\(\int (g+h x) \sqrt {a+b x+c x^2} (d+e x+f x^2) \, dx\) [188]

Optimal result

Integrand size = 30, antiderivative size = 322 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {\left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\left (35 b^2 f h^2-16 c^2 \left (3 f g^2-5 h (e g+d h)\right )-2 c h (16 a f h+25 b (f g+e h))-6 c h (6 c f g-10 c e h+7 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3 h}-\frac {\left (b^2-4 a c\right ) \left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2}} \]

[Out]

1/5*f*(h*x+g)^2*(c*x^2+b*x+a)^(3/2)/c/h+1/240*(35*b^2*f*h^2-16*c^2*(3*f*g^2-5*h*(d*h+e*g))-2*c*h*(16*a*f*h+25*
b*(e*h+f*g))-6*c*h*(7*b*f*h-10*c*e*h+6*c*f*g)*x)*(c*x^2+b*x+a)^(3/2)/c^3/h-1/256*(-4*a*c+b^2)*(32*c^3*d*g-7*b^
3*f*h-8*c^2*(a*e*h+a*f*g+2*b*d*h+2*b*e*g)+2*b*c*(6*a*f*h+5*b*(e*h+f*g)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+
b*x+a)^(1/2))/c^(9/2)+1/128*(32*c^3*d*g-7*b^3*f*h-8*c^2*(a*e*h+a*f*g+2*b*d*h+2*b*e*g)+2*b*c*(6*a*f*h+5*b*(e*h+
f*g)))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^4

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1667, 793, 626, 635, 212} \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 c^2 (a e h+a f g+2 b d h+2 b e g)+2 b c (6 a f h+5 b (e h+f g))-7 b^3 f h+32 c^3 d g\right )}{256 c^{9/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-8 c^2 (a e h+a f g+2 b d h+2 b e g)+2 b c (6 a f h+5 b (e h+f g))-7 b^3 f h+32 c^3 d g\right )}{128 c^4}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c h (16 a f h+25 b (e h+f g))+35 b^2 f h^2-6 c h x (7 b f h-10 c e h+6 c f g)-\left (c^2 \left (48 f g^2-80 h (d h+e g)\right )\right )\right )}{240 c^3 h}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h} \]

[In]

Int[(g + h*x)*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2),x]

[Out]

((32*c^3*d*g - 7*b^3*f*h - 8*c^2*(2*b*e*g + a*f*g + 2*b*d*h + a*e*h) + 2*b*c*(6*a*f*h + 5*b*(f*g + e*h)))*(b +
2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^4) + (f*(g + h*x)^2*(a + b*x + c*x^2)^(3/2))/(5*c*h) + ((35*b^2*f*h^2 -
c^2*(48*f*g^2 - 80*h*(e*g + d*h)) - 2*c*h*(16*a*f*h + 25*b*(f*g + e*h)) - 6*c*h*(6*c*f*g - 10*c*e*h + 7*b*f*h)
*x)*(a + b*x + c*x^2)^(3/2))/(240*c^3*h) - ((b^2 - 4*a*c)*(32*c^3*d*g - 7*b^3*f*h - 8*c^2*(2*b*e*g + a*f*g + 2
*b*d*h + a*e*h) + 2*b*c*(6*a*f*h + 5*b*(f*g + e*h)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(
256*c^(9/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 793

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 1667

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m
+ q + 2*p + 1))), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; 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]))

Rubi steps \begin{align*} \text {integral}& = \frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\int (g+h x) \left (-\frac {1}{2} h (3 b f g-10 c d h+4 a f h)-\frac {1}{2} h (6 c f g-10 c e h+7 b f h) x\right ) \sqrt {a+b x+c x^2} \, dx}{5 c h^2} \\ & = \frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\left (35 b^2 f h^2-c^2 \left (48 f g^2-80 h (e g+d h)\right )-2 c h (16 a f h+25 b (f g+e h))-6 c h (6 c f g-10 c e h+7 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3 h}+\frac {\left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) \int \sqrt {a+b x+c x^2} \, dx}{32 c^3} \\ & = \frac {\left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\left (35 b^2 f h^2-c^2 \left (48 f g^2-80 h (e g+d h)\right )-2 c h (16 a f h+25 b (f g+e h))-6 c h (6 c f g-10 c e h+7 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3 h}-\frac {\left (\left (b^2-4 a c\right ) \left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^4} \\ & = \frac {\left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\left (35 b^2 f h^2-c^2 \left (48 f g^2-80 h (e g+d h)\right )-2 c h (16 a f h+25 b (f g+e h))-6 c h (6 c f g-10 c e h+7 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3 h}-\frac {\left (\left (b^2-4 a c\right ) \left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{128 c^4} \\ & = \frac {\left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\left (35 b^2 f h^2-c^2 \left (48 f g^2-80 h (e g+d h)\right )-2 c h (16 a f h+25 b (f g+e h))-6 c h (6 c f g-10 c e h+7 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3 h}-\frac {\left (b^2-4 a c\right ) \left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.46 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.07 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^4 f h+10 b^3 c (15 f g+15 e h+7 f h x)-4 b^2 c \left (-115 a f h+c \left (60 e g+60 d h+25 f g x+25 e h x+14 f h x^2\right )\right )+8 b c^2 (20 c d (3 g+h x)-a (65 f g+65 e h+29 f h x)+2 c x (5 e (2 g+h x)+f x (5 g+3 h x)))+16 c^2 \left (-16 a^2 f h+a c (40 d h+5 e (8 g+3 h x)+f x (15 g+8 h x))+2 c^2 x (10 d (3 g+2 h x)+x (5 e (4 g+3 h x)+3 f x (5 g+4 h x)))\right )\right )+15 \left (b^2-4 a c\right ) \left (-32 c^3 d g+7 b^3 f h+8 c^2 (2 b e g+a f g+2 b d h+a e h)-2 b c (6 a f h+5 b (f g+e h))\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{1920 c^{9/2}} \]

[In]

Integrate[(g + h*x)*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2),x]

[Out]

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^4*f*h + 10*b^3*c*(15*f*g + 15*e*h + 7*f*h*x) - 4*b^2*c*(-115*a*f*h + c*
(60*e*g + 60*d*h + 25*f*g*x + 25*e*h*x + 14*f*h*x^2)) + 8*b*c^2*(20*c*d*(3*g + h*x) - a*(65*f*g + 65*e*h + 29*
f*h*x) + 2*c*x*(5*e*(2*g + h*x) + f*x*(5*g + 3*h*x))) + 16*c^2*(-16*a^2*f*h + a*c*(40*d*h + 5*e*(8*g + 3*h*x)
+ f*x*(15*g + 8*h*x)) + 2*c^2*x*(10*d*(3*g + 2*h*x) + x*(5*e*(4*g + 3*h*x) + 3*f*x*(5*g + 4*h*x))))) + 15*(b^2
- 4*a*c)*(-32*c^3*d*g + 7*b^3*f*h + 8*c^2*(2*b*e*g + a*f*g + 2*b*d*h + a*e*h) - 2*b*c*(6*a*f*h + 5*b*(f*g + e
*h)))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(1920*c^(9/2))

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.52

method	result	size
risch	\(-\frac {\left (-384 h f \,c^{4} x^{4}-48 b \,c^{3} f h \,x^{3}-480 c^{4} e h \,x^{3}-480 c^{4} f g \,x^{3}-128 a \,c^{3} f h \,x^{2}+56 b^{2} c^{2} f h \,x^{2}-80 b \,c^{3} e h \,x^{2}-80 b \,c^{3} f g \,x^{2}-640 c^{4} d h \,x^{2}-640 c^{4} e g \,x^{2}+232 a b \,c^{2} f h x -240 a \,c^{3} e h x -240 a \,c^{3} f g x -70 b^{3} c f h x +100 b^{2} c^{2} e h x +100 b^{2} c^{2} f g x -160 b \,c^{3} d h x -160 b \,c^{3} e g x -960 c^{4} d g x +256 a^{2} c^{2} f h -460 a \,b^{2} c f h +520 a b \,c^{2} e h +520 a b \,c^{2} f g -640 a \,c^{3} d h -640 a \,c^{3} e g +105 b^{4} f h -150 b^{3} c e h -150 b^{3} c f g +240 b^{2} c^{2} d h +240 b^{2} c^{2} e g -480 b \,c^{3} d g \right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{4}}+\frac {\left (48 a^{2} b \,c^{2} f h -32 a^{2} c^{3} e h -32 a^{2} c^{3} f g -40 a \,b^{3} c f h +48 a \,b^{2} c^{2} e h +48 a \,b^{2} c^{2} f g -64 a b \,c^{3} d h -64 a b \,c^{3} e g +128 a \,c^{4} d g +7 b^{5} f h -10 b^{4} c e h -10 b^{4} c f g +16 b^{3} c^{2} d h +16 b^{3} c^{2} e g -32 b^{2} c^{3} d g \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {9}{2}}}\)	\(488\)
default	\(d g \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )+h f \left (\frac {x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{5 c}-\frac {7 b \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )}{10 c}-\frac {2 a \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{5 c}\right )+\left (e h +f g \right ) \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )+\left (d h +e g \right ) \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\)	\(663\)

[In]

int((h*x+g)*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/1920*(-384*c^4*f*h*x^4-48*b*c^3*f*h*x^3-480*c^4*e*h*x^3-480*c^4*f*g*x^3-128*a*c^3*f*h*x^2+56*b^2*c^2*f*h*x^
2-80*b*c^3*e*h*x^2-80*b*c^3*f*g*x^2-640*c^4*d*h*x^2-640*c^4*e*g*x^2+232*a*b*c^2*f*h*x-240*a*c^3*e*h*x-240*a*c^
3*f*g*x-70*b^3*c*f*h*x+100*b^2*c^2*e*h*x+100*b^2*c^2*f*g*x-160*b*c^3*d*h*x-160*b*c^3*e*g*x-960*c^4*d*g*x+256*a
^2*c^2*f*h-460*a*b^2*c*f*h+520*a*b*c^2*e*h+520*a*b*c^2*f*g-640*a*c^3*d*h-640*a*c^3*e*g+105*b^4*f*h-150*b^3*c*e
*h-150*b^3*c*f*g+240*b^2*c^2*d*h+240*b^2*c^2*e*g-480*b*c^3*d*g)/c^4*(c*x^2+b*x+a)^(1/2)+1/256*(48*a^2*b*c^2*f*
h-32*a^2*c^3*e*h-32*a^2*c^3*f*g-40*a*b^3*c*f*h+48*a*b^2*c^2*e*h+48*a*b^2*c^2*f*g-64*a*b*c^3*d*h-64*a*b*c^3*e*g
+128*a*c^4*d*g+7*b^5*f*h-10*b^4*c*e*h-10*b^4*c*f*g+16*b^3*c^2*d*h+16*b^3*c^2*e*g-32*b^2*c^3*d*g)/c^(9/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 1009, normalized size of antiderivative = 3.13 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]

[In]

integrate((h*x+g)*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/7680*(15*(2*(16*(b^2*c^3 - 4*a*c^4)*d - 8*(b^3*c^2 - 4*a*b*c^3)*e + (5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*
f)*g - (16*(b^3*c^2 - 4*a*b*c^3)*d - 2*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*e + (7*b^5 - 40*a*b^3*c + 48*a^2*
b*c^2)*f)*h)*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
*(384*c^5*f*h*x^4 + 48*(10*c^5*f*g + (10*c^5*e + b*c^4*f)*h)*x^3 + 8*(10*(8*c^5*e + b*c^4*f)*g + (80*c^5*d + 1
0*b*c^4*e - (7*b^2*c^3 - 16*a*c^4)*f)*h)*x^2 + 10*(48*b*c^4*d - 8*(3*b^2*c^3 - 8*a*c^4)*e + (15*b^3*c^2 - 52*a
*b*c^3)*f)*g - (80*(3*b^2*c^3 - 8*a*c^4)*d - 10*(15*b^3*c^2 - 52*a*b*c^3)*e + (105*b^4*c - 460*a*b^2*c^2 + 256
*a^2*c^3)*f)*h + 2*(10*(48*c^5*d + 8*b*c^4*e - (5*b^2*c^3 - 12*a*c^4)*f)*g + (80*b*c^4*d - 10*(5*b^2*c^3 - 12*
a*c^4)*e + (35*b^3*c^2 - 116*a*b*c^3)*f)*h)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/3840*(15*(2*(16*(b^2*c^3 - 4*a*c^
4)*d - 8*(b^3*c^2 - 4*a*b*c^3)*e + (5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*f)*g - (16*(b^3*c^2 - 4*a*b*c^3)*d -
2*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*e + (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*f)*h)*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*(384*c^5*f*h*x^4 + 48*(10*c^5*f*g + (10*c^
5*e + b*c^4*f)*h)*x^3 + 8*(10*(8*c^5*e + b*c^4*f)*g + (80*c^5*d + 10*b*c^4*e - (7*b^2*c^3 - 16*a*c^4)*f)*h)*x^
2 + 10*(48*b*c^4*d - 8*(3*b^2*c^3 - 8*a*c^4)*e + (15*b^3*c^2 - 52*a*b*c^3)*f)*g - (80*(3*b^2*c^3 - 8*a*c^4)*d
- 10*(15*b^3*c^2 - 52*a*b*c^3)*e + (105*b^4*c - 460*a*b^2*c^2 + 256*a^2*c^3)*f)*h + 2*(10*(48*c^5*d + 8*b*c^4*
e - (5*b^2*c^3 - 12*a*c^4)*f)*g + (80*b*c^4*d - 10*(5*b^2*c^3 - 12*a*c^4)*e + (35*b^3*c^2 - 116*a*b*c^3)*f)*h)
*x)*sqrt(c*x^2 + b*x + a))/c^5]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 993 vs. \(2 (332) = 664\).

Time = 1.04 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.08 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\begin {cases} \sqrt {a + b x + c x^{2}} \left (\frac {f h x^{4}}{5} + \frac {x^{3} \left (\frac {b f h}{10} + c e h + c f g\right )}{4 c} + \frac {x^{2} \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{3 c} + \frac {x \left (a e h + a f g - \frac {3 a \left (\frac {b f h}{10} + c e h + c f g\right )}{4 c} + b d h + b e g - \frac {5 b \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{6 c} + c d g\right )}{2 c} + \frac {a d h + a e g - \frac {2 a \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{3 c} + b d g - \frac {3 b \left (a e h + a f g - \frac {3 a \left (\frac {b f h}{10} + c e h + c f g\right )}{4 c} + b d h + b e g - \frac {5 b \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{6 c} + c d g\right )}{4 c}}{c}\right ) + \left (a d g - \frac {a \left (a e h + a f g - \frac {3 a \left (\frac {b f h}{10} + c e h + c f g\right )}{4 c} + b d h + b e g - \frac {5 b \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{6 c} + c d g\right )}{2 c} - \frac {b \left (a d h + a e g - \frac {2 a \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{3 c} + b d g - \frac {3 b \left (a e h + a f g - \frac {3 a \left (\frac {b f h}{10} + c e h + c f g\right )}{4 c} + b d h + b e g - \frac {5 b \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{6 c} + c d g\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {f h \left (a + b x\right )^{\frac {9}{2}}}{9 b^{3}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 3 a f h + b e h + b f g\right )}{7 b^{3}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} f h - 2 a b e h - 2 a b f g + b^{2} d h + b^{2} e g\right )}{5 b^{3}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- a^{3} f h + a^{2} b e h + a^{2} b f g - a b^{2} d h - a b^{2} e g + b^{3} d g\right )}{3 b^{3}}\right )}{b} & \text {for}\: b \neq 0 \\\sqrt {a} \left (d g x + \frac {f h x^{4}}{4} + \frac {x^{3} \left (e h + f g\right )}{3} + \frac {x^{2} \left (d h + e g\right )}{2}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((h*x+g)*(f*x**2+e*x+d)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Piecewise((sqrt(a + b*x + c*x**2)*(f*h*x**4/5 + x**3*(b*f*h/10 + c*e*h + c*f*g)/(4*c) + x**2*(a*f*h/5 + b*e*h
+ b*f*g - 7*b*(b*f*h/10 + c*e*h + c*f*g)/(8*c) + c*d*h + c*e*g)/(3*c) + x*(a*e*h + a*f*g - 3*a*(b*f*h/10 + c*e
*h + c*f*g)/(4*c) + b*d*h + b*e*g - 5*b*(a*f*h/5 + b*e*h + b*f*g - 7*b*(b*f*h/10 + c*e*h + c*f*g)/(8*c) + c*d*
h + c*e*g)/(6*c) + c*d*g)/(2*c) + (a*d*h + a*e*g - 2*a*(a*f*h/5 + b*e*h + b*f*g - 7*b*(b*f*h/10 + c*e*h + c*f*
g)/(8*c) + c*d*h + c*e*g)/(3*c) + b*d*g - 3*b*(a*e*h + a*f*g - 3*a*(b*f*h/10 + c*e*h + c*f*g)/(4*c) + b*d*h +
b*e*g - 5*b*(a*f*h/5 + b*e*h + b*f*g - 7*b*(b*f*h/10 + c*e*h + c*f*g)/(8*c) + c*d*h + c*e*g)/(6*c) + c*d*g)/(4
*c))/c) + (a*d*g - a*(a*e*h + a*f*g - 3*a*(b*f*h/10 + c*e*h + c*f*g)/(4*c) + b*d*h + b*e*g - 5*b*(a*f*h/5 + b*
e*h + b*f*g - 7*b*(b*f*h/10 + c*e*h + c*f*g)/(8*c) + c*d*h + c*e*g)/(6*c) + c*d*g)/(2*c) - b*(a*d*h + a*e*g -
2*a*(a*f*h/5 + b*e*h + b*f*g - 7*b*(b*f*h/10 + c*e*h + c*f*g)/(8*c) + c*d*h + c*e*g)/(3*c) + b*d*g - 3*b*(a*e*
h + a*f*g - 3*a*(b*f*h/10 + c*e*h + c*f*g)/(4*c) + b*d*h + b*e*g - 5*b*(a*f*h/5 + b*e*h + b*f*g - 7*b*(b*f*h/1
0 + c*e*h + c*f*g)/(8*c) + c*d*h + c*e*g)/(6*c) + c*d*g)/(4*c))/(2*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(a + b
*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c), 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2
), True)), Ne(c, 0)), (2*(f*h*(a + b*x)**(9/2)/(9*b**3) + (a + b*x)**(7/2)*(-3*a*f*h + b*e*h + b*f*g)/(7*b**3)
+ (a + b*x)**(5/2)*(3*a**2*f*h - 2*a*b*e*h - 2*a*b*f*g + b**2*d*h + b**2*e*g)/(5*b**3) + (a + b*x)**(3/2)*(-a
**3*f*h + a**2*b*e*h + a**2*b*f*g - a*b**2*d*h - a*b**2*e*g + b**3*d*g)/(3*b**3))/b, Ne(b, 0)), (sqrt(a)*(d*g*
x + f*h*x**4/4 + x**3*(e*h + f*g)/3 + x**2*(d*h + e*g)/2), True))

Maxima [F(-2)]

Exception generated. \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((h*x+g)*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.48 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, f h x + \frac {10 \, c^{4} f g + 10 \, c^{4} e h + b c^{3} f h}{c^{4}}\right )} x + \frac {80 \, c^{4} e g + 10 \, b c^{3} f g + 80 \, c^{4} d h + 10 \, b c^{3} e h - 7 \, b^{2} c^{2} f h + 16 \, a c^{3} f h}{c^{4}}\right )} x + \frac {480 \, c^{4} d g + 80 \, b c^{3} e g - 50 \, b^{2} c^{2} f g + 120 \, a c^{3} f g + 80 \, b c^{3} d h - 50 \, b^{2} c^{2} e h + 120 \, a c^{3} e h + 35 \, b^{3} c f h - 116 \, a b c^{2} f h}{c^{4}}\right )} x + \frac {480 \, b c^{3} d g - 240 \, b^{2} c^{2} e g + 640 \, a c^{3} e g + 150 \, b^{3} c f g - 520 \, a b c^{2} f g - 240 \, b^{2} c^{2} d h + 640 \, a c^{3} d h + 150 \, b^{3} c e h - 520 \, a b c^{2} e h - 105 \, b^{4} f h + 460 \, a b^{2} c f h - 256 \, a^{2} c^{2} f h}{c^{4}}\right )} + \frac {{\left (32 \, b^{2} c^{3} d g - 128 \, a c^{4} d g - 16 \, b^{3} c^{2} e g + 64 \, a b c^{3} e g + 10 \, b^{4} c f g - 48 \, a b^{2} c^{2} f g + 32 \, a^{2} c^{3} f g - 16 \, b^{3} c^{2} d h + 64 \, a b c^{3} d h + 10 \, b^{4} c e h - 48 \, a b^{2} c^{2} e h + 32 \, a^{2} c^{3} e h - 7 \, b^{5} f h + 40 \, a b^{3} c f h - 48 \, a^{2} b c^{2} f h\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {9}{2}}} \]

[In]

integrate((h*x+g)*(f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*f*h*x + (10*c^4*f*g + 10*c^4*e*h + b*c^3*f*h)/c^4)*x + (80*c^4*e*g +
10*b*c^3*f*g + 80*c^4*d*h + 10*b*c^3*e*h - 7*b^2*c^2*f*h + 16*a*c^3*f*h)/c^4)*x + (480*c^4*d*g + 80*b*c^3*e*g
- 50*b^2*c^2*f*g + 120*a*c^3*f*g + 80*b*c^3*d*h - 50*b^2*c^2*e*h + 120*a*c^3*e*h + 35*b^3*c*f*h - 116*a*b*c^2*
f*h)/c^4)*x + (480*b*c^3*d*g - 240*b^2*c^2*e*g + 640*a*c^3*e*g + 150*b^3*c*f*g - 520*a*b*c^2*f*g - 240*b^2*c^2
*d*h + 640*a*c^3*d*h + 150*b^3*c*e*h - 520*a*b*c^2*e*h - 105*b^4*f*h + 460*a*b^2*c*f*h - 256*a^2*c^2*f*h)/c^4)
+ 1/256*(32*b^2*c^3*d*g - 128*a*c^4*d*g - 16*b^3*c^2*e*g + 64*a*b*c^3*e*g + 10*b^4*c*f*g - 48*a*b^2*c^2*f*g +
32*a^2*c^3*f*g - 16*b^3*c^2*d*h + 64*a*b*c^3*d*h + 10*b^4*c*e*h - 48*a*b^2*c^2*e*h + 32*a^2*c^3*e*h - 7*b^5*f
*h + 40*a*b^3*c*f*h - 48*a^2*b*c^2*f*h)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(9/2)

Mupad [B] (verification not implemented)

Time = 15.10 (sec) , antiderivative size = 877, normalized size of antiderivative = 2.72 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=d\,g\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}-\frac {2\,a\,f\,h\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}-\frac {5\,b\,e\,h\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {5\,b\,f\,g\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}+\frac {d\,h\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}+\frac {e\,g\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}+\frac {e\,h\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {f\,g\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {7\,b\,f\,h\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{10\,c}+\frac {f\,h\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}-\frac {a\,e\,h\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}-\frac {a\,f\,g\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}+\frac {d\,g\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}+\frac {d\,h\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {e\,g\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}} \]

[In]

int((g + h*x)*(a + b*x + c*x^2)^(1/2)*(d + e*x + f*x^2),x)

[Out]

d*g*(x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) - (2*a*f*h*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*
(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(5*c)
- (5*b*e*h*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a +
c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (5*b*f*g*((log((b + 2*c*x)/c^(1/2) + 2*(
a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)
^(1/2))/(24*c^2)))/(8*c) + (d*h*(8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2) + (e*g*(
8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2) + (e*h*x*(a + b*x + c*x^2)^(3/2))/(4*c) +
(f*g*x*(a + b*x + c*x^2)^(3/2))/(4*c) + (7*b*f*h*((5*b*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))
*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c
) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2)
+ (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(10*c) + (f*h*x^2*(a + b*x + c*x^2)^(3/2))/(5
*c) - (a*e*h*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a
*c - b^2/4))/(2*c^(3/2))))/(4*c) - (a*f*g*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2)
+ (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c) + (d*g*log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^
2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2)) + (d*h*log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*
c))/(16*c^(5/2)) + (e*g*log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2))