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

Optimal. Leaf size=1015 \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+b x+a} (d+e x)^3}{9 e}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^4+g \left (3 a g \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e g f+21 d^2 g^2\right )\right ) c^3+3 g^2 \left (-\left (e^2 f^2-5 d e g f-7 d^2 g^2\right ) b^2+a e g (5 e f+29 d g) b+7 a^2 e^2 g^2\right ) c^2-4 b^2 e g^3 (b e f+6 b d g+9 a e g) c+8 b^4 e^2 g^4\right ) \sqrt{f+g x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c f^2-b g f+a g^2\right ) \left (-2 f \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^3-3 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^2 g^3\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{c x^2+b x+a}}{63 c g^3}-\frac{4 \left (\left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^2+e g (4 b e f-9 b d g-7 a e g) c+3 b^2 e^2 g^2\right ) (f+g x)^{3/2} \sqrt{c x^2+b x+a}}{315 c^2 g^3}+\frac{2 \left (\left (19 e^3 f^3-57 d e^2 g f^2+63 d^2 e g^2 f-35 d^3 g^3\right ) c^3-3 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e^2 g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^3 g^3\right ) \sqrt{f+g x} \sqrt{c x^2+b x+a}}{315 c^3 e g^3} \]

[Out]

(2*(8*b^3*e^3*g^3 + 3*b*c*e^2*g^2*(b*e*f - 8*b*d*g - 9*a*e*g) + c^3*(19*e^3*f^3 - 57*d*e^2*f^2*g + 63*d^2*e*f*
g^2 - 35*d^3*g^3) - 3*c^2*e*g^2*(2*a*e*(e*f - 10*d*g) + b*d*(2*e*f - 7*d*g)))*Sqrt[f + g*x]*Sqrt[a + b*x + c*x
^2])/(315*c^3*e*g^3) + (2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(9*e) - (4*(3*b^2*e^2*g^2 + c*e*g*(
4*b*e*f - 9*b*d*g - 7*a*e*g) + c^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2
])/(315*c^2*g^3) + (2*e*(c*e*f - 3*c*d*g + b*e*g)*(f + g*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(63*c*g^3) - (2*Sqrt[
2]*Sqrt[b^2 - 4*a*c]*(8*b^4*e^2*g^4 - 4*b^2*c*e*g^3*(b*e*f + 6*b*d*g + 9*a*e*g) + c^4*f^2*(8*e^2*f^2 - 24*d*e*
f*g + 21*d^2*g^2) + 3*c^2*g^2*(7*a^2*e^2*g^2 + a*b*e*g*(5*e*f + 29*d*g) - b^2*(e^2*f^2 - 5*d*e*f*g - 7*d^2*g^2
)) + c^3*g*(3*a*g*(3*e^2*f^2 - 16*d*e*f*g - 21*d^2*g^2) - b*f*(4*e^2*f^2 - 15*d*e*f*g + 21*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)])/(315*c^4*g^4*Sqrt[(c*(f
+ g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*f^2 - b
*f*g + a*g^2)*(8*b^3*e^2*g^3 + 3*b*c*e*g^2*(b*e*f - 8*b*d*g - 9*a*e*g) - 2*c^3*f*(8*e^2*f^2 - 24*d*e*f*g + 21*
d^2*g^2) - 3*c^2*g^2*(2*a*e*(e*f - 10*d*g) + b*d*(2*e*f - 7*d*g)))*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)])/(315*c^4*g^4*Sqr
t[f + g*x]*Sqrt[a + b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 3.77533, antiderivative size = 1015, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {918, 1653, 843, 718, 424, 419} \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+b x+a} (d+e x)^3}{9 e}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^4+g \left (3 a g \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e g f+21 d^2 g^2\right )\right ) c^3+3 g^2 \left (-\left (e^2 f^2-5 d e g f-7 d^2 g^2\right ) b^2+a e g (5 e f+29 d g) b+7 a^2 e^2 g^2\right ) c^2-4 b^2 e g^3 (b e f+6 b d g+9 a e g) c+8 b^4 e^2 g^4\right ) \sqrt{f+g x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c f^2-b g f+a g^2\right ) \left (-2 f \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^3-3 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^2 g^3\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{c x^2+b x+a}}{63 c g^3}-\frac{4 \left (\left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^2+e g (4 b e f-9 b d g-7 a e g) c+3 b^2 e^2 g^2\right ) (f+g x)^{3/2} \sqrt{c x^2+b x+a}}{315 c^2 g^3}+\frac{2 \left (\left (19 e^3 f^3-57 d e^2 g f^2+63 d^2 e g^2 f-35 d^3 g^3\right ) c^3-3 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e^2 g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^3 g^3\right ) \sqrt{f+g x} \sqrt{c x^2+b x+a}}{315 c^3 e g^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(8*b^3*e^3*g^3 + 3*b*c*e^2*g^2*(b*e*f - 8*b*d*g - 9*a*e*g) + c^3*(19*e^3*f^3 - 57*d*e^2*f^2*g + 63*d^2*e*f*
g^2 - 35*d^3*g^3) - 3*c^2*e*g^2*(2*a*e*(e*f - 10*d*g) + b*d*(2*e*f - 7*d*g)))*Sqrt[f + g*x]*Sqrt[a + b*x + c*x
^2])/(315*c^3*e*g^3) + (2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(9*e) - (4*(3*b^2*e^2*g^2 + c*e*g*(
4*b*e*f - 9*b*d*g - 7*a*e*g) + c^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2
])/(315*c^2*g^3) + (2*e*(c*e*f - 3*c*d*g + b*e*g)*(f + g*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(63*c*g^3) - (2*Sqrt[
2]*Sqrt[b^2 - 4*a*c]*(8*b^4*e^2*g^4 - 4*b^2*c*e*g^3*(b*e*f + 6*b*d*g + 9*a*e*g) + c^4*f^2*(8*e^2*f^2 - 24*d*e*
f*g + 21*d^2*g^2) + 3*c^2*g^2*(7*a^2*e^2*g^2 + a*b*e*g*(5*e*f + 29*d*g) - b^2*(e^2*f^2 - 5*d*e*f*g - 7*d^2*g^2
)) + c^3*g*(3*a*g*(3*e^2*f^2 - 16*d*e*f*g - 21*d^2*g^2) - b*f*(4*e^2*f^2 - 15*d*e*f*g + 21*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)])/(315*c^4*g^4*Sqrt[(c*(f
+ g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*f^2 - b
*f*g + a*g^2)*(8*b^3*e^2*g^3 + 3*b*c*e*g^2*(b*e*f - 8*b*d*g - 9*a*e*g) - 2*c^3*f*(8*e^2*f^2 - 24*d*e*f*g + 21*
d^2*g^2) - 3*c^2*g^2*(2*a*e*(e*f - 10*d*g) + b*d*(2*e*f - 7*d*g)))*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)])/(315*c^4*g^4*Sqr
t[f + g*x]*Sqrt[a + b*x + c*x^2])

Rule 918

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :>
Simp[(2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(e*(2*m + 5)), x] - Dist[1/(e*(2*m + 5)), Int[(
(d + e*x)^m*Simp[b*d*f - 3*a*e*f + a*d*g + 2*(c*d*f - b*e*f + b*d*g - a*e*g)*x - (c*e*f - 3*c*d*g + b*e*g)*x^2
, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, 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] &&  !LtQ[m, -1]

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 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2} \, dx &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 e}-\frac{\int \frac{(d+e x)^2 \left (b d f-3 a e f+a d g+2 (c d f-b e f+b d g-a e g) x-(c e f-3 c d g+b e g) x^2\right )}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{9 e}\\ &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 e}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^3}-\frac{2 \int \frac{\frac{1}{2} g \left (b^2 e^3 f^3 g+a c g \left (5 e^3 f^3-15 d e^2 f^2 g-21 d^2 e f g^2+7 d^3 g^3\right )+b f \left (5 a e^3 f g^2+c \left (e^3 f^3-3 d e^2 f^2 g+7 d^3 g^3\right )\right )\right )+g \left (b e^3 f g^2 (4 b f+5 a g)+c^2 \left (e^3 f^4-3 d e^2 f^3 g+7 d^3 f g^3\right )+c g \left (a e^2 f g (5 e f-36 d g)+b \left (5 e^3 f^3-12 d e^2 f^2 g+7 d^3 g^3\right )\right )\right ) x+\frac{1}{2} g^2 \left (b e^3 g^2 (13 b f+5 a g)+c^2 \left (11 e^3 f^3-33 d e^2 f^2 g+21 d^2 e f g^2+21 d^3 g^3\right )-c e g \left (4 a e g (4 e f+9 d g)-3 b \left (8 e^2 f^2-20 d e f g+7 d^2 g^2\right )\right )\right ) x^2+e g^3 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) x^3}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{63 c e g^4}\\ &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 e}-\frac{4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{315 c^2 g^3}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^3}-\frac{4 \int \frac{-\frac{1}{4} g^4 \left (6 b^3 e^3 f^2 g^2+3 b^2 e^2 f g \left (6 a e g^2+c f (e f-6 d g)\right )-b c f \left (3 a e^2 g^2 (5 e f+18 d g)-c \left (11 e^3 f^3-33 d e^2 f^2 g+42 d^2 e f g^2-35 d^3 g^3\right )\right )-a c g \left (42 a e^3 f g^2-c \left (23 e^3 f^3-69 d e^2 f^2 g+231 d^2 e f g^2-35 d^3 g^3\right )\right )\right )-\frac{1}{2} g^4 \left (3 b^2 e^3 g^3 (5 b f+3 a g)+c^3 f \left (11 e^3 f^3-33 d e^2 f^2 g+42 d^2 e f g^2-35 d^3 g^3\right )-3 c e^2 g^2 \left (7 a^2 e g^2-b^2 f (2 e f-15 d g)+a b g (16 e f+9 d g)\right )-c^2 g \left (3 a e g \left (5 e^2 f^2-36 d e f g-21 d^2 g^2\right )-b \left (23 e^3 f^3-78 d e^2 f^2 g+105 d^2 e f g^2-35 d^3 g^3\right )\right )\right ) x-\frac{3}{4} g^5 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) x^2}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{315 c^2 e g^7}\\ &=\frac{2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{315 c^3 e g^3}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 e}-\frac{4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{315 c^2 g^3}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^3}-\frac{8 \int \frac{\frac{3}{8} e g^6 \left (8 b^4 e^2 f g^3+b^3 e g^2 \left (8 a e g^2-3 c f (e f+8 d g)\right )-3 b^2 c g \left (2 a e g^2 (7 e f+4 d g)+c f \left (e^2 f^2-4 d e f g-7 d^2 g^2\right )\right )-b c \left (27 a^2 e^2 g^4-3 a c g^2 \left (3 e^2 f^2+36 d e f g+7 d^2 g^2\right )-c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )+4 a c^2 g \left (3 a e g^2 (3 e f+5 d g)-c f \left (e^2 f^2-3 d e f g+42 d^2 g^2\right )\right )\right )+\frac{3}{4} e g^6 \left (8 b^4 e^2 g^4-4 b^2 c e g^3 (b e f+6 b d g+9 a e g)+c^4 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )+3 c^2 g^2 \left (7 a^2 e^2 g^2+a b e g (5 e f+29 d g)-b^2 \left (e^2 f^2-5 d e f g-7 d^2 g^2\right )\right )+c^3 g \left (3 a g \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )\right ) x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{945 c^3 e g^9}\\ &=\frac{2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{315 c^3 e g^3}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 e}-\frac{4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{315 c^2 g^3}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^3}-\frac{\left (\left (c f^2-b f g+a g^2\right ) \left (8 b^3 e^2 g^3+3 b c e g^2 (b e f-8 b d g-9 a e g)-2 c^3 f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 c^2 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{315 c^3 g^4}-\frac{\left (2 \left (8 b^4 e^2 g^4-4 b^2 c e g^3 (b e f+6 b d g+9 a e g)+c^4 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )+3 c^2 g^2 \left (7 a^2 e^2 g^2+a b e g (5 e f+29 d g)-b^2 \left (e^2 f^2-5 d e f g-7 d^2 g^2\right )\right )+c^3 g \left (3 a g \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )\right )\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{315 c^3 g^4}\\ &=\frac{2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{315 c^3 e g^3}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 e}-\frac{4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{315 c^2 g^3}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^3}-\frac{\left (2 \sqrt{2} \sqrt{b^2-4 a c} \left (8 b^4 e^2 g^4-4 b^2 c e g^3 (b e f+6 b d g+9 a e g)+c^4 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )+3 c^2 g^2 \left (7 a^2 e^2 g^2+a b e g (5 e f+29 d g)-b^2 \left (e^2 f^2-5 d e f g-7 d^2 g^2\right )\right )+c^3 g \left (3 a g \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e f g+21 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 )}{315 c^4 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 (2 \sqrt{2} \sqrt{b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (8 b^3 e^2 g^3+3 b c e g^2 (b e f-8 b d g-9 a e g)-2 c^3 f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 c^2 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\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 )}{315 c^4 g^4 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{315 c^3 e g^3}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 e}-\frac{4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{315 c^2 g^3}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^3}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (8 b^4 e^2 g^4-4 b^2 c e g^3 (b e f+6 b d g+9 a e g)+c^4 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )+3 c^2 g^2 \left (7 a^2 e^2 g^2+a b e g (5 e f+29 d g)-b^2 \left (e^2 f^2-5 d e f g-7 d^2 g^2\right )\right )+c^3 g \left (3 a g \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e f g+21 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 )}{315 c^4 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{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (8 b^3 e^2 g^3+3 b c e g^2 (b e f-8 b d g-9 a e g)-2 c^3 f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 c^2 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\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 )}{315 c^4 g^4 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 15.8663, size = 15781, normalized size = 15.55 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.393, size = 20224, normalized size = 19.9 \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*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(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 \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*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(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 ({\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*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right )^{2} \sqrt{f + g x} \sqrt{a + b x + c x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out