3.9.94 \(\int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx\) [894]

Optimal. Leaf size=755 \[ -\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]

[Out]

-2/35*e*(-b*e*g-4*c*d*g+6*c*e*f)*(g*x+f)^(3/2)*(c*x^2+b*x+a)^(1/2)/c/g^3-4/105*(2*b^2*e^2*g^2+c*e*g*(-5*a*e*g-
7*b*d*g+4*b*e*f)-c^2*(10*d^2*g^2-34*d*e*f*g+21*e^2*f^2))*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2/g^3+2/7*(e*x+d)
^2*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/g+1/105*(8*b^3*e^2*g^3+b*c*e*g^2*(-29*a*e*g-28*b*d*g+9*b*e*f)-2*c^3*f*(35
*d^2*g^2-56*d*e*f*g+24*e^2*f^2)-c^2*g*(2*a*e*g*(-42*d*g+13*e*f)-b*(35*d^2*g^2-42*d*e*f*g+16*e^2*f^2)))*Ellipti
cE(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^3/
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)+4/105*(a*g^2-b*f*g+c*f^2)*(2*b^2*e^
2*g^2+c*e*g*(-5*a*e*g-7*b*d*g+4*b*e*f)+c^2*(35*d^2*g^2-56*d*e*f*g+24*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^3/g^4/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.32, antiderivative size = 755, normalized size of antiderivative = 1.00, 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 = {934, 1667, 857, 732, 435, 430} \begin {gather*} \frac {4 \sqrt {2} \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 (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2+c^2 \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) F\left (\text {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^3 g^4 \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 g \left (2 a e g (13 e f-42 d g)-b \left (35 d^2 g^2-42 d e f g+16 e^2 f^2\right )\right )+b c e g^2 (-29 a e g-28 b d g+9 b e f)+8 b^3 e^2 g^3-2 c^3 f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) E\left (\text {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^3 g^4 \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 {4 \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2-\left (c^2 \left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )\right )}{105 c^2 g^3}-\frac {2 e (f+g x)^{3/2} \sqrt {a+b x+c x^2} (-b e g-4 c d g+6 c e f)}{35 c g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(2*b^2*e^2*g^2 + c*e*g*(4*b*e*f - 7*b*d*g - 5*a*e*g) - c^2*(21*e^2*f^2 - 34*d*e*f*g + 10*d^2*g^2))*Sqrt[f
+ g*x]*Sqrt[a + b*x + c*x^2])/(105*c^2*g^3) + (2*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(7*g) - (2*e
*(6*c*e*f - 4*c*d*g - b*e*g)*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(35*c*g^3) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8
*b^3*e^2*g^3 + b*c*e*g^2*(9*b*e*f - 28*b*d*g - 29*a*e*g) - 2*c^3*f*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2) - c^
2*g*(2*a*e*g*(13*e*f - 42*d*g) - b*(16*e^2*f^2 - 42*d*e*f*g + 35*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^3*g^4*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqr
t[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*f^2 - b*f*g + a*g^2)*(2*b^2*e^2*g
^2 + c*e*g*(4*b*e*f - 7*b*d*g - 5*a*e*g) + c^2*(24*e^2*f^2 - 56*d*e*f*g + 35*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^3*g^4*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 934

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)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*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], 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 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*} \int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx &=\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {\int \frac {(d+e x) \left (b d f+4 a e f-6 a d g+(2 c d f+5 b e f-5 b d g-2 a e g) x+(6 c e f-4 c d g-b e g) x^2\right )}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{7 g}\\ &=\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}-\frac {2 \int \frac {\frac {1}{2} g \left (b^2 e^2 f^2 g-2 a c g \left (9 e^2 f^2-16 d e f g+15 d^2 g^2\right )+b f \left (3 a e^2 g^2-c \left (6 e^2 f^2-4 d e f g-5 d^2 g^2\right )\right )\right )+\frac {1}{2} g \left (b e^2 g^2 (5 b f+3 a g)-2 c^2 f \left (6 e^2 f^2-4 d e f g-5 d^2 g^2\right )+c g \left (2 a e g (e f-14 d g)-b \left (28 e^2 f^2-50 d e f g+25 d^2 g^2\right )\right )\right ) x+g^2 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{35 c g^4}\\ &=-\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}-\frac {4 \int \frac {-\frac {1}{4} g^3 \left (4 b^3 e^2 f g^2+b^2 e g \left (4 a e g^2+c f (5 e f-14 d g)\right )-b c \left (a e g^2 (11 e f+14 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right )-2 a c g \left (5 a e^2 g^2-c \left (6 e^2 f^2-14 d e f g+35 d^2 g^2\right )\right )\right )-\frac {1}{4} g^3 \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^6}\\ &=-\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\left (2 \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^4}+\frac {\left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^4}\\ &=-\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 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^3 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 (4 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 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^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=-\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 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^3 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 {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 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^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 33.32, size = 10030, normalized size = 13.28 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(12921\) vs. \(2(685)=1370\).
time = 0.15, size = 12922, normalized size = 17.12

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 e^{2} x^{2} \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a x g +b f x +f a}}{7 g}+\frac {2 \left (e^{2} b +2 c d e -\frac {2 e^{2} \left (3 b g +3 c f \right )}{7 g}\right ) x \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a x g +b f x +f a}}{5 c g}+\frac {2 \left (a \,e^{2}+2 d e b +c \,d^{2}-\frac {2 e^{2} \left (\frac {5 a g}{2}+\frac {5 b f}{2}\right )}{7 g}-\frac {2 \left (e^{2} b +2 c d e -\frac {2 e^{2} \left (3 b g +3 c f \right )}{7 g}\right ) \left (2 b g +2 c f \right )}{5 c g}\right ) \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a x g +b f x +f a}}{3 c g}+\frac {2 \left (a \,d^{2}-\frac {2 \left (e^{2} b +2 c d e -\frac {2 e^{2} \left (3 b g +3 c f \right )}{7 g}\right ) f a}{5 c g}-\frac {2 \left (a \,e^{2}+2 d e b +c \,d^{2}-\frac {2 e^{2} \left (\frac {5 a g}{2}+\frac {5 b f}{2}\right )}{7 g}-\frac {2 \left (e^{2} b +2 c d e -\frac {2 e^{2} \left (3 b g +3 c f \right )}{7 g}\right ) \left (2 b g +2 c f \right )}{5 c g}\right ) \left (\frac {a g}{2}+\frac {b f}{2}\right )}{3 c g}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a x g +b f x +f a}}+\frac {2 \left (2 a d e +d^{2} b -\frac {4 f a \,e^{2}}{7 g}-\frac {2 \left (e^{2} b +2 c d e -\frac {2 e^{2} \left (3 b g +3 c f \right )}{7 g}\right ) \left (\frac {3 a g}{2}+\frac {3 b f}{2}\right )}{5 c g}-\frac {2 \left (a \,e^{2}+2 d e b +c \,d^{2}-\frac {2 e^{2} \left (\frac {5 a g}{2}+\frac {5 b f}{2}\right )}{7 g}-\frac {2 \left (e^{2} b +2 c d e -\frac {2 e^{2} \left (3 b g +3 c f \right )}{7 g}\right ) \left (2 b g +2 c f \right )}{5 c g}\right ) \left (b g +c f \right )}{3 c g}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a x g +b f x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}\)	\(1272\)
risch	\(\text {Expression too large to display}\)	\(4566\)
default	\(\text {Expression too large to display}\)	\(12922\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(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)*(x*e + d)^2/sqrt(g*x + f), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.36, size = 840, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left ({\left (70 \, c^{4} d^{2} f^{2} g^{2} - 70 \, b c^{3} d^{2} f g^{3} - 35 \, {\left (b^{2} c^{2} - 6 \, a c^{3}\right )} d^{2} g^{4} + {\left (48 \, c^{4} f^{4} - 40 \, b c^{3} f^{3} g - 2 \, {\left (5 \, b^{2} c^{2} - 31 \, a c^{3}\right )} f^{2} g^{2} - {\left (5 \, b^{3} c - 22 \, a b c^{2}\right )} f g^{3} - {\left (8 \, b^{4} - 41 \, a b^{2} c + 30 \, a^{2} c^{2}\right )} g^{4}\right )} e^{2} - 14 \, {\left (8 \, c^{4} d f^{3} g - 7 \, b c^{3} d f^{2} g^{2} - 2 \, {\left (b^{2} c^{2} - 6 \, a c^{3}\right )} d f g^{3} - {\left (2 \, b^{3} c - 9 \, a b c^{2}\right )} d g^{4}\right )} e\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 (70 \, c^{4} d^{2} f g^{3} - 35 \, b c^{3} d^{2} g^{4} + {\left (48 \, c^{4} f^{3} g - 16 \, b c^{3} f^{2} g^{2} - {\left (9 \, b^{2} c^{2} - 26 \, a c^{3}\right )} f g^{3} - {\left (8 \, b^{3} c - 29 \, a b c^{2}\right )} g^{4}\right )} e^{2} - 14 \, {\left (8 \, c^{4} d f^{2} g^{2} - 3 \, b c^{3} d f g^{3} - 2 \, {\left (b^{2} c^{2} - 3 \, a c^{3}\right )} d g^{4}\right )} e\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 (35 \, c^{4} d^{2} g^{4} + {\left (15 \, c^{4} g^{4} x^{2} + 24 \, c^{4} f^{2} g^{2} - 5 \, b c^{3} f g^{3} - 2 \, {\left (2 \, b^{2} c^{2} - 5 \, a c^{3}\right )} g^{4} - 3 \, {\left (6 \, c^{4} f g^{3} - b c^{3} g^{4}\right )} x\right )} e^{2} + 14 \, {\left (3 \, c^{4} d g^{4} x - 4 \, c^{4} d f g^{3} + b c^{3} d g^{4}\right )} e\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{315 \, c^{4} g^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*((70*c^4*d^2*f^2*g^2 - 70*b*c^3*d^2*f*g^3 - 35*(b^2*c^2 - 6*a*c^3)*d^2*g^4 + (48*c^4*f^4 - 40*b*c^3*f^3*
g - 2*(5*b^2*c^2 - 31*a*c^3)*f^2*g^2 - (5*b^3*c - 22*a*b*c^2)*f*g^3 - (8*b^4 - 41*a*b^2*c + 30*a^2*c^2)*g^4)*e
^2 - 14*(8*c^4*d*f^3*g - 7*b*c^3*d*f^2*g^2 - 2*(b^2*c^2 - 6*a*c^3)*d*f*g^3 - (2*b^3*c - 9*a*b*c^2)*d*g^4)*e)*s
qrt(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*(7
0*c^4*d^2*f*g^3 - 35*b*c^3*d^2*g^4 + (48*c^4*f^3*g - 16*b*c^3*f^2*g^2 - (9*b^2*c^2 - 26*a*c^3)*f*g^3 - (8*b^3*
c - 29*a*b*c^2)*g^4)*e^2 - 14*(8*c^4*d*f^2*g^2 - 3*b*c^3*d*f*g^3 - 2*(b^2*c^2 - 3*a*c^3)*d*g^4)*e)*sqrt(c*g)*w
eierstrassZeta(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^4*d^2*g^4 + (15*c^4*g^4*x^2 + 24*c^4*f^2*g^2 - 5*b*c^3*f*
g^3 - 2*(2*b^2*c^2 - 5*a*c^3)*g^4 - 3*(6*c^4*f*g^3 - b*c^3*g^4)*x)*e^2 + 14*(3*c^4*d*g^4*x - 4*c^4*d*f*g^3 + b
*c^3*d*g^4)*e)*sqrt(c*x^2 + b*x + a)*sqrt(g*x + f))/(c^4*g^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{2} \sqrt {a + b x + c x^{2}}}{\sqrt {f + g x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^2\,\sqrt {c\,x^2+b\,x+a}}{\sqrt {f+g\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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