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

Optimal. Leaf size=636 \[ -\frac {2 x \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (4 a^2 d^2-a e (2 b d-5 c e)-3 a e x (a d-4 b e)+4 b^2 e^2\right )}{105 a^2 e^2}-\frac {2 \sqrt {2} x \sqrt {b^2-4 a c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (8 a^2 d^2-a e (b d-10 c e)-4 b^2 e^2\right ) \left (a d^2-e (b d-c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \sqrt {d+e x} \left (a x^2+b x+c\right )}+\frac {\sqrt {2} x \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (8 a^3 d^3-a^2 d e (5 b d-16 c e)-a b e^2 (5 b d+29 c e)+8 b^3 e^3\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \left (a x^2+b x+c\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 x \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (a x^2+b x+c\right )}{7 a} \]

[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]  time = 0.99, antiderivative size = 636, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1573, 832, 814, 843, 718, 424, 419} \[ -\frac {2 x \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (4 a^2 d^2-a e (2 b d-5 c e)-3 a e x (a d-4 b e)+4 b^2 e^2\right )}{105 a^2 e^2}-\frac {2 \sqrt {2} x \sqrt {b^2-4 a c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (8 a^2 d^2-a e (b d-10 c e)-4 b^2 e^2\right ) \left (a d^2-e (b d-c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \sqrt {d+e x} \left (a x^2+b x+c\right )}+\frac {\sqrt {2} x \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (-a^2 d e (5 b d-16 c e)+8 a^3 d^3-a b e^2 (5 b d+29 c e)+8 b^3 e^3\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \left (a x^2+b x+c\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 x \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (a x^2+b x+c\right )}{7 a} \]

Antiderivative was successfully verified.

[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 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)])

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 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 814

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 832

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 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 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]

Rubi steps

\begin {align*} \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^2 \sqrt {d+e x} \, dx &=\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 ) \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 )}{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 ) \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 )}{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*}

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Mathematica [C]  time = 13.05, size = 5350, normalized size = 8.41 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [F]  time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {e x + d} x^{2} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.05, size = 6302, normalized size = 9.91 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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