Integrand size = 20, antiderivative size = 297 \[ \int \frac {x^3 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\frac {a (d-e x) (d+e x)^{1+n}}{2 c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (\sqrt {-a} d e n-\frac {2 c d^2+a e^2 (2+n)}{\sqrt {c}}\right ) (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 c \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}-\frac {\left (2 c d^2+\sqrt {-a} \sqrt {c} d e n+a e^2 (2+n)\right ) (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 c^{3/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)} \]
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Time = 0.27 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1663, 845, 70} \[ \int \frac {x^3 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=-\frac {(d+e x)^{n+1} \left (\sqrt {-a} \sqrt {c} d e n+a e^2 (n+2)+2 c d^2\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 c^{3/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}+\frac {(d+e x)^{n+1} \left (\sqrt {-a} d e n-\frac {a e^2 (n+2)+2 c d^2}{\sqrt {c}}\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 c (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {a (d-e x) (d+e x)^{n+1}}{2 c \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]
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Rule 70
Rule 845
Rule 1663
Rubi steps \begin{align*} \text {integral}& = \frac {a (d-e x) (d+e x)^{1+n}}{2 c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \frac {(d+e x)^n \left (\frac {a^2 d e n}{c}-\frac {a \left (2 c d^2+a e^2 (2+n)\right ) x}{c}\right )}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )} \\ & = \frac {a (d-e x) (d+e x)^{1+n}}{2 c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \left (\frac {\left (\frac {\sqrt {-a} a^2 d e n}{c}+\frac {a^2 \left (2 c d^2+a e^2 (2+n)\right )}{c^{3/2}}\right ) (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\left (\frac {\sqrt {-a} a^2 d e n}{c}-\frac {a^2 \left (2 c d^2+a e^2 (2+n)\right )}{c^{3/2}}\right ) (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{2 a \left (c d^2+a e^2\right )} \\ & = \frac {a (d-e x) (d+e x)^{1+n}}{2 c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\left (2 c d^2+\sqrt {-a} \sqrt {c} d e n+a e^2 (2+n)\right ) \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{4 c^{3/2} \left (c d^2+a e^2\right )}-\frac {\left (\sqrt {-a} d e n-\frac {2 c d^2+a e^2 (2+n)}{\sqrt {c}}\right ) \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{4 c \left (c d^2+a e^2\right )} \\ & = \frac {a (d-e x) (d+e x)^{1+n}}{2 c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (\sqrt {-a} d e n-\frac {2 c d^2+a e^2 (2+n)}{\sqrt {c}}\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 c \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}-\frac {\left (2 c d^2+\sqrt {-a} \sqrt {c} d e n+a e^2 (2+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 c^{3/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=-\frac {(d+e x)^{1+n} \left (\frac {2 a \sqrt {c} (-d+e x)}{a+c x^2}+\frac {\left (2 c d^2-\sqrt {-a} \sqrt {c} d e n+a e^2 (2+n)\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}+\frac {\left (2 c d^2+\sqrt {-a} \sqrt {c} d e n+a e^2 (2+n)\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}\right )}{4 c^{3/2} \left (c d^2+a e^2\right )} \]
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\[\int \frac {x^{3} \left (e x +d \right )^{n}}{\left (c \,x^{2}+a \right )^{2}}d x\]
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\[ \int \frac {x^3 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n} x^{3}}{{\left (c x^{2} + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^3 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x^3 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n} x^{3}}{{\left (c x^{2} + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^3 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n} x^{3}}{{\left (c x^{2} + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^3 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\int \frac {x^3\,{\left (d+e\,x\right )}^n}{{\left (c\,x^2+a\right )}^2} \,d x \]
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