Integrand size = 20, antiderivative size = 164 \[ \int (g x)^m (d+e x)^n \left (a+c x^2\right ) \, dx=-\frac {c d (2+m) (g x)^{1+m} (d+e x)^{1+n}}{e^2 g (2+m+n) (3+m+n)}+\frac {c (g x)^{2+m} (d+e x)^{1+n}}{e g^2 (3+m+n)}+\frac {\left (c d^2 (1+m) (2+m)+a e^2 (2+m+n) (3+m+n)\right ) (g x)^{1+m} (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {e x}{d}\right )}{e^2 g (1+m) (2+m+n) (3+m+n)} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {966, 81, 68, 66} \[ \int (g x)^m (d+e x)^n \left (a+c x^2\right ) \, dx=\frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} \left (\frac {a}{m+1}+\frac {c d^2 (m+2)}{e^2 (m+n+2) (m+n+3)}\right ) \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {e x}{d}\right )}{g}-\frac {c d (m+2) (g x)^{m+1} (d+e x)^{n+1}}{e^2 g (m+n+2) (m+n+3)}+\frac {c (g x)^{m+2} (d+e x)^{n+1}}{e g^2 (m+n+3)} \]
[In]
[Out]
Rule 66
Rule 68
Rule 81
Rule 966
Rubi steps \begin{align*} \text {integral}& = \frac {c (g x)^{2+m} (d+e x)^{1+n}}{e g^2 (3+m+n)}+\frac {\int (g x)^m (d+e x)^n \left (a e g^2 (3+m+n)-c d g^2 (2+m) x\right ) \, dx}{e g^2 (3+m+n)} \\ & = -\frac {c d (2+m) (g x)^{1+m} (d+e x)^{1+n}}{e^2 g (2+m+n) (3+m+n)}+\frac {c (g x)^{2+m} (d+e x)^{1+n}}{e g^2 (3+m+n)}+\left (a+\frac {c d^2 (1+m) (2+m)}{e^2 (2+m+n) (3+m+n)}\right ) \int (g x)^m (d+e x)^n \, dx \\ & = -\frac {c d (2+m) (g x)^{1+m} (d+e x)^{1+n}}{e^2 g (2+m+n) (3+m+n)}+\frac {c (g x)^{2+m} (d+e x)^{1+n}}{e g^2 (3+m+n)}+\left (\left (a+\frac {c d^2 (1+m) (2+m)}{e^2 (2+m+n) (3+m+n)}\right ) (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n}\right ) \int (g x)^m \left (1+\frac {e x}{d}\right )^n \, dx \\ & = -\frac {c d (2+m) (g x)^{1+m} (d+e x)^{1+n}}{e^2 g (2+m+n) (3+m+n)}+\frac {c (g x)^{2+m} (d+e x)^{1+n}}{e g^2 (3+m+n)}+\frac {\left (a+\frac {c d^2 (1+m) (2+m)}{e^2 (2+m+n) (3+m+n)}\right ) (g x)^{1+m} (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {e x}{d}\right )}{g (1+m)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.69 \[ \int (g x)^m (d+e x)^n \left (a+c x^2\right ) \, dx=\frac {x (g x)^m (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n} \left (c d^2 \operatorname {Hypergeometric2F1}\left (1+m,-2-n,2+m,-\frac {e x}{d}\right )-2 c d^2 \operatorname {Hypergeometric2F1}\left (1+m,-1-n,2+m,-\frac {e x}{d}\right )+\left (c d^2+a e^2\right ) \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {e x}{d}\right )\right )}{e^2 (1+m)} \]
[In]
[Out]
\[\int \left (g x \right )^{m} \left (e x +d \right )^{n} \left (c \,x^{2}+a \right )d x\]
[In]
[Out]
\[ \int (g x)^m (d+e x)^n \left (a+c x^2\right ) \, dx=\int { {\left (c x^{2} + a\right )} {\left (e x + d\right )}^{n} \left (g x\right )^{m} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 6.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.49 \[ \int (g x)^m (d+e x)^n \left (a+c x^2\right ) \, dx=\frac {a d^{n} g^{m} x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {e x e^{i \pi }}{d}} \right )}}{\Gamma \left (m + 2\right )} + \frac {c d^{n} g^{m} x^{m + 3} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {e x e^{i \pi }}{d}} \right )}}{\Gamma \left (m + 4\right )} \]
[In]
[Out]
\[ \int (g x)^m (d+e x)^n \left (a+c x^2\right ) \, dx=\int { {\left (c x^{2} + a\right )} {\left (e x + d\right )}^{n} \left (g x\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (g x)^m (d+e x)^n \left (a+c x^2\right ) \, dx=\int { {\left (c x^{2} + a\right )} {\left (e x + d\right )}^{n} \left (g x\right )^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int (g x)^m (d+e x)^n \left (a+c x^2\right ) \, dx=\int {\left (g\,x\right )}^m\,\left (c\,x^2+a\right )\,{\left (d+e\,x\right )}^n \,d x \]
[In]
[Out]