Integrand size = 17, antiderivative size = 167 \[ \int \frac {(d+e x)^n}{a+c x^2} \, dx=\frac {(d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 \sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {(d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 \sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)} \]
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Time = 0.06 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {726, 70} \[ \int \frac {(d+e x)^n}{a+c x^2} \, dx=\frac {(d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 \sqrt {-a} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {(d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 \sqrt {-a} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]
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Rule 70
Rule 726
Rubi steps \begin{align*} \text {integral}& = \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 \\ & = -\frac {\int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{2 \sqrt {-a}}-\frac {\int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{2 \sqrt {-a}} \\ & = \frac {(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 \sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {(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 \sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^n}{a+c x^2} \, dx=\frac {(d+e x)^{1+n} \left (\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {c} d-\sqrt {-a} e}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 \sqrt {-a} (1+n)} \]
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\[\int \frac {\left (e x +d \right )^{n}}{c \,x^{2}+a}d x\]
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\[ \int \frac {(d+e x)^n}{a+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n}}{c x^{2} + a} \,d x } \]
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\[ \int \frac {(d+e x)^n}{a+c x^2} \, dx=\int \frac {\left (d + e x\right )^{n}}{a + c x^{2}}\, dx \]
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\[ \int \frac {(d+e x)^n}{a+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n}}{c x^{2} + a} \,d x } \]
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\[ \int \frac {(d+e x)^n}{a+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n}}{c x^{2} + a} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^n}{a+c x^2} \, dx=\int \frac {{\left (d+e\,x\right )}^n}{c\,x^2+a} \,d x \]
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