\(\int \frac {(f+g x)^2 \sqrt {a+b x+c x^2}}{d+e x} \, dx\) [855]

Optimal result

Integrand size = 29, antiderivative size = 325 \[ \int \frac {(f+g x)^2 \sqrt {a+b x+c x^2}}{d+e x} \, dx=-\frac {\left (b^2 e^2 g^2-8 c^2 (e f-d g)^2-2 b c e g (2 e f-d g)-2 c e g (4 c e f-2 c d g-b e g) x\right ) \sqrt {a+b x+c x^2}}{8 c^2 e^3}+\frac {g^2 \left (a+b x+c x^2\right )^{3/2}}{3 c e}+\frac {\left (\left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) g (4 c e f-2 c d g-b e g)-4 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} e^4}+\frac {\sqrt {c d^2-b d e+a e^2} (e f-d g)^2 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1667, 828, 857, 635, 212, 738} \[ \int \frac {(f+g x)^2 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (g \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right ) (-b e g-2 c d g+4 c e f)-4 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )}{16 c^{5/2} e^4}+\frac {(e f-d g)^2 \sqrt {a e^2-b d e+c d^2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^4}-\frac {\sqrt {a+b x+c x^2} \left (b^2 e^2 g^2-2 c e g x (-b e g-2 c d g+4 c e f)-2 b c e g (2 e f-d g)-8 c^2 (e f-d g)^2\right )}{8 c^2 e^3}+\frac {g^2 \left (a+b x+c x^2\right )^{3/2}}{3 c e} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 1.42 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.97 \[ \int \frac {(f+g x)^2 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {\frac {2 e \sqrt {a+x (b+c x)} \left (-3 b^2 e^2 g^2+2 c e g (4 a e g+b (6 e f-3 d g+e g x))+4 c^2 \left (6 d^2 g^2-3 d e g (4 f+g x)+2 e^2 \left (3 f^2+3 f g x+g^2 x^2\right )\right )\right )}{c^2}+96 \sqrt {-c d^2+b d e-a e^2} (e f-d g)^2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )+\frac {3 \left (-b^3 e^3 g^2+16 c^3 d (e f-d g)^2+2 b c e^2 g (2 b e f-b d g+2 a e g)-8 c^2 e \left (b (e f-d g)^2+a e g (2 e f-d g)\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{c^{5/2}}}{48 e^4} \]

[In]

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

[Out]

((2*e*Sqrt[a + x*(b + c*x)]*(-3*b^2*e^2*g^2 + 2*c*e*g*(4*a*e*g + b*(6*e*f - 3*d*g + e*g*x)) + 4*c^2*(6*d^2*g^2
- 3*d*e*g*(4*f + g*x) + 2*e^2*(3*f^2 + 3*f*g*x + g^2*x^2))))/c^2 + 96*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(e*f - d
*g)^2*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]] + (3*(-(b^3*e^3*g^2
) + 16*c^3*d*(e*f - d*g)^2 + 2*b*c*e^2*g*(2*b*e*f - b*d*g + 2*a*e*g) - 8*c^2*e*(b*(e*f - d*g)^2 + a*e*g*(2*e*f
- d*g)))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(5/2))/(48*e^4)

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.75

method	result	size
risch	\(\frac {\left (8 c^{2} e^{2} g^{2} x^{2}+2 b c \,e^{2} g^{2} x -12 c^{2} d e \,g^{2} x +24 c^{2} e^{2} f g x +8 a c \,e^{2} g^{2}-3 b^{2} e^{2} g^{2}-6 b c d e \,g^{2}+12 b c \,e^{2} f g +24 c^{2} d^{2} g^{2}-48 c^{2} d e f g +24 c^{2} e^{2} f^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{24 c^{2} e^{3}}-\frac {\frac {16 \left (a \,d^{2} e^{2} g^{2}-2 a d \,e^{3} f g +f^{2} a \,e^{4}-b \,d^{3} e \,g^{2}+2 b \,d^{2} e^{2} f g -b d \,e^{3} f^{2}+d^{4} g^{2} c -2 c \,d^{3} e f g +c \,d^{2} e^{2} f^{2}\right ) c^{2} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}+\frac {\left (4 a b c \,e^{3} g^{2}+8 a \,c^{2} d \,e^{2} g^{2}-16 a \,c^{2} e^{3} f g -b^{3} e^{3} g^{2}-2 b^{2} c d \,e^{2} g^{2}+4 b^{2} c \,e^{3} f g -8 b \,c^{2} d^{2} e \,g^{2}+16 b \,c^{2} d \,e^{2} f g -8 b \,c^{2} e^{3} f^{2}+16 c^{3} d^{3} g^{2}-32 c^{3} d^{2} e f g +16 c^{3} d \,e^{2} f^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}}{16 c^{2} e^{3}}\)	\(570\)
default	\(-\frac {g \left (d g \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )-2 e f \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )-e g \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\right )}{e^{2}}+\frac {\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \left (\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\)	\(585\)

[In]

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

[Out]

1/24*(8*c^2*e^2*g^2*x^2+2*b*c*e^2*g^2*x-12*c^2*d*e*g^2*x+24*c^2*e^2*f*g*x+8*a*c*e^2*g^2-3*b^2*e^2*g^2-6*b*c*d*
e*g^2+12*b*c*e^2*f*g+24*c^2*d^2*g^2-48*c^2*d*e*f*g+24*c^2*e^2*f^2)*(c*x^2+b*x+a)^(1/2)/c^2/e^3-1/16/c^2/e^3*(1
6*(a*d^2*e^2*g^2-2*a*d*e^3*f*g+a*e^4*f^2-b*d^3*e*g^2+2*b*d^2*e^2*f*g-b*d*e^3*f^2+c*d^4*g^2-2*c*d^3*e*f*g+c*d^2
*e^2*f^2)*c^2/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^
2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))+(4*a*b*c
*e^3*g^2+8*a*c^2*d*e^2*g^2-16*a*c^2*e^3*f*g-b^3*e^3*g^2-2*b^2*c*d*e^2*g^2+4*b^2*c*e^3*f*g-8*b*c^2*d^2*e*g^2+16
*b*c^2*d*e^2*f*g-8*b*c^2*e^3*f^2+16*c^3*d^3*g^2-32*c^3*d^2*e*f*g+16*c^3*d*e^2*f^2)/e*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. \[ \int \frac {(f+g x)^2 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((g*x+f)^2*(c*x^2+b*x+a)^(1/2)/(e*x+d),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(e>0)', see `assume?` for more
details)Is e

Giac [F(-2)]

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

[In]

integrate((g*x+f)^2*(c*x^2+b*x+a)^(1/2)/(e*x+d),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. \[ \int \frac {(f+g x)^2 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\sqrt {c\,x^2+b\,x+a}}{d+e\,x} \,d x \]

[In]

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

[Out]

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