Integrand size = 26, antiderivative size = 510 \[ \int (d+e x)^{-5-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=-\frac {\left (a e^2 f (2+p)-c d \left (d f-e^2 (3+p)\right )\right ) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (3+2 p)}+\frac {c \left (c d^2 \left (d f-e^2 \left (9+8 p+2 p^2\right )\right )+a e^2 \left (e^2 (3+2 p)-d f \left (11+10 p+2 p^2\right )\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right )^3 (1+p) (2+p) (3+2 p)}-\frac {\left (e^2-d f\right ) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (2+p)}+\frac {c e \left (a^2 e^2 f-c^2 d^3 (3+2 p)+a c d \left (3 e^2-d f (5+2 p)\right )\right ) \left (\sqrt {-a}-\sqrt {c} x\right ) \left (-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {-a}+\sqrt {c} x\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right )^3 (1+2 p) (3+2 p)} \] Output:
-(a*e^2*f*(2+p)-c*d*(d*f-e^2*(3+p)))*(e*x+d)^(-3-2*p)*(c*x^2+a)^(p+1)/(a*e ^2+c*d^2)^2/(2+p)/(3+2*p)+1/2*c*(c*d^2*(d*f-e^2*(2*p^2+8*p+9))+a*e^2*(e^2* (3+2*p)-d*f*(2*p^2+10*p+11)))*(c*x^2+a)^(p+1)/(a*e^2+c*d^2)^3/(p+1)/(2+p)/ (3+2*p)/((e*x+d)^(2*p+2))-1/2*(-d*f+e^2)*(c*x^2+a)^(p+1)/(a*e^2+c*d^2)/(2+ p)/((e*x+d)^(4+2*p))+c*e*(a^2*e^2*f-c^2*d^3*(3+2*p)+a*c*d*(3*e^2-d*f*(5+2* p)))*((-a)^(1/2)-c^(1/2)*x)*(e*x+d)^(-1-2*p)*(c*x^2+a)^p*hypergeom([-p, -1 -2*p],[-2*p],2*(-a)^(1/2)*c^(1/2)*(e*x+d)/(c^(1/2)*d-(-a)^(1/2)*e)/((-a)^( 1/2)-c^(1/2)*x))/(c^(1/2)*d+(-a)^(1/2)*e)/(a*e^2+c*d^2)^3/(1+2*p)/(3+2*p)/ ((-(c^(1/2)*d+(-a)^(1/2)*e)*((-a)^(1/2)+c^(1/2)*x)/(c^(1/2)*d-(-a)^(1/2)*e )/((-a)^(1/2)-c^(1/2)*x))^p)
\[ \int (d+e x)^{-5-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=\int (d+e x)^{-5-2 p} (e+f x) \left (a+c x^2\right )^p \, dx \] Input:
Integrate[(d + e*x)^(-5 - 2*p)*(e + f*x)*(a + c*x^2)^p,x]
Output:
Integrate[(d + e*x)^(-5 - 2*p)*(e + f*x)*(a + c*x^2)^p, x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (e+f x) \left (a+c x^2\right )^p (d+e x)^{-2 p-5} \, dx\) |
| \(\Big \downarrow \) 689 |
| \(\displaystyle -\frac {\int -2 (d+e x)^{-2 (p+2)} \left (e (c d+a f) (p+2)-c \left (e^2-d f\right ) x\right ) \left (c x^2+a\right )^pdx}{2 (p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d+e x)^{-2 (p+2)} \left (e (c d+a f) (p+2)-c \left (e^2-d f\right ) x\right ) \left (c x^2+a\right )^pdx}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 689 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (c d^2 (p+2)-a \left (e^2-d f (p+3)\right )\right )-\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -c (d+e x)^{-2 p-3} \left (e (2 p+3) \left (-c (p+2) d^2-a f (p+3) d+a e^2\right )+\left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right ) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (a e^2 f (p+2)-c d \left (d f-e^2 (p+3)\right )\right )}{(2 p+3) \left (a e^2+c d^2\right )}}{(p+2) \left (a e^2+c d^2\right )}-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
Input:
Int[(d + e*x)^(-5 - 2*p)*(e + f*x)*(a + c*x^2)^p,x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && ILtQ[Sim plify[m + 2*p + 3], 0] && NeQ[m, -1]
\[\int \left (e x +d \right )^{-5-2 p} \left (f x +e \right ) \left (c \,x^{2}+a \right )^{p}d x\]
Input:
int((e*x+d)^(-5-2*p)*(f*x+e)*(c*x^2+a)^p,x)
Output:
int((e*x+d)^(-5-2*p)*(f*x+e)*(c*x^2+a)^p,x)
\[ \int (d+e x)^{-5-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=\int { {\left (f x + e\right )} {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 5} \,d x } \] Input:
integrate((e*x+d)^(-5-2*p)*(f*x+e)*(c*x^2+a)^p,x, algorithm="fricas")
Output:
integral((f*x + e)*(c*x^2 + a)^p*(e*x + d)^(-2*p - 5), x)
Timed out. \[ \int (d+e x)^{-5-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(-5-2*p)*(f*x+e)*(c*x**2+a)**p,x)
Output:
Timed out
\[ \int (d+e x)^{-5-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=\int { {\left (f x + e\right )} {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 5} \,d x } \] Input:
integrate((e*x+d)^(-5-2*p)*(f*x+e)*(c*x^2+a)^p,x, algorithm="maxima")
Output:
integrate((f*x + e)*(c*x^2 + a)^p*(e*x + d)^(-2*p - 5), x)
\[ \int (d+e x)^{-5-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=\int { {\left (f x + e\right )} {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 5} \,d x } \] Input:
integrate((e*x+d)^(-5-2*p)*(f*x+e)*(c*x^2+a)^p,x, algorithm="giac")
Output:
integrate((f*x + e)*(c*x^2 + a)^p*(e*x + d)^(-2*p - 5), x)
Timed out. \[ \int (d+e x)^{-5-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=\int \frac {\left (e+f\,x\right )\,{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+5}} \,d x \] Input:
int(((e + f*x)*(a + c*x^2)^p)/(d + e*x)^(2*p + 5),x)
Output:
int(((e + f*x)*(a + c*x^2)^p)/(d + e*x)^(2*p + 5), x)
\[ \int (d+e x)^{-5-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=\left (\int \frac {\left (c \,x^{2}+a \right )^{p}}{\left (e x +d \right )^{2 p} d^{5}+5 \left (e x +d \right )^{2 p} d^{4} e x +10 \left (e x +d \right )^{2 p} d^{3} e^{2} x^{2}+10 \left (e x +d \right )^{2 p} d^{2} e^{3} x^{3}+5 \left (e x +d \right )^{2 p} d \,e^{4} x^{4}+\left (e x +d \right )^{2 p} e^{5} x^{5}}d x \right ) e +\left (\int \frac {\left (c \,x^{2}+a \right )^{p} x}{\left (e x +d \right )^{2 p} d^{5}+5 \left (e x +d \right )^{2 p} d^{4} e x +10 \left (e x +d \right )^{2 p} d^{3} e^{2} x^{2}+10 \left (e x +d \right )^{2 p} d^{2} e^{3} x^{3}+5 \left (e x +d \right )^{2 p} d \,e^{4} x^{4}+\left (e x +d \right )^{2 p} e^{5} x^{5}}d x \right ) f \] Input:
int((e*x+d)^(-5-2*p)*(f*x+e)*(c*x^2+a)^p,x)
Output:
int((a + c*x**2)**p/((d + e*x)**(2*p)*d**5 + 5*(d + e*x)**(2*p)*d**4*e*x + 10*(d + e*x)**(2*p)*d**3*e**2*x**2 + 10*(d + e*x)**(2*p)*d**2*e**3*x**3 + 5*(d + e*x)**(2*p)*d*e**4*x**4 + (d + e*x)**(2*p)*e**5*x**5),x)*e + int(( (a + c*x**2)**p*x)/((d + e*x)**(2*p)*d**5 + 5*(d + e*x)**(2*p)*d**4*e*x + 10*(d + e*x)**(2*p)*d**3*e**2*x**2 + 10*(d + e*x)**(2*p)*d**2*e**3*x**3 + 5*(d + e*x)**(2*p)*d*e**4*x**4 + (d + e*x)**(2*p)*e**5*x**5),x)*f