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

Optimal. Leaf size=755 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2+c^2 \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 g \sqrt{b^2-4 a c}}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}\right )}{105 c^3 g^4 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{f+g x} \sqrt{a+b x+c x^2} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2+c^2 \left (-\left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )\right )}{105 c^2 g^3}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (35 d^2 g^2-42 d e f g+16 e^2 f^2\right )\right )+b c e g^2 (-29 a e g-28 b d g+9 b e f)+8 b^3 e^2 g^3-2 c^3 f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 e (f+g x)^{3/2} \sqrt{a+b x+c x^2} (-b e g-4 c d g+6 c e f)}{35 c g^3}+\frac{2 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 g} \]

[Out]

(-4*(2*b^2*e^2*g^2 + c*e*g*(4*b*e*f - 7*b*d*g - 5*a*e*g) - c^2*(21*e^2*f^2 - 34*d*e*f*g + 10*d^2*g^2))*Sqrt[f
+ g*x]*Sqrt[a + b*x + c*x^2])/(105*c^2*g^3) + (2*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(7*g) - (2*e
*(6*c*e*f - 4*c*d*g - b*e*g)*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(35*c*g^3) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8
*b^3*e^2*g^3 + b*c*e*g^2*(9*b*e*f - 28*b*d*g - 29*a*e*g) - 2*c^3*f*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2) - c^
2*g*(2*a*e*g*(13*e*f - 42*d*g) - b*(16*e^2*f^2 - 42*d*e*f*g + 35*d^2*g^2)))*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x +
c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-
2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(105*c^3*g^4*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqr
t[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*f^2 - b*f*g + a*g^2)*(2*b^2*e^2*g
^2 + c*e*g*(4*b*e*f - 7*b*d*g - 5*a*e*g) + c^2*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2))*Sqrt[(c*(f + g*x))/(2*c
*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[
b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g
)])/(105*c^3*g^4*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.93068, antiderivative size = 755, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {920, 1653, 843, 718, 424, 419} \[ -\frac{4 \sqrt{f+g x} \sqrt{a+b x+c x^2} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2+c^2 \left (-\left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )\right )}{105 c^2 g^3}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2+c^2 \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (35 d^2 g^2-42 d e f g+16 e^2 f^2\right )\right )+b c e g^2 (-29 a e g-28 b d g+9 b e f)+8 b^3 e^2 g^3-2 c^3 f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 e (f+g x)^{3/2} \sqrt{a+b x+c x^2} (-b e g-4 c d g+6 c e f)}{35 c g^3}+\frac{2 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(2*b^2*e^2*g^2 + c*e*g*(4*b*e*f - 7*b*d*g - 5*a*e*g) - c^2*(21*e^2*f^2 - 34*d*e*f*g + 10*d^2*g^2))*Sqrt[f
+ g*x]*Sqrt[a + b*x + c*x^2])/(105*c^2*g^3) + (2*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(7*g) - (2*e
*(6*c*e*f - 4*c*d*g - b*e*g)*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(35*c*g^3) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8
*b^3*e^2*g^3 + b*c*e*g^2*(9*b*e*f - 28*b*d*g - 29*a*e*g) - 2*c^3*f*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2) - c^
2*g*(2*a*e*g*(13*e*f - 42*d*g) - b*(16*e^2*f^2 - 42*d*e*f*g + 35*d^2*g^2)))*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x +
c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-
2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(105*c^3*g^4*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqr
t[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*f^2 - b*f*g + a*g^2)*(2*b^2*e^2*g
^2 + c*e*g*(4*b*e*f - 7*b*d*g - 5*a*e*g) + c^2*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2))*Sqrt[(c*(f + g*x))/(2*c
*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[
b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g
)])/(105*c^3*g^4*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

Rule 920

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

Rule 1653

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]))

Rule 843

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 718

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

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{(d+e x)^2 \sqrt{a+b x+c x^2}}{\sqrt{f+g x}} \, dx &=\frac{2 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 g}-\frac{\int \frac{(d+e x) \left (b d f+4 a e f-6 a d g+(2 c d f+5 b e f-5 b d g-2 a e g) x+(6 c e f-4 c d g-b e g) x^2\right )}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{7 g}\\ &=\frac{2 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 g}-\frac{2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c g^3}-\frac{2 \int \frac{\frac{1}{2} g \left (b^2 e^2 f^2 g-2 a c g \left (9 e^2 f^2-16 d e f g+15 d^2 g^2\right )+b f \left (3 a e^2 g^2-c \left (6 e^2 f^2-4 d e f g-5 d^2 g^2\right )\right )\right )+\frac{1}{2} g \left (b e^2 g^2 (5 b f+3 a g)-2 c^2 f \left (6 e^2 f^2-4 d e f g-5 d^2 g^2\right )+c g \left (2 a e g (e f-14 d g)-b \left (28 e^2 f^2-50 d e f g+25 d^2 g^2\right )\right )\right ) x+g^2 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) x^2}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{35 c g^4}\\ &=-\frac{4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^2 g^3}+\frac{2 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 g}-\frac{2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c g^3}-\frac{4 \int \frac{-\frac{1}{4} g^3 \left (4 b^3 e^2 f g^2+b^2 e g \left (4 a e g^2+c f (5 e f-14 d g)\right )-b c \left (a e g^2 (11 e f+14 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right )-2 a c g \left (5 a e^2 g^2-c \left (6 e^2 f^2-14 d e f g+35 d^2 g^2\right )\right )\right )-\frac{1}{4} g^3 \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{105 c^2 g^6}\\ &=-\frac{4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^2 g^3}+\frac{2 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 g}-\frac{2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c g^3}+\frac{\left (2 \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{105 c^2 g^4}+\frac{\left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{105 c^2 g^4}\\ &=-\frac{4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^2 g^3}+\frac{2 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 g}-\frac{2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c g^3}+\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{105 c^3 g^4 \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \sqrt{a+b x+c x^2}}+\frac{\left (4 \sqrt{2} \sqrt{b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{105 c^3 g^4 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=-\frac{4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^2 g^3}+\frac{2 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 g}-\frac{2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c g^3}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 14.3634, size = 10030, normalized size = 13.28 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.426, size = 12923, normalized size = 17.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{2}}{\sqrt{g x + f}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{c x^{2} + b x + a}}{\sqrt{g x + f}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out