\(\int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx\) [80]

Optimal result

Integrand size = 29, antiderivative size = 778 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\frac {2 \left (19 a^3 d^3-6 a^2 c d e^2+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{315 a^3 e^3}+\frac {2}{9} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x}-\frac {4 \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{315 a^2 e^3}+\frac {2 (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{5/2}}{63 a e^3}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (8 a^4 d^4+8 b^4 e^4-a^3 d^2 e (4 b d-9 c e)-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (16 a^3 d^3+6 a^2 c d e^2-8 b^3 e^3-3 a b e^2 (b d-9 c e)\right ) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \sqrt {d+e x} \left (c+b x+a x^2\right )} \]

[Out]

-4/315*(8*a^2*d^2+3*b^2*e^2+a*e*(4*b*d-7*c*e))*x*(e*x+d)^(3/2)*(a+c/x^2+b/x)^(1/2)/a^2/e^3+2/63*(a*d+b*e)*x*(e
*x+d)^(5/2)*(a+c/x^2+b/x)^(1/2)/a/e^3+2/315*(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))*x*(a+c/
x^2+b/x)^(1/2)*(e*x+d)^(1/2)/a^3/e^3+2/9*x^4*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)-2/315*(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))*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^4/e^4/(a*x^2+b*x+c)/(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)+2/315*(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))*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^4/e^4/(a*x^2+b*x+c)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 778, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1587, 932, 1667, 857, 732, 435, 430} \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=-\frac {4 x (d+e x)^{3/2} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right )}{315 a^2 e^3}+\frac {2 x \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (19 a^3 d^3-6 a^2 c d e^2+3 a b e^2 (b d-9 c e)+8 b^3 e^3\right )}{315 a^3 e^3}-\frac {2 \sqrt {2} x \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (8 a^4 d^4-a^3 d^2 e (4 b d-9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )-4 a b^2 e^3 (b d+9 c e)+8 b^4 e^4\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \left (a x^2+b x+c\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {2} x \sqrt {b^2-4 a c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (16 a^3 d^3+6 a^2 c d e^2-3 a b e^2 (b d-9 c e)-8 b^3 e^3\right ) \left (a d^2-e (b d-c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \sqrt {d+e x} \left (a x^2+b x+c\right )}+\frac {2 x (d+e x)^{5/2} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} (a d+b e)}{63 a e^3}+\frac {2}{9} x^4 \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \]

[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 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*} \text {integral}& = \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 ) \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 )}{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 ) \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 )}{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] (verified)

Result contains complex when optimal does not.

Time = 34.88 (sec) , antiderivative size = 7531, normalized size of antiderivative = 9.68 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3660\) vs. \(2(702)=1404\).

Time = 2.06 (sec) , antiderivative size = 3661, normalized size of antiderivative = 4.71

method	result	size
risch	\(\text {Expression too large to display}\)	\(3661\)
default	\(\text {Expression too large to display}\)	\(9182\)

[In]

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

[Out]

2/315*(35*a^3*e^3*x^3+5*a^3*d*e^2*x^2+5*a^2*b*e^3*x^2-6*a^3*d^2*e*x+2*a^2*b*d*e^2*x+14*a^2*c*e^3*x-6*a*b^2*e^3
*x+8*a^3*d^3-3*a^2*b*d^2*e+8*a^2*c*d*e^2-3*a*b^2*d*e^2-27*a*b*c*e^3+8*b^3*e^3)*(e*x+d)^(1/2)/a^3/e^3*((a*x^2+b
*x+c)/x^2)^(1/2)*x-1/315/a^3/e^3*(16*a^3*b*d^4*(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+1/e*d)/(1/e*d-1/2*(b+(
-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*
((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+
c*e*x+c*d)^(1/2)*EllipticF(((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-1/e*d+1/2*(b+(-4*a*c+b^2)
^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))+16*b^4*d*e^3*(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((
x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*
c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^
3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)*EllipticF(((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-1
/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))+16*b^3*c*e^4*(1/e*d-1/2*(b+(
-4*a*c+b^2)^(1/2))/a)*((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a
)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d+1/2*(b+(-4*a*c+b^2)^
(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)*EllipticF(((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^
2)^(1/2))/a))^(1/2),((-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))-8*a^
3*c*d^3*e*(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*
(-b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1
/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)*EllipticF(((x+1/e*d)
/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^
2)^(1/2))/a))^(1/2))-6*a^2*b^2*d^3*e*(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2
)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b
+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)
^(1/2)*EllipticF(((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a)
/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))-6*a*b^3*d^2*e^2*(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+1/e*d
)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^
(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x
^2+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)*EllipticF(((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-1/e*d+1/
2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))+72*a^2*c^2*d*e^3*(1/e*d-1/2*(b+(-4*
a*c+b^2)^(1/2))/a)*((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(
-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/
2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)*EllipticF(((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^
(1/2))/a))^(1/2),((-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))-54*a*b*
c^2*e^4*(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-
b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e
*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)*EllipticF(((x+1/e*d)/(
1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)
^(1/2))/a))^(1/2))+18*a^2*b*c*d^2*e^2*(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^
2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(
b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d
)^(1/2)*EllipticF(((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a
)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))-84*b^2*c*d*e^3*a*(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+1/e
*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2
)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d
*x^2+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)*EllipticF(((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-1/e*d+
1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))+2*(16*a^4*d^4-8*a^3*b*d^3*e+18*a^
3*c*d^2*e^2-6*a^2*b^2*d^2*e^2+30*a^2*b*c*d*e^3+42*a^2*c^2*e^4-8*a*b^3*d*e^3-72*a*b^2*c*e^4+16*b^4*e^4)*(1/e*d-
1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^
(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d+1/2*(b+(-4*a
*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)*((-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/
a)*EllipticE(((x+1/e*d)/(1/e*d-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1
/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))+1/2*(-b+(-4*a*c+b^2)^(1/2))/a*EllipticF(((x+1/e*d)/(1/e*d-1/2*(b+(
-4*a*c+b^2)^(1/2))/a))^(1/2),((-1/e*d+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-1/e*d-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1
/2))))*((a*x^2+b*x+c)/x^2)^(1/2)*x/(a*x^2+b*x+c)*((a*x^2+b*x+c)*(e*x+d))^(1/2)/(e*x+d)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.13 (sec) , antiderivative size = 734, normalized size of antiderivative = 0.94 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\frac {2 \, {\left ({\left (16 \, a^{5} d^{5} - 16 \, a^{4} b d^{4} e - 5 \, {\left (a^{3} b^{2} - 6 \, a^{4} c\right )} d^{3} e^{2} - {\left (5 \, a^{2} b^{3} - 21 \, a^{3} b c\right )} d^{2} e^{3} - 2 \, {\left (8 \, a b^{4} - 42 \, a^{2} b^{2} c + 33 \, a^{3} c^{2}\right )} d e^{4} + {\left (16 \, b^{5} - 96 \, a b^{3} c + 123 \, a^{2} b c^{2}\right )} e^{5}\right )} \sqrt {a e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{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 )}}{27 \, a^{3} e^{3}}, \frac {3 \, a e x + a d + b e}{3 \, a e}\right ) + 6 \, {\left (8 \, a^{5} d^{4} e - 4 \, a^{4} b d^{3} e^{2} - 3 \, {\left (a^{3} b^{2} - 3 \, a^{4} c\right )} d^{2} e^{3} - {\left (4 \, a^{2} b^{3} - 15 \, a^{3} b c\right )} d e^{4} + {\left (8 \, a b^{4} - 36 \, a^{2} b^{2} c + 21 \, a^{3} c^{2}\right )} e^{5}\right )} \sqrt {a e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{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 )}}{27 \, a^{3} e^{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 )}}{3 \, a^{2} e^{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 )}}{27 \, a^{3} e^{3}}, \frac {3 \, a e x + a d + b e}{3 \, a e}\right )\right ) + 3 \, {\left (35 \, a^{5} e^{5} x^{4} + 5 \, {\left (a^{5} d e^{4} + a^{4} b e^{5}\right )} x^{3} - 2 \, {\left (3 \, a^{5} d^{2} e^{3} - a^{4} b d e^{4} + {\left (3 \, a^{3} b^{2} - 7 \, a^{4} c\right )} e^{5}\right )} x^{2} + {\left (8 \, a^{5} d^{3} e^{2} - 3 \, a^{4} b d^{2} e^{3} - {\left (3 \, a^{3} b^{2} - 8 \, a^{4} c\right )} d e^{4} + {\left (8 \, a^{2} b^{3} - 27 \, a^{3} b c\right )} e^{5}\right )} x\right )} \sqrt {e x + d} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right )}}{945 \, a^{5} e^{5}} \]

[In]

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

[Out]

2/945*((16*a^5*d^5 - 16*a^4*b*d^4*e - 5*(a^3*b^2 - 6*a^4*c)*d^3*e^2 - (5*a^2*b^3 - 21*a^3*b*c)*d^2*e^3 - 2*(8*
a*b^4 - 42*a^2*b^2*c + 33*a^3*c^2)*d*e^4 + (16*b^5 - 96*a*b^3*c + 123*a^2*b*c^2)*e^5)*sqrt(a*e)*weierstrassPIn
verse(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 3*a*c)*e^2)/(a^2*e^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)/(a^3*e^3), 1/3*(3*a*e*x + a*d + b*e)/(a*e)) + 6*(8*a^5*d^4*e - 4*a^4*b*d^
3*e^2 - 3*(a^3*b^2 - 3*a^4*c)*d^2*e^3 - (4*a^2*b^3 - 15*a^3*b*c)*d*e^4 + (8*a*b^4 - 36*a^2*b^2*c + 21*a^3*c^2)
*e^5)*sqrt(a*e)*weierstrassZeta(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 3*a*c)*e^2)/(a^2*e^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)/(a^3*e^3), weierstrassPInverse(4/3*(a^2*d^2 - a
*b*d*e + (b^2 - 3*a*c)*e^2)/(a^2*e^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)/(a^3*e^3), 1/3*(3*a*e*x + a*d + b*e)/(a*e))) + 3*(35*a^5*e^5*x^4 + 5*(a^5*d*e^4 + a^4*b*e^5)*x^
3 - 2*(3*a^5*d^2*e^3 - a^4*b*d*e^4 + (3*a^3*b^2 - 7*a^4*c)*e^5)*x^2 + (8*a^5*d^3*e^2 - 3*a^4*b*d^2*e^3 - (3*a^
3*b^2 - 8*a^4*c)*d*e^4 + (8*a^2*b^3 - 27*a^3*b*c)*e^5)*x)*sqrt(e*x + d)*sqrt((a*x^2 + b*x + c)/x^2))/(a^5*e^5)

Sympy [F]

\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\int x^{3} \sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x^{3} \,d x } \]

[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)

Giac [F]

\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x^{3} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\int x^3\,\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \,d x \]

[In]

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

[Out]

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