3.893 \(\int \frac{(d+e x)^3 \sqrt{a+b x+c x^2}}{\sqrt{f+g x}} \, dx\)

Optimal. Leaf size=1098 \[ \text{result too large to display} \]

[Out]

(2*(8*b^3*e^3*g^3 + 3*b*c*e^2*g^2*(5*b*e*f - 12*b*d*g - 9*a*e*g) - c^3*(152*e^3*f^3 - 408*d*e^2*f^2*g + 336*d^
2*e*f*g^2 - 70*d^3*g^3) - 3*c^2*e*g*(6*a*e*g*(2*e*f - 5*d*g) - b*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2)))*Sqrt[
f + g*x]*Sqrt[a + b*x + c*x^2])/(315*c^3*g^4) + (2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(9*g) - (2
*e*(6*b^2*e^2*g^2 + c*e*g*(17*b*e*f - 27*b*d*g - 14*a*e*g) - 2*c^2*(64*e^2*f^2 - 111*d*e*f*g + 42*d^2*g^2))*(f
+ g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(315*c^2*g^4) - (2*e^2*(8*c*e*f - 6*c*d*g - b*e*g)*(f + g*x)^(5/2)*Sqrt[a
+ b*x + c*x^2])/(63*c*g^4) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*b^4*e^3*g^4 + 8*b^2*c*e^2*g^3*(2*b*e*f - 9*b*d*g
- 9*a*e*g) - 2*c^4*f*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3) + 3*c^2*e*g^2*(14*a^2*e^2*
g^2 - a*b*e*g*(19*e*f - 87*d*g) + b^2*(7*e^2*f^2 - 27*d*e*f*g + 42*d^2*g^2)) - c^3*g*(6*a*e*g*(10*e^2*f^2 - 39
*d*e*f*g + 63*d^2*g^2) - b*(40*e^3*f^3 - 144*d*e^2*f^2*g + 189*d^2*e*f*g^2 - 105*d^3*g^3)))*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^5*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^3*g^3 + 3*b*c*e^2*g^2*(5*b*e*f - 12*b*d*g - 9*a*e*g) + 2*c^3*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*
d^2*e*f*g^2 - 105*d^3*g^3) - 3*c^2*e*g*(6*a*e*g*(2*e*f - 5*d*g) - b*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2)))*Sq
rt[(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[A
rcSin[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^5*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 5.76396, antiderivative size = 1098, 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 = {920, 1653, 843, 718, 424, 419} \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+b x+a} (d+e x)^3}{9 g}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (-2 f \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right ) c^4-g \left (6 a e g \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 g f^2+189 d^2 e g^2 f-105 d^3 g^3\right )\right ) c^3+3 e g^2 \left (\left (7 e^2 f^2-27 d e g f+42 d^2 g^2\right ) b^2-a e g (19 e f-87 d g) b+14 a^2 e^2 g^2\right ) c^2+8 b^2 e^2 g^3 (2 b e f-9 b d g-9 a e g) c+16 b^4 e^3 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^5 \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 \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right ) c^3-3 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) c^2+3 b e^2 g^2 (5 b e f-12 b d g-9 a e g) c+8 b^3 e^3 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^5 \sqrt{f+g x} \sqrt{c x^2+b x+a}}-\frac{2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt{c x^2+b x+a}}{63 c g^4}-\frac{2 e \left (-2 \left (64 e^2 f^2-111 d e g f+42 d^2 g^2\right ) c^2+e g (17 b e f-27 b d g-14 a e g) c+6 b^2 e^2 g^2\right ) (f+g x)^{3/2} \sqrt{c x^2+b x+a}}{315 c^2 g^4}+\frac{2 \left (-\left (152 e^3 f^3-408 d e^2 g f^2+336 d^2 e g^2 f-70 d^3 g^3\right ) c^3-3 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) c^2+3 b e^2 g^2 (5 b e f-12 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 g^4} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*Sqrt[a + b*x + c*x^2])/Sqrt[f + g*x],x]

[Out]

(2*(8*b^3*e^3*g^3 + 3*b*c*e^2*g^2*(5*b*e*f - 12*b*d*g - 9*a*e*g) - c^3*(152*e^3*f^3 - 408*d*e^2*f^2*g + 336*d^
2*e*f*g^2 - 70*d^3*g^3) - 3*c^2*e*g*(6*a*e*g*(2*e*f - 5*d*g) - b*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2)))*Sqrt[
f + g*x]*Sqrt[a + b*x + c*x^2])/(315*c^3*g^4) + (2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(9*g) - (2
*e*(6*b^2*e^2*g^2 + c*e*g*(17*b*e*f - 27*b*d*g - 14*a*e*g) - 2*c^2*(64*e^2*f^2 - 111*d*e*f*g + 42*d^2*g^2))*(f
+ g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(315*c^2*g^4) - (2*e^2*(8*c*e*f - 6*c*d*g - b*e*g)*(f + g*x)^(5/2)*Sqrt[a
+ b*x + c*x^2])/(63*c*g^4) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*b^4*e^3*g^4 + 8*b^2*c*e^2*g^3*(2*b*e*f - 9*b*d*g
- 9*a*e*g) - 2*c^4*f*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3) + 3*c^2*e*g^2*(14*a^2*e^2*
g^2 - a*b*e*g*(19*e*f - 87*d*g) + b^2*(7*e^2*f^2 - 27*d*e*f*g + 42*d^2*g^2)) - c^3*g*(6*a*e*g*(10*e^2*f^2 - 39
*d*e*f*g + 63*d^2*g^2) - b*(40*e^3*f^3 - 144*d*e^2*f^2*g + 189*d^2*e*f*g^2 - 105*d^3*g^3)))*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^5*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^3*g^3 + 3*b*c*e^2*g^2*(5*b*e*f - 12*b*d*g - 9*a*e*g) + 2*c^3*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*
d^2*e*f*g^2 - 105*d^3*g^3) - 3*c^2*e*g*(6*a*e*g*(2*e*f - 5*d*g) - b*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2)))*Sq
rt[(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[A
rcSin[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^5*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)^3 \sqrt{a+b x+c x^2}}{\sqrt{f+g x}} \, dx &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 g}-\frac{\int \frac{(d+e x)^2 \left (b d f+6 a e f-8 a d g+(2 c d f+7 b e f-7 b d g-2 a e g) x+(8 c e f-6 c d g-b e g) x^2\right )}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{9 g}\\ &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 g}-\frac{2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^4}-\frac{2 \int \frac{\frac{1}{2} g \left (b^2 e^3 f^3 g-2 a c g \left (20 e^3 f^3-15 d e^2 f^2 g-21 d^2 e f g^2+28 d^3 g^3\right )+b f \left (5 a e^3 f g^2-c \left (8 e^3 f^3-6 d e^2 f^2 g-7 d^3 g^3\right )\right )\right )+\frac{1}{2} g \left (2 b e^3 f g^2 (4 b f+5 a g)-2 c^2 \left (8 e^3 f^4-6 d e^2 f^3 g-7 d^3 f g^3\right )-c g \left (2 a e g \left (40 e^2 f^2-72 d e f g+63 d^2 g^2\right )+b \left (62 e^3 f^3-48 d e^2 f^2 g-63 d^2 e f g^2+49 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 (88 e^3 f^3-66 d e^2 f^2 g-84 d^2 e f g^2+42 d^3 g^3\right )+c e g \left (2 a e g (e f-27 d g)-3 b \left (31 e^2 f^2-61 d e f g+35 d^2 g^2\right )\right )\right ) x^2+\frac{1}{2} e g^3 \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) x^3}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{63 c g^5}\\ &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 g}-\frac{2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{315 c^2 g^4}-\frac{2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^4}-\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 (4 e f-9 d g)\right )-2 a c g \left (21 a e^3 f g^2+c \left (92 e^3 f^3-258 d e^2 f^2 g+231 d^2 e f g^2-140 d^3 g^3\right )\right )+b c f \left (3 a e^2 g^2 (4 e f-27 d g)-c \left (88 e^3 f^3-192 d e^2 f^2 g+84 d^2 e f g^2+35 d^3 g^3\right )\right )\right )-\frac{1}{4} g^4 \left (6 b^2 e^3 g^3 (5 b f+3 a g)-2 c^3 f \left (88 e^3 f^3-192 d e^2 f^2 g+84 d^2 e f g^2+35 d^3 g^3\right )-3 c e^2 g^2 \left (14 a^2 e g^2-b^2 f (19 e f-45 d g)+a b g (23 e f+27 d g)\right )-c^2 g \left (6 a e g \left (2 e^2 f^2+9 d e f g-63 d^2 g^2\right )+b \left (296 e^3 f^3-816 d e^2 f^2 g+735 d^2 e f g^2-245 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 (5 b e f-12 b d g-9 a e g)-2 c^3 \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) x^2}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{315 c^2 g^8}\\ &=\frac{2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{315 c^3 g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 g}-\frac{2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{315 c^2 g^4}-\frac{2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^4}-\frac{8 \int \frac{\frac{3}{8} g^6 \left (8 b^4 e^3 f g^3+b^3 e^2 g^2 \left (8 a e g^2+9 c f (e f-4 d g)\right )-3 b^2 c e g \left (2 a e g^2 (5 e f+6 d g)-c f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )+2 a c^2 g \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-b c \left (27 a^2 e^3 g^4+3 a c e g^2 \left (8 e^2 f^2-33 d e f g-21 d^2 g^2\right )+c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right )+\frac{3}{8} g^6 \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{945 c^3 g^{10}}\\ &=\frac{2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{315 c^3 g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 g}-\frac{2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{315 c^2 g^4}-\frac{2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^4}-\frac{\left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{315 c^3 g^5}-\frac{\left (8 \left (-\frac{3}{8} f g^6 \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right )+\frac{3}{8} g^7 \left (8 b^4 e^3 f g^3+b^3 e^2 g^2 \left (8 a e g^2+9 c f (e f-4 d g)\right )-3 b^2 c e g \left (2 a e g^2 (5 e f+6 d g)-c f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )+2 a c^2 g \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-b c \left (27 a^2 e^3 g^4+3 a c e g^2 \left (8 e^2 f^2-33 d e f g-21 d^2 g^2\right )+c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right )\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{945 c^3 g^{11}}\\ &=\frac{2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{315 c^3 g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 g}-\frac{2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{315 c^2 g^4}-\frac{2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^4}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\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^5 \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 (16 \sqrt{2} \sqrt{b^2-4 a c} \left (-\frac{3}{8} f g^6 \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right )+\frac{3}{8} g^7 \left (8 b^4 e^3 f g^3+b^3 e^2 g^2 \left (8 a e g^2+9 c f (e f-4 d g)\right )-3 b^2 c e g \left (2 a e g^2 (5 e f+6 d g)-c f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )+2 a c^2 g \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-b c \left (27 a^2 e^3 g^4+3 a c e g^2 \left (8 e^2 f^2-33 d e f g-21 d^2 g^2\right )+c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\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 )}{945 c^4 g^{11} \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 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{315 c^3 g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{9 g}-\frac{2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{315 c^2 g^4}-\frac{2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}{63 c g^4}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\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^5 \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 (128 c^3 e^3 f^3-432 c^3 d e^2 f^2 g+24 b c^2 e^3 f^2 g+504 c^3 d^2 e f g^2-72 b c^2 d e^2 f g^2+15 b^2 c e^3 f g^2-36 a c^2 e^3 f g^2-210 c^3 d^3 g^3+63 b c^2 d^2 e g^3-36 b^2 c d e^2 g^3+90 a c^2 d e^2 g^3+8 b^3 e^3 g^3-27 a b c e^3 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 (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^5 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 16.3749, size = 17771, normalized size = 16.18 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d + e*x)^3*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 = 0.395, size = 22215, normalized size = 20.2 \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)^3*(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 )}^{3}}{\sqrt{g x + f}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(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)^3/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^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\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)^3*(c*x^2+b*x+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*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)**3*(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)^3*(c*x^2+b*x+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

Timed out