Integrand size = 21, antiderivative size = 436 \[ \int (d+e x)^{-5-2 p} \left (a+c x^2\right )^p \, dx=-\frac {c d e (3+p) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (3+2 p)}+\frac {c e \left (a e^2 (3+2 p)-c d^2 \left (9+8 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right )^3 (1+p) (2+p) (3+2 p)}-\frac {e (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (2+p)}+\frac {c^2 d \left (3 a e^2-c d^2 (3+2 p)\right ) \left (\sqrt {-a}-\sqrt {c} x\right ) \left (-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {-a}+\sqrt {c} x\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right )^3 (1+2 p) (3+2 p)} \]
-c*d*e*(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*e*(a*e^2*(3+2*p)-c*d^2*(2*p^2+8*p+9))*(c*x^2+a)^(p+1)/(a*e^2+c*d^2 )^3/(p+1)/(2+p)/(3+2*p)/((e*x+d)^(2+2*p))-1/2*e*(c*x^2+a)^(p+1)/(a*e^2+c*d ^2)/(2+p)/((e*x+d)^(4+2*p))+c^2*d*(3*a*e^2-c*d^2*(3+2*p))*(e*x+d)^(-1-2*p) *(c*x^2+a)^p*hypergeom([-p, -1-2*p],[-2*p],2*(e*x+d)*(-a)^(1/2)*c^(1/2)/(- e*(-a)^(1/2)+d*c^(1/2))/((-a)^(1/2)-x*c^(1/2)))*((-a)^(1/2)-x*c^(1/2))/(a* e^2+c*d^2)^3/(1+2*p)/(3+2*p)/(e*(-a)^(1/2)+d*c^(1/2))/((-(e*(-a)^(1/2)+d*c ^(1/2))*((-a)^(1/2)+x*c^(1/2))/(-e*(-a)^(1/2)+d*c^(1/2))/((-a)^(1/2)-x*c^( 1/2)))^p)
\[ \int (d+e x)^{-5-2 p} \left (a+c x^2\right )^p \, dx=\int (d+e x)^{-5-2 p} \left (a+c x^2\right )^p \, dx \]
Time = 0.57 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {498, 27, 689, 679, 489}
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.
| \(\displaystyle \int \left (a+c x^2\right )^p (d+e x)^{-2 p-5} \, dx\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -\frac {c \int -2 (d (p+2)-e x) (d+e x)^{-2 (p+2)} \left (c x^2+a\right )^pdx}{2 (p+2) \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \int (d (p+2)-e x) (d+e x)^{-2 (p+2)} \left (c x^2+a\right )^pdx}{(p+2) \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 689 |
| \(\displaystyle \frac {c \left (-\frac {\int (d+e x)^{-2 p-3} \left ((2 p+3) \left (a e^2-c d^2 (p+2)\right )+c d e (p+3) x\right ) \left (c x^2+a\right )^pdx}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(2 p+3) \left (a e^2+c d^2\right )}\right )}{(p+2) \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {c \left (-\frac {\frac {c d (p+2) \left (3 a e^2-c d^2 (2 p+3)\right ) \int (d+e x)^{-2 (p+1)} \left (c x^2+a\right )^pdx}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (2 p+3)-c d^2 \left (2 p^2+8 p+9\right )\right )}{2 (p+1) \left (a e^2+c d^2\right )}}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(2 p+3) \left (a e^2+c d^2\right )}\right )}{(p+2) \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 489 |
| \(\displaystyle \frac {c \left (-\frac {-\frac {c d (p+2) \left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a e^2-c d^2 (2 p+3)\right ) \left (-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (2 p+3)-c d^2 \left (2 p^2+8 p+9\right )\right )}{2 (p+1) \left (a e^2+c d^2\right )}}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(2 p+3) \left (a e^2+c d^2\right )}\right )}{(p+2) \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )}\) |
-1/2*(e*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(2 + p)*(d + e*x)^(2*(2 + p) )) + (c*(-((d*e*(3 + p)*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(3 + 2*p))) - (-1/2*(e*(a*e^2*(3 + 2*p) - c*d^2*(9 + 8*p + 2*p^2) )*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(1 + p)*(d + e*x)^(2*(1 + p))) - ( c*d*(2 + p)*(3*a*e^2 - c*d^2*(3 + 2*p))*(Sqrt[-a] - Sqrt[c]*x)*(d + e*x)^( -1 - 2*p)*(a + c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a]* Sqrt[c]*(d + e*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))])/((S qrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + 2*p)*(-(((Sqrt[c]*d + Sqrt[-a] *e)*(Sqrt[-a] + Sqrt[c]*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]* x))))^p))/((c*d^2 + a*e^2)*(3 + 2*p))))/((c*d^2 + a*e^2)*(2 + p))
3.8.44.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Rt[(-a)*b, 2]}, Simp[(q - b*x)*(c + d*x)^(n + 1)*((a + b*x^2)^p/((n + 1)*(b*c + d*q)*((b*c + d*q)*((q + b*x)/((b*c - d*q)*(-q + b*x))))^p))*Hyper geometric2F1[n + 1, -p, n + 2, 2*b*q*((c + d*x)/((b*c - d*q)*(q - b*x)))], x]] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 2*p + 2, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
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 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
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]
\[\int \left (e x +d \right )^{-5-2 p} \left (c \,x^{2}+a \right )^{p}d x\]
\[ \int (d+e x)^{-5-2 p} \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 5} \,d x } \]
Timed out. \[ \int (d+e x)^{-5-2 p} \left (a+c x^2\right )^p \, dx=\text {Timed out} \]
\[ \int (d+e x)^{-5-2 p} \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 5} \,d x } \]
\[ \int (d+e x)^{-5-2 p} \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 5} \,d x } \]
Timed out. \[ \int (d+e x)^{-5-2 p} \left (a+c x^2\right )^p \, dx=\int \frac {{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+5}} \,d x \]