Integrand size = 34, antiderivative size = 590 \[ \int (g+h x)^{-3-2 p} \left (a+b x+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx=-\frac {\left (f g^2-h (e g-d h)\right ) (g+h x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{2 h \left (c g^2-b g h+a h^2\right ) (1+p)}-\frac {f (g+h x)^{-2 p} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h}\right )^{-p} \left (1-\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h},\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )}{2 h^3 p}-\frac {\left (2 c \left (f g^3-d g h^2\right )+h \left (2 a h (2 f g-e h)-b \left (3 f g^2-e g h-d h^2\right )\right )\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (g+h x)^{-1-2 p} \left (a+b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (g+h x)}{\left (2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{2 h^2 \left (2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h\right ) \left (c g^2-b g h+a h^2\right ) (1+2 p)} \]
-1/2*(f*g^2-h*(-d*h+e*g))*(c*x^2+b*x+a)^(p+1)/h/(a*h^2-b*g*h+c*g^2)/(p+1)/ ((h*x+g)^(2+2*p))-1/2*(2*c*(-d*g*h^2+f*g^3)+h*(2*a*h*(-e*h+2*f*g)-b*(-d*h^ 2-e*g*h+3*f*g^2)))*(h*x+g)^(-1-2*p)*(c*x^2+b*x+a)^p*hypergeom([-p, -1-2*p] ,[-2*p],-4*c*(h*x+g)*(-4*a*c+b^2)^(1/2)/(b+2*c*x-(-4*a*c+b^2)^(1/2))/(2*c* g-h*(b+(-4*a*c+b^2)^(1/2))))*(b+2*c*x-(-4*a*c+b^2)^(1/2))/h^2/(a*h^2-b*g*h +c*g^2)/(1+2*p)/(2*c*g-h*(b-(-4*a*c+b^2)^(1/2)))/(((2*c*g-h*(b-(-4*a*c+b^2 )^(1/2)))*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+2*c*x-(-4*a*c+b^2)^(1/2))/(2*c*g -h*(b+(-4*a*c+b^2)^(1/2))))^p)-1/2*f*(c*x^2+b*x+a)^p*AppellF1(-2*p,-p,-p,1 -2*p,2*c*(h*x+g)/(2*c*g-h*(b-(-4*a*c+b^2)^(1/2))),2*c*(h*x+g)/(2*c*g-h*(b+ (-4*a*c+b^2)^(1/2))))/h^3/p/((h*x+g)^(2*p))/((1-2*c*(h*x+g)/(2*c*g-h*(b-(- 4*a*c+b^2)^(1/2))))^p)/((1-2*c*(h*x+g)/(2*c*g-h*(b+(-4*a*c+b^2)^(1/2))))^p )
\[ \int (g+h x)^{-3-2 p} \left (a+b x+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx=\int (g+h x)^{-3-2 p} \left (a+b x+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx \]
Time = 0.88 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2186, 25, 1179, 150, 1228, 1155}
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.
| \(\displaystyle \int \left (d+e x+f x^2\right ) (g+h x)^{-2 p-3} \left (a+b x+c x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 2186 |
| \(\displaystyle \frac {\int -(g+h x)^{-2 p-3} \left (f g^2-d h^2+h (2 f g-e h) x\right ) \left (c x^2+b x+a\right )^pdx}{h^2}+\frac {f \int (g+h x)^{-2 p-1} \left (c x^2+b x+a\right )^pdx}{h^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f \int (g+h x)^{-2 p-1} \left (c x^2+b x+a\right )^pdx}{h^2}-\frac {\int (g+h x)^{-2 p-3} \left (f g^2-d h^2+h (2 f g-e h) x\right ) \left (c x^2+b x+a\right )^pdx}{h^2}\) |
| \(\Big \downarrow \) 1179 |
| \(\displaystyle \frac {f \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (g+h x)}{2 c g-h \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (g+h x)}{2 c g-h \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \int (g+h x)^{-2 p-1} \left (1-\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h}\right )^p \left (1-\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )^pd(g+h x)}{h^3}-\frac {\int (g+h x)^{-2 p-3} \left (f g^2-d h^2+h (2 f g-e h) x\right ) \left (c x^2+b x+a\right )^pdx}{h^2}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle -\frac {\int (g+h x)^{-2 p-3} \left (f g^2-d h^2+h (2 f g-e h) x\right ) \left (c x^2+b x+a\right )^pdx}{h^2}-\frac {f (g+h x)^{-2 p} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (g+h x)}{2 c g-h \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (g+h x)}{2 c g-h \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h},\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )}{2 h^3 p}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {\frac {\left (2 c \left (f g^3-d g h^2\right )-h \left (-2 a h (2 f g-e h)-b h (d h+e g)+3 b f g^2\right )\right ) \int (g+h x)^{-2 (p+1)} \left (c x^2+b x+a\right )^pdx}{2 \left (a h^2-b g h+c g^2\right )}+\frac {h (g+h x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1} \left (f g^2-h (e g-d h)\right )}{2 (p+1) \left (a h^2-b g h+c g^2\right )}}{h^2}-\frac {f (g+h x)^{-2 p} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (g+h x)}{2 c g-h \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (g+h x)}{2 c g-h \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h},\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )}{2 h^3 p}\) |
| \(\Big \downarrow \) 1155 |
| \(\displaystyle -\frac {f (g+h x)^{-2 p} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (g+h x)}{2 c g-h \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (g+h x)}{2 c g-h \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h},\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )}{2 h^3 p}-\frac {\frac {\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) (g+h x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c g-h \left (b-\sqrt {b^2-4 a c}\right )\right )}{\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c g-h \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )^{-p} \left (2 c \left (f g^3-d g h^2\right )-h \left (-2 a h (2 f g-e h)-b h (d h+e g)+3 b f g^2\right )\right ) \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (g+h x)}{\left (2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )}{2 (2 p+1) \left (2 c g-h \left (b-\sqrt {b^2-4 a c}\right )\right ) \left (a h^2-b g h+c g^2\right )}+\frac {h (g+h x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1} \left (f g^2-h (e g-d h)\right )}{2 (p+1) \left (a h^2-b g h+c g^2\right )}}{h^2}\) |
-1/2*(f*(a + b*x + c*x^2)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (2*c*(g + h*x) )/(2*c*g - (b - Sqrt[b^2 - 4*a*c])*h), (2*c*(g + h*x))/(2*c*g - (b + Sqrt[ b^2 - 4*a*c])*h)])/(h^3*p*(g + h*x)^(2*p)*(1 - (2*c*(g + h*x))/(2*c*g - (b - Sqrt[b^2 - 4*a*c])*h))^p*(1 - (2*c*(g + h*x))/(2*c*g - (b + Sqrt[b^2 - 4*a*c])*h))^p) - ((h*(f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(1 + p))/(2 *(c*g^2 - b*g*h + a*h^2)*(1 + p)*(g + h*x)^(2*(1 + p))) + ((2*c*(f*g^3 - d *g*h^2) - h*(3*b*f*g^2 - b*h*(e*g + d*h) - 2*a*h*(2*f*g - e*h)))*(b - Sqrt [b^2 - 4*a*c] + 2*c*x)*(g + h*x)^(-1 - 2*p)*(a + b*x + c*x^2)^p*Hypergeome tric2F1[-1 - 2*p, -p, -2*p, (-4*c*Sqrt[b^2 - 4*a*c]*(g + h*x))/((2*c*g - ( b + Sqrt[b^2 - 4*a*c])*h)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))])/(2*(2*c*g - ( b - Sqrt[b^2 - 4*a*c])*h)*(c*g^2 - b*g*h + a*h^2)*(1 + 2*p)*(((2*c*g - (b - Sqrt[b^2 - 4*a*c])*h)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*g - (b + Sq rt[b^2 - 4*a*c])*h)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))^p))/h^2
3.3.74.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(b - q + 2*c*x))*(d + e*x)^ (m + 1)*((a + b*x + c*x^2)^p/((m + 1)*(2*c*d - b*e + e*q)*((2*c*d - b*e + e *q)*((b + q + 2*c*x)/((2*c*d - b*e - e*q)*(b - q + 2*c*x))))^p))*Hypergeome tric2F1[m + 1, -p, m + 2, -4*c*q*((d + e*x)/((2*c*d - b*e - e*q)*(b - q + 2 *c*x)))], x]] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) ^p) Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d - e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m , p}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> With[{q = Expon[Pq, x]}, Simp[Coeff[Pq, x, q]/e^q Int[( d + e*x)^(m + q)*(a + b*x + c*x^2)^p, x], x] + Simp[1/e^q Int[(d + e*x)^m *(a + b*x + c*x^2)^p*ExpandToSum[e^q*Pq - Coeff[Pq, x, q]*(d + e*x)^q, x], x], x]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a *c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
\[\int \left (h x +g \right )^{-3-2 p} \left (c \,x^{2}+b x +a \right )^{p} \left (f \,x^{2}+e x +d \right )d x\]
\[ \int (g+h x)^{-3-2 p} \left (a+b x+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx=\int { {\left (f x^{2} + e x + d\right )} {\left (c x^{2} + b x + a\right )}^{p} {\left (h x + g\right )}^{-2 \, p - 3} \,d x } \]
Timed out. \[ \int (g+h x)^{-3-2 p} \left (a+b x+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx=\text {Timed out} \]
\[ \int (g+h x)^{-3-2 p} \left (a+b x+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx=\int { {\left (f x^{2} + e x + d\right )} {\left (c x^{2} + b x + a\right )}^{p} {\left (h x + g\right )}^{-2 \, p - 3} \,d x } \]
\[ \int (g+h x)^{-3-2 p} \left (a+b x+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx=\int { {\left (f x^{2} + e x + d\right )} {\left (c x^{2} + b x + a\right )}^{p} {\left (h x + g\right )}^{-2 \, p - 3} \,d x } \]
Timed out. \[ \int (g+h x)^{-3-2 p} \left (a+b x+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^p\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^{2\,p+3}} \,d x \]