Integrand size = 32, antiderivative size = 44 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=\frac {(d+e x)^{1+p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {658, 32} \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=\frac {(d+e x)^{p+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]
[In]
[Out]
Rule 32
Rule 658
Rubi steps \begin{align*} \text {integral}& = \left ((d+e x)^{2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}\right ) \int (d+e x)^{-p} \, dx \\ & = \frac {(d+e x)^{1+p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.75 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=\frac {(d+e x)^{1+p} \left (c (d+e x)^2\right )^{-p}}{e (1-p)} \]
[In]
[Out]
Time = 2.95 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(-\frac {\left (e x +d \right )^{1+p} \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{-p}}{e \left (p -1\right )}\) | \(44\) |
parallelrisch | \(\frac {\left (-x \left (e x +d \right )^{p} e -\left (e x +d \right )^{p} d \right ) {\left (c \left (x^{2} e^{2}+2 d e x +d^{2}\right )\right )}^{-p}}{e \left (p -1\right )}\) | \(54\) |
norman | \(\left (-\frac {x \,{\mathrm e}^{p \ln \left (e x +d \right )}}{p -1}-\frac {d \,{\mathrm e}^{p \ln \left (e x +d \right )}}{e \left (p -1\right )}\right ) {\mathrm e}^{-p \ln \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )}\) | \(66\) |
risch | \(-\frac {\left (e x +d \right ) \left (e x +d \right )^{p} \left (e x +d \right )^{-2 p} c^{-p} {\mathrm e}^{\frac {i \pi p \left (\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right )^{3}-2 \operatorname {csgn}\left (i \left (e x +d \right )^{2}\right )^{2} \operatorname {csgn}\left (i \left (e x +d \right )\right )+\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i \left (e x +d \right )\right )^{2}-\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right )^{2}+\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i c \right )+\operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right )^{3}-\operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right )^{2} \operatorname {csgn}\left (i c \right )\right )}{2}}}{e \left (p -1\right )}\) | \(197\) |
[In]
[Out]
none
Time = 0.39 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.68 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=-\frac {e x + d}{{\left (e p - e\right )} {\left (e x + d\right )}^{p} c^{p}} \]
[In]
[Out]
Exception generated. \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=-\frac {e x + d}{{\left (e x + d\right )}^{p} c^{p} e {\left (p - 1\right )}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.43 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=-\frac {{\left (e x + d\right )}^{p} e x e^{\left (-2 \, p \log \left (e x + d\right ) - p \log \left (c\right )\right )} + {\left (e x + d\right )}^{p} d e^{\left (-2 \, p \log \left (e x + d\right ) - p \log \left (c\right )\right )}}{e p - e} \]
[In]
[Out]
Time = 9.82 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=-\frac {{\left (d+e\,x\right )}^{p+1}}{e\,\left (p-1\right )\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p} \]
[In]
[Out]