\(\int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^2 \sqrt {d+e x} \, dx\) [81]

Optimal result

Integrand size = 29, antiderivative size = 636 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^2 \sqrt {d+e x} \, dx=-\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (4 a^2 d^2+4 b^2 e^2-a e (2 b d-5 c e)-3 a e (a d-4 b e) x\right )}{105 a^2 e^2}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (c+b x+a x^2\right )}{7 a}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 a^3 d^3+8 b^3 e^3-a^2 d e (5 b d-16 c e)-a b e^2 (5 b d+29 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (8 a^2 d^2-4 b^2 e^2-a e (b d-10 c e)\right ) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \sqrt {d+e x} \left (c+b x+a x^2\right )} \]

[Out]

-2/105*x*(4*a^2*d^2+4*b^2*e^2-a*e*(2*b*d-5*c*e)-3*a*e*(a*d-4*b*e)*x)*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/a^2/e^2
+2/7*x*(a*x^2+b*x+c)*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/a+1/105*(8*a^3*d^3+8*b^3*e^3-a^2*d*e*(5*b*d-16*c*e)-a*b
*e^2*(5*b*d+29*c*e))*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^3/e^3/(a*x^2+b*x+c)/(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))
))^(1/2)-2/105*(8*a^2*d^2-4*b^2*e^2-a*e*(b*d-10*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^3/e^3/(a*x^2+b*x+c)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1587, 846, 828, 857, 732, 435, 430} \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^2 \sqrt {d+e x} \, dx=-\frac {2 x \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (4 a^2 d^2-a e (2 b d-5 c e)-3 a e x (a d-4 b e)+4 b^2 e^2\right )}{105 a^2 e^2}-\frac {2 \sqrt {2} x \sqrt {b^2-4 a c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (8 a^2 d^2-a e (b d-10 c e)-4 b^2 e^2\right ) \left (a d^2-e (b d-c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \sqrt {d+e x} \left (a x^2+b x+c\right )}+\frac {\sqrt {2} x \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (8 a^3 d^3-a^2 d e (5 b d-16 c e)-a b e^2 (5 b d+29 c e)+8 b^3 e^3\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \left (a x^2+b x+c\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 x \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (a x^2+b x+c\right )}{7 a} \]

[In]

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

[Out]

(-2*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x]*(4*a^2*d^2 + 4*b^2*e^2 - a*e*(2*b*d - 5*c*e) - 3*a*e*(a*d - 4*b*e)*x
))/(105*a^2*e^2) + (2*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x]*(c + b*x + a*x^2))/(7*a) + (Sqrt[2]*Sqrt[b^2 - 4*a
*c]*(8*a^3*d^3 + 8*b^3*e^3 - a^2*d*e*(5*b*d - 16*c*e) - a*b*e^2*(5*b*d + 29*c*e))*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)/S
qrt[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)])/(105*a^3*e^3*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]*(8*a^2*d^2
- 4*b^2*e^2 - a*e*(b*d - 10*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)])/
(105*a^3*e^3*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 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 846

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

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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int x \sqrt {d+e x} \sqrt {c+b x+a x^2} \, dx}{\sqrt {c+b x+a x^2}} \\ & = \frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (c+b x+a x^2\right )}{7 a}+\frac {\left (2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\left (\frac {1}{2} (-3 b d-c e)+\frac {1}{2} (a d-4 b e) x\right ) \sqrt {c+b x+a x^2}}{\sqrt {d+e x}} \, dx}{7 a \sqrt {c+b x+a x^2}} \\ & = -\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (4 a^2 d^2+4 b^2 e^2-a e (2 b d-5 c e)-3 a e (a d-4 b e) x\right )}{105 a^2 e^2}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (c+b x+a x^2\right )}{7 a}-\frac {\left (4 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\frac {1}{2} \left (-a^2 d^2 (2 b d-c e)-2 b^2 e^2 (b d+c e)+a e \left (b^2 d^2+9 b c d e+5 c^2 e^2\right )\right )-\frac {1}{4} \left (8 a^3 d^3+8 b^3 e^3-a^2 d e (5 b d-16 c e)-a b e^2 (5 b d+29 c e)\right ) x}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{105 a^2 e^2 \sqrt {c+b x+a x^2}} \\ & = -\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (4 a^2 d^2+4 b^2 e^2-a e (2 b d-5 c e)-3 a e (a d-4 b e) x\right )}{105 a^2 e^2}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (c+b x+a x^2\right )}{7 a}-\frac {\left (\left (-8 a^3 d^3-8 b^3 e^3+a^2 d e (5 b d-16 c e)+a b e^2 (5 b d+29 c e)\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}{105 a^2 e^3 \sqrt {c+b x+a x^2}}-\frac {\left (4 \left (-\frac {1}{4} d \left (-8 a^3 d^3-8 b^3 e^3+a^2 d e (5 b d-16 c e)+a b e^2 (5 b d+29 c e)\right )+\frac {1}{2} e \left (-a^2 d^2 (2 b d-c e)-2 b^2 e^2 (b d+c e)+a e \left (b^2 d^2+9 b c d e+5 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}{105 a^2 e^3 \sqrt {c+b x+a x^2}} \\ & = -\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (4 a^2 d^2+4 b^2 e^2-a e (2 b d-5 c e)-3 a e (a d-4 b e) x\right )}{105 a^2 e^2}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (c+b x+a x^2\right )}{7 a}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (-8 a^3 d^3-8 b^3 e^3+a^2 d e (5 b d-16 c e)+a b e^2 (5 b d+29 c e)\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 )}{105 a^3 e^3 \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 (8 \sqrt {2} \sqrt {b^2-4 a c} \left (-\frac {1}{4} d \left (-8 a^3 d^3-8 b^3 e^3+a^2 d e (5 b d-16 c e)+a b e^2 (5 b d+29 c e)\right )+\frac {1}{2} e \left (-a^2 d^2 (2 b d-c e)-2 b^2 e^2 (b d+c e)+a e \left (b^2 d^2+9 b c d e+5 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 )}{105 a^3 e^3 \sqrt {d+e x} \left (c+b x+a x^2\right )} \\ & = -\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (4 a^2 d^2+4 b^2 e^2-a e (2 b d-5 c e)-3 a e (a d-4 b e) x\right )}{105 a^2 e^2}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \left (c+b x+a x^2\right )}{7 a}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 a^3 d^3+8 b^3 e^3-a^2 d e (5 b d-16 c e)-a b e^2 (5 b d+29 c e)\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 )}{105 a^3 e^3 \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 (8 a^2 d^2-a b d e-4 b^2 e^2+10 a c e^2\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 )}{105 a^3 e^3 \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 = 34.03 (sec) , antiderivative size = 1314, normalized size of antiderivative = 2.07 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^2 \sqrt {d+e x} \, dx=x \sqrt {d+e x} \left (\frac {4 \left (-2 a^2 d^2+a b d e-2 b^2 e^2+5 a c e^2\right )}{105 a^2 e^2}+\frac {2 (a d+b e) x}{35 a e}+\frac {2 x^2}{7}\right ) \sqrt {a+\frac {c+b x}{x^2}}+\frac {x (d+e x)^{3/2} \sqrt {a+\frac {c+b x}{x^2}} \left (4 \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (8 a^3 d^3+8 b^3 e^3+a^2 d e (-5 b d+16 c e)-a b e^2 (5 b d+29 c e)\right ) \left (a \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {c e}{d+e x}\right )}{d+e x}\right )-\frac {i \sqrt {2} \left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (8 a^3 d^3+8 b^3 e^3+a^2 d e (-5 b d+16 c e)-a b e^2 (5 b d+29 c e)\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 c e^2}{d+e x}-2 a d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 c e^2}{d+e x}+2 a d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}+\frac {i \sqrt {2} \left (8 b^3 e^3 \left (-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+a^3 \left (-4 c d^2 e^2+8 d^3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )+a b e^2 \left (13 b^2 d e+37 b c e^2-5 b d \sqrt {\left (b^2-4 a c\right ) e^2}-29 c e \sqrt {\left (b^2-4 a c\right ) e^2}\right )+a^2 e \left (b^2 d^2 e-4 c e \left (5 c e^2-4 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )-b d \left (52 c e^2+5 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 c e^2}{d+e x}-2 a d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 c e^2}{d+e x}+2 a d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}\right )}{210 a^3 e^4 \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {c+b x+a x^2} \sqrt {\frac {(d+e x)^2 \left (a \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {c e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \]

[In]

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

[Out]

x*Sqrt[d + e*x]*((4*(-2*a^2*d^2 + a*b*d*e - 2*b^2*e^2 + 5*a*c*e^2))/(105*a^2*e^2) + (2*(a*d + b*e)*x)/(35*a*e)
+ (2*x^2)/7)*Sqrt[a + (c + b*x)/x^2] + (x*(d + e*x)^(3/2)*Sqrt[a + (c + b*x)/x^2]*(4*Sqrt[(a*d^2 + e*(-(b*d)
+ c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(8*a^3*d^3 + 8*b^3*e^3 + a^2*d*e*(-5*b*d + 16*c*e) - a*b*e^2
*(5*b*d + 29*c*e))*(a*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (c*e)/(d + e*x)))/(d + e*x)) - (I*Sqrt[
2]*(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(8*a^3*d^3 + 8*b^3*e^3 + a^2*d*e*(-5*b*d + 16*c*e) - a*b*e^2*(5*b*d
+ 29*c*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*c*e^2)/(d + e*x) - 2*a*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(
d + e*x)))/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*e^2)/(d + e*x) + 2*a*
d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSin
h[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d +
b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] + (I*Sqrt[2]*(8*b^3*e
^3*(-(b*e) + Sqrt[(b^2 - 4*a*c)*e^2]) + a^3*(-4*c*d^2*e^2 + 8*d^3*Sqrt[(b^2 - 4*a*c)*e^2]) + a*b*e^2*(13*b^2*d
*e + 37*b*c*e^2 - 5*b*d*Sqrt[(b^2 - 4*a*c)*e^2] - 29*c*e*Sqrt[(b^2 - 4*a*c)*e^2]) + a^2*e*(b^2*d^2*e - 4*c*e*(
5*c*e^2 - 4*d*Sqrt[(b^2 - 4*a*c)*e^2]) - b*d*(52*c*e^2 + 5*d*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(Sqrt[(b^2 - 4*a*
c)*e^2] - (2*c*e^2)/(d + e*x) - 2*a*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*a*d - b*e + Sqrt[(b^
2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*e^2)/(d + e*x) + 2*a*d*(-1 + d/(d + e*x)) + b*(e - (2*d
*e)/(d + e*x)))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c
*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2
*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(210*a^3*e^4*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a*d
+ b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[c + b*x + a*x^2]*Sqrt[((d + e*x)^2*(a*(-1 + d/(d + e*x))^2 + (e*(b - (b
*d)/(d + e*x) + (c*e)/(d + e*x)))/(d + e*x)))/e^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2661\) vs. \(2(572)=1144\).

Time = 1.84 (sec) , antiderivative size = 2662, normalized size of antiderivative = 4.19

method	result	size
risch	\(\text {Expression too large to display}\)	\(2662\)
default	\(\text {Expression too large to display}\)	\(6302\)

[In]

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

[Out]

-2/105*(-15*a^2*e^2*x^2-3*a^2*d*e*x-3*a*b*e^2*x+4*a^2*d^2-2*a*b*d*e-10*a*c*e^2+4*b^2*e^2)*(e*x+d)^(1/2)/a^2/e^
2*((a*x^2+b*x+c)/x^2)^(1/2)*x+1/105/e^2/a^2*(8*a^2*b*d^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))+8*b^3*d*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))+8*b^2*c*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))-20*a*c^2*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))-4*a^2*c*d^2*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))-4*a*b^2*d^2*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))-36*a*b*c*d*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))+2*(8*a
^3*d^3-5*a^2*b*d^2*e+16*a^2*c*d*e^2-5*a*b^2*d*e^2-29*a*b*c*e^3+8*b^3*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)*((-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 = 0.11 (sec) , antiderivative size = 598, normalized size of antiderivative = 0.94 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^2 \sqrt {d+e x} \, dx=-\frac {2 \, {\left ({\left (8 \, a^{4} d^{4} - 9 \, a^{3} b d^{3} e - 2 \, {\left (2 \, a^{2} b^{2} - 11 \, a^{3} c\right )} d^{2} e^{2} - {\left (9 \, a b^{3} - 41 \, a^{2} b c\right )} d e^{3} + {\left (8 \, b^{4} - 41 \, a b^{2} c + 30 \, a^{2} c^{2}\right )} e^{4}\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 ) + 3 \, {\left (8 \, a^{4} d^{3} e - 5 \, a^{3} b d^{2} e^{2} - {\left (5 \, a^{2} b^{2} - 16 \, a^{3} c\right )} d e^{3} + {\left (8 \, a b^{3} - 29 \, a^{2} b c\right )} e^{4}\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 (15 \, a^{4} e^{4} x^{3} + 3 \, {\left (a^{4} d e^{3} + a^{3} b e^{4}\right )} x^{2} - 2 \, {\left (2 \, a^{4} d^{2} e^{2} - a^{3} b d e^{3} + {\left (2 \, a^{2} b^{2} - 5 \, a^{3} c\right )} e^{4}\right )} x\right )} \sqrt {e x + d} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right )}}{315 \, a^{4} e^{4}} \]

[In]

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

[Out]

-2/315*((8*a^4*d^4 - 9*a^3*b*d^3*e - 2*(2*a^2*b^2 - 11*a^3*c)*d^2*e^2 - (9*a*b^3 - 41*a^2*b*c)*d*e^3 + (8*b^4
- 41*a*b^2*c + 30*a^2*c^2)*e^4)*sqrt(a*e)*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*(8*a^4*d^3*e - 5*a^3*b*d^2*e^2 - (5*a^2*b^2 - 16*a^3*c)*d*e^3 + (8*a*b^3 - 29*a
^2*b*c)*e^4)*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*(15*a^4*e^4*x^3 + 3*(a^4*d*e^3 + a^3*b*
e^4)*x^2 - 2*(2*a^4*d^2*e^2 - a^3*b*d*e^3 + (2*a^2*b^2 - 5*a^3*c)*e^4)*x)*sqrt(e*x + d)*sqrt((a*x^2 + b*x + c)
/x^2))/(a^4*e^4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(x^2*(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^2, x)

Giac [F]

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

[In]

integrate(x^2*(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^2, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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