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

Optimal. Leaf size=778 \[ \frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (6 a^2 c d e^2+16 a^3 d^3-3 a b e^2 (b d-9 c e)-8 b^3 e^3\right ) \left (a d^2-e (b d-c e)\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+2 a x+b}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{315 a^4 e^4 \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )-a^3 d^2 e (4 b d-9 c e)+8 a^4 d^4-4 a b^2 e^3 (b d+9 c e)+8 b^4 e^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \left (a x^2+b x+c\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{4 x (d+e x)^{3/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right )}{315 a^2 e^3}+\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (-6 a^2 c d e^2+19 a^3 d^3+3 a b e^2 (b d-9 c e)+8 b^3 e^3\right )}{315 a^3 e^3}+\frac{2 x (d+e x)^{5/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (a d+b e)}{63 a e^3}+\frac{2}{9} x^4 \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \]

[Out]

(2*(19*a^3*d^3 - 6*a^2*c*d*e^2 + 8*b^3*e^3 + 3*a*b*e^2*(b*d - 9*c*e))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x])/(
315*a^3*e^3) + (2*Sqrt[a + c/x^2 + b/x]*x^4*Sqrt[d + e*x])/9 - (4*(8*a^2*d^2 + 3*b^2*e^2 + a*e*(4*b*d - 7*c*e)
)*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(3/2))/(315*a^2*e^3) + (2*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(5
/2))/(63*a*e^3) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8*a^4*d^4 + 8*b^4*e^4 - a^3*d^2*e*(4*b*d - 9*c*e) - 4*a*b^2*e^
3*(b*d + 9*c*e) - 3*a^2*e^2*(b^2*d^2 - 5*b*c*d*e - 7*c^2*e^2))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x]*Sqrt[-((a
*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/S
qrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(315*a^4*e^4*Sqrt[(a*(d + e*x))/(2*a*d
- (b + Sqrt[b^2 - 4*a*c])*e)]*(c + b*x + a*x^2)) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*a^3*d^3 + 6*a^2*c*d*e^2 -
8*b^3*e^3 - 3*a*b*e^2*(b*d - 9*c*e))*(a*d^2 - e*(b*d - c*e))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*x))/(2*a*
d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b
^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)
])/(315*a^4*e^4*Sqrt[d + e*x]*(c + b*x + a*x^2))

________________________________________________________________________________________

Rubi [A]  time = 2.3492, antiderivative size = 778, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {1573, 918, 1653, 843, 718, 424, 419} \[ -\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )-a^3 d^2 e (4 b d-9 c e)+8 a^4 d^4-4 a b^2 e^3 (b d+9 c e)+8 b^4 e^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \left (a x^2+b x+c\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{4 x (d+e x)^{3/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right )}{315 a^2 e^3}+\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (-6 a^2 c d e^2+19 a^3 d^3+3 a b e^2 (b d-9 c e)+8 b^3 e^3\right )}{315 a^3 e^3}+\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (6 a^2 c d e^2+16 a^3 d^3-3 a b e^2 (b d-9 c e)-8 b^3 e^3\right ) \left (a d^2-e (b d-c e)\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \sqrt{d+e x} \left (a x^2+b x+c\right )}+\frac{2 x (d+e x)^{5/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (a d+b e)}{63 a e^3}+\frac{2}{9} x^4 \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(19*a^3*d^3 - 6*a^2*c*d*e^2 + 8*b^3*e^3 + 3*a*b*e^2*(b*d - 9*c*e))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x])/(
315*a^3*e^3) + (2*Sqrt[a + c/x^2 + b/x]*x^4*Sqrt[d + e*x])/9 - (4*(8*a^2*d^2 + 3*b^2*e^2 + a*e*(4*b*d - 7*c*e)
)*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(3/2))/(315*a^2*e^3) + (2*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(5
/2))/(63*a*e^3) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8*a^4*d^4 + 8*b^4*e^4 - a^3*d^2*e*(4*b*d - 9*c*e) - 4*a*b^2*e^
3*(b*d + 9*c*e) - 3*a^2*e^2*(b^2*d^2 - 5*b*c*d*e - 7*c^2*e^2))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x]*Sqrt[-((a
*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/S
qrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(315*a^4*e^4*Sqrt[(a*(d + e*x))/(2*a*d
- (b + Sqrt[b^2 - 4*a*c])*e)]*(c + b*x + a*x^2)) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*a^3*d^3 + 6*a^2*c*d*e^2 -
8*b^3*e^3 - 3*a*b*e^2*(b*d - 9*c*e))*(a*d^2 - e*(b*d - c*e))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*x))/(2*a*
d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b
^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)
])/(315*a^4*e^4*Sqrt[d + e*x]*(c + b*x + a*x^2))

Rule 1573

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol
] :> Dist[(x^(2*n*FracPart[p])*(a + b/x^n + c/x^(2*n))^FracPart[p])/(c + b*x^n + a*x^(2*n))^FracPart[p], Int[x
^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[m
n, -n] && EqQ[mn2, 2*mn] &&  !IntegerQ[p] &&  !IntegerQ[q] && PosQ[n]

Rule 918

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*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])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), 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 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 424

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

Rule 419

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

Rubi steps

\begin{align*} \int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^3 \sqrt{d+e x} \, dx &=\frac{\left (\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int x^2 \sqrt{d+e x} \sqrt{c+b x+a x^2} \, dx}{\sqrt{c+b x+a x^2}}\\ &=\frac{2}{9} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^4 \sqrt{d+e x}-\frac{\left (\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{x^2 \left (-3 c d-2 (b d+c e) x-(a d+b e) x^2\right )}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{9 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{9} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^4 \sqrt{d+e x}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{63 a e^3}-\frac{\left (2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{\frac{1}{2} d^2 e (a d+b e) (b d+5 c e)+d e (a d+b e) \left (a d^2+e (4 b d+5 c e)\right ) x+\frac{1}{2} e^2 \left (11 a^2 d^3+8 a d e (3 b d-2 c e)+b e^2 (13 b d+5 c e)\right ) x^2+e^3 \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) x^3}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{63 a e^4 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{9} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^4 \sqrt{d+e x}-\frac{4 \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{3/2}}{315 a^2 e^3}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{63 a e^3}-\frac{\left (4 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{-\frac{1}{4} d e^4 \left (6 b^2 e^2 (b d+3 c e)+a^2 d^2 (11 b d+23 c e)+3 a e \left (b^2 d^2-5 b c d e-14 c^2 e^2\right )\right )-\frac{1}{2} e^4 \left (11 a^3 d^4+a^2 d^2 e (23 b d-15 c e)+3 b^2 e^3 (5 b d+3 c e)+3 a e^2 \left (2 b^2 d^2-16 b c d e-7 c^2 e^2\right )\right ) x-\frac{3}{4} e^5 \left (19 a^3 d^3-6 a^2 c d e^2+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) x^2}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{315 a^2 e^7 \sqrt{c+b x+a x^2}}\\ &=\frac{2 \left (19 a^3 d^3-6 a^2 c d e^2+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}}{315 a^3 e^3}+\frac{2}{9} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^4 \sqrt{d+e x}-\frac{4 \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{3/2}}{315 a^2 e^3}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{63 a e^3}-\frac{\left (8 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{\frac{3}{8} e^6 \left (4 a^3 d^3 (2 b d-c e)+8 b^3 e^3 (b d+c e)-3 a^2 d e \left (b^2 d^2-3 b c d e-12 c^2 e^2\right )-3 a b e^2 \left (b^2 d^2+14 b c d e+9 c^2 e^2\right )\right )+\frac{3}{4} e^6 \left (8 a^4 d^4+8 b^4 e^4-a^3 d^2 e (4 b d-9 c e)-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{945 a^3 e^9 \sqrt{c+b x+a x^2}}\\ &=\frac{2 \left (19 a^3 d^3-6 a^2 c d e^2+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}}{315 a^3 e^3}+\frac{2}{9} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^4 \sqrt{d+e x}-\frac{4 \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{3/2}}{315 a^2 e^3}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{63 a e^3}-\frac{\left (2 \left (8 a^4 d^4+8 b^4 e^4-a^3 d^2 e (4 b d-9 c e)-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{\sqrt{d+e x}}{\sqrt{c+b x+a x^2}} \, dx}{315 a^3 e^4 \sqrt{c+b x+a x^2}}-\frac{\left (8 \left (-\frac{3}{4} d e^6 \left (8 a^4 d^4+8 b^4 e^4-a^3 d^2 e (4 b d-9 c e)-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )\right )+\frac{3}{8} e^7 \left (4 a^3 d^3 (2 b d-c e)+8 b^3 e^3 (b d+c e)-3 a^2 d e \left (b^2 d^2-3 b c d e-12 c^2 e^2\right )-3 a b e^2 \left (b^2 d^2+14 b c d e+9 c^2 e^2\right )\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{945 a^3 e^{10} \sqrt{c+b x+a x^2}}\\ &=\frac{2 \left (19 a^3 d^3-6 a^2 c d e^2+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}}{315 a^3 e^3}+\frac{2}{9} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^4 \sqrt{d+e x}-\frac{4 \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{3/2}}{315 a^2 e^3}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{63 a e^3}-\frac{\left (2 \sqrt{2} \sqrt{b^2-4 a c} \left (8 a^4 d^4+8 b^4 e^4-a^3 d^2 e (4 b d-9 c e)-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 a d-b e-\sqrt{b^2-4 a c} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{315 a^4 e^4 \sqrt{\frac{a (d+e x)}{2 a d-b e-\sqrt{b^2-4 a c} e}} \left (c+b x+a x^2\right )}-\frac{\left (16 \sqrt{2} \sqrt{b^2-4 a c} \left (-\frac{3}{4} d e^6 \left (8 a^4 d^4+8 b^4 e^4-a^3 d^2 e (4 b d-9 c e)-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )\right )+\frac{3}{8} e^7 \left (4 a^3 d^3 (2 b d-c e)+8 b^3 e^3 (b d+c e)-3 a^2 d e \left (b^2 d^2-3 b c d e-12 c^2 e^2\right )-3 a b e^2 \left (b^2 d^2+14 b c d e+9 c^2 e^2\right )\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 a d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{945 a^4 e^{10} \sqrt{d+e x} \left (c+b x+a x^2\right )}\\ &=\frac{2 \left (19 a^3 d^3-6 a^2 c d e^2+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}}{315 a^3 e^3}+\frac{2}{9} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^4 \sqrt{d+e x}-\frac{4 \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{3/2}}{315 a^2 e^3}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{63 a e^3}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (8 a^4 d^4+8 b^4 e^4-a^3 d^2 e (4 b d-9 c e)-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (a d^2-b d e+c e^2\right ) \left (16 a^3 d^3-3 a b^2 d e^2+6 a^2 c d e^2-8 b^3 e^3+27 a b c e^3\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \sqrt{d+e x} \left (c+b x+a x^2\right )}\\ \end{align*}

Mathematica [C]  time = 13.7562, size = 7531, normalized size = 9.68 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.062, size = 9182, normalized size = 11.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x + d} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{e x + d} x^{3} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out