3.9.92 \(\int \frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx\) [892]

Optimal. Leaf size=1034 \[ -\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 e \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {b^2-4 a c} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{4 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {\sqrt {b^2-4 a c} (-c d (2 e f+3 d g)+e (b e f+4 b d g-5 a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} g}{-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{2 \sqrt {2} e^3 \left (c d^2+e (-b d+a e)\right ) \sqrt {f+g x} \sqrt {a+x (b+c x)}}+\frac {\sqrt {2 c f-b g+\sqrt {b^2-4 a c} g} \left (b^2 e^4 f^2+a^2 e^4 g^2+c^2 d^3 g (4 e f-3 d g)-2 a c e^2 \left (2 e^2 f^2-6 d e f g+3 d^2 g^2\right )-2 b e g \left (a e^3 f+c d^2 (3 e f-2 d g)\right )\right ) \sqrt {\frac {g \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c f+\left (-b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {\frac {g \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {2 c e f-b e g+\sqrt {b^2-4 a c} e g}{2 c e f-2 c d g};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-b g+\sqrt {b^2-4 a c} g}}\right )|\frac {2 c f+\left (-b+\sqrt {b^2-4 a c}\right ) g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{4 \sqrt {2} \sqrt {c} e^3 \left (c d^2+e (-b d+a e)\right ) (e f-d g)^2 \sqrt {a+x (b+c x)}} \]

[Out]

-1/2*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/e/(e*x+d)^2+1/4*(c*d*(-3*d*g+2*e*f)-e*(a*e*g-2*b*d*g+b*e*f))*(g*x+f)^(1
/2)*(c*x^2+b*x+a)^(1/2)/e/(a*e^2-b*d*e+c*d^2)/(-d*g+e*f)/(e*x+d)-1/8*(c*d*(-3*d*g+2*e*f)-e*(a*e*g-2*b*d*g+b*e*
f))*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*g*(-4*a*c+b^2)^(1/2)/(2*
c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*(-4*a*c+b^2)^(1/2)*(g*x+f)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/e
^2/(a*e^2-b*d*e+c*d^2)/(-d*g+e*f)*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*(g*x+f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/
2)-1/4*(-c*d*(3*d*g+2*e*f)+e*(-5*a*e*g+4*b*d*g+b*e*f))*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2
)^(1/2))^(1/2)*2^(1/2),2^(1/2)*(g*(-4*a*c+b^2)^(1/2)/(-2*c*f+g*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*(-4*a*c+b^2)^(1
/2)*(c*(a+x*(c*x+b))/(4*a*c-b^2))^(1/2)*(c*(g*x+f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/e^3/(c*d^2+e*(a*e-b
*d))*2^(1/2)/(g*x+f)^(1/2)/(a+x*(c*x+b))^(1/2)+1/8*(b^2*e^4*f^2+a^2*e^4*g^2+c^2*d^3*g*(-3*d*g+4*e*f)-2*a*c*e^2
*(3*d^2*g^2-6*d*e*f*g+2*e^2*f^2)-2*b*e*g*(a*e^3*f+c*d^2*(-2*d*g+3*e*f)))*EllipticPi(2^(1/2)*c^(1/2)*(g*x+f)^(1
/2)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2))^(1/2),(2*c*e*f-b*e*g+e*g*(-4*a*c+b^2)^(1/2))/(-2*c*d*g+2*c*e*f),((2*c*f+g
*(-b+(-4*a*c+b^2)^(1/2)))/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2))^(1/2)*(g*(
-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*f+g*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(-2*c*f+
g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/e^3/(c*d^2+e*(a*e-b*d))/(-d*g+e*f)^2*2^(1/2)/c^(1/2)/(a+x*(c*x+b))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 5.41, antiderivative size = 1705, normalized size of antiderivative = 1.65, number of steps used = 25, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {930, 6874, 732, 430, 953, 857, 435, 948, 175, 552, 551} \begin {gather*} \frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\text {ArcSin}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right ) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2}{4 \sqrt {2} \sqrt {c} e^3 \left (c d^2-b e d+a e^2\right ) (e f-d g)^2 \sqrt {c x^2+b x+a}}-\frac {\sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right ) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))}{4 \sqrt {2} e^2 \left (c d^2-b e d+a e^2\right ) (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}+\frac {\sqrt {b^2-4 a c} f \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right ) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))}{2 \sqrt {2} e^2 \left (c d^2-b e d+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {b^2-4 a c} d g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right ) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))}{2 \sqrt {2} e^3 \left (c d^2-b e d+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))}{4 e \left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)}+\frac {3 \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {2} e^3 \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c e f-3 c d g+b e g) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\text {ArcSin}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {2} \sqrt {c} e^3 (e f-d g) \sqrt {c x^2+b x+a}}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 175

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] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

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 732

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

Rule 857

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

Rule 930

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :>
Simp[(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + b*x + 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 + b*x + c*x^2]))*Simp[b*f + a*g + 2*(c*f + b*g)*x + 3*c*g*x^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0
] && IntegerQ[2*m] && LtQ[m, -1]

Rule 948

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[Sqrt[b - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2]), Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 953

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :>
Simp[e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + b*x + c*x^2]/((m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2))
), x] + Dist[1/(2*(m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2)), Int[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a +
b*x + c*x^2]))*Simp[2*d*(c*e*f - c*d*g + b*e*g)*(m + 1) - e^2*(b*f + a*g)*(2*m + 3) + 2*e*(c*d*g*(m + 1) - e*
(c*f + b*g)*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g
, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + 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+b x+c x^2}}{(d+e x)^3} \, dx &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {\int \frac {b f+a g+2 (c f+b g) x+3 c g x^2}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{4 e}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {\int \left (\frac {3 c g}{e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {-c d (2 e f-3 d g)+e (b e f-2 b d g+a e g)}{e^2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {2 (c e f-3 c d g+b e g)}{e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}\right ) \, dx}{4 e}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(3 c g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{4 e^3}+\frac {(c e f-3 c d g+b e g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{2 e^3}-\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{4 e^3}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 e \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \int \frac {-2 c d (e f-d g)+e (b e f-2 b d g+a e g)-2 c d e g x-c e^2 g x^2}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{8 e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g)}+\frac {\left ((c e f-3 c d g+b e g) \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \int \frac {1}{\sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x} (d+e x) \sqrt {f+g x}} \, dx}{2 e^3 \sqrt {a+b x+c x^2}}+\frac {\left (3 \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{\sqrt {2} e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 e \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)}+\frac {3 \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {2} e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \int \left (-\frac {c d g}{\sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {c e g x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {-c d (2 e f-3 d g)+e (b e f-2 b d g+a e g)}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}\right ) \, dx}{8 e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g)}-\frac {\left ((c e f-3 c d g+b e g) \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}}} \, dx,x,\sqrt {f+g x}\right )}{e^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 e \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)}+\frac {3 \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {2} e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {(c d g (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{8 e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g)}-\frac {(c g (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))) \int \frac {x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)}-\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{8 e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g)}-\frac {\left ((c e f-3 c d g+b e g) \sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{e^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 e \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)}+\frac {3 \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {2} e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {(c (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)}+\frac {(c f (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)}-\frac {\left ((c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \int \frac {1}{\sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x} (d+e x) \sqrt {f+g x}} \, dx}{8 e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {a+b x+c x^2}}-\frac {\left (\sqrt {b^2-4 a c} d g (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{2 \sqrt {2} e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\left ((c e f-3 c d g+b e g) \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c (f+g x)}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c x^2}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{e^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 e \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)}+\frac {3 \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {2} e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {b^2-4 a c} d g (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{2 \sqrt {2} e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c e f-3 c d g+b e g) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {2} \sqrt {c} e^3 (e f-d g) \sqrt {a+b x+c x^2}}+\frac {\left ((c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}}} \, dx,x,\sqrt {f+g x}\right )}{4 e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {a+b x+c x^2}}-\frac {\left (\sqrt {b^2-4 a c} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{4 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}+\frac {\left (\sqrt {b^2-4 a c} f (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{2 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 e \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {b^2-4 a c} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{4 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {3 \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {2} e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {b^2-4 a c} f (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{2 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {b^2-4 a c} d g (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{2 \sqrt {2} e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c e f-3 c d g+b e g) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {2} \sqrt {c} e^3 (e f-d g) \sqrt {a+b x+c x^2}}+\frac {\left ((c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{4 e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 e \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {b^2-4 a c} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{4 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {3 \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {2} e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {b^2-4 a c} f (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{2 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {b^2-4 a c} d g (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{2 \sqrt {2} e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c e f-3 c d g+b e g) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {2} \sqrt {c} e^3 (e f-d g) \sqrt {a+b x+c x^2}}+\frac {\left ((c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c (f+g x)}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c x^2}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{4 e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 e \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {b^2-4 a c} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{4 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {3 \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {2} e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {b^2-4 a c} f (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{2 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {b^2-4 a c} d g (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{2 \sqrt {2} e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c e f-3 c d g+b e g) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {2} \sqrt {c} e^3 (e f-d g) \sqrt {a+b x+c x^2}}+\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{4 \sqrt {2} \sqrt {c} e^3 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 34.71, size = 33765, normalized size = 32.65 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(55359\) vs. \(2(934)=1868\).
time = 0.21, size = 55360, normalized size = 53.54

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1593\)
default	\(\text {Expression too large to display}\)	\(55360\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integrate(sqrt(c*x^2 + b*x + 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+b\,x+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 + b*x + c*x^2)^(1/2))/(d + e*x)^3,x)

[Out]

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

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