Integrand size = 32, antiderivative size = 497 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\frac {\left (16 c^2 d g^2+16 a^2 f h^2-8 a b h (2 f g+e h)+b^2 \left (5 f g^2+3 e g h+5 d h^2\right )-4 c \left (2 b g (e g+2 d h)+a \left (f g^2-5 e g h+d h^2\right )\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{64 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{3/2}}{4 h \left (c g^2-b g h+a h^2\right ) (g+h x)^4}+\frac {\left (2 c g \left (3 f g^2+h (e g-5 d h)\right )+h \left (8 a h (2 f g-e h)-b \left (11 f g^2-3 e g h-5 d h^2\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{24 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^3}-\frac {\left (b^2-4 a c\right ) \left (16 c^2 d g^2+16 a^2 f h^2-8 a b h (2 f g+e h)+b^2 \left (5 f g^2+3 e g h+5 d h^2\right )-4 c \left (2 b g (e g+2 d h)+a \left (f g^2-5 e g h+d h^2\right )\right )\right ) \text {arctanh}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{128 \left (c g^2-b g h+a h^2\right )^{7/2}} \]
-1/4*(f*g^2-h*(-d*h+e*g))*(c*x^2+b*x+a)^(3/2)/h/(a*h^2-b*g*h+c*g^2)/(h*x+g )^4+1/24*(2*c*g*(3*f*g^2+h*(-5*d*h+e*g))+h*(8*a*h*(-e*h+2*f*g)-b*(-5*d*h^2 -3*e*g*h+11*f*g^2)))*(c*x^2+b*x+a)^(3/2)/h/(a*h^2-b*g*h+c*g^2)^2/(h*x+g)^3 -1/128*(-4*a*c+b^2)*(16*c^2*d*g^2+16*a^2*f*h^2-8*a*b*h*(e*h+2*f*g)+b^2*(5* d*h^2+3*e*g*h+5*f*g^2)-4*c*(2*b*g*(2*d*h+e*g)+a*(d*h^2-5*e*g*h+f*g^2)))*ar ctanh(1/2*(b*g-2*a*h+(-b*h+2*c*g)*x)/(a*h^2-b*g*h+c*g^2)^(1/2)/(c*x^2+b*x+ a)^(1/2))/(a*h^2-b*g*h+c*g^2)^(7/2)+1/64*(16*c^2*d*g^2+16*a^2*f*h^2-8*a*b* h*(e*h+2*f*g)+b^2*(5*d*h^2+3*e*g*h+5*f*g^2)-4*c*(2*b*g*(2*d*h+e*g)+a*(d*h^ 2-5*e*g*h+f*g^2)))*(b*g-2*a*h+(-b*h+2*c*g)*x)*(c*x^2+b*x+a)^(1/2)/(a*h^2-b *g*h+c*g^2)^3/(h*x+g)^2
Time = 13.35 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\frac {-48 h \left (c g^2+h (-b g+a h)\right )^{5/2} \left (f g^2+h (-e g+d h)\right ) (a+x (b+c x))^{3/2}+64 h (2 f g-e h) \left (c g^2+h (-b g+a h)\right )^{5/2} (g+h x) (a+x (b+c x))^{3/2}+48 f \left (c g^2+h (-b g+a h)\right )^{5/2} (g+h x)^2 \sqrt {a+x (b+c x)} (-2 a h+2 c g x+b (g-h x))-\left (f g^2+h (-e g+d h)\right ) (g+h x) \left (40 h (2 c g-b h) \left (c g^2+h (-b g+a h)\right )^{3/2} (a+x (b+c x))^{3/2}+3 \left (8 c^2 g^2+\frac {5 b^2 h^2}{2}-2 c h (4 b g+a h)\right ) (g+h x) \left (-2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)} (-2 a h+2 c g x+b (g-h x))-\left (b^2-4 a c\right ) (g+h x)^2 \text {arctanh}\left (\frac {-b g+2 a h-2 c g x+b h x}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )\right )\right )-24 \left (b^2-4 a c\right ) f \left (c g^2+h (-b g+a h)\right )^2 (g+h x)^4 \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )+12 (2 c g-b h) (2 f g-e h) \left (c g^2+h (-b g+a h)\right ) (g+h x)^2 \left (-2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)} (-2 a h+2 c g x+b (g-h x))+\left (b^2-4 a c\right ) (g+h x)^2 \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )\right )}{192 h^2 \left (c g^2+h (-b g+a h)\right )^{7/2} (g+h x)^4} \]
(-48*h*(c*g^2 + h*(-(b*g) + a*h))^(5/2)*(f*g^2 + h*(-(e*g) + d*h))*(a + x* (b + c*x))^(3/2) + 64*h*(2*f*g - e*h)*(c*g^2 + h*(-(b*g) + a*h))^(5/2)*(g + h*x)*(a + x*(b + c*x))^(3/2) + 48*f*(c*g^2 + h*(-(b*g) + a*h))^(5/2)*(g + h*x)^2*Sqrt[a + x*(b + c*x)]*(-2*a*h + 2*c*g*x + b*(g - h*x)) - (f*g^2 + h*(-(e*g) + d*h))*(g + h*x)*(40*h*(2*c*g - b*h)*(c*g^2 + h*(-(b*g) + a*h) )^(3/2)*(a + x*(b + c*x))^(3/2) + 3*(8*c^2*g^2 + (5*b^2*h^2)/2 - 2*c*h*(4* b*g + a*h))*(g + h*x)*(-2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c *x)]*(-2*a*h + 2*c*g*x + b*(g - h*x)) - (b^2 - 4*a*c)*(g + h*x)^2*ArcTanh[ (-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[ a + x*(b + c*x)])])) - 24*(b^2 - 4*a*c)*f*(c*g^2 + h*(-(b*g) + a*h))^2*(g + h*x)^4*ArcTanh[(-2*a*h + 2*c*g*x + b*(g - h*x))/(2*Sqrt[c*g^2 + h*(-(b*g ) + a*h)]*Sqrt[a + x*(b + c*x)])] + 12*(2*c*g - b*h)*(2*f*g - e*h)*(c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^2*(-2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)]*(-2*a*h + 2*c*g*x + b*(g - h*x)) + (b^2 - 4*a*c)*(g + h*x)^ 2*ArcTanh[(-2*a*h + 2*c*g*x + b*(g - h*x))/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h )]*Sqrt[a + x*(b + c*x)])]))/(192*h^2*(c*g^2 + h*(-(b*g) + a*h))^(7/2)*(g + h*x)^4)
Time = 0.83 (sec) , antiderivative size = 456, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2181, 27, 1228, 1152, 1154, 219}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {\int -\frac {\left (\frac {3 b f g^2}{h}+8 c d g-3 b e g-8 a f g-5 b d h+8 a e h+2 \left (\frac {3 c f g^2}{h}+c e g-4 b f g-c d h+4 a f h\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (g+h x)^4}dx}{4 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (8 c d g-8 a f g+8 a e h-b \left (-\frac {3 f g^2}{h}+3 e g+5 d h\right )-2 \left (4 b f g-4 a f h-c \left (\frac {3 f g^2}{h}+e g-d h\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{(g+h x)^4}dx}{8 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\frac {\left (16 a^2 f h^2-4 c \left (-a h (5 e g-d h)+a f g^2+2 b g (2 d h+e g)\right )-8 a b h (e h+2 f g)+b^2 \left (h (5 d h+3 e g)+5 f g^2\right )+16 c^2 d g^2\right ) \int \frac {\sqrt {c x^2+b x+a}}{(g+h x)^3}dx}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (8 a h^2 (2 f g-e h)-b h \left (11 f g^2-h (5 d h+3 e g)\right )+2 c g h (e g-5 d h)+6 c f g^3\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}}{8 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {\frac {\left (16 a^2 f h^2-4 c \left (-a h (5 e g-d h)+a f g^2+2 b g (2 d h+e g)\right )-8 a b h (e h+2 f g)+b^2 \left (h (5 d h+3 e g)+5 f g^2\right )+16 c^2 d g^2\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{8 \left (a h^2-b g h+c g^2\right )}\right )}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (8 a h^2 (2 f g-e h)-b h \left (11 f g^2-h (5 d h+3 e g)\right )+2 c g h (e g-5 d h)+6 c f g^3\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}}{8 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {\left (16 a^2 f h^2-4 c \left (-a h (5 e g-d h)+a f g^2+2 b g (2 d h+e g)\right )-8 a b h (e h+2 f g)+b^2 \left (h (5 d h+3 e g)+5 f g^2\right )+16 c^2 d g^2\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c g^2-b h g+a h^2\right )-\frac {(b g-2 a h+(2 c g-b h) x)^2}{c x^2+b x+a}}d\left (-\frac {b g-2 a h+(2 c g-b h) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a h^2-b g h+c g^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\right )}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (8 a h^2 (2 f g-e h)-b h \left (11 f g^2-h (5 d h+3 e g)\right )+2 c g h (e g-5 d h)+6 c f g^3\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}}{8 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (\frac {\sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right )}{8 \left (a h^2-b g h+c g^2\right )^{3/2}}\right ) \left (16 a^2 f h^2-4 c \left (-a h (5 e g-d h)+a f g^2+2 b g (2 d h+e g)\right )-8 a b h (e h+2 f g)+b^2 \left (h (5 d h+3 e g)+5 f g^2\right )+16 c^2 d g^2\right )}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (8 a h^2 (2 f g-e h)-b h \left (11 f g^2-h (5 d h+3 e g)\right )+2 c g h (e g-5 d h)+6 c f g^3\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}}{8 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}\) |
-1/4*((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(3/2))/(h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^4) + (((6*c*f*g^3 + 2*c*g*h*(e*g - 5*d*h) + 8*a*h^2*(2*f *g - e*h) - b*h*(11*f*g^2 - h*(3*e*g + 5*d*h)))*(a + b*x + c*x^2)^(3/2))/( 3*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^3) + ((16*c^2*d*g^2 + 16*a^2*f*h^2 - 8*a*b*h*(2*f*g + e*h) - 4*c*(a*f*g^2 - a*h*(5*e*g - d*h) + 2*b*g*(e*g + 2 *d*h)) + b^2*(5*f*g^2 + h*(3*e*g + 5*d*h)))*(((b*g - 2*a*h + (2*c*g - b*h) *x)*Sqrt[a + b*x + c*x^2])/(4*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) - ((b^2 - 4*a*c)*ArcTanh[(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(8*(c*g^2 - b*g*h + a*h^2)^(3/2))))/(2*(c* g^2 - b*g*h + a*h^2)))/(8*(c*g^2 - b*g*h + a*h^2))
3.2.94.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, 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[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(4939\) vs. \(2(475)=950\).
Time = 1.40 (sec) , antiderivative size = 4940, normalized size of antiderivative = 9.94
f/h^5*(-1/2/(a*h^2-b*g*h+c*g^2)*h^2/(x+1/h*g)^2*((x+1/h*g)^2*c+(b*h-2*c*g) /h*(x+1/h*g)+(a*h^2-b*g*h+c*g^2)/h^2)^(3/2)-1/4*(b*h-2*c*g)*h/(a*h^2-b*g*h +c*g^2)*(-1/(a*h^2-b*g*h+c*g^2)*h^2/(x+1/h*g)*((x+1/h*g)^2*c+(b*h-2*c*g)/h *(x+1/h*g)+(a*h^2-b*g*h+c*g^2)/h^2)^(3/2)+1/2*(b*h-2*c*g)*h/(a*h^2-b*g*h+c *g^2)*(((x+1/h*g)^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/ 2)+1/2*(b*h-2*c*g)/h*ln((1/2*(b*h-2*c*g)/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g) ^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/c^(1/2)-(a*h^ 2-b*g*h+c*g^2)/h^2/((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2-b*g*h+c*g^ 2)/h^2+(b*h-2*c*g)/h*(x+1/h*g)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+1/h*g )^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+1/h*g)))+ 2*c/(a*h^2-b*g*h+c*g^2)*h^2*(1/4*(2*c*(x+1/h*g)+(b*h-2*c*g)/h)/c*((x+1/h*g )^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)+1/8*(4*c*(a*h ^2-b*g*h+c*g^2)/h^2-(b*h-2*c*g)^2/h^2)/c^(3/2)*ln((1/2*(b*h-2*c*g)/h+c*(x+ 1/h*g))/c^(1/2)+((x+1/h*g)^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a*h^2-b*g*h+c*g^2) /h^2)^(1/2))))+1/2*c/(a*h^2-b*g*h+c*g^2)*h^2*(((x+1/h*g)^2*c+(b*h-2*c*g)/h *(x+1/h*g)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)+1/2*(b*h-2*c*g)/h*ln((1/2*(b*h-2 *c*g)/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a*h^2 -b*g*h+c*g^2)/h^2)^(1/2))/c^(1/2)-(a*h^2-b*g*h+c*g^2)/h^2/((a*h^2-b*g*h+c* g^2)/h^2)^(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+1/h*g)+2*(( a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+1/h*g)^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a...
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\int \frac {\sqrt {a + b x + c x^{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{5}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*h^2-b*g*h>0)', see `assume?` f or more de
\[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (f x^{2} + e x + d\right )}}{{\left (h x + g\right )}^{5}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^5} \,d x \]