3.9.93 \(\int \frac {(d+e x)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx\) [893]

Optimal. Leaf size=1098 \[ \frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{315 c^4 g^5 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)+2 c^3 \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{315 c^4 g^5 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]

[Out]

-2/315*e*(6*b^2*e^2*g^2+c*e*g*(-14*a*e*g-27*b*d*g+17*b*e*f)-2*c^2*(42*d^2*g^2-111*d*e*f*g+64*e^2*f^2))*(g*x+f)
^(3/2)*(c*x^2+b*x+a)^(1/2)/c^2/g^4-2/63*e^2*(-b*e*g-6*c*d*g+8*c*e*f)*(g*x+f)^(5/2)*(c*x^2+b*x+a)^(1/2)/c/g^4+2
/315*(8*b^3*e^3*g^3+3*b*c*e^2*g^2*(-9*a*e*g-12*b*d*g+5*b*e*f)-c^3*(-70*d^3*g^3+336*d^2*e*f*g^2-408*d*e^2*f^2*g
+152*e^3*f^3)-3*c^2*e*g*(6*a*e*g*(-5*d*g+2*e*f)-b*(21*d^2*g^2-24*d*e*f*g+8*e^2*f^2)))*(g*x+f)^(1/2)*(c*x^2+b*x
+a)^(1/2)/c^3/g^4+2/9*(e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/g-1/315*(16*b^4*e^3*g^4+8*b^2*c*e^2*g^3*(-9*
a*e*g-9*b*d*g+2*b*e*f)-2*c^4*f*(-105*d^3*g^3+252*d^2*e*f*g^2-216*d*e^2*f^2*g+64*e^3*f^3)+3*c^2*e*g^2*(14*a^2*e
^2*g^2-a*b*e*g*(-87*d*g+19*e*f)+b^2*(42*d^2*g^2-27*d*e*f*g+7*e^2*f^2))-c^3*g*(6*a*e*g*(63*d^2*g^2-39*d*e*f*g+1
0*e^2*f^2)-b*(-105*d^3*g^3+189*d^2*e*f*g^2-144*d*e^2*f^2*g+40*e^3*f^3)))*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^5/(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^3*g^3+3*b*c*e^2*g^2*(-9*a*e*g-
12*b*d*g+5*b*e*f)+2*c^3*(-105*d^3*g^3+252*d^2*e*f*g^2-216*d*e^2*f^2*g+64*e^3*f^3)-3*c^2*e*g*(6*a*e*g*(-5*d*g+2
*e*f)-b*(21*d^2*g^2-24*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^5/(g*x+f)^(1/2)/(c*x
^2+b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 4.41, antiderivative size = 1098, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {934, 1667, 857, 732, 435, 430} \begin {gather*} \frac {2 \sqrt {f+g x} \sqrt {c x^2+b x+a} (d+e x)^3}{9 g}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (-2 f \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right ) c^4-g \left (6 a e g \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 g f^2+189 d^2 e g^2 f-105 d^3 g^3\right )\right ) c^3+3 e g^2 \left (\left (7 e^2 f^2-27 d e g f+42 d^2 g^2\right ) b^2-a e g (19 e f-87 d g) b+14 a^2 e^2 g^2\right ) c^2+8 b^2 e^2 g^3 (2 b e f-9 b d g-9 a e g) c+16 b^4 e^3 g^4\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\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 )}{315 c^4 g^5 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b g f+a g^2\right ) \left (2 \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right ) c^3-3 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) c^2+3 b e^2 g^2 (5 b e f-12 b d g-9 a e g) c+8 b^3 e^3 g^3\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\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 )}{315 c^4 g^5 \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{63 c g^4}-\frac {2 e \left (-2 \left (64 e^2 f^2-111 d e g f+42 d^2 g^2\right ) c^2+e g (17 b e f-27 b d g-14 a e g) c+6 b^2 e^2 g^2\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{315 c^2 g^4}+\frac {2 \left (-\left (\left (152 e^3 f^3-408 d e^2 g f^2+336 d^2 e g^2 f-70 d^3 g^3\right ) c^3\right )-3 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) c^2+3 b e^2 g^2 (5 b e f-12 b d g-9 a e g) c+8 b^3 e^3 g^3\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}{315 c^3 g^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 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)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx &=\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {\int \frac {(d+e x)^2 \left (b d f+6 a e f-8 a d g+(2 c d f+7 b e f-7 b d g-2 a e g) x+(8 c e f-6 c d g-b e g) x^2\right )}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{9 g}\\ &=\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {2 \int \frac {\frac {1}{2} g \left (b^2 e^3 f^3 g-2 a c g \left (20 e^3 f^3-15 d e^2 f^2 g-21 d^2 e f g^2+28 d^3 g^3\right )+b f \left (5 a e^3 f g^2-c \left (8 e^3 f^3-6 d e^2 f^2 g-7 d^3 g^3\right )\right )\right )+\frac {1}{2} g \left (2 b e^3 f g^2 (4 b f+5 a g)-2 c^2 \left (8 e^3 f^4-6 d e^2 f^3 g-7 d^3 f g^3\right )-c g \left (2 a e g \left (40 e^2 f^2-72 d e f g+63 d^2 g^2\right )+b \left (62 e^3 f^3-48 d e^2 f^2 g-63 d^2 e f g^2+49 d^3 g^3\right )\right )\right ) x+\frac {1}{2} g^2 \left (b e^3 g^2 (13 b f+5 a g)-c^2 \left (88 e^3 f^3-66 d e^2 f^2 g-84 d^2 e f g^2+42 d^3 g^3\right )+c e g \left (2 a e g (e f-27 d g)-3 b \left (31 e^2 f^2-61 d e f g+35 d^2 g^2\right )\right )\right ) x^2+\frac {1}{2} e g^3 \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) x^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{63 c g^5}\\ &=\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {4 \int \frac {-\frac {1}{4} g^4 \left (6 b^3 e^3 f^2 g^2+3 b^2 e^2 f g \left (6 a e g^2+c f (4 e f-9 d g)\right )-2 a c g \left (21 a e^3 f g^2+c \left (92 e^3 f^3-258 d e^2 f^2 g+231 d^2 e f g^2-140 d^3 g^3\right )\right )+b c f \left (3 a e^2 g^2 (4 e f-27 d g)-c \left (88 e^3 f^3-192 d e^2 f^2 g+84 d^2 e f g^2+35 d^3 g^3\right )\right )\right )-\frac {1}{4} g^4 \left (6 b^2 e^3 g^3 (5 b f+3 a g)-2 c^3 f \left (88 e^3 f^3-192 d e^2 f^2 g+84 d^2 e f g^2+35 d^3 g^3\right )-3 c e^2 g^2 \left (14 a^2 e g^2-b^2 f (19 e f-45 d g)+a b g (23 e f+27 d g)\right )-c^2 g \left (6 a e g \left (2 e^2 f^2+9 d e f g-63 d^2 g^2\right )+b \left (296 e^3 f^3-816 d e^2 f^2 g+735 d^2 e f g^2-245 d^3 g^3\right )\right )\right ) x-\frac {3}{4} g^5 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-2 c^3 \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{315 c^2 g^8}\\ &=\frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {8 \int \frac {\frac {3}{8} g^6 \left (8 b^4 e^3 f g^3+b^3 e^2 g^2 \left (8 a e g^2+9 c f (e f-4 d g)\right )-3 b^2 c e g \left (2 a e g^2 (5 e f+6 d g)-c f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )+2 a c^2 g \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-b c \left (27 a^2 e^3 g^4+3 a c e g^2 \left (8 e^2 f^2-33 d e f g-21 d^2 g^2\right )+c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right )+\frac {3}{8} g^6 \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{945 c^3 g^{10}}\\ &=\frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {\left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{315 c^3 g^5}-\frac {\left (8 \left (-\frac {3}{8} f g^6 \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right )+\frac {3}{8} g^7 \left (8 b^4 e^3 f g^3+b^3 e^2 g^2 \left (8 a e g^2+9 c f (e f-4 d g)\right )-3 b^2 c e g \left (2 a e g^2 (5 e f+6 d g)-c f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )+2 a c^2 g \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-b c \left (27 a^2 e^3 g^4+3 a c e g^2 \left (8 e^2 f^2-33 d e f g-21 d^2 g^2\right )+c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{945 c^3 g^{11}}\\ &=\frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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^5 \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}-\frac {\left (16 \sqrt {2} \sqrt {b^2-4 a c} \left (-\frac {3}{8} f g^6 \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right )+\frac {3}{8} g^7 \left (8 b^4 e^3 f g^3+b^3 e^2 g^2 \left (8 a e g^2+9 c f (e f-4 d g)\right )-3 b^2 c e g \left (2 a e g^2 (5 e f+6 d g)-c f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )+2 a c^2 g \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-b c \left (27 a^2 e^3 g^4+3 a c e g^2 \left (8 e^2 f^2-33 d e f g-21 d^2 g^2\right )+c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 )}{945 c^4 g^{11} \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{315 c^4 g^5 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (128 c^3 e^3 f^3-432 c^3 d e^2 f^2 g+24 b c^2 e^3 f^2 g+504 c^3 d^2 e f g^2-72 b c^2 d e^2 f g^2+15 b^2 c e^3 f g^2-36 a c^2 e^3 f g^2-210 c^3 d^3 g^3+63 b c^2 d^2 e g^3-36 b^2 c d e^2 g^3+90 a c^2 d e^2 g^3+8 b^3 e^3 g^3-27 a b c e^3 g^3\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{315 c^4 g^5 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 34.28, size = 17771, normalized size = 16.18 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(22214\) vs. \(2(1022)=2044\).
time = 0.16, size = 22215, normalized size = 20.23

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1845\)
risch	\(\text {Expression too large to display}\)	\(7892\)
default	\(\text {Expression too large to display}\)	\(22215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*x^2 + b*x + a)*(x*e + d)^3/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 = 0.62, size = 1267, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left ({\left (210 \, c^{5} d^{3} f^{2} g^{3} - 210 \, b c^{4} d^{3} f g^{4} - 105 \, {\left (b^{2} c^{3} - 6 \, a c^{4}\right )} d^{3} g^{5} - {\left (128 \, c^{5} f^{5} - 104 \, b c^{4} f^{4} g - {\left (25 \, b^{2} c^{3} - 156 \, a c^{4}\right )} f^{3} g^{2} - 5 \, {\left (2 \, b^{3} c^{2} - 9 \, a b c^{3}\right )} f^{2} g^{3} - {\left (8 \, b^{4} c - 39 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} f g^{4} - {\left (16 \, b^{5} - 96 \, a b^{3} c + 123 \, a^{2} b c^{2}\right )} g^{5}\right )} e^{3} + 9 \, {\left (48 \, c^{5} d f^{4} g - 40 \, b c^{4} d f^{3} g^{2} - 2 \, {\left (5 \, b^{2} c^{3} - 31 \, a c^{4}\right )} d f^{2} g^{3} - {\left (5 \, b^{3} c^{2} - 22 \, a b c^{3}\right )} d f g^{4} - {\left (8 \, b^{4} c - 41 \, a b^{2} c^{2} + 30 \, a^{2} c^{3}\right )} d g^{5}\right )} e^{2} - 63 \, {\left (8 \, c^{5} d^{2} f^{3} g^{2} - 7 \, b c^{4} d^{2} f^{2} g^{3} - 2 \, {\left (b^{2} c^{3} - 6 \, a c^{4}\right )} d^{2} f g^{4} - {\left (2 \, b^{3} c^{2} - 9 \, a b c^{3}\right )} d^{2} g^{5}\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 (210 \, c^{5} d^{3} f g^{4} - 105 \, b c^{4} d^{3} g^{5} - {\left (128 \, c^{5} f^{4} g - 40 \, b c^{4} f^{3} g^{2} - 3 \, {\left (7 \, b^{2} c^{3} - 20 \, a c^{4}\right )} f^{2} g^{3} - {\left (16 \, b^{3} c^{2} - 57 \, a b c^{3}\right )} f g^{4} - 2 \, {\left (8 \, b^{4} c - 36 \, a b^{2} c^{2} + 21 \, a^{2} c^{3}\right )} g^{5}\right )} e^{3} + 9 \, {\left (48 \, c^{5} d f^{3} g^{2} - 16 \, b c^{4} d f^{2} g^{3} - {\left (9 \, b^{2} c^{3} - 26 \, a c^{4}\right )} d f g^{4} - {\left (8 \, b^{3} c^{2} - 29 \, a b c^{3}\right )} d g^{5}\right )} e^{2} - 63 \, {\left (8 \, c^{5} d^{2} f^{2} g^{3} - 3 \, b c^{4} d^{2} f g^{4} - 2 \, {\left (b^{2} c^{3} - 3 \, a c^{4}\right )} d^{2} g^{5}\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 (105 \, c^{5} d^{3} g^{5} + {\left (35 \, c^{5} g^{5} x^{3} - 64 \, c^{5} f^{3} g^{2} + 12 \, b c^{4} f^{2} g^{3} + {\left (9 \, b^{2} c^{3} - 22 \, a c^{4}\right )} f g^{4} + {\left (8 \, b^{3} c^{2} - 27 \, a b c^{3}\right )} g^{5} - 5 \, {\left (8 \, c^{5} f g^{4} - b c^{4} g^{5}\right )} x^{2} + {\left (48 \, c^{5} f^{2} g^{3} - 7 \, b c^{4} f g^{4} - 2 \, {\left (3 \, b^{2} c^{3} - 7 \, a c^{4}\right )} g^{5}\right )} x\right )} e^{3} + 9 \, {\left (15 \, c^{5} d g^{5} x^{2} + 24 \, c^{5} d f^{2} g^{3} - 5 \, b c^{4} d f g^{4} - 2 \, {\left (2 \, b^{2} c^{3} - 5 \, a c^{4}\right )} d g^{5} - 3 \, {\left (6 \, c^{5} d f g^{4} - b c^{4} d g^{5}\right )} x\right )} e^{2} + 63 \, {\left (3 \, c^{5} d^{2} g^{5} x - 4 \, c^{5} d^{2} f g^{4} + b c^{4} d^{2} g^{5}\right )} e\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{945 \, c^{5} g^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/945*((210*c^5*d^3*f^2*g^3 - 210*b*c^4*d^3*f*g^4 - 105*(b^2*c^3 - 6*a*c^4)*d^3*g^5 - (128*c^5*f^5 - 104*b*c^4
*f^4*g - (25*b^2*c^3 - 156*a*c^4)*f^3*g^2 - 5*(2*b^3*c^2 - 9*a*b*c^3)*f^2*g^3 - (8*b^4*c - 39*a*b^2*c^2 + 24*a
^2*c^3)*f*g^4 - (16*b^5 - 96*a*b^3*c + 123*a^2*b*c^2)*g^5)*e^3 + 9*(48*c^5*d*f^4*g - 40*b*c^4*d*f^3*g^2 - 2*(5
*b^2*c^3 - 31*a*c^4)*d*f^2*g^3 - (5*b^3*c^2 - 22*a*b*c^3)*d*f*g^4 - (8*b^4*c - 41*a*b^2*c^2 + 30*a^2*c^3)*d*g^
5)*e^2 - 63*(8*c^5*d^2*f^3*g^2 - 7*b*c^4*d^2*f^2*g^3 - 2*(b^2*c^3 - 6*a*c^4)*d^2*f*g^4 - (2*b^3*c^2 - 9*a*b*c^
3)*d^2*g^5)*e)*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*(210*c^5*d^3*f*g^4 - 105*b*c^4*d^3*g^5 - (128*c^5*f^4*g - 40*b*c^4*f^3*g^2 - 3*(7*b^2*c^3 - 20*a
*c^4)*f^2*g^3 - (16*b^3*c^2 - 57*a*b*c^3)*f*g^4 - 2*(8*b^4*c - 36*a*b^2*c^2 + 21*a^2*c^3)*g^5)*e^3 + 9*(48*c^5
*d*f^3*g^2 - 16*b*c^4*d*f^2*g^3 - (9*b^2*c^3 - 26*a*c^4)*d*f*g^4 - (8*b^3*c^2 - 29*a*b*c^3)*d*g^5)*e^2 - 63*(8
*c^5*d^2*f^2*g^3 - 3*b*c^4*d^2*f*g^4 - 2*(b^2*c^3 - 3*a*c^4)*d^2*g^5)*e)*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), -4/2
7*(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*(105*c^5*d^3*g^5 + (35*c^5*g^5*x^3 - 64*c^5*f^3*g^2 + 12*b*c^4*f^2*g^3 + (9*b^2*c^3 - 22*
a*c^4)*f*g^4 + (8*b^3*c^2 - 27*a*b*c^3)*g^5 - 5*(8*c^5*f*g^4 - b*c^4*g^5)*x^2 + (48*c^5*f^2*g^3 - 7*b*c^4*f*g^
4 - 2*(3*b^2*c^3 - 7*a*c^4)*g^5)*x)*e^3 + 9*(15*c^5*d*g^5*x^2 + 24*c^5*d*f^2*g^3 - 5*b*c^4*d*f*g^4 - 2*(2*b^2*
c^3 - 5*a*c^4)*d*g^5 - 3*(6*c^5*d*f*g^4 - b*c^4*d*g^5)*x)*e^2 + 63*(3*c^5*d^2*g^5*x - 4*c^5*d^2*f*g^4 + b*c^4*
d^2*g^5)*e)*sqrt(c*x^2 + b*x + a)*sqrt(g*x + f))/(c^5*g^6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{3} \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)**3*(c*x**2+b*x+a)**(1/2)/(g*x+f)**(1/2),x)

[Out]

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

[Out]

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

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