3.7.28 \(\int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx\) [628]

Optimal. Leaf size=1205 \[ -\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 e^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 {a+c x^2}}-\frac {3 \sqrt {-a} \sqrt {c} 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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \left (a e^2 g-c d (2 e f-3 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 )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (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}} \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}}+\frac {\left (a e^2 g-c d (2 e f-3 d g)\right )^2 \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 )}{4 e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 2.76, antiderivative size = 1205, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {931, 6874, 733, 430, 954, 858, 435, 947, 174, 552, 551} \begin {gather*} \frac {\sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\text {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 ) \left (a e^2 g-c d (2 e f-3 d g)\right )^2}{4 e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {\sqrt {-a} \sqrt {c} \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\text {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 ) \left (a e^2 g-c d (2 e f-3 d g)\right )}{4 e^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 {c x^2+a}}+\frac {\sqrt {-a} \sqrt {c} f \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} F\left (\text {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 ) \left (a e^2 g-c d (2 e f-3 d g)\right )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {\sqrt {-a} \sqrt {c} d g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} F\left (\text {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 ) \left (a e^2 g-c d (2 e f-3 d g)\right )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {\sqrt {f+g x} \sqrt {c x^2+a} \left (a e^2 g-c d (2 e f-3 d g)\right )}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {3 \sqrt {-a} \sqrt {c} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} F\left (\text {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 )}{2 e^3 \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {c (e f-3 d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\text {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 {c x^2+a}}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x)^3,x]

[Out]

-1/2*(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(e*(d + e*x)^2) - ((a*e^2*g - c*d*(2*e*f - 3*d*g))*Sqrt[f + g*x]*Sqrt[a +
c*x^2])/(4*e*(c*d^2 + a*e^2)*(e*f - d*g)*(d + e*x)) - (Sqrt[-a]*Sqrt[c]*(a*e^2*g - c*d*(2*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)])/(4*e^2*(c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a +
c*x^2]) - (3*Sqrt[-a]*Sqrt[c]*g*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*Ellipti
cF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(2*e^3*Sqrt[f + g*x]*
Sqrt[a + c*x^2]) + (Sqrt[-a]*Sqrt[c]*f*(a*e^2*g - c*d*(2*e*f - 3*d*g))*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + S
qrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*S
qrt[c]*f - a*g)])/(4*e^2*(c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2]) - (Sqrt[-a]*Sqrt[c]*d*g*(a
*e^2*g - c*d*(2*e*f - 3*d*g))*Sqrt[(Sqrt[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)])/(4*e^3*(c*d^2 + a*e^2)*
(e*f - d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2]) - (c*(e*f - 3*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*Sqrt[-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])
+ ((a*e^2*g - c*d*(2*e*f - 3*d*g))^2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*E
llipticPi[(2*e)/((Sqrt[c]*d)/Sqrt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (2*Sqrt[-a]*g)/(Sq
rt[c]*f + Sqrt[-a]*g)])/(4*e^3*((Sqrt[c]*d)/Sqrt[-a] + e)*(c*d^2 + a*e^2)*(e*f - d*g)*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 931

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Simp[(d + e*x)
^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/(e*(m + 1))), x] - Dist[1/(2*e*(m + 1)), Int[((d + e*x)^(m + 1)/(Sqrt[
f + g*x]*Sqrt[a + c*x^2]))*Simp[a*g + 2*c*f*x + 3*c*g*x^2, 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] && 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 954

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[e^2*(d +
e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/((m + 1)*(e*f - d*g)*(c*d^2 + a*e^2))), x] + Dist[1/(2*(m + 1)*(e
*f - d*g)*(c*d^2 + a*e^2)), Int[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]))*Simp[2*d*(c*e*f - c*d*g)*(
m + 1) - a*e^2*g*(2*m + 3) + 2*e*(c*d*g*(m + 1) - c*e*f*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x], x] /; Free
Q[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]

Rule 6874

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

Rubi steps

\begin {align*} \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}+\frac {\int \frac {a g+2 c f x+3 c g x^2}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 e}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}+\frac {\int \left (\frac {3 c g}{e^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {a e^2 g-c d (2 e f-3 d g)}{e^2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {2 c (e f-3 d g)}{e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx}{4 e}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}+\frac {(3 c g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 e^3}+\frac {(c (e f-3 d g)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 e^3}+\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 e^3}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {a e^2 g-2 c d (e f-d g)-2 c d e g x-c e^2 g x^2}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e^3 \left (c d^2+a e^2\right ) (e f-d g)}+\frac {\left (c (e f-3 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}{2 e^3 \sqrt {a+c x^2}}+\frac {\left (3 a \sqrt {c} 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 )}{2 \sqrt {-a} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {3 \sqrt {-a} \sqrt {c} 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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \left (-\frac {c d g}{\sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c e g x}{\sqrt {f+g x} \sqrt {a+c x^2}}+\frac {a e^2 g-c d (2 e f-3 d g)}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx}{8 e^3 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (c (e f-3 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 {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {3 \sqrt {-a} \sqrt {c} 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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (c d g \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e^3 \left (c d^2+a e^2\right ) (e f-d g)}+\frac {\left (c g \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac {x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e^2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right )^2 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e^3 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (c (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}}\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 {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {3 \sqrt {-a} \sqrt {c} 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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (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}} \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}}+\frac {\left (c \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{8 e^2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (c f \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e^2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (\left (a e^2 g-c d (2 e f-3 d g)\right )^2 \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}{8 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {a+c x^2}}+\frac {\left (a \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 d g)\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 )}{4 \sqrt {-a} e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {3 \sqrt {-a} \sqrt {c} 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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (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}} \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}}+\frac {\left (\left (a e^2 g-c d (2 e f-3 d g)\right )^2 \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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {a+c x^2}}+\frac {\left (a \sqrt {c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \sqrt {-a} e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (a \sqrt {c} f \left (a e^2 g-c d (2 e f-3 d g)\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 )}{4 \sqrt {-a} e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 e^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 {a+c x^2}}-\frac {3 \sqrt {-a} \sqrt {c} 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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \left (a e^2 g-c d (2 e f-3 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 )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (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}} \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}}+\frac {\left (\left (a e^2 g-c d (2 e f-3 d g)\right )^2 \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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 e^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 {a+c x^2}}-\frac {3 \sqrt {-a} \sqrt {c} 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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \left (a e^2 g-c d (2 e f-3 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 )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (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}} \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}}+\frac {\left (a e^2 g-c d (2 e f-3 d g)\right )^2 \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 )}{4 e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 27.49, size = 2526, normalized size = 2.10 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x)^3,x]

[Out]

(Sqrt[f + g*x]*Sqrt[a + c*x^2]*(-2 + ((a*e^2*g + c*d*(-2*e*f + 3*d*g))*(d + e*x))/((c*d^2 + a*e^2)*(-(e*f) + d
*g))))/(4*e*(d + e*x)^2) + (-2*c^2*d*e^3*f^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 5*c^2*d^2*e^2*f^3*g*Sqrt[-f -
(I*Sqrt[a]*g)/Sqrt[c]] + a*c*e^4*f^3*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*c^2*d^3*e*f^2*g^2*Sqrt[-f - (I*Sqr
t[a]*g)/Sqrt[c]] - 3*a*c*d*e^3*f^2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 5*a*c*d^2*e^2*f*g^3*Sqrt[-f - (I*Sqr
t[a]*g)/Sqrt[c]] + a^2*e^4*f*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*a*c*d^3*e*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sq
rt[c]] - a^2*d*e^3*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 4*c^2*d*e^3*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f
+ g*x) - 10*c^2*d^2*e^2*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 2*a*c*e^4*f^2*g*Sqrt[-f - (I*Sqrt[a
]*g)/Sqrt[c]]*(f + g*x) + 6*c^2*d^3*e*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) + 2*a*c*d*e^3*f*g^2*Sqr
t[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 2*c^2*d*e^3*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + 5*c^2
*d^2*e^2*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + a*c*e^4*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f +
g*x)^2 - 3*c^2*d^3*e*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 - a*c*d*e^3*g^2*Sqrt[-f - (I*Sqrt[a]*g)/
Sqrt[c]]*(f + g*x)^2 + I*Sqrt[c]*e*(Sqrt[c]*f + I*Sqrt[a]*g)*(-(e*f) + d*g)*(a*e^2*g + c*d*(-2*e*f + 3*d*g))*S
qrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*
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)] + e*(Sqrt[c]*d - I*Sqrt[a]*e)*g*(I*Sqrt[c]*f - Sqrt[a]*g)*(a*e^2*g + c*d*(4*e*f - 3*d*g) + (2*I)*S
qrt[a]*Sqrt[c]*e*(e*f - d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*
x)/(f + g*x))]*(f + g*x)^(3/2)*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)] + (4*I)*a*c*e^4*f^2*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]
*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*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)] - (4*I)*c^2*d^3*e*f*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]
*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*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
)] - (12*I)*a*c*d*e^3*f*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)
/(f + g*x))]*(f + g*x)^(3/2)*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)] + (3*I)*c^2*d^4
*g^3*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^
(3/2)*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)] + (6*I)*a*c*d^2*e^2*g^3*Sqrt[(g*((I*Sq
rt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(S
qrt[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)] - I*a^2*e^4*g^3*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g
*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*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)])/(4*e^3*(c*d^2 + a*e^2)*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)^2*Sqrt[f + g*
x]*Sqrt[a + c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(19180\) vs. \(2(1009)=2018\).
time = 0.10, size = 19181, normalized size = 15.92

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{2 e \left (e x +d \right )^{2}}+\frac {\left (a \,e^{2} g +3 c \,d^{2} g -2 c d e f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{4 e \left (a d \,e^{2} g -a \,e^{3} f +c \,d^{3} g -c \,d^{2} e f \right ) \left (e x +d \right )}+\frac {2 \left (\frac {c g}{e^{3}}-\frac {c g \left (3 a d \,e^{2} g -2 a \,e^{3} f +5 c \,d^{3} g -4 c \,d^{2} e f \right )}{8 e^{3} \left (a d \,e^{2} g -a \,e^{3} f +c \,d^{3} g -c \,d^{2} e f \right )}\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}}}\, \EllipticF \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 {c g \left (a \,e^{2} g +3 c \,d^{2} g -2 c d 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}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \EllipticE \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}\, \EllipticF \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 )}{4 e^{2} \left (a d \,e^{2} g -a \,e^{3} f +c \,d^{3} g -c \,d^{2} e f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {\left (a^{2} e^{4} g^{2}-6 a c \,d^{2} e^{2} g^{2}+12 a c d \,e^{3} f g -4 a c \,e^{4} f^{2}-3 c^{2} d^{4} g^{2}+4 c^{2} d^{3} e f g \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}}}\, \EllipticPi \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 )}{4 \left (a d \,e^{2} g -a \,e^{3} f +c \,d^{3} g -c \,d^{2} e f \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}}\)	\(1161\)
default	\(\text {Expression too large to display}\)	\(19181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + c x^{2}} \sqrt {f + g x}}{\left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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