Integrand size = 32, antiderivative size = 225 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (b^2 d h-b (c d g+a f g+a e h)+2 a (c e g-c d h+a f h)-\left (2 c^2 d g+b f (b g-a h)-c (b e g+2 a f g+b d h-2 a e h)\right ) x\right )}{\left (b^2-4 a c\right ) \left (c g^2-b g h+a h^2\right ) \sqrt {a+b x+c x^2}}+\frac {\left (f g^2-h (e g-d h)\right ) \text {arctanh}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{\left (c g^2-b g h+a h^2\right )^{3/2}} \]
(f*g^2-h*(-d*h+e*g))*arctanh(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)^(3/2)+2*(b^2*d*h-b*(a *e*h+a*f*g+c*d*g)+2*a*(a*f*h-c*d*h+c*e*g)-(2*c^2*d*g+b*f*(-a*h+b*g)-c*(-2* a*e*h+2*a*f*g+b*d*h+b*e*g))*x)/(-4*a*c+b^2)/(a*h^2-b*g*h+c*g^2)/(c*x^2+b*x +a)^(1/2)
Time = 1.02 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.04 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+b x+c x^2\right )^{3/2}} \, dx=2 \left (\frac {-2 a^2 f h+2 c^2 d g x+b^2 (-d h+f g x)+2 a c (-e g+d h-f g x+e h x)+b c (-e g x+d (g-h x))+a b (e h+f (g-h x))}{\left (b^2-4 a c\right ) \left (-c g^2+h (b g-a h)\right ) \sqrt {a+x (b+c x)}}+\frac {\sqrt {-c g^2+b g h-a h^2} \left (f g^2+h (-e g+d h)\right ) \arctan \left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+x (b+c x)}}{\sqrt {-c g^2+h (b g-a h)}}\right )}{\left (c g^2+h (-b g+a h)\right )^2}\right ) \]
2*((-2*a^2*f*h + 2*c^2*d*g*x + b^2*(-(d*h) + f*g*x) + 2*a*c*(-(e*g) + d*h - f*g*x + e*h*x) + b*c*(-(e*g*x) + d*(g - h*x)) + a*b*(e*h + f*(g - h*x))) /((b^2 - 4*a*c)*(-(c*g^2) + h*(b*g - a*h))*Sqrt[a + x*(b + c*x)]) + (Sqrt[ -(c*g^2) + b*g*h - a*h^2]*(f*g^2 + h*(-(e*g) + d*h))*ArcTan[(Sqrt[c]*(g + h*x) - h*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*g^2) + h*(b*g - a*h)]])/(c*g^2 + h*(-(b*g) + a*h))^2)
Time = 0.47 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2177, 27, 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.
| \(\displaystyle \int \frac {d+e x+f x^2}{(g+h x) \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {2 \left (-x \left (-c (-2 a e h+2 a f g+b d h+b e g)+b f (b g-a h)+2 c^2 d g\right )-b (a e h+a f g+c d g)+2 a (a f h-c d h+c e g)+b^2 d h\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a h^2-b g h+c g^2\right )}-\frac {2 \int -\frac {\left (b^2-4 a c\right ) \left (f g^2-h (e g-d h)\right )}{2 \left (c g^2-b h g+a h^2\right ) (g+h x) \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (f g^2-h (e g-d h)\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{a h^2-b g h+c g^2}+\frac {2 \left (-x \left (-c (-2 a e h+2 a f g+b d h+b e g)+b f (b g-a h)+2 c^2 d g\right )-b (a e h+a f g+c d g)+2 a (a f h-c d h+c e g)+b^2 d h\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {2 \left (-x \left (-c (-2 a e h+2 a f g+b d h+b e g)+b f (b g-a h)+2 c^2 d g\right )-b (a e h+a f g+c d g)+2 a (a f h-c d h+c e g)+b^2 d h\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a h^2-b g h+c g^2\right )}-\frac {2 \left (f g^2-h (e g-d h)\right ) \int \frac {1}{4 \left (c g^2-b h g+a h^2\right )-\frac {(b g-2 a h+(2 c g-b h) x)^2}{c x^2+b x+a}}d\left (-\frac {b g-2 a h+(2 c g-b h) x}{\sqrt {c x^2+b x+a}}\right )}{a h^2-b g h+c g^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (f g^2-h (e g-d h)\right ) \text {arctanh}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right )}{\left (a h^2-b g h+c g^2\right )^{3/2}}+\frac {2 \left (-x \left (-c (-2 a e h+2 a f g+b d h+b e g)+b f (b g-a h)+2 c^2 d g\right )-b (a e h+a f g+c d g)+2 a (a f h-c d h+c e g)+b^2 d h\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a h^2-b g h+c g^2\right )}\) |
(2*(b^2*d*h - b*(c*d*g + a*f*g + a*e*h) + 2*a*(c*e*g - c*d*h + a*f*h) - (2 *c^2*d*g + b*f*(b*g - a*h) - c*(b*e*g + 2*a*f*g + b*d*h - 2*a*e*h))*x))/(( b^2 - 4*a*c)*(c*g^2 - b*g*h + a*h^2)*Sqrt[a + b*x + c*x^2]) + ((f*g^2 - h* (e*g - d*h))*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])])/(c*g^2 - b*g*h + a*h^2)^(3/2)
3.3.37.3.1 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[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[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x + c* x^2, x], R = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + (2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Qx)/(d + e*x )^m - ((2*p + 3)*(2*c*R - b*S))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(546\) vs. \(2(215)=430\).
Time = 0.76 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.43
| method | result | size |
| default | \(\frac {\frac {2 e h \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+f h \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )-\frac {2 f g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}}{h^{2}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (\frac {h^{2}}{\left (a \,h^{2}-b g h +c \,g^{2}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c +\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}-\frac {\left (b h -2 c g \right ) h \left (2 c \left (x +\frac {g}{h}\right )+\frac {b h -2 c g}{h}\right )}{\left (a \,h^{2}-b g h +c \,g^{2}\right ) \left (\frac {4 c \left (a \,h^{2}-b g h +c \,g^{2}\right )}{h^{2}}-\frac {\left (b h -2 c g \right )^{2}}{h^{2}}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c +\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}-\frac {h^{2} \ln \left (\frac {\frac {2 a \,h^{2}-2 b g h +2 c \,g^{2}}{h^{2}}+\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c +\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}-b g h +c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}\right )}{h^{3}}\) | \(547\) |
1/h^2*(2*e*h*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+f*h*(-1/c/(c*x^2+b* x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))-2*f*g*(2*c*x+b)/ (4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+(d*h^2-e*g*h+f*g^2)/h^3*(1/(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)-(b*h-2*c*g)*h/(a*h^2-b*g*h+c*g^2)*(2*c*(x+1/h*g)+(b*h-2*c*g)/h)/(4*c *(a*h^2-b*g*h+c*g^2)/h^2-(b*h-2*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/(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)))
Leaf count of result is larger than twice the leaf count of optimal. 931 vs. \(2 (215) = 430\).
Time = 9.79 (sec) , antiderivative size = 1905, normalized size of antiderivative = 8.47 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/2*(((a*b^2 - 4*a^2*c)*f*g^2 - (a*b^2 - 4*a^2*c)*e*g*h + (a*b^2 - 4*a^2* c)*d*h^2 + ((b^2*c - 4*a*c^2)*f*g^2 - (b^2*c - 4*a*c^2)*e*g*h + (b^2*c - 4 *a*c^2)*d*h^2)*x^2 + ((b^3 - 4*a*b*c)*f*g^2 - (b^3 - 4*a*b*c)*e*g*h + (b^3 - 4*a*b*c)*d*h^2)*x)*sqrt(c*g^2 - b*g*h + a*h^2)*log((8*a*b*g*h - 8*a^2*h ^2 - (b^2 + 4*a*c)*g^2 - (8*c^2*g^2 - 8*b*c*g*h + (b^2 + 4*a*c)*h^2)*x^2 - 4*sqrt(c*g^2 - b*g*h + a*h^2)*sqrt(c*x^2 + b*x + a)*(b*g - 2*a*h + (2*c*g - b*h)*x) - 2*(4*b*c*g^2 + 4*a*b*h^2 - (3*b^2 + 4*a*c)*g*h)*x)/(h^2*x^2 + 2*g*h*x + g^2)) - 4*((b*c^2*d - 2*a*c^2*e + a*b*c*f)*g^3 + (3*a*b*c*e - 2 *(b^2*c - a*c^2)*d - (a*b^2 + 2*a^2*c)*f)*g^2*h + (3*a^2*b*f + (b^3 - a*b* c)*d - (a*b^2 + 2*a^2*c)*e)*g*h^2 + (a^2*b*e - 2*a^3*f - (a*b^2 - 2*a^2*c) *d)*h^3 + ((2*c^3*d - b*c^2*e + (b^2*c - 2*a*c^2)*f)*g^3 - (3*b*c^2*d - (b ^2*c + 2*a*c^2)*e + (b^3 - a*b*c)*f)*g^2*h - (3*a*b*c*e - (b^2*c + 2*a*c^2 )*d - 2*(a*b^2 - a^2*c)*f)*g*h^2 - (a*b*c*d - 2*a^2*c*e + a^2*b*f)*h^3)*x) *sqrt(c*x^2 + b*x + a))/((a*b^2*c^2 - 4*a^2*c^3)*g^4 - 2*(a*b^3*c - 4*a^2* b*c^2)*g^3*h + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*g^2*h^2 - 2*(a^2*b^3 - 4* a^3*b*c)*g*h^3 + (a^3*b^2 - 4*a^4*c)*h^4 + ((b^2*c^3 - 4*a*c^4)*g^4 - 2*(b ^3*c^2 - 4*a*b*c^3)*g^3*h + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*g^2*h^2 - 2* (a*b^3*c - 4*a^2*b*c^2)*g*h^3 + (a^2*b^2*c - 4*a^3*c^2)*h^4)*x^2 + ((b^3*c ^2 - 4*a*b*c^3)*g^4 - 2*(b^4*c - 4*a*b^2*c^2)*g^3*h + (b^5 - 2*a*b^3*c - 8 *a^2*b*c^2)*g^2*h^2 - 2*(a*b^4 - 4*a^2*b^2*c)*g*h^3 + (a^2*b^3 - 4*a^3*...
\[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {d + e x + f x^{2}}{\left (g + h x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((b/h-(2*c*g)/h^2)^2>0)', see `as sume?` for
Leaf count of result is larger than twice the leaf count of optimal. 708 vs. \(2 (215) = 430\).
Time = 0.31 (sec) , antiderivative size = 708, normalized size of antiderivative = 3.15 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {{\left (2 \, c^{3} d g^{3} - b c^{2} e g^{3} + b^{2} c f g^{3} - 2 \, a c^{2} f g^{3} - 3 \, b c^{2} d g^{2} h + b^{2} c e g^{2} h + 2 \, a c^{2} e g^{2} h - b^{3} f g^{2} h + a b c f g^{2} h + b^{2} c d g h^{2} + 2 \, a c^{2} d g h^{2} - 3 \, a b c e g h^{2} + 2 \, a b^{2} f g h^{2} - 2 \, a^{2} c f g h^{2} - a b c d h^{3} + 2 \, a^{2} c e h^{3} - a^{2} b f h^{3}\right )} x}{b^{2} c^{2} g^{4} - 4 \, a c^{3} g^{4} - 2 \, b^{3} c g^{3} h + 8 \, a b c^{2} g^{3} h + b^{4} g^{2} h^{2} - 2 \, a b^{2} c g^{2} h^{2} - 8 \, a^{2} c^{2} g^{2} h^{2} - 2 \, a b^{3} g h^{3} + 8 \, a^{2} b c g h^{3} + a^{2} b^{2} h^{4} - 4 \, a^{3} c h^{4}} + \frac {b c^{2} d g^{3} - 2 \, a c^{2} e g^{3} + a b c f g^{3} - 2 \, b^{2} c d g^{2} h + 2 \, a c^{2} d g^{2} h + 3 \, a b c e g^{2} h - a b^{2} f g^{2} h - 2 \, a^{2} c f g^{2} h + b^{3} d g h^{2} - a b c d g h^{2} - a b^{2} e g h^{2} - 2 \, a^{2} c e g h^{2} + 3 \, a^{2} b f g h^{2} - a b^{2} d h^{3} + 2 \, a^{2} c d h^{3} + a^{2} b e h^{3} - 2 \, a^{3} f h^{3}}{b^{2} c^{2} g^{4} - 4 \, a c^{3} g^{4} - 2 \, b^{3} c g^{3} h + 8 \, a b c^{2} g^{3} h + b^{4} g^{2} h^{2} - 2 \, a b^{2} c g^{2} h^{2} - 8 \, a^{2} c^{2} g^{2} h^{2} - 2 \, a b^{3} g h^{3} + 8 \, a^{2} b c g h^{3} + a^{2} b^{2} h^{4} - 4 \, a^{3} c h^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {2 \, {\left (f g^{2} - e g h + d h^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} h + \sqrt {c} g}{\sqrt {-c g^{2} + b g h - a h^{2}}}\right )}{{\left (c g^{2} - b g h + a h^{2}\right )} \sqrt {-c g^{2} + b g h - a h^{2}}} \]
-2*((2*c^3*d*g^3 - b*c^2*e*g^3 + b^2*c*f*g^3 - 2*a*c^2*f*g^3 - 3*b*c^2*d*g ^2*h + b^2*c*e*g^2*h + 2*a*c^2*e*g^2*h - b^3*f*g^2*h + a*b*c*f*g^2*h + b^2 *c*d*g*h^2 + 2*a*c^2*d*g*h^2 - 3*a*b*c*e*g*h^2 + 2*a*b^2*f*g*h^2 - 2*a^2*c *f*g*h^2 - a*b*c*d*h^3 + 2*a^2*c*e*h^3 - a^2*b*f*h^3)*x/(b^2*c^2*g^4 - 4*a *c^3*g^4 - 2*b^3*c*g^3*h + 8*a*b*c^2*g^3*h + b^4*g^2*h^2 - 2*a*b^2*c*g^2*h ^2 - 8*a^2*c^2*g^2*h^2 - 2*a*b^3*g*h^3 + 8*a^2*b*c*g*h^3 + a^2*b^2*h^4 - 4 *a^3*c*h^4) + (b*c^2*d*g^3 - 2*a*c^2*e*g^3 + a*b*c*f*g^3 - 2*b^2*c*d*g^2*h + 2*a*c^2*d*g^2*h + 3*a*b*c*e*g^2*h - a*b^2*f*g^2*h - 2*a^2*c*f*g^2*h + b ^3*d*g*h^2 - a*b*c*d*g*h^2 - a*b^2*e*g*h^2 - 2*a^2*c*e*g*h^2 + 3*a^2*b*f*g *h^2 - a*b^2*d*h^3 + 2*a^2*c*d*h^3 + a^2*b*e*h^3 - 2*a^3*f*h^3)/(b^2*c^2*g ^4 - 4*a*c^3*g^4 - 2*b^3*c*g^3*h + 8*a*b*c^2*g^3*h + b^4*g^2*h^2 - 2*a*b^2 *c*g^2*h^2 - 8*a^2*c^2*g^2*h^2 - 2*a*b^3*g*h^3 + 8*a^2*b*c*g*h^3 + a^2*b^2 *h^4 - 4*a^3*c*h^4))/sqrt(c*x^2 + b*x + a) + 2*(f*g^2 - e*g*h + d*h^2)*arc tan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*h + sqrt(c)*g)/sqrt(-c*g^2 + b*g *h - a*h^2))/((c*g^2 - b*g*h + a*h^2)*sqrt(-c*g^2 + b*g*h - a*h^2))
Timed out. \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {f\,x^2+e\,x+d}{\left (g+h\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]