\frac {-1-m+n}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{a \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m)}+\frac {2^{-\frac {1+m-3 n}{n}} c^2 \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-\frac {1+m}{n}} \operatorname {Hypergeometric2F1}\left (-\frac {1+m}{n},-\frac {1+m}{n},\frac {-1-m+n}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m)}+\frac {2^{\frac {-1-m+n}{n}} b^2 c \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-\frac {1+m}{n}} \operatorname {Hypergeometric2F1}\left (-\frac {1+m}{n},-\frac {1+m}{n},\frac {-1-m+n}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{a \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m) n}-\frac {2^{-\frac {1+m-2 n}{n}} c^2 \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-\frac {1+m}{n}} \operatorname {Hypergeometric2F1}\left (-\frac {1+m}{n},-\frac {1+m}{n},\frac {-1-m+n}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m) n}+\frac {2^{\frac {-1-m+n}{n}} b^2 c m \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-\frac {1+m}{n}} \operatorname {Hypergeometric2F1}\left (-\frac {1+m}{n},-\frac {1+m}{n},\frac {-1-m+n}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{a \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m) n}-\frac {2^{-\frac {1+m-2 n}{n}} c^2 m \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-\frac {1+m}{n}} \operatorname {Hypergeometric2F1}\left (-\frac {1+m}{n},-\frac {1+m}{n},\frac {-1-m+n}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m) n}\right )}{-b^2+4 a c} \]
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\[\int \frac {\left (d x \right )^{m}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{2}}d x\]
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\[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^2} \,d x \]
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