Integrand size = 18, antiderivative size = 279 \[ \int \frac {x (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=-\frac {(d-e x) (d+e x)^{1+n}}{2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (\sqrt {c} d+\sqrt {-a} e\right ) n (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {c} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {e \left (\sqrt {-a} \sqrt {c} d+a e\right ) n (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 a \sqrt {c} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)} \]
[Out]
Time = 0.23 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {837, 845, 70} \[ \int \frac {x (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\frac {e n \left (\sqrt {-a} e+\sqrt {c} d\right ) (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 \sqrt {-a} \sqrt {c} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {e n \left (\sqrt {-a} \sqrt {c} d+a e\right ) (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 a \sqrt {c} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {(d-e x) (d+e x)^{n+1}}{2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]
[In]
[Out]
Rule 70
Rule 837
Rule 845
Rubi steps \begin{align*} \text {integral}& = -\frac {(d-e x) (d+e x)^{1+n}}{2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \frac {(d+e x)^n \left (-a c d e n+a c e^2 n x\right )}{a+c x^2} \, dx}{2 a c \left (c d^2+a e^2\right )} \\ & = -\frac {(d-e x) (d+e x)^{1+n}}{2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \left (\frac {\left (-\sqrt {-a} a c d e n-a^2 \sqrt {c} e^2 n\right ) (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\left (-\sqrt {-a} a c d e n+a^2 \sqrt {c} e^2 n\right ) (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{2 a c \left (c d^2+a e^2\right )} \\ & = -\frac {(d-e x) (d+e x)^{1+n}}{2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (e \left (\sqrt {-a} \sqrt {c} d-a e\right ) n\right ) \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{4 a \sqrt {c} \left (c d^2+a e^2\right )}+\frac {\left (e \left (\sqrt {-a} d+\frac {a e}{\sqrt {c}}\right ) n\right ) \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{4 a \left (c d^2+a e^2\right )} \\ & = -\frac {(d-e x) (d+e x)^{1+n}}{2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (\sqrt {c} d+\sqrt {-a} e\right ) n (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 \sqrt {-a} \sqrt {c} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {e \left (\sqrt {-a} \sqrt {c} d+a e\right ) n (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 \sqrt {c} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.82 \[ \int \frac {x (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\frac {(d+e x)^{1+n} \left (-\frac {2 a c (d-e x)}{a+c x^2}-\frac {\left (\sqrt {-a} c d e n-a \sqrt {c} e^2 n\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}+\frac {\left (\sqrt {-a} c d e n+a \sqrt {c} e^2 n\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}\right )}{4 a c \left (c d^2+a e^2\right )} \]
[In]
[Out]
\[\int \frac {x \left (e x +d \right )^{n}}{\left (c \,x^{2}+a \right )^{2}}d x\]
[In]
[Out]
\[ \int \frac {x (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n} x}{{\left (c x^{2} + a\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {x (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n} x}{{\left (c x^{2} + a\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n} x}{{\left (c x^{2} + a\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x (d+e x)^n}{\left (a+c x^2\right )^2} \, dx=\int \frac {x\,{\left (d+e\,x\right )}^n}{{\left (c\,x^2+a\right )}^2} \,d x \]
[In]
[Out]