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

Optimal. Leaf size=774 \[ -\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 g \sqrt{b^2-4 a c}}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}\right )}{105 c^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (-\left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )\right )}{105 c^3 g^2}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e g \left (a e g (189 d g+19 e f)-b \left (-210 d^2 g^2-63 d e f g+9 e^2 f^2\right )\right )-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3+c^3 \left (-\left (105 d^2 e f g^2+105 d^3 g^3-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e^2 (f+g x)^{3/2} \sqrt{a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c} \]

[Out]

(2*e*(24*b^2*e^2*g^2 + c*e*g*(13*b*e*f - 84*b*d*g - 25*a*e*g) - c^2*(7*e^2*f^2 + 12*d*e*f*g - 90*d^2*g^2))*Sqr
t[f + g*x]*Sqrt[a + b*x + c*x^2])/(105*c^3*g^2) + (2*e*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(7*c)
+ (2*e^2*(c*e*f + 11*c*d*g - 6*b*e*g)*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(35*c^2*g^2) - (Sqrt[2]*Sqrt[b^2
- 4*a*c]*(48*b^3*e^3*g^3 - 8*b*c*e^2*g^2*(2*b*e*f + 21*b*d*g + 13*a*e*g) - c^3*(8*e^3*f^3 - 42*d*e^2*f^2*g + 1
05*d^2*e*f*g^2 + 105*d^3*g^3) + c^2*e*g*(a*e*g*(19*e*f + 189*d*g) - b*(9*e^2*f^2 - 63*d*e*f*g - 210*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^4*g^3*S
qrt[(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]*e
*(c*f^2 - b*f*g + a*g^2)*(24*b^2*e^2*g^2 + c*e*g*(13*b*e*f - 84*b*d*g - 25*a*e*g) + c^2*(8*e^2*f^2 - 42*d*e*f*
g + 105*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^4*g^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 2.10553, antiderivative size = 774, 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 = {941, 1653, 843, 718, 424, 419} \[ \frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (-\left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )\right )}{105 c^3 g^2}-\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e g \left (a e g (189 d g+19 e f)-b \left (-210 d^2 g^2-63 d e f g+9 e^2 f^2\right )\right )-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3+c^3 \left (-\left (105 d^2 e f g^2+105 d^3 g^3-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e^2 (f+g x)^{3/2} \sqrt{a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(24*b^2*e^2*g^2 + c*e*g*(13*b*e*f - 84*b*d*g - 25*a*e*g) - c^2*(7*e^2*f^2 + 12*d*e*f*g - 90*d^2*g^2))*Sqr
t[f + g*x]*Sqrt[a + b*x + c*x^2])/(105*c^3*g^2) + (2*e*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(7*c)
+ (2*e^2*(c*e*f + 11*c*d*g - 6*b*e*g)*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(35*c^2*g^2) - (Sqrt[2]*Sqrt[b^2
- 4*a*c]*(48*b^3*e^3*g^3 - 8*b*c*e^2*g^2*(2*b*e*f + 21*b*d*g + 13*a*e*g) - c^3*(8*e^3*f^3 - 42*d*e^2*f^2*g + 1
05*d^2*e*f*g^2 + 105*d^3*g^3) + c^2*e*g*(a*e*g*(19*e*f + 189*d*g) - b*(9*e^2*f^2 - 63*d*e*f*g - 210*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^4*g^3*S
qrt[(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]*e
*(c*f^2 - b*f*g + a*g^2)*(24*b^2*e^2*g^2 + c*e*g*(13*b*e*f - 84*b*d*g - 25*a*e*g) + c^2*(8*e^2*f^2 - 42*d*e*f*
g + 105*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^4*g^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

Rule 941

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :>
Simp[(2*e*(d + e*x)^(m - 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(c*(2*m + 1)), x] - Dist[1/(c*(2*m + 1)), In
t[((d + e*x)^(m - 2)*Simp[e*(b*d*f + a*(d*g + 2*e*f*(m - 1))) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(
4*e*f*m + d*g*(2*m + 1)) + b*e*(2*d*g + e*f*(2*m - 1)))*x + e*(2*b*e*g*m - c*(e*f + d*g*(4*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] && GtQ[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 \frac{(d+e x)^3 \sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx &=\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}-\frac{\int \frac{(d+e x) \left (-7 c d^2 f+e (b d f+4 a e f+a d g)-(c d (12 e f+7 d g)-e (5 b e f+2 b d g+5 a e g)) x-e (c e f+11 c d g-6 b e g) x^2\right )}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{7 c}\\ &=\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 g^2}-\frac{2 \int \frac{-\frac{1}{2} g \left (6 b^2 e^3 f^2 g+b e f \left (18 a e^2 g^2-c \left (e^2 f^2+11 d e f g+5 d^2 g^2\right )\right )+c g \left (35 c d^3 f g-a e \left (3 e^2 f^2+53 d e f g+5 d^2 g^2\right )\right )\right )-\frac{1}{2} g \left (6 b e^3 g^2 (5 b f+3 a g)-c^2 \left (2 e^3 f^3+22 d e^2 f^2 g-95 d^2 e f g^2-35 d^3 g^3\right )-c e g \left (a e g (23 e f+63 d g)-b \left (7 e^2 f^2-85 d e f g-10 d^2 g^2\right )\right )\right ) x-\frac{1}{2} e g^2 \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) x^2}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{35 c^2 g^3}\\ &=\frac{2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^3 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 g^2}-\frac{4 \int \frac{\frac{1}{4} g^3 \left (24 b^3 e^3 f g^2+b^2 e^2 g \left (24 a e g^2-c f (5 e f+84 d g)\right )-b c e \left (6 a e g^2 (11 e f+14 d g)+c f \left (4 e^2 f^2-21 d e f g-105 d^2 g^2\right )\right )-c g \left (105 c^2 d^3 f g+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e f g+105 d^2 g^2\right )\right )\right )+\frac{1}{4} g^3 \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3 g^5}\\ &=\frac{2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^3 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 g^2}-\frac{\left (e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3 g^3}-\frac{\left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{105 c^3 g^3}\\ &=\frac{2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^3 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 g^2}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 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^4 g^3 \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} e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 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^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^3 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 g^2}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 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^4 g^3 \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} e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 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^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 14.9232, size = 10649, normalized size = 13.76 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d + e*x)^3*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 = 0.395, size = 14978, normalized size = 19.4 \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*(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 \frac{{\left (e x + d\right )}^{3} \sqrt{g x + f}}{\sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x + d)^3*sqrt(g*x + f)/sqrt(c*x^2 + b*x + a), 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{g x + f}}{\sqrt{c x^{2} + b x + a}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)^(1/2)/(c*x^2+b*x+a)^(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(g*x + f)/sqrt(c*x^2 + b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{3} \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)**3*(g*x+f)**(1/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

[Out]

Timed out