3.1.79 \(\int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx\) [79]

Optimal. Leaf size=981 \[ -\frac {2 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{3465 a^4 e^4}+\frac {2}{11} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^5 \sqrt {d+e x}+\frac {2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac {2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac {2 (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{7/2}}{99 a e^4}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 c^2 e^2\right )\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3465 a^5 e^5 \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right ) \left (128 a^4 d^4-64 b^4 e^4-4 a b^2 e^3 (7 b d-69 c e)+4 a^3 d^2 e (2 b d+3 c e)-3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3465 a^5 e^5 \sqrt {d+e x} \left (c+b x+a x^2\right )} \]

[Out]

2/3465*(233*a^3*d^3+48*b^3*e^3+a*b*e^2*(67*b*d-157*c*e)+4*a^2*d*e*(18*b*d-37*c*e))*x*(e*x+d)^(3/2)*(a+c/x^2+b/
x)^(1/2)/a^3/e^4-2/693*(29*a^2*d^2+8*b^2*e^2+a*e*(19*b*d-18*c*e))*x*(e*x+d)^(5/2)*(a+c/x^2+b/x)^(1/2)/a^2/e^4+
2/99*(a*d+b*e)*x*(e*x+d)^(7/2)*(a+c/x^2+b/x)^(1/2)/a/e^4-2/3465*(187*a^4*d^4+64*b^4*e^4+4*a*b^2*e^3*(7*b*d-69*
c*e)-4*a^3*d^2*e*(2*b*d+3*c*e)+3*a^2*e^2*(3*b^2*d^2-29*b*c*d*e+50*c^2*e^2))*x*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2
)/a^4/e^4+2/11*x^5*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)+1/3465*(128*a^5*d^5+128*b^5*e^5-4*a^4*d^3*e*(14*b*d-27*c*
e)-8*a*b^3*e^4*(7*b*d+87*c*e)-a^2*b*e^3*(37*b^2*d^2-258*b*c*d*e-771*c^2*e^2)-a^3*d*e^2*(37*b^2*d^2-135*b*c*d*e
+156*c^2*e^2))*x*EllipticE(1/2*((b+2*a*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b
^2)^(1/2)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2
)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)/a^5/e^5/(a*x^2+b*x+c)/(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/
2)-2/3465*(a*d^2-e*(b*d-c*e))*(128*a^4*d^4-64*b^4*e^4-4*a*b^2*e^3*(7*b*d-69*c*e)+4*a^3*d^2*e*(2*b*d+3*c*e)-3*a
^2*e^2*(3*b^2*d^2-29*b*c*d*e+50*c^2*e^2))*x*EllipticF(1/2*((b+2*a*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1
/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a+c/
x^2+b/x)^(1/2)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)*(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/a^5/e^
5/(a*x^2+b*x+c)/(e*x+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 3.92, antiderivative size = 981, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1587, 932, 1667, 857, 732, 435, 430} \begin {gather*} \frac {2}{11} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x} x^5+\frac {2 (a d+b e) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} (d+e x)^{7/2} x}{99 a e^4}-\frac {2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} (d+e x)^{5/2} x}{693 a^2 e^4}+\frac {2 \left (233 a^3 d^3+4 a^2 e (18 b d-37 c e) d+48 b^3 e^3+a b e^2 (67 b d-157 c e)\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} (d+e x)^{3/2} x}{3465 a^3 e^4}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (128 a^5 d^5-4 a^4 e (14 b d-27 c e) d^3-a^3 e^2 \left (37 b^2 d^2-135 b c e d+156 c^2 e^2\right ) d+128 b^5 e^5-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c e d-771 c^2 e^2\right )\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\text {ArcSin}\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 ) x}{3465 a^5 e^5 \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right ) \left (128 a^4 d^4+4 a^3 e (2 b d+3 c e) d^2-64 b^4 e^4-4 a b^2 e^3 (7 b d-69 c e)-3 a^2 e^2 \left (3 b^2 d^2-29 b c e d+50 c^2 e^2\right )\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\text {ArcSin}\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 ) x}{3465 a^5 e^5 \sqrt {d+e x} \left (a x^2+b x+c\right )}-\frac {2 \left (187 a^4 d^4-4 a^3 e (2 b d+3 c e) d^2+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c e d+50 c^2 e^2\right )\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x} x}{3465 a^4 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c/x^2 + b/x]*x^4*Sqrt[d + e*x],x]

[Out]

(-2*(187*a^4*d^4 + 64*b^4*e^4 + 4*a*b^2*e^3*(7*b*d - 69*c*e) - 4*a^3*d^2*e*(2*b*d + 3*c*e) + 3*a^2*e^2*(3*b^2*
d^2 - 29*b*c*d*e + 50*c^2*e^2))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x])/(3465*a^4*e^4) + (2*Sqrt[a + c/x^2 + b/
x]*x^5*Sqrt[d + e*x])/11 + (2*(233*a^3*d^3 + 48*b^3*e^3 + a*b*e^2*(67*b*d - 157*c*e) + 4*a^2*d*e*(18*b*d - 37*
c*e))*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(3/2))/(3465*a^3*e^4) - (2*(29*a^2*d^2 + 8*b^2*e^2 + a*e*(19*b*d - 18*
c*e))*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(5/2))/(693*a^2*e^4) + (2*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*(d + e*x
)^(7/2))/(99*a*e^4) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(128*a^5*d^5 + 128*b^5*e^5 - 4*a^4*d^3*e*(14*b*d - 27*c*e) -
8*a*b^3*e^4*(7*b*d + 87*c*e) - a^2*b*e^3*(37*b^2*d^2 - 258*b*c*d*e - 771*c^2*e^2) - a^3*d*e^2*(37*b^2*d^2 - 13
5*b*c*d*e + 156*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))]*E
llipticE[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)])/(3465*a^5*e^5*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]*(a*d^2 - e*(b*d - c*e))*(128*a^4*d^4 - 64*b^4*e^4 - 4*a*b^2*e^3*(
7*b*d - 69*c*e) + 4*a^3*d^2*e*(2*b*d + 3*c*e) - 3*a^2*e^2*(3*b^2*d^2 - 29*b*c*d*e + 50*c^2*e^2))*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)])/(3465*a^5*e^5*Sqrt[d + e*x]*(c + b*x + a*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 1587

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 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 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx &=\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int x^3 \sqrt {d+e x} \sqrt {c+b x+a x^2} \, dx}{\sqrt {c+b x+a x^2}}\\ &=\frac {2}{11} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^5 \sqrt {d+e x}-\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {x^3 \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}{11 \sqrt {c+b x+a x^2}}\\ &=\frac {2}{11} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^5 \sqrt {d+e x}+\frac {2 (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{7/2}}{99 a e^4}-\frac {\left (2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\frac {1}{2} d^3 e (a d+b e) (b d+7 c e)+\frac {1}{2} d^2 e (a d+b e) \left (2 a d^2+e (11 b d+21 c e)\right ) x+\frac {3}{2} d e^2 (a d+b e) \left (5 a d^2+e (9 b d+7 c e)\right ) x^2+\frac {1}{2} e^3 \left (33 a^2 d^3+2 a d e (29 b d-10 c e)+b e^2 (25 b d+7 c e)\right ) x^3+\frac {1}{2} e^4 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) x^4}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{99 a e^5 \sqrt {c+b x+a x^2}}\\ &=\frac {2}{11} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^5 \sqrt {d+e x}-\frac {2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac {2 (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{7/2}}{99 a e^4}-\frac {\left (4 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {-\frac {1}{2} d^2 e^5 \left (4 b^2 e^2 (b d+5 c e)+a^2 d^2 (11 b d+48 c e)+a e \left (6 b^2 d^2+14 b c d e-45 c^2 e^2\right )\right )-\frac {1}{4} d e^5 \left (44 a^3 d^4+16 b^2 e^3 (4 b d+5 c e)+a^2 d^2 e (179 b d+107 c e)+a e^2 \left (91 b^2 d^2-101 b c d e-180 c^2 e^2\right )\right ) x-\frac {1}{2} e^6 \left (107 a^3 d^4+2 a^2 d^2 e (73 b d-50 c e)+4 b^2 e^3 (13 b d+5 c e)+a e^2 \left (73 b^2 d^2-143 b c d e-45 c^2 e^2\right )\right ) x^2-\frac {1}{4} e^7 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) x^3}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{693 a^2 e^9 \sqrt {c+b x+a x^2}}\\ &=\frac {2}{11} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^5 \sqrt {d+e x}+\frac {2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac {2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac {2 (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{7/2}}{99 a e^4}-\frac {\left (8 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\frac {3}{8} d e^8 \left (16 b^3 e^3 (b d+3 c e)+a^3 d^3 (41 b d+73 c e)+a b e^2 \left (9 b^2 d^2-52 b c d e-157 c^2 e^2\right )+2 a^2 d e \left (2 b^2 d^2-12 b c d e+c^2 e^2\right )\right )+\frac {3}{8} e^8 \left (82 a^4 d^5+2 a^3 d^3 e (69 b d-22 c e)+16 b^3 e^4 (5 b d+3 c e)+a b e^3 \left (37 b^2 d^2-328 b c d e-157 c^2 e^2\right )+a^2 d e^2 \left (13 b^2 d^2-111 b c d e+152 c^2 e^2\right )\right ) x+\frac {3}{8} e^9 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) x^2}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{3465 a^3 e^{12} \sqrt {c+b x+a x^2}}\\ &=-\frac {2 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{3465 a^4 e^4}+\frac {2}{11} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^5 \sqrt {d+e x}+\frac {2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac {2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac {2 (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{7/2}}{99 a e^4}-\frac {\left (16 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {-\frac {3}{8} e^{10} \left (16 a^4 d^4 (2 b d-c e)+32 b^4 e^4 (b d+c e)-a^3 d^2 e \left (10 b^2 d^2-26 b c d e+9 c^2 e^2\right )-2 a b^2 e^3 \left (5 b^2 d^2+98 b c d e+69 c^2 e^2\right )-3 a^2 e^2 \left (3 b^3 d^3-13 b^2 c d^2 e-89 b c^2 d e^2-25 c^3 e^3\right )\right )-\frac {3}{16} e^{10} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 c^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{10395 a^4 e^{14} \sqrt {c+b x+a x^2}}\\ &=-\frac {2 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{3465 a^4 e^4}+\frac {2}{11} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^5 \sqrt {d+e x}+\frac {2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac {2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac {2 (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{7/2}}{99 a e^4}+\frac {\left (\left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 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}{3465 a^4 e^5 \sqrt {c+b x+a x^2}}-\frac {\left (16 \left (\frac {3}{16} d e^{10} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 c^2 e^2\right )\right )-\frac {3}{8} e^{11} \left (16 a^4 d^4 (2 b d-c e)+32 b^4 e^4 (b d+c e)-a^3 d^2 e \left (10 b^2 d^2-26 b c d e+9 c^2 e^2\right )-2 a b^2 e^3 \left (5 b^2 d^2+98 b c d e+69 c^2 e^2\right )-3 a^2 e^2 \left (3 b^3 d^3-13 b^2 c d^2 e-89 b c^2 d e^2-25 c^3 e^3\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}{10395 a^4 e^{15} \sqrt {c+b x+a x^2}}\\ &=-\frac {2 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{3465 a^4 e^4}+\frac {2}{11} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^5 \sqrt {d+e x}+\frac {2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac {2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac {2 (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{7/2}}{99 a e^4}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 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 ) \text {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 )}{3465 a^5 e^5 \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 (32 \sqrt {2} \sqrt {b^2-4 a c} \left (\frac {3}{16} d e^{10} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 c^2 e^2\right )\right )-\frac {3}{8} e^{11} \left (16 a^4 d^4 (2 b d-c e)+32 b^4 e^4 (b d+c e)-a^3 d^2 e \left (10 b^2 d^2-26 b c d e+9 c^2 e^2\right )-2 a b^2 e^3 \left (5 b^2 d^2+98 b c d e+69 c^2 e^2\right )-3 a^2 e^2 \left (3 b^3 d^3-13 b^2 c d^2 e-89 b c^2 d e^2-25 c^3 e^3\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 ) \text {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 )}{10395 a^5 e^{15} \sqrt {d+e x} \left (c+b x+a x^2\right )}\\ &=-\frac {2 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{3465 a^4 e^4}+\frac {2}{11} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^5 \sqrt {d+e x}+\frac {2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac {2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac {2 (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{7/2}}{99 a e^4}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 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 )}{3465 a^5 e^5 \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 (128 a^4 d^4+8 a^3 b d^3 e-9 a^2 b^2 d^2 e^2+12 a^3 c d^2 e^2-28 a b^3 d e^3+87 a^2 b c d e^3-64 b^4 e^4+276 a b^2 c e^4-150 a^2 c^2 e^4\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 )}{3465 a^5 e^5 \sqrt {d+e x} \left (c+b x+a x^2\right )}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 33.37, size = 10904, normalized size = 11.12 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c/x^2 + b/x]*x^4*Sqrt[d + e*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. \(11937\) vs. \(2(899)=1798\).
time = 0.28, size = 11938, normalized size = 12.17

method	result	size
risch	\(\text {Expression too large to display}\)	\(5004\)
default	\(\text {Expression too large to display}\)	\(11938\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(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(x^4*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x*e + d)*sqrt(a + b/x + c/x^2)*x^4, x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.16, size = 894, normalized size = 0.91 \begin {gather*} -\frac {2 \, {\left ({\left (128 \, a^{6} d^{6} - 120 \, a^{5} b d^{5} e - 3 \, {\left (11 \, a^{4} b^{2} - 68 \, a^{5} c\right )} d^{4} e^{2} - {\left (20 \, a^{3} b^{3} - 87 \, a^{4} b c\right )} d^{3} e^{3} - 3 \, {\left (11 \, a^{2} b^{4} - 53 \, a^{3} b^{2} c + 34 \, a^{4} c^{2}\right )} d^{2} e^{4} - 3 \, {\left (40 \, a b^{5} - 246 \, a^{2} b^{3} c + 329 \, a^{3} b c^{2}\right )} d e^{5} + {\left (128 \, b^{6} - 888 \, a b^{4} c + 1599 \, a^{2} b^{2} c^{2} - 450 \, a^{3} c^{3}\right )} e^{6}\right )} \sqrt {a} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, a^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, a^{3}}, \frac {{\left (a d + {\left (3 \, a x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, a}\right ) + 3 \, {\left (128 \, a^{6} d^{5} e - 56 \, a^{5} b d^{4} e^{2} - {\left (37 \, a^{4} b^{2} - 108 \, a^{5} c\right )} d^{3} e^{3} - {\left (37 \, a^{3} b^{3} - 135 \, a^{4} b c\right )} d^{2} e^{4} - 2 \, {\left (28 \, a^{2} b^{4} - 129 \, a^{3} b^{2} c + 78 \, a^{4} c^{2}\right )} d e^{5} + {\left (128 \, a b^{5} - 696 \, a^{2} b^{3} c + 771 \, a^{3} b c^{2}\right )} e^{6}\right )} \sqrt {a} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, a^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, a^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, a^{3}}, \frac {{\left (a d + {\left (3 \, a x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, a}\right )\right ) + 3 \, {\left (64 \, a^{6} d^{4} x e^{2} - {\left (315 \, a^{6} x^{5} + 35 \, a^{5} b x^{4} - 10 \, {\left (4 \, a^{4} b^{2} - 9 \, a^{5} c\right )} x^{3} + {\left (48 \, a^{3} b^{3} - 157 \, a^{4} b c\right )} x^{2} - 2 \, {\left (32 \, a^{2} b^{4} - 138 \, a^{3} b^{2} c + 75 \, a^{4} c^{2}\right )} x\right )} e^{6} - {\left (35 \, a^{6} d x^{4} + 10 \, a^{5} b d x^{3} - {\left (13 \, a^{4} b^{2} - 32 \, a^{5} c\right )} d x^{2} + 10 \, {\left (2 \, a^{3} b^{3} - 7 \, a^{4} b c\right )} d x\right )} e^{5} + {\left (40 \, a^{6} d^{2} x^{3} + 13 \, a^{5} b d^{2} x^{2} - 2 \, {\left (9 \, a^{4} b^{2} - 23 \, a^{5} c\right )} d^{2} x\right )} e^{4} - 4 \, {\left (12 \, a^{6} d^{3} x^{2} + 5 \, a^{5} b d^{3} x\right )} e^{3}\right )} \sqrt {x e + d} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right )} e^{\left (-6\right )}}{10395 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-2/10395*((128*a^6*d^6 - 120*a^5*b*d^5*e - 3*(11*a^4*b^2 - 68*a^5*c)*d^4*e^2 - (20*a^3*b^3 - 87*a^4*b*c)*d^3*e
^3 - 3*(11*a^2*b^4 - 53*a^3*b^2*c + 34*a^4*c^2)*d^2*e^4 - 3*(40*a*b^5 - 246*a^2*b^3*c + 329*a^3*b*c^2)*d*e^5 +
(128*b^6 - 888*a*b^4*c + 1599*a^2*b^2*c^2 - 450*a^3*c^3)*e^6)*sqrt(a)*e^(1/2)*weierstrassPInverse(4/3*(a^2*d^
2 - a*b*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/a^2, -4/27*(2*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2
*b^3 - 9*a*b*c)*e^3)*e^(-3)/a^3, 1/3*(a*d + (3*a*x + b)*e)*e^(-1)/a) + 3*(128*a^6*d^5*e - 56*a^5*b*d^4*e^2 - (
37*a^4*b^2 - 108*a^5*c)*d^3*e^3 - (37*a^3*b^3 - 135*a^4*b*c)*d^2*e^4 - 2*(28*a^2*b^4 - 129*a^3*b^2*c + 78*a^4*
c^2)*d*e^5 + (128*a*b^5 - 696*a^2*b^3*c + 771*a^3*b*c^2)*e^6)*sqrt(a)*e^(1/2)*weierstrassZeta(4/3*(a^2*d^2 - a
*b*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/a^2, -4/27*(2*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2*b^3
- 9*a*b*c)*e^3)*e^(-3)/a^3, weierstrassPInverse(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/a^2, -4/27*
(2*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/a^3, 1/3*(a*d + (3*a*x
+ b)*e)*e^(-1)/a)) + 3*(64*a^6*d^4*x*e^2 - (315*a^6*x^5 + 35*a^5*b*x^4 - 10*(4*a^4*b^2 - 9*a^5*c)*x^3 + (48*a^
3*b^3 - 157*a^4*b*c)*x^2 - 2*(32*a^2*b^4 - 138*a^3*b^2*c + 75*a^4*c^2)*x)*e^6 - (35*a^6*d*x^4 + 10*a^5*b*d*x^3
- (13*a^4*b^2 - 32*a^5*c)*d*x^2 + 10*(2*a^3*b^3 - 7*a^4*b*c)*d*x)*e^5 + (40*a^6*d^2*x^3 + 13*a^5*b*d^2*x^2 -
2*(9*a^4*b^2 - 23*a^5*c)*d^2*x)*e^4 - 4*(12*a^6*d^3*x^2 + 5*a^5*b*d^3*x)*e^3)*sqrt(x*e + d)*sqrt((a*x^2 + b*x
+ c)/x^2))*e^(-6)/a^6

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+c/x**2+b/x)**(1/2)*(e*x+d)**(1/2),x)

[Out]

Timed out

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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(x^4*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(x*e + d)*sqrt(a + b/x + c/x^2)*x^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d + e*x)^(1/2)*(a + b/x + c/x^2)^(1/2),x)

[Out]

int(x^4*(d + e*x)^(1/2)*(a + b/x + c/x^2)^(1/2), x)

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