\(\int (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx\) [887]

Optimal result

Integrand size = 31, antiderivative size = 1015 \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\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 (\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 )}{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}} \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 )}{315 c^4 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]

[Out]

-4/315*(3*b^2*e^2*g^2+c*e*g*(-7*a*e*g-9*b*d*g+4*b*e*f)+c^2*(21*d^2*g^2-24*d*e*f*g+8*e^2*f^2))*(g*x+f)^(3/2)*(c
*x^2+b*x+a)^(1/2)/c^2/g^3+2/63*e*(b*e*g-3*c*d*g+c*e*f)*(g*x+f)^(5/2)*(c*x^2+b*x+a)^(1/2)/c/g^3+2/315*(8*b^3*e^
3*g^3+3*b*c*e^2*g^2*(-9*a*e*g-8*b*d*g+b*e*f)+c^3*(-35*d^3*g^3+63*d^2*e*f*g^2-57*d*e^2*f^2*g+19*e^3*f^3)-3*c^2*
e*g^2*(2*a*e*(-10*d*g+e*f)+b*d*(-7*d*g+2*e*f)))*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3/e/g^3+2/9*(e*x+d)^3*(g*x
+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/e-2/315*(8*b^4*e^2*g^4-4*b^2*c*e*g^3*(9*a*e*g+6*b*d*g+b*e*f)+c^4*f^2*(21*d^2*g^2
-24*d*e*f*g+8*e^2*f^2)+3*c^2*g^2*(7*a^2*e^2*g^2+a*b*e*g*(29*d*g+5*e*f)-b^2*(-7*d^2*g^2-5*d*e*f*g+e^2*f^2))+c^3
*g*(3*a*g*(-21*d^2*g^2-16*d*e*f*g+3*e^2*f^2)-b*f*(21*d^2*g^2-15*d*e*f*g+4*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^4/(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)-2/315*(a*g^2-b*f*g+c*f^2)*(8*b^3*e^2*g^3+3*b*c*e*g^2*
(-9*a*e*g-8*b*d*g+b*e*f)-2*c^3*f*(21*d^2*g^2-24*d*e*f*g+8*e^2*f^2)-3*c^2*g^2*(2*a*e*(-10*d*g+e*f)+b*d*(-7*d*g+
2*e*f)))*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^4/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 2.42 (sec) , antiderivative size = 1015, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {932, 1667, 857, 732, 435, 430} \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\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 (\left (e^2 f^2-5 d e g f-7 d^2 g^2\right ) b^2\right )+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 (\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 )}{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}} \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 )}{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} \]

[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 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 932

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/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*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], 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 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 (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 ) \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 )}{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 ) \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 )}{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] (verified)

Result contains complex when optimal does not.

Time = 35.97 (sec) , antiderivative size = 15781, normalized size of antiderivative = 15.55 \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1935\) vs. \(2(939)=1878\).

Time = 3.52 (sec) , antiderivative size = 1936, normalized size of antiderivative = 1.91

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1936\)
risch	\(\text {Expression too large to display}\)	\(7219\)
default	\(\text {Expression too large to display}\)	\(20224\)

[In]

int((e*x+d)^2*(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)*(2/9*e^2*x^3*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*
f*x+a*f)^(1/2)+2/7*(b*e^2*g+2*c*d*e*g+c*e^2*f-2/9*e^2*(4*b*g+4*c*f))/c/g*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*(a*e^2*g+2*b*d*e*g+b*e^2*f+c*d^2*g+2*c*d*e*f-2/9*e^2*(7/2*a*g+7/2*b*f)-2/7*(b*e^2*g+2*c*d*e
*g+c*e^2*f-2/9*e^2*(4*b*g+4*c*f))/c/g*(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
*(2*a*d*e*g+1/3*a*e^2*f+b*d^2*g+2*b*d*e*f+c*d^2*f-2/7*(b*e^2*g+2*c*d*e*g+c*e^2*f-2/9*e^2*(4*b*g+4*c*f))/c/g*(5
/2*a*g+5/2*b*f)-2/5*(a*e^2*g+2*b*d*e*g+b*e^2*f+c*d^2*g+2*c*d*e*f-2/9*e^2*(7/2*a*g+7/2*b*f)-2/7*(b*e^2*g+2*c*d*
e*g+c*e^2*f-2/9*e^2*(4*b*g+4*c*f))/c/g*(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*(a*d^2*f-2/5*(a*e^2*g+2*b*d*e*g+b*e^2*f+c*d^2*g+2*c*d*e*f-2/9*e^2*(7/2*a*g+7/2*b*f)-2/7*(b*e^
2*g+2*c*d*e*g+c*e^2*f-2/9*e^2*(4*b*g+4*c*f))/c/g*(3*b*g+3*c*f))/c/g*f*a-2/3*(2*a*d*e*g+1/3*a*e^2*f+b*d^2*g+2*b
*d*e*f+c*d^2*f-2/7*(b*e^2*g+2*c*d*e*g+c*e^2*f-2/9*e^2*(4*b*g+4*c*f))/c/g*(5/2*a*g+5/2*b*f)-2/5*(a*e^2*g+2*b*d*
e*g+b*e^2*f+c*d^2*g+2*c*d*e*f-2/9*e^2*(7/2*a*g+7/2*b*f)-2/7*(b*e^2*g+2*c*d*e*g+c*e^2*f-2/9*e^2*(4*b*g+4*c*f))/
c/g*(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*(a*d^2*g+2*a*d*e*f+b*d^2*f-4/7*(b*e^2*g+2*c*d*e*g+c*e^
2*f-2/9*e^2*(4*b*g+4*c*f))/c/g*f*a-2/5*(a*e^2*g+2*b*d*e*g+b*e^2*f+c*d^2*g+2*c*d*e*f-2/9*e^2*(7/2*a*g+7/2*b*f)-
2/7*(b*e^2*g+2*c*d*e*g+c*e^2*f-2/9*e^2*(4*b*g+4*c*f))/c/g*(3*b*g+3*c*f))/c/g*(3/2*a*g+3/2*b*f)-2/3*(2*a*d*e*g+
1/3*a*e^2*f+b*d^2*g+2*b*d*e*f+c*d^2*f-2/7*(b*e^2*g+2*c*d*e*g+c*e^2*f-2/9*e^2*(4*b*g+4*c*f))/c/g*(5/2*a*g+5/2*b
*f)-2/5*(a*e^2*g+2*b*d*e*g+b*e^2*f+c*d^2*g+2*c*d*e*f-2/9*e^2*(7/2*a*g+7/2*b*f)-2/7*(b*e^2*g+2*c*d*e*g+c*e^2*f-
2/9*e^2*(4*b*g+4*c*f))/c/g*(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 = 0.14 (sec) , antiderivative size = 1132, normalized size of antiderivative = 1.12 \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/945*((16*c^5*e^2*f^5 - 16*(3*c^5*d*e + b*c^4*e^2)*f^4*g + (42*c^5*d^2 + 54*b*c^4*d*e - 5*(b^2*c^3 - 6*a*c^4)
*e^2)*f^3*g^2 - (63*b*c^4*d^2 - 12*(2*b^2*c^3 - 11*a*c^4)*d*e + (5*b^3*c^2 - 21*a*b*c^3)*e^2)*f^2*g^3 - (63*(b
^2*c^3 - 6*a*c^4)*d^2 - 6*(9*b^3*c^2 - 41*a*b*c^3)*d*e + 2*(8*b^4*c - 42*a*b^2*c^2 + 33*a^2*c^3)*e^2)*f*g^4 +
(21*(2*b^3*c^2 - 9*a*b*c^3)*d^2 - 6*(8*b^4*c - 41*a*b^2*c^2 + 30*a^2*c^3)*d*e + (16*b^5 - 96*a*b^3*c + 123*a^2
*b*c^2)*e^2)*g^5)*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)) + 6*(8*c^5*e^2*f^4*g - 4*(6*c^5*d*e + b*c^4*e^2)*f^3*g^2 + 3*(7*c^5*d^2 + 5*b*c^4*d*e - (b^2*c^3
- 3*a*c^4)*e^2)*f^2*g^3 - (21*b*c^4*d^2 - 3*(5*b^2*c^3 - 16*a*c^4)*d*e + (4*b^3*c^2 - 15*a*b*c^3)*e^2)*f*g^4 +
(21*(b^2*c^3 - 3*a*c^4)*d^2 - 3*(8*b^3*c^2 - 29*a*b*c^3)*d*e + (8*b^4*c - 36*a*b^2*c^2 + 21*a^2*c^3)*e^2)*g^5
)*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*(35*c^5*e^2*g^5*x^3 + 8*c^5*e^2*f^3*g^2 - 3*(8*c^5
*d*e + b*c^4*e^2)*f^2*g^3 + (21*c^5*d^2 + 12*b*c^4*d*e - (3*b^2*c^3 - 8*a*c^4)*e^2)*f*g^4 + (21*b*c^4*d^2 - 12
*(2*b^2*c^3 - 5*a*c^4)*d*e + (8*b^3*c^2 - 27*a*b*c^3)*e^2)*g^5 + 5*(c^5*e^2*f*g^4 + (18*c^5*d*e + b*c^4*e^2)*g
^5)*x^2 - (6*c^5*e^2*f^2*g^3 - 2*(9*c^5*d*e + b*c^4*e^2)*f*g^4 - (63*c^5*d^2 + 18*b*c^4*d*e - 2*(3*b^2*c^3 - 7
*a*c^4)*e^2)*g^5)*x)*sqrt(c*x^2 + b*x + a)*sqrt(g*x + f))/(c^5*g^5)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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