Integrand size = 20, antiderivative size = 207 \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )} \, dx=\frac {c (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 (-a)^{3/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {c (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 (-a)^{3/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {e (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {e x}{d}\right )}{a d^2 (1+n)} \]
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Time = 0.14 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {975, 67, 726, 70} \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )} \, dx=\frac {c (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 (-a)^{3/2} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {c (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 (-a)^{3/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}+\frac {e (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {e x}{d}+1\right )}{a d^2 (n+1)} \]
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Rule 67
Rule 70
Rule 726
Rule 975
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(d+e x)^n}{a x^2}-\frac {c (d+e x)^n}{a \left (a+c x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {(d+e x)^n}{x^2} \, dx}{a}-\frac {c \int \frac {(d+e x)^n}{a+c x^2} \, dx}{a} \\ & = \frac {e (d+e x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {e x}{d}\right )}{a 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} \\ & = \frac {e (d+e x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {e x}{d}\right )}{a d^2 (1+n)}-\frac {c \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{2 (-a)^{3/2}}-\frac {c \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{2 (-a)^{3/2}} \\ & = \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)^{3/2} \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)^{3/2} \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 d^2 (1+n)} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )} \, dx=\frac {(d+e x)^{1+n} \left (-\frac {c \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {-a} \sqrt {c} d+a e}+\frac {c \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {-a} \sqrt {c} d-a e}+\frac {2 e \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {e x}{d}\right )}{d^2}\right )}{2 a (1+n)} \]
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\[\int \frac {\left (e x +d \right )^{n}}{x^{2} \left (c \,x^{2}+a \right )}d x\]
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\[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )} \, dx=\int { \frac {{\left (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )} \, dx=\int { \frac {{\left (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )} x^{2}} \,d x } \]
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\[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )} \, dx=\int { \frac {{\left (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )} \, dx=\int \frac {{\left (d+e\,x\right )}^n}{x^2\,\left (c\,x^2+a\right )} \,d x \]
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