Integrand size = 30, antiderivative size = 186 \[ \int \frac {(g+h x) \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (c \left (2 a e-b \left (d+\frac {a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right ) (g+h x)}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (4 c^2 d-2 b c e+3 b^2 f-8 a c f\right ) h \sqrt {a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}-\frac {(3 b f h-2 c (f g+e h)) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2}} \]
-1/2*(3*b*f*h-2*c*(e*h+f*g))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^( 1/2))/c^(5/2)+2*(c*(2*a*e-b*(d+a*f/c))-(-2*a*c*f+b^2*f-b*c*e+2*c^2*d)*x)*( h*x+g)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)+(4*c^2*d+3*b^2*f-2*c*(4*a*f+b*e) )*h*(c*x^2+b*x+a)^(1/2)/c^2/(-4*a*c+b^2)
Time = 0.91 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.04 \[ \int \frac {(g+h x) \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-3 b^3 f h x+2 b c (a e h-c e g x+c d (g-h x)+a f (g+5 h x))+b^2 (-3 a f h+c x (2 f g+2 e h-f h x))+4 c \left (2 a^2 f h+c^2 d g x-a c (d h+f x (g-h x)+e (g+h x))\right )}{c^2 \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {(-3 b f h+2 c (f g+e h)) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{c^{5/2}} \]
(-3*b^3*f*h*x + 2*b*c*(a*e*h - c*e*g*x + c*d*(g - h*x) + a*f*(g + 5*h*x)) + b^2*(-3*a*f*h + c*x*(2*f*g + 2*e*h - f*h*x)) + 4*c*(2*a^2*f*h + c^2*d*g* x - a*c*(d*h + f*x*(g - h*x) + e*(g + h*x))))/(c^2*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)]) + ((-3*b*f*h + 2*c*(f*g + e*h))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[ a] + Sqrt[a + x*(b + c*x)])])/c^(5/2)
Time = 0.44 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2175, 27, 1160, 1092, 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.
| \(\displaystyle \int \frac {(g+h x) \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2175 |
| \(\displaystyle \frac {2 (g+h x) \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {f g b^2+2 (c d+a f) h b-4 a c (f g+e h)+c \left (\frac {3 f b^2}{c}-2 e b+4 c d-8 a f\right ) h x}{2 c \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {f g b^2+2 (c d+a f) h b-4 a c (f g+e h)+\left (3 f b^2-2 c e b+4 c^2 d-8 a c f\right ) h x}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x) \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {\frac {h \sqrt {a+b x+c x^2} \left (-8 a c f+3 b^2 f-2 b c e+4 c^2 d\right )}{c}-\frac {\left (b^2-4 a c\right ) (3 b f h-2 c (e h+f g)) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 c}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x) \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {h \sqrt {a+b x+c x^2} \left (-8 a c f+3 b^2 f-2 b c e+4 c^2 d\right )}{c}-\frac {\left (b^2-4 a c\right ) (3 b f h-2 c (e h+f g)) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x) \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {h \sqrt {a+b x+c x^2} \left (-8 a c f+3 b^2 f-2 b c e+4 c^2 d\right )}{c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (3 b f h-2 c (e h+f g))}{2 c^{3/2}}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x) \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
(2*(c*(2*a*e - b*(d + (a*f)/c)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x)*( g + h*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (((4*c^2*d - 2*b*c*e + 3*b^2*f - 8*a*c*f)*h*Sqrt[a + b*x + c*x^2])/c - ((b^2 - 4*a*c)*(3*b*f*h - 2*c*(f*g + e*h))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/ (2*c^(3/2)))/(c*(b^2 - 4*a*c))
3.3.35.3.1 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/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[Polyno mialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*((R*b - 2*a*S + (2*c*R - b*S)*x)/((p + 1)*(b^2 - 4*a*c))), x ] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2 )^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Qx + S*(2*a*e*m + b*d *(2*p + 3)) - R*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*R - b*S)*(m + 2*p + 3)*x , 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] && GtQ[m, 0] && (Inte gerQ[p] || !IntegerQ[m] || !RationalQ[a, b, c, d, e]) && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.80 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.67
| method | result | size |
| risch | \(\frac {f h \sqrt {c \,x^{2}+b x +a}}{c^{2}}-\frac {\frac {2 a b f h \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {4 c^{2} d g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (3 b c f h -2 c^{2} e h -2 c^{2} f g \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (2 a c f h +b^{2} f h -2 c^{2} d h -2 c^{2} e g \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c^{2}}\) | \(311\) |
| default | \(\frac {2 d g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+f h \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+\left (e h +f g \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (d h +e g \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\) | \(411\) |
f*h/c^2*(c*x^2+b*x+a)^(1/2)-1/2/c^2*(2*a*b*f*h*(2*c*x+b)/(4*a*c-b^2)/(c*x^ 2+b*x+a)^(1/2)-4*c^2*d*g*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+(3*b*c* f*h-2*c^2*e*h-2*c^2*f*g)*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-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))+1/c^(3/2)*ln((1/ 2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+(2*a*c*f*h+b^2*f*h-2*c^2*d*h-2*c^2* e*g)*(-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)))
Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (171) = 342\).
Time = 3.87 (sec) , antiderivative size = 905, normalized size of antiderivative = 4.87 \[ \int \frac {(g+h x) \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (2 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} f g + {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} f g + {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e - 3 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} f\right )} h\right )} x^{2} + {\left (2 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e - 3 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} f\right )} h + {\left (2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} f g + {\left (2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} e - 3 \, {\left (b^{4} - 4 \, a b^{2} c\right )} f\right )} h\right )} x\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} f h x^{2} - 2 \, {\left (b c^{3} d - 2 \, a c^{3} e + a b c^{2} f\right )} g + {\left (4 \, a c^{3} d - 2 \, a b c^{2} e + {\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} f\right )} h - {\left (2 \, {\left (2 \, c^{4} d - b c^{3} e + {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} f\right )} g - {\left (2 \, b c^{3} d - 2 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} e + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} f\right )} h\right )} x\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )}}, -\frac {{\left (2 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} f g + {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} f g + {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e - 3 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} f\right )} h\right )} x^{2} + {\left (2 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e - 3 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} f\right )} h + {\left (2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} f g + {\left (2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} e - 3 \, {\left (b^{4} - 4 \, a b^{2} c\right )} f\right )} h\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} f h x^{2} - 2 \, {\left (b c^{3} d - 2 \, a c^{3} e + a b c^{2} f\right )} g + {\left (4 \, a c^{3} d - 2 \, a b c^{2} e + {\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} f\right )} h - {\left (2 \, {\left (2 \, c^{4} d - b c^{3} e + {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} f\right )} g - {\left (2 \, b c^{3} d - 2 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} e + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} f\right )} h\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )}}\right ] \]
[-1/4*((2*(a*b^2*c - 4*a^2*c^2)*f*g + (2*(b^2*c^2 - 4*a*c^3)*f*g + (2*(b^2 *c^2 - 4*a*c^3)*e - 3*(b^3*c - 4*a*b*c^2)*f)*h)*x^2 + (2*(a*b^2*c - 4*a^2* c^2)*e - 3*(a*b^3 - 4*a^2*b*c)*f)*h + (2*(b^3*c - 4*a*b*c^2)*f*g + (2*(b^3 *c - 4*a*b*c^2)*e - 3*(b^4 - 4*a*b^2*c)*f)*h)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*( (b^2*c^2 - 4*a*c^3)*f*h*x^2 - 2*(b*c^3*d - 2*a*c^3*e + a*b*c^2*f)*g + (4*a *c^3*d - 2*a*b*c^2*e + (3*a*b^2*c - 8*a^2*c^2)*f)*h - (2*(2*c^4*d - b*c^3* e + (b^2*c^2 - 2*a*c^3)*f)*g - (2*b*c^3*d - 2*(b^2*c^2 - 2*a*c^3)*e + (3*b ^3*c - 10*a*b*c^2)*f)*h)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x), -1/2*((2*(a*b^2*c - 4*a^2*c^2)*f*g + (2*(b^2*c^2 - 4*a*c^3)*f*g + (2*(b^2*c^2 - 4*a*c^3)*e - 3 *(b^3*c - 4*a*b*c^2)*f)*h)*x^2 + (2*(a*b^2*c - 4*a^2*c^2)*e - 3*(a*b^3 - 4 *a^2*b*c)*f)*h + (2*(b^3*c - 4*a*b*c^2)*f*g + (2*(b^3*c - 4*a*b*c^2)*e - 3 *(b^4 - 4*a*b^2*c)*f)*h)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c *x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*((b^2*c^2 - 4*a*c^3)*f*h*x^2 - 2*(b*c^3*d - 2*a*c^3*e + a*b*c^2*f)*g + (4*a*c^3*d - 2*a*b*c^2*e + (3*a *b^2*c - 8*a^2*c^2)*f)*h - (2*(2*c^4*d - b*c^3*e + (b^2*c^2 - 2*a*c^3)*f)* g - (2*b*c^3*d - 2*(b^2*c^2 - 2*a*c^3)*e + (3*b^3*c - 10*a*b*c^2)*f)*h)*x) *sqrt(c*x^2 + b*x + a))/(a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)]
\[ \int \frac {(g+h x) \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (g + h x\right ) \left (d + e x + f x^{2}\right )}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(g+h x) \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.36 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.41 \[ \int \frac {(g+h x) \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left (\frac {{\left (b^{2} c f h - 4 \, a c^{2} f h\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} - \frac {4 \, c^{3} d g - 2 \, b c^{2} e g + 2 \, b^{2} c f g - 4 \, a c^{2} f g - 2 \, b c^{2} d h + 2 \, b^{2} c e h - 4 \, a c^{2} e h - 3 \, b^{3} f h + 10 \, a b c f h}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x - \frac {2 \, b c^{2} d g - 4 \, a c^{2} e g + 2 \, a b c f g - 4 \, a c^{2} d h + 2 \, a b c e h - 3 \, a b^{2} f h + 8 \, a^{2} c f h}{b^{2} c^{2} - 4 \, a c^{3}}}{\sqrt {c x^{2} + b x + a}} - \frac {{\left (2 \, c f g + 2 \, c e h - 3 \, b f h\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2 \, c^{\frac {5}{2}}} \]
(((b^2*c*f*h - 4*a*c^2*f*h)*x/(b^2*c^2 - 4*a*c^3) - (4*c^3*d*g - 2*b*c^2*e *g + 2*b^2*c*f*g - 4*a*c^2*f*g - 2*b*c^2*d*h + 2*b^2*c*e*h - 4*a*c^2*e*h - 3*b^3*f*h + 10*a*b*c*f*h)/(b^2*c^2 - 4*a*c^3))*x - (2*b*c^2*d*g - 4*a*c^2 *e*g + 2*a*b*c*f*g - 4*a*c^2*d*h + 2*a*b*c*e*h - 3*a*b^2*f*h + 8*a^2*c*f*h )/(b^2*c^2 - 4*a*c^3))/sqrt(c*x^2 + b*x + a) - 1/2*(2*c*f*g + 2*c*e*h - 3* b*f*h)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(5/2)
Timed out. \[ \int \frac {(g+h x) \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (g+h\,x\right )\,\left (f\,x^2+e\,x+d\right )}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]