\(\int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx\) [626]

Optimal result

Integrand size = 28, antiderivative size = 683 \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx=\frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 e}-\frac {2 \sqrt {-a} \sqrt {c} (e f-3 d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 e^2 g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} f (e f-3 d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 e^2 g \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \left (2 a e^2 g-3 c d (e f-d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

2/3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/e-2/3*(-3*d*g+e*f)*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*
a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*c^(1/2)*(g*x+f)^(1/2)*(1+c*x^2/a)^(1/2)/e^2/g/(c*x^2+a)^(1/
2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)-2/3*(2*a*e^2*g-3*c*d*(-d*g+e*f))*EllipticF(1/2*(1-x*c^(1/2
)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(1+c*x^2/a)^(1/2)*((g*x+f)*
c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/e^3/c^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)+2/3*f*(-3*d*g+e*f)*EllipticF
(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*c^(1/2)*(1+
c*x^2/a)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/e^2/g/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)-2*(a*e^2+c
*d^2)*(-d*g+e*f)*EllipticPi(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),2*e/(e+d*c^(1/2)/(-a)^(1/2)),2^(1/2)*(g
*(-a)^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2))*(1+c*x^2/a)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2
)/e^3/(e+d*c^(1/2)/(-a)^(1/2))/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 1.74 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {933, 6874, 733, 430, 858, 435, 947, 174, 552, 551} \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx=-\frac {2 \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^3 \sqrt {a+c x^2} \sqrt {f+g x} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right )}+\frac {2 \sqrt {-a} \sqrt {c} f \sqrt {\frac {c x^2}{a}+1} (e f-3 d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 e^2 g \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {2 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} (e f-3 d g) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 e^2 g \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (2 a e^2 g-3 c d (e f-d g)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e^3 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e} \]

[In]

Int[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x),x]

[Out]

(2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*e) - (2*Sqrt[-a]*Sqrt[c]*(e*f - 3*d*g)*Sqrt[f + g*x]*Sqrt[1 + (c*x^2)/a]*
EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3*e^2*g*Sqrt[
(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*Sqrt[c]*f*(e*f - 3*d*g)*Sqrt[(Sqr
t[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/
Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3*e^2*g*Sqrt[f + g*x]*Sqrt[a + c*x^2]) - (2*Sqrt[-a]*(2*a*e^2
*g - 3*c*d*(e*f - d*g))*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSi
n[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3*Sqrt[c]*e^3*Sqrt[f + g*x]*
Sqrt[a + c*x^2]) - (2*(c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 +
(c*x^2)/a]*EllipticPi[(2*e)/((Sqrt[c]*d)/Sqrt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (2*Sqr
t[-a]*g)/(Sqrt[c]*f + Sqrt[-a]*g)])/(e^3*((Sqrt[c]*d)/Sqrt[-a] + e)*Sqrt[f + g*x]*Sqrt[a + c*x^2])

Rule 174

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d
*g - c*h)/d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

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 551

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

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d/c)*x^2]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 733

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

Rule 858

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

Rule 933

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2*(d + e*
x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + 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 + c*x^2]))*Simp[3*a*e*f - a*d*g - 2*(c*d*f - a*e*g)*x + (c*e*f - 3*c*d*g)*x^2, x], x], x] /; Fre
eQ[{a, c, d, e, f, g, m}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] &&  !LtQ[m, -1]

Rule 947

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-c/
a, 2]}, Dist[Sqrt[1 + c*(x^2/a)]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]),
x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] &&  !GtQ[a, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 e}+\frac {\int \frac {a (3 e f-d g)-2 (c d f-a e g) x+c (e f-3 d g) x^2}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 e} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 e}+\frac {\int \left (\frac {2 a e^2 g-3 c d (e f-d g)}{e^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {c (e f-3 d g) x}{e \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {3 \left (c d^2+a e^2\right ) (e f-d g)}{e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx}{3 e} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 e}+\frac {(c (e f-3 d g)) \int \frac {x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 e^2}+\frac {\left (\left (c d^2+a e^2\right ) (e f-d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^3}+\frac {\left (2 a g-\frac {3 c d (e f-d g)}{e^2}\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 e} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 e}+\frac {(c (e f-3 d g)) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{3 e^2 g}-\frac {(c f (e f-3 d g)) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 e^2 g}+\frac {\left (\left (c d^2+a e^2\right ) (e f-d g) \sqrt {1+\frac {c x^2}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}} \sqrt {1+\frac {\sqrt {c} x}{\sqrt {-a}}} (d+e x) \sqrt {f+g x}} \, dx}{e^3 \sqrt {a+c x^2}}+\frac {\left (2 a \left (2 a g-\frac {3 c d (e f-d g)}{e^2}\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} \sqrt {c} e \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 e}-\frac {2 \sqrt {-a} \left (2 a e^2 g-3 c d (e f-d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {f+\frac {\sqrt {-a} g}{\sqrt {c}}-\frac {\sqrt {-a} g x^2}{\sqrt {c}}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{e^3 \sqrt {a+c x^2}}+\frac {\left (2 a \sqrt {c} (e f-3 d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} e^2 g \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (2 a \sqrt {c} f (e f-3 d g) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} e^2 g \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 e}-\frac {2 \sqrt {-a} \sqrt {c} (e f-3 d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 e^2 g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} f (e f-3 d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 e^2 g \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \left (2 a e^2 g-3 c d (e f-d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {1-\frac {\sqrt {-a} g x^2}{\sqrt {c} \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{e^3 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 e}-\frac {2 \sqrt {-a} \sqrt {c} (e f-3 d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 e^2 g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} f (e f-3 d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 e^2 g \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \left (2 a e^2 g-3 c d (e f-d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 29.08 (sec) , antiderivative size = 1216, normalized size of antiderivative = 1.78 \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx=\frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 e}+\frac {(f+g x)^{3/2} \left (2 c e^2 f \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-6 c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+\frac {2 c e^2 f^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{(f+g x)^2}-\frac {6 c d e f^2 g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{(f+g x)^2}+\frac {2 a e^2 f g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{(f+g x)^2}-\frac {6 a d e g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{(f+g x)^2}-\frac {4 c e^2 f^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{f+g x}+\frac {12 c d e f g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{f+g x}+\frac {2 \sqrt {c} e \left (-i \sqrt {c} f+\sqrt {a} g\right ) (e f-3 d g) \sqrt {1-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \sqrt {1-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {2 e \left (3 \sqrt {c} d-i \sqrt {a} e\right ) g \left (-i \sqrt {c} f+\sqrt {a} g\right ) \sqrt {1-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \sqrt {1-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {6 i c d^2 g^2 \sqrt {1-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \sqrt {1-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \operatorname {EllipticPi}\left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )},i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {6 i a e^2 g^2 \sqrt {1-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \sqrt {1-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \operatorname {EllipticPi}\left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )},i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}\right )}{3 e^3 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \sqrt {a+\frac {c (f+g x)^2 \left (-1+\frac {f}{f+g x}\right )^2}{g^2}}} \]

[In]

Integrate[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x),x]

[Out]

(2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*e) + ((f + g*x)^(3/2)*(2*c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 6*c*d
*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + (2*c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (6*c*d*e*
f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (2*a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*
x)^2 - (6*a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (4*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[
c]])/(f + g*x) + (12*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (2*Sqrt[c]*e*((-I)*Sqrt[c]*f + Sq
rt[a]*g)*(e*f - 3*d*g)*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqr
t[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f -
I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + (2*e*(3*Sqrt[c]*d - I*Sqrt[a]*e)*g*((-I)*Sqrt[c]*f +
Sqrt[a]*g)*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sq
rt[c]*(f + g*x))]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*
g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((6*I)*c*d^2*g^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*
(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqr
t[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(
Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((6*I)*a*e^2*g^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f +
g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]
*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt
[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x]))/(3*e^3*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*Sqrt[a + (c*(f + g*x)^2*(-1
+ f/(f + g*x))^2)/g^2])

Maple [A] (verified)

Time = 2.61 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.35

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 e}+\frac {2 \left (\frac {a \,e^{2} g +c \,d^{2} g -c d e f}{e^{3}}-\frac {a g}{3 e}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (-\frac {\left (d g -e f \right ) c}{e^{2}}-\frac {2 c f}{3 e}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}-\frac {2 \left (a \,e^{2} g d -a \,e^{3} f +c \,d^{3} g -c \,d^{2} e f \right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \Pi \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{e^{4} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\)	\(922\)
risch	\(\text {Expression too large to display}\)	\(1323\)
default	\(\text {Expression too large to display}\)	\(2496\)

[In]

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

[Out]

((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/3/e*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2*((a*e^2*g+c
*d^2*g-c*d*e*f)/e^3-1/3/e*a*g)*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(
-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)
*EllipticF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+2*(-(d*g-
e*f)/e^2*c-2/3/e*c*f)*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-f/g-(-a*
c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)*((-f/g-(
-a*c)^(1/2)/c)*EllipticE(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1
/2))+(-a*c)^(1/2)/c*EllipticF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c
))^(1/2)))-2*(a*d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)/e^4*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/
2)*((x-(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+
c*f*x^2+a*g*x+a*f)^(1/2)/(-f/g+d/e)*EllipticPi(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),(-f/g+(-a*c)^(1/2)/c)/(-f/
g+d/e),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(c*x^2 + a)*sqrt(g*x + f)/(e*x + d), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(c*x^2 + a)*sqrt(g*x + f)/(e*x + d), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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