\(\int \frac {\sqrt {a+b x+c x^2} (d+e x+f x^2)}{g+h x} \, dx\) [190]

Optimal result

Integrand size = 32, antiderivative size = 321 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{g+h x} \, dx=-\frac {(4 c h (b f g-2 c d h)-(4 c g-b h) (2 c f g-2 c e h+b f h)+2 c h (2 c f g-2 c e h+b f h) x) \sqrt {a+b x+c x^2}}{8 c^2 h^3}+\frac {f \left (a+b x+c x^2\right )^{3/2}}{3 c h}+\frac {\left (4 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (8 c^2 g^2-b^2 h^2-4 c h (b g-a h)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} h^4}+\frac {\sqrt {c g^2-b g h+a h^2} \left (f g^2-e g h+d h^2\right ) \text {arctanh}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{h^4} \]

[Out]

1/3*f*(c*x^2+b*x+a)^(3/2)/c/h+1/16*(4*c*h*(-b*h+2*c*g)*(b*f*g-2*c*d*h)-(b*f*h-2*c*e*h+2*c*f*g)*(8*c^2*g^2-b^2*
h^2-4*c*h*(-a*h+b*g)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(5/2)/h^4+(d*h^2-e*g*h+f*g^2)*arct
anh(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)^(1/2)/h^
4-1/8*(4*c*h*(b*f*g-2*c*d*h)-(-b*h+4*c*g)*(b*f*h-2*c*e*h+2*c*f*g)+2*c*h*(b*f*h-2*c*e*h+2*c*f*g)*x)*(c*x^2+b*x+
a)^(1/2)/c^2/h^3

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1667, 828, 857, 635, 212, 738} \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{g+h x} \, dx=\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c h (2 c g-b h) (b f g-2 c d h)-\left (-4 c h (b g-a h)-b^2 h^2+8 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )}{16 c^{5/2} h^4}+\frac {\sqrt {a h^2-b g h+c g^2} \left (d h^2-e g h+f g^2\right ) \text {arctanh}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right )}{h^4}-\frac {\sqrt {a+b x+c x^2} (4 c h (b f g-2 c d h)+2 c h x (b f h-2 c e h+2 c f g)-(4 c g-b h) (b f h-2 c e h+2 c f g))}{8 c^2 h^3}+\frac {f \left (a+b x+c x^2\right )^{3/2}}{3 c h} \]

[In]

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

[Out]

-1/8*((4*c*h*(b*f*g - 2*c*d*h) - (4*c*g - b*h)*(2*c*f*g - 2*c*e*h + b*f*h) + 2*c*h*(2*c*f*g - 2*c*e*h + b*f*h)
*x)*Sqrt[a + b*x + c*x^2])/(c^2*h^3) + (f*(a + b*x + c*x^2)^(3/2))/(3*c*h) + ((4*c*h*(2*c*g - b*h)*(b*f*g - 2*
c*d*h) - (2*c*f*g - 2*c*e*h + b*f*h)*(8*c^2*g^2 - b^2*h^2 - 4*c*h*(b*g - a*h)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]
*Sqrt[a + b*x + c*x^2])])/(16*c^(5/2)*h^4) + (Sqrt[c*g^2 - b*g*h + a*h^2]*(f*g^2 - e*g*h + d*h^2)*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])])/h^4

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 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 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-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] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 857

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

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 \left (a+b x+c x^2\right )^{3/2}}{3 c h}+\frac {\int \frac {\left (-\frac {3}{2} h (b f g-2 c d h)-\frac {3}{2} h (2 c f g-2 c e h+b f h) x\right ) \sqrt {a+b x+c x^2}}{g+h x} \, dx}{3 c h^2} \\ & = -\frac {(4 c h (b f g-2 c d h)-(4 c g-b h) (2 c f g-2 c e h+b f h)+2 c h (2 c f g-2 c e h+b f h) x) \sqrt {a+b x+c x^2}}{8 c^2 h^3}+\frac {f \left (a+b x+c x^2\right )^{3/2}}{3 c h}-\frac {\int \frac {-\frac {3}{4} h \left (4 c h (b g-2 a h) (b f g-2 c d h)-g \left (4 b c g-b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-\frac {3}{4} h \left (4 c h (2 c g-b h) (b f g-2 c d h)-2 (2 c f g-2 c e h+b f h) \left (4 c^2 g^2-\frac {b^2 h^2}{2}-2 c h (b g-a h)\right )\right ) x}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{12 c^2 h^4} \\ & = -\frac {(4 c h (b f g-2 c d h)-(4 c g-b h) (2 c f g-2 c e h+b f h)+2 c h (2 c f g-2 c e h+b f h) x) \sqrt {a+b x+c x^2}}{8 c^2 h^3}+\frac {f \left (a+b x+c x^2\right )^{3/2}}{3 c h}+\frac {\left (\left (c g^2-b g h+a h^2\right ) \left (f g^2-h (e g-d h)\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{h^4}+\frac {\left (4 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (8 c^2 g^2-b^2 h^2-4 c h (b g-a h)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^2 h^4} \\ & = -\frac {(4 c h (b f g-2 c d h)-(4 c g-b h) (2 c f g-2 c e h+b f h)+2 c h (2 c f g-2 c e h+b f h) x) \sqrt {a+b x+c x^2}}{8 c^2 h^3}+\frac {f \left (a+b x+c x^2\right )^{3/2}}{3 c h}-\frac {\left (2 \left (c g^2-b g h+a h^2\right ) \left (f g^2-h (e g-d h)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c g^2-4 b g h+4 a h^2-x^2} \, dx,x,\frac {-b g+2 a h-(2 c g-b h) x}{\sqrt {a+b x+c x^2}}\right )}{h^4}+\frac {\left (4 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (8 c^2 g^2-b^2 h^2-4 c h (b g-a 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 )}{8 c^2 h^4} \\ & = -\frac {(4 c h (b f g-2 c d h)-(4 c g-b h) (2 c f g-2 c e h+b f h)+2 c h (2 c f g-2 c e h+b f h) x) \sqrt {a+b x+c x^2}}{8 c^2 h^3}+\frac {f \left (a+b x+c x^2\right )^{3/2}}{3 c h}+\frac {\left (4 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (8 c^2 g^2-b^2 h^2-4 c h (b g-a h)\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} h^4}+\frac {\sqrt {c g^2-b g h+a h^2} \left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\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 )}{h^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.21 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{g+h x} \, dx=\frac {\frac {h \sqrt {a+x (b+c x)} \left (-3 b^2 f h^2+2 c h (4 a f h+b (-3 f g+3 e h+f h x))+4 c^2 \left (3 h (-2 e g+2 d h+e h x)+f \left (6 g^2-3 g h x+2 h^2 x^2\right )\right )\right )}{c^2}+48 \sqrt {-c g^2+h (b g-a h)} \left (f g^2+h (-e g+d h)\right ) \arctan \left (\frac {\sqrt {-c g^2+h (b g-a h)} x}{\sqrt {a} (g+h x)-g \sqrt {a+x (b+c x)}}\right )-\frac {3 \left (-b^3 f h^3+2 b c h^2 (-b f g+b e h+2 a f h)+16 c^3 \left (f g^3+g h (-e g+d h)\right )-8 c^2 h \left (b f g^2+b h (-e g+d h)+a h (-f g+e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{c^{5/2}}}{24 h^4} \]

[In]

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

[Out]

((h*Sqrt[a + x*(b + c*x)]*(-3*b^2*f*h^2 + 2*c*h*(4*a*f*h + b*(-3*f*g + 3*e*h + f*h*x)) + 4*c^2*(3*h*(-2*e*g +
2*d*h + e*h*x) + f*(6*g^2 - 3*g*h*x + 2*h^2*x^2))))/c^2 + 48*Sqrt[-(c*g^2) + h*(b*g - a*h)]*(f*g^2 + h*(-(e*g)
+ d*h))*ArcTan[(Sqrt[-(c*g^2) + h*(b*g - a*h)]*x)/(Sqrt[a]*(g + h*x) - g*Sqrt[a + x*(b + c*x)])] - (3*(-(b^3*
f*h^3) + 2*b*c*h^2*(-(b*f*g) + b*e*h + 2*a*f*h) + 16*c^3*(f*g^3 + g*h*(-(e*g) + d*h)) - 8*c^2*h*(b*f*g^2 + b*h
*(-(e*g) + d*h) + a*h*(-(f*g) + e*h)))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/c^(5/2))/(24*h
^4)

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.60

method	result	size
risch	\(\frac {\left (8 f \,h^{2} c^{2} x^{2}+2 b c f \,h^{2} x +12 c^{2} e \,h^{2} x -12 c^{2} f g h x +8 a c f \,h^{2}-3 b^{2} f \,h^{2}+6 b c e \,h^{2}-6 b c f g h +24 c^{2} d \,h^{2}-24 c^{2} e g h +24 c^{2} f \,g^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{24 c^{2} h^{3}}-\frac {\frac {16 \left (a d \,h^{4}-a e g \,h^{3}+a f \,g^{2} h^{2}-b d g \,h^{3}+b e \,g^{2} h^{2}-b f \,g^{3} h +c d \,g^{2} h^{2}-g^{3} c e h +g^{4} c f \right ) c^{2} \ln \left (\frac {\frac {2 a \,h^{2}-2 b g h +2 c \,g^{2}}{h^{2}}+\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c +\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{h^{2} \sqrt {\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}+\frac {\left (4 a b c f \,h^{3}-8 a \,c^{2} e \,h^{3}+8 a \,c^{2} f g \,h^{2}-b^{3} f \,h^{3}+2 b^{2} c e \,h^{3}-2 b^{2} c f g \,h^{2}-8 b \,c^{2} d \,h^{3}+8 b \,c^{2} e g \,h^{2}-8 b \,c^{2} f \,g^{2} h +16 c^{3} d g \,h^{2}-16 c^{3} e \,g^{2} h +16 c^{3} f \,g^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{h \sqrt {c}}}{16 c^{2} h^{3}}\)	\(515\)
default	\(\frac {e h \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )+f h \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )-f g \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{h^{2}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (\sqrt {\left (x +\frac {g}{h}\right )^{2} c +\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}+\frac {\left (b h -2 c g \right ) \ln \left (\frac {\frac {b h -2 c g}{2 h}+c \left (x +\frac {g}{h}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {g}{h}\right )^{2} c +\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}\right )}{2 h \sqrt {c}}-\frac {\left (a \,h^{2}-b g h +c \,g^{2}\right ) \ln \left (\frac {\frac {2 a \,h^{2}-2 b g h +2 c \,g^{2}}{h^{2}}+\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c +\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{h^{2} \sqrt {\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}\right )}{h^{3}}\)	\(577\)

[In]

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

[Out]

1/24*(8*c^2*f*h^2*x^2+2*b*c*f*h^2*x+12*c^2*e*h^2*x-12*c^2*f*g*h*x+8*a*c*f*h^2-3*b^2*f*h^2+6*b*c*e*h^2-6*b*c*f*
g*h+24*c^2*d*h^2-24*c^2*e*g*h+24*c^2*f*g^2)/c^2*(c*x^2+b*x+a)^(1/2)/h^3-1/16/c^2/h^3*(16*(a*d*h^4-a*e*g*h^3+a*
f*g^2*h^2-b*d*g*h^3+b*e*g^2*h^2-b*f*g^3*h+c*d*g^2*h^2-c*e*g^3*h+c*f*g^4)*c^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))+(4*a*b*c*f*h^3-8*a*c^2*e*h^3+8*a*c^2*f*g*h^2-b^
3*f*h^3+2*b^2*c*e*h^3-2*b^2*c*f*g*h^2-8*b*c^2*d*h^3+8*b*c^2*e*g*h^2-8*b*c^2*f*g^2*h+16*c^3*d*g*h^2-16*c^3*e*g^
2*h+16*c^3*f*g^3)/h*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2))

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{g+h x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2)/(h*x+g),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(b*h-2*c*g>0)', see `assume?` f
or more deta

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{g+h x} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}\,\left (f\,x^2+e\,x+d\right )}{g+h\,x} \,d x \]

[In]

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

[Out]

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