Integrand size = 20, antiderivative size = 513 \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx=-\frac {c (a e+c d x) (d+e x)^{1+n}}{2 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {c (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 (-a)^{5/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {c \left (c d^2+a e^2 (1-n)+\sqrt {-a} \sqrt {c} d e n\right ) (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{5/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {c (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 (-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {c \left (c d^2+a e^2 (1-n)-\sqrt {-a} \sqrt {c} d e n\right ) (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 (-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {e (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)} \]
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Time = 0.46 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {975, 67, 755, 845, 70, 726} \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx=-\frac {c (d+e x)^{n+1} (a e+c d x)}{2 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac {e (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {e x}{d}+1\right )}{a^2 d^2 (n+1)}-\frac {c (d+e x)^{n+1} \left (\sqrt {-a} \sqrt {c} d e n+a e^2 (1-n)+c d^2\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {c (d+e x)^{n+1} \left (-\sqrt {-a} \sqrt {c} d e n+a e^2 (1-n)+c d^2\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {c (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}+\frac {c (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]
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Rule 67
Rule 70
Rule 726
Rule 755
Rule 845
Rule 975
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(d+e x)^n}{a^2 x^2}-\frac {c (d+e x)^n}{a \left (a+c x^2\right )^2}-\frac {c (d+e x)^n}{a^2 \left (a+c x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {(d+e x)^n}{x^2} \, dx}{a^2}-\frac {c \int \frac {(d+e x)^n}{a+c x^2} \, dx}{a^2}-\frac {c \int \frac {(d+e x)^n}{\left (a+c x^2\right )^2} \, dx}{a} \\ & = -\frac {c (a e+c d x) (d+e x)^{1+n}}{2 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e (d+e x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)}-\frac {c \int \left (\frac {\sqrt {-a} (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\sqrt {-a} (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{a^2}+\frac {c \int \frac {(d+e x)^n \left (-c d^2-a e^2 (1-n)+c d e n x\right )}{a+c x^2} \, dx}{2 a^2 \left (c d^2+a e^2\right )} \\ & = -\frac {c (a e+c d x) (d+e x)^{1+n}}{2 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e (d+e x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)}+\frac {c \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{2 (-a)^{5/2}}+\frac {c \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{2 (-a)^{5/2}}+\frac {c \int \left (\frac {\left (\sqrt {-a} \left (-c d^2-a e^2 (1-n)\right )-a \sqrt {c} d e n\right ) (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\left (\sqrt {-a} \left (-c d^2-a e^2 (1-n)\right )+a \sqrt {c} d e n\right ) (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{2 a^2 \left (c d^2+a e^2\right )} \\ & = -\frac {c (a e+c d x) (d+e x)^{1+n}}{2 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {c (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 (-a)^{5/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}+\frac {c (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 (-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {e (d+e x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)}+\frac {\left (c \left (c d^2+a e^2 (1-n)-\sqrt {-a} \sqrt {c} d e n\right )\right ) \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{4 (-a)^{5/2} \left (c d^2+a e^2\right )}+\frac {\left (c \left (c d^2+a e^2 (1-n)+\sqrt {-a} \sqrt {c} d e n\right )\right ) \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{4 (-a)^{5/2} \left (c d^2+a e^2\right )} \\ & = -\frac {c (a e+c d x) (d+e x)^{1+n}}{2 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {c (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 (-a)^{5/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {c \left (c d^2+a e^2 (1-n)+\sqrt {-a} \sqrt {c} d e n\right ) (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{5/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {c (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 (-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {c \left (c d^2+a e^2 (1-n)-\sqrt {-a} \sqrt {c} d e n\right ) (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 (-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {e (d+e x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx=\frac {1}{4} (d+e x)^{1+n} \left (-\frac {2 c (a e+c d x)}{a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {2 c \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{(-a)^{5/2} \left (-\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {2 c \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{(-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {a c \left (\frac {\left (c d^2-a e^2 (-1+n)+\sqrt {-a} \sqrt {c} d e n\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {c} d-\sqrt {-a} e}-\frac {\left (c d^2-a e^2 (-1+n)-\sqrt {-a} \sqrt {c} d e n\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {c} d+\sqrt {-a} e}\right )}{(-a)^{7/2} \left (c d^2+a e^2\right ) (1+n)}+\frac {4 e \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)}\right ) \]
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\[\int \frac {\left (e x +d \right )^{n}}{x^{2} \left (c \,x^{2}+a \right )^{2}}d x\]
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\[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )}^{2} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )}^{2} x^{2}} \,d x } \]
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\[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )}^{2} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx=\int \frac {{\left (d+e\,x\right )}^n}{x^2\,{\left (c\,x^2+a\right )}^2} \,d x \]
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