\(\int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx\) [651]

Optimal result

Integrand size = 28, antiderivative size = 1257 \[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac {3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac {3 \sqrt {-a} \sqrt {c} e \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 (\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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\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}} \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 )}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {3 \sqrt {-a} \sqrt {c} e 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}} \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {3 \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}} \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \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}} \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 )}{\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}}-\frac {3 \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}} \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 )}{4 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 3.28 (sec) , antiderivative size = 1257, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {954, 6874, 733, 430, 858, 435, 947, 174, 552, 551} \[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {3 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {c x^2+a} e^2}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}-\frac {\sqrt {f+g x} \sqrt {c x^2+a} e^2}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac {3 \sqrt {-a} \sqrt {c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} 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 ) e}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}-\frac {3 \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 {\frac {c x^2}{a}+1} \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 ) e}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {3 \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 {\frac {c x^2}{a}+1} \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {\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} \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 )}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {3 \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 {\frac {c x^2}{a}+1} \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 )}{4 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right )^2 (e f-d g)^2 \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} \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 )}{\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}} \]

[In]

Int[1/((d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]

[Out]

-1/2*(e^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(e*f - d*g)*(d + e*x)^2) + (3*e^2*(a*e^2*g - c*d*(2*
e*f - 3*d*g))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)^2*(d + e*x)) + (3*Sqrt[-a]*Sqrt[
c]*e*(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)/S
qrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)^2*Sqrt[(Sqrt[c]*(f +
g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) + (Sqrt[-a]*Sqrt[c]*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)])/(2*(c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2]) - (3*Sqrt[-a]*Sqrt[c]*e*f*(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*(c*d^2 + a*e^2)^2*(e*
f - d*g)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2]) + (3*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]]/S
qrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)^2*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)])/(((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]) - (3*(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]*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)])/(4*((Sqrt[c]*d)/Sqrt[-a] + e)*(c*d^2 + a*e^2)^2*(e*f - d*g)^2*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 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*} \text {integral}& = -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}-\frac {\int \frac {3 a e^2 g-4 c d (e f-d g)+2 c e (e f-2 d g) x+c e^2 g x^2}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 \left (c d^2+a e^2\right ) (e f-d g)} \\ & = -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}-\frac {\int \left (\frac {c g}{\sqrt {f+g x} \sqrt {a+c x^2}}+\frac {3 \left (a e^2 g-c d (2 e f-3 d g)\right )}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {2 c (e f-3 d g)}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx}{4 \left (c d^2+a e^2\right ) (e f-d g)} \\ & = -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}-\frac {(c g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {(c (e f-3 d g)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 \left (c d^2+a e^2\right ) (e f-d g)} \\ & = -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac {3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac {\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )\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 \left (c d^2+a e^2\right )^2 (e f-d g)^2}-\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 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {a+c x^2}}-\frac {\left (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} \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac {3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac {\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 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )\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 \left (c d^2+a e^2\right )^2 (e f-d g)^2}+\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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {a+c x^2}} \\ & = -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac {3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac {\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 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (3 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 \left (c d^2+a e^2\right )^2 (e f-d g)^2}-\frac {\left (3 c e 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 \left (c d^2+a e^2\right )^2 (e f-d g)^2}+\frac {\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )^2\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)^2}+\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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac {3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac {\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 \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 )}{\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}}-\frac {\left (3 c e \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 \left (c d^2+a e^2\right )^2 (e f-d g)^2}+\frac {\left (3 c e 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 \left (c d^2+a e^2\right )^2 (e f-d g)^2}+\frac {\left (3 \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 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {a+c x^2}}-\frac {\left (3 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} \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac {3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac {\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 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {3 \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 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \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 )}{\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}}-\frac {\left (3 \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 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {a+c x^2}}-\frac {\left (3 a \sqrt {c} e \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} \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (3 a \sqrt {c} e 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} \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac {3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac {3 \sqrt {-a} \sqrt {c} e \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 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\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 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {3 \sqrt {-a} \sqrt {c} e 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 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {3 \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 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \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 )}{\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}}-\frac {\left (3 \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 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac {3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac {3 \sqrt {-a} \sqrt {c} e \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 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\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 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {3 \sqrt {-a} \sqrt {c} e 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 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {3 \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 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \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 )}{\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}}-\frac {3 \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 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 29.75 (sec) , antiderivative size = 2491, normalized size of antiderivative = 1.98 \[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[1/((d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]

[Out]

(-15*c^2*d^2*e^2*f^3 - 3*a*c*e^4*f^3 + (6*c^2*d*e^3*f^4)/g + 9*c^2*d^3*e*f^2*g + 9*a*c*d*e^3*f^2*g - 15*a*c*d^
2*e^2*f*g^2 - 3*a^2*e^4*f*g^2 + 9*a*c*d^3*e*g^3 + 3*a^2*d*e^3*g^3 + 30*c^2*d^2*e^2*f^2*(f + g*x) + 6*a*c*e^4*f
^2*(f + g*x) - (12*c^2*d*e^3*f^3*(f + g*x))/g - 18*c^2*d^3*e*f*g*(f + g*x) - 6*a*c*d*e^3*f*g*(f + g*x) - 15*c^
2*d^2*e^2*f*(f + g*x)^2 - 3*a*c*e^4*f*(f + g*x)^2 + (6*c^2*d*e^3*f^2*(f + g*x)^2)/g + 9*c^2*d^3*e*g*(f + g*x)^
2 + 3*a*c*d*e^3*g*(f + g*x)^2 - (e^2*(e*f - d*g)*(f + g*x)*(a + c*x^2)*(2*(c*d^2 + a*e^2)*(e*f - d*g) - 3*(a*e
^2*g + c*d*(-2*e*f + 3*d*g))*(d + e*x)))/(d + e*x)^2 + (3*Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-(e*f) + d*g
)*(a*e^2*g + c*d*(-2*e*f + 3*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)*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)])/(g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + ((Sqrt[c]*d - I*Sqrt[a
]*e)*(3*a^(3/2)*e^3*g^2 + (3*I)*a*Sqrt[c]*e^2*g*(e*f - 2*d*g) - Sqrt[a]*c*e*(2*e^2*f^2 - 6*d*e*f*g + d^2*g^2)
- I*c^(3/2)*d*(4*e^2*f^2 - 9*d*e*f*g + 8*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sq
rt[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)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((8*I)*c^2
*d^2*e^2*f^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)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[
c]] - ((4*I)*a*c*e^4*f^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)])/Sqrt[-f - (I*Sqrt
[a]*g)/Sqrt[c]] - ((20*I)*c^2*d^3*e*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sq
rt[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*Arc
Sinh[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)])/Sq
rt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((4*I)*a*c*d*e^3*f*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*S
qrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((15*I)*c^2*d^4*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)/(Sq
rt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((6*I)*a*c*d^2*e^2*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)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((3*I)*a^2*e^4*g^2*Sqrt[(g*((I*Sqr
t[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[(Sq
rt[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 - (I*Sqrt[a]*g)/Sqrt[c]])/(4*(c*d^2 + a*e^2)^2*(e
*f - d*g)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])

Maple [A] (verified)

Time = 3.46 (sec) , antiderivative size = 1192, normalized size of antiderivative = 0.95

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1192\)
default	\(\text {Expression too large to display}\)	\(20366\)

[In]

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

[Out]

((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(1/2*e^2/(a*d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)*(c*g*x^
3+c*f*x^2+a*g*x+a*f)^(1/2)/(e*x+d)^2+3/4*e^2*(a*e^2*g+3*c*d^2*g-2*c*d*e*f)/(a*d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*
f)^2*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)/(e*x+d)-1/4*c*g*(a*d*e^2*g+2*a*e^3*f+7*c*d^3*g-4*c*d^2*e*f)/(a*d*e^2*g-
a*e^3*f+c*d^3*g-c*d^2*e*f)^2*(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)*E
llipticF(((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))-3/4*c*e*g*(
a*e^2*g+3*c*d^2*g-2*c*d*e*f)/(a*d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)^2*(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)))+1/4*(3*a^2*e^4*g^2+6*a*c*d^2*e^2*g^2+4*a*c*d*e^3*f*g
-4*a*c*e^4*f^2+15*c^2*d^4*g^2-20*c^2*d^3*e*f*g+8*c^2*d^2*e^2*f^2)/(a*d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)^2/e*(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. \[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3} \sqrt {g x + f}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3} \sqrt {g x + f}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3} \,d x \]

[In]

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

[Out]

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