\(\int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx\) [900]

Optimal result

Integrand size = 31, antiderivative size = 774 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^3 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]

[Out]

2/35*e^2*(-6*b*e*g+11*c*d*g+c*e*f)*(g*x+f)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^2/g^2+2/105*e*(24*b^2*e^2*g^2+c*e*g*(-2
5*a*e*g-84*b*d*g+13*b*e*f)-c^2*(-90*d^2*g^2+12*d*e*f*g+7*e^2*f^2))*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3/g^2+2
/7*e*(e*x+d)^2*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c-1/105*(48*b^3*e^3*g^3-8*b*c*e^2*g^2*(13*a*e*g+21*b*d*g+2*b*
e*f)-c^3*(105*d^3*g^3+105*d^2*e*f*g^2-42*d*e^2*f^2*g+8*e^3*f^3)+c^2*e*g*(a*e*g*(189*d*g+19*e*f)-b*(-210*d^2*g^
2-63*d*e*f*g+9*e^2*f^2)))*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*g*
(-4*a*c+b^2)^(1/2)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(g*x+f)^(1/2)*(-c*(c*x^
2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^4/g^3/(c*x^2+b*x+a)^(1/2)/(c*(g*x+f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)-2/
105*e*(a*g^2-b*f*g+c*f^2)*(24*b^2*e^2*g^2+c*e*g*(-25*a*e*g-84*b*d*g+13*b*e*f)+c^2*(105*d^2*g^2-42*d*e*f*g+8*e^
2*f^2))*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*g*(-4*a*c+b^2)^(1/2)
/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*
(g*x+f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/c^4/g^3/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 774, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {955, 1667, 857, 732, 435, 430} \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=-\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e g \left (a e g (189 d g+19 e f)-b \left (-210 d^2 g^2-63 d e f g+9 e^2 f^2\right )\right )-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3-\left (c^3 \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 e \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2-\left (c^2 \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )\right )}{105 c^3 g^2}+\frac {2 e^2 (f+g x)^{3/2} \sqrt {a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c} \]

[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 430

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

Rule 435

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

Rule 732

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 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 955

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)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^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], 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 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 {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 ) \text {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 ) \text {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] (verified)

Result contains complex when optimal does not.

Time = 34.65 (sec) , antiderivative size = 1402, normalized size of antiderivative = 1.81 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {f+g x} \left (a+b x+c x^2\right ) \left (-\frac {2 e \left (4 c^2 e^2 f^2-21 c^2 d e f g+5 b c e^2 f g-105 c^2 d^2 g^2+84 b c d e g^2-24 b^2 e^2 g^2+25 a c e^2 g^2\right )}{105 c^3 g^2}-\frac {2 e^2 (-c e f-21 c d g+6 b e g) x}{35 c^2 g}+\frac {2 e^3 x^2}{7 c}\right )}{\sqrt {a+x (b+c x)}}-\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2} \left (-\left (\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 ) \left (c \left (-1+\frac {f}{f+g x}\right )^2+\frac {g \left (b-\frac {b f}{f+g x}+\frac {a g}{f+g x}\right )}{f+g x}\right )\right )-\frac {i \sqrt {1-\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {1+\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \left (\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \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 ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )+\left (48 b^4 e^3 g^4-8 b^3 e^2 g^3 \left (8 c e f+21 c d g+6 e \sqrt {\left (b^2-4 a c\right ) g^2}\right )+b^2 c e g^2 \left (-152 a e^2 g^2+8 e \sqrt {\left (b^2-4 a c\right ) g^2} (2 e f+21 d g)+c \left (e^2 f^2+231 d e f g+210 d^2 g^2\right )\right )-b \left (-104 a c e^3 g^3 \sqrt {\left (b^2-4 a c\right ) g^2}+105 c^3 d^2 g^3 (3 e f+d g)+c^2 e g \left (-a e g^2 (151 e f+357 d g)+3 \sqrt {\left (b^2-4 a c\right ) g^2} \left (-3 e^2 f^2+21 d e f g+70 d^2 g^2\right )\right )\right )+c^2 \left (50 a^2 e^3 g^4-a e g^2 \left (e \sqrt {\left (b^2-4 a c\right ) g^2} (19 e f+189 d g)+c \left (4 e^2 f^2+294 d e f g+210 d^2 g^2\right )\right )+c \left (210 c d^3 f g^3+\sqrt {\left (b^2-4 a c\right ) g^2} \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 )\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right ),-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c f^2+g (-b f+a g)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {f+g x}}\right )}{105 c^4 g^4 \sqrt {a+x (b+c x)} \sqrt {\frac {(f+g x)^2 \left (c \left (-1+\frac {f}{f+g x}\right )^2+\frac {g \left (b-\frac {b f}{f+g x}+\frac {a g}{f+g x}\right )}{f+g x}\right )}{g^2}}} \]

[In]

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

[Out]

(Sqrt[f + g*x]*(a + b*x + c*x^2)*((-2*e*(4*c^2*e^2*f^2 - 21*c^2*d*e*f*g + 5*b*c*e^2*f*g - 105*c^2*d^2*g^2 + 84
*b*c*d*e*g^2 - 24*b^2*e^2*g^2 + 25*a*c*e^2*g^2))/(105*c^3*g^2) - (2*e^2*(-(c*e*f) - 21*c*d*g + 6*b*e*g)*x)/(35
*c^2*g) + (2*e^3*x^2)/(7*c)))/Sqrt[a + x*(b + c*x)] - (2*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2]*(-((-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 + 105*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)))*(c*(-1 + f/(f + g*
x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x))) - ((I/2)*Sqrt[1 - (2*(c*f^2 + g*(-(b*f) + a*g)
))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[1 + (2*(c*f^2 + g*(-(b*f) + a*g)))/((-2*c*f + b*g
+ Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(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 + 105*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)))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c
*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 -
4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] + (48*b^4*e^3*g^4 - 8*b^3*e^2*g^3*(8*c*e*f + 21*c*d*g
+ 6*e*Sqrt[(b^2 - 4*a*c)*g^2]) + b^2*c*e*g^2*(-152*a*e^2*g^2 + 8*e*Sqrt[(b^2 - 4*a*c)*g^2]*(2*e*f + 21*d*g) +
c*(e^2*f^2 + 231*d*e*f*g + 210*d^2*g^2)) - b*(-104*a*c*e^3*g^3*Sqrt[(b^2 - 4*a*c)*g^2] + 105*c^3*d^2*g^3*(3*e*
f + d*g) + c^2*e*g*(-(a*e*g^2*(151*e*f + 357*d*g)) + 3*Sqrt[(b^2 - 4*a*c)*g^2]*(-3*e^2*f^2 + 21*d*e*f*g + 70*d
^2*g^2))) + c^2*(50*a^2*e^3*g^4 - a*e*g^2*(e*Sqrt[(b^2 - 4*a*c)*g^2]*(19*e*f + 189*d*g) + c*(4*e^2*f^2 + 294*d
*e*f*g + 210*d^2*g^2)) + c*(210*c*d^3*f*g^3 + Sqrt[(b^2 - 4*a*c)*g^2]*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*
f*g^2 + 105*d^3*g^3))))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 -
4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^
2]))]))/(Sqrt[2]*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[f + g*x])))/(1
05*c^4*g^4*Sqrt[a + x*(b + c*x)]*Sqrt[((f + g*x)^2*(c*(-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(
f + g*x)))/(f + g*x)))/g^2])

Maple [A] (verified)

Time = 3.62 (sec) , antiderivative size = 1283, normalized size of antiderivative = 1.66

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1283\)
risch	\(\text {Expression too large to display}\)	\(4891\)
default	\(\text {Expression too large to display}\)	\(14978\)

[In]

int((e*x+d)^3*(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((g*x+f)*(c*x^2+b*x+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/7*e^3/c*x^2*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+
b*f*x+a*f)^(1/2)+2/5*(3*d*e^2*g+e^3*f-2/7*e^3/c*(3*b*g+3*c*f))/c/g*x*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)
^(1/2)+2/3*(3*d^2*e*g+3*d*e^2*f-2/7*e^3/c*(5/2*a*g+5/2*b*f)-2/5*(3*d*e^2*g+e^3*f-2/7*e^3/c*(3*b*g+3*c*f))/c/g*
(2*b*g+2*c*f))/c/g*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)+2*(d^3*f-2/5*(3*d*e^2*g+e^3*f-2/7*e^3/c*(3*
b*g+3*c*f))/c/g*f*a-2/3*(3*d^2*e*g+3*d*e^2*f-2/7*e^3/c*(5/2*a*g+5/2*b*f)-2/5*(3*d*e^2*g+e^3*f-2/7*e^3/c*(3*b*g
+3*c*f))/c/g*(2*b*g+2*c*f))/c/g*(1/2*a*g+1/2*b*f))*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-1/2*(b+(-4
*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+
1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+
a*f)^(1/2)*EllipticF(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(
-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2*(d^3*g+3*d^2*e*f-4/7*a/c*f*e^3-2/5*(3*d*e^2*g+e^3*f-2/7*e^3/c*(3
*b*g+3*c*f))/c/g*(3/2*a*g+3/2*b*f)-2/3*(3*d^2*e*g+3*d*e^2*f-2/7*e^3/c*(5/2*a*g+5/2*b*f)-2/5*(3*d*e^2*g+e^3*f-2
/7*e^3/c*(3*b*g+3*c*f))/c/g*(2*b*g+2*c*f))/c/g*(b*g+c*f))*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-1/2
*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/
2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x
+b*f*x+a*f)^(1/2)*((-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))
^(1/2),((-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2
)^(1/2))*EllipticF(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f
/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.19 (sec) , antiderivative size = 879, normalized size of antiderivative = 1.14 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=-\frac {2 \, {\left ({\left (8 \, c^{4} e^{3} f^{4} - {\left (42 \, c^{4} d e^{2} - 5 \, b c^{3} e^{3}\right )} f^{3} g + {\left (105 \, c^{4} d^{2} e - 42 \, b c^{3} d e^{2} + {\left (10 \, b^{2} c^{2} - 13 \, a c^{3}\right )} e^{3}\right )} f^{2} g^{2} - {\left (210 \, c^{4} d^{3} - 210 \, b c^{3} d^{2} e + 21 \, {\left (7 \, b^{2} c^{2} - 12 \, a c^{3}\right )} d e^{2} - {\left (40 \, b^{3} c - 113 \, a b c^{2}\right )} e^{3}\right )} f g^{3} + {\left (105 \, b c^{3} d^{3} - 105 \, {\left (2 \, b^{2} c^{2} - 3 \, a c^{3}\right )} d^{2} e + 21 \, {\left (8 \, b^{3} c - 21 \, a b c^{2}\right )} d e^{2} - {\left (48 \, b^{4} - 176 \, a b^{2} c + 75 \, a^{2} c^{2}\right )} e^{3}\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right ) + 3 \, {\left (8 \, c^{4} e^{3} f^{3} g - 3 \, {\left (14 \, c^{4} d e^{2} - 3 \, b c^{3} e^{3}\right )} f^{2} g^{2} + {\left (105 \, c^{4} d^{2} e - 63 \, b c^{3} d e^{2} + {\left (16 \, b^{2} c^{2} - 19 \, a c^{3}\right )} e^{3}\right )} f g^{3} + {\left (105 \, c^{4} d^{3} - 210 \, b c^{3} d^{2} e + 21 \, {\left (8 \, b^{2} c^{2} - 9 \, a c^{3}\right )} d e^{2} - 8 \, {\left (6 \, b^{3} c - 13 \, a b c^{2}\right )} e^{3}\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right ) - 3 \, {\left (15 \, c^{4} e^{3} g^{4} x^{2} - 4 \, c^{4} e^{3} f^{2} g^{2} + {\left (21 \, c^{4} d e^{2} - 5 \, b c^{3} e^{3}\right )} f g^{3} + {\left (105 \, c^{4} d^{2} e - 84 \, b c^{3} d e^{2} + {\left (24 \, b^{2} c^{2} - 25 \, a c^{3}\right )} e^{3}\right )} g^{4} + 3 \, {\left (c^{4} e^{3} f g^{3} + 3 \, {\left (7 \, c^{4} d e^{2} - 2 \, b c^{3} e^{3}\right )} g^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{315 \, c^{5} g^{4}} \]

[In]

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

[Out]

-2/315*((8*c^4*e^3*f^4 - (42*c^4*d*e^2 - 5*b*c^3*e^3)*f^3*g + (105*c^4*d^2*e - 42*b*c^3*d*e^2 + (10*b^2*c^2 -
13*a*c^3)*e^3)*f^2*g^2 - (210*c^4*d^3 - 210*b*c^3*d^2*e + 21*(7*b^2*c^2 - 12*a*c^3)*d*e^2 - (40*b^3*c - 113*a*
b*c^2)*e^3)*f*g^3 + (105*b*c^3*d^3 - 105*(2*b^2*c^2 - 3*a*c^3)*d^2*e + 21*(8*b^3*c - 21*a*b*c^2)*d*e^2 - (48*b
^4 - 176*a*b^2*c + 75*a^2*c^2)*e^3)*g^4)*sqrt(c*g)*weierstrassPInverse(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*
g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3
), 1/3*(3*c*g*x + c*f + b*g)/(c*g)) + 3*(8*c^4*e^3*f^3*g - 3*(14*c^4*d*e^2 - 3*b*c^3*e^3)*f^2*g^2 + (105*c^4*d
^2*e - 63*b*c^3*d*e^2 + (16*b^2*c^2 - 19*a*c^3)*e^3)*f*g^3 + (105*c^4*d^3 - 210*b*c^3*d^2*e + 21*(8*b^2*c^2 -
9*a*c^3)*d*e^2 - 8*(6*b^3*c - 13*a*b*c^2)*e^3)*g^4)*sqrt(c*g)*weierstrassZeta(4/3*(c^2*f^2 - b*c*f*g + (b^2 -
3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(
c^3*g^3), weierstrassPInverse(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b*c^
2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), 1/3*(3*c*g*x + c*f + b*g)/(c*g))) - 3*
(15*c^4*e^3*g^4*x^2 - 4*c^4*e^3*f^2*g^2 + (21*c^4*d*e^2 - 5*b*c^3*e^3)*f*g^3 + (105*c^4*d^2*e - 84*b*c^3*d*e^2
+ (24*b^2*c^2 - 25*a*c^3)*e^3)*g^4 + 3*(c^4*e^3*f*g^3 + 3*(7*c^4*d*e^2 - 2*b*c^3*e^3)*g^4)*x)*sqrt(c*x^2 + b*
x + a)*sqrt(g*x + f))/(c^5*g^4)

Sympy [F]

\[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{3} \sqrt {f + g x}}{\sqrt {a + b x + c x^{2}}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+b\,x+a}} \,d x \]

[In]

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

[Out]

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