\(\int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx\) [915]

Optimal result

Integrand size = 31, antiderivative size = 1114 \[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac {3 e^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 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac {3 \sqrt {b^2-4 a c} e (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 (\arcsin \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} \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 \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 (-6 e f+7 d g)+e (3 b e f-4 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 (a+x (b+c x))}{-b^2+4 a c}} \operatorname {EllipticF}\left (\arcsin \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} \left (c d^2+e (-b d+a e)\right )^2 (e f-d g) \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 (c^2 d^2 \left (8 e^2 f^2-20 d e f g+15 d^2 g^2\right )+2 c e \left (b d \left (-4 e^2 f^2+11 d e f g-10 d^2 g^2\right )+a e \left (-2 e^2 f^2+2 d e f g+3 d^2 g^2\right )\right )+e^2 \left (3 a^2 e^2 g^2+2 a b e g (e f-4 d g)+b^2 \left (3 e^2 f^2-8 d e f g+8 d^2 g^2\right )\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}} \operatorname {EllipticPi}\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},\arcsin \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} \left (c d^2+e (-b d+a e)\right )^2 (-e f+d g)^3 \sqrt {a+x (b+c x)}} \]

[Out]

-1/2*e^2*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(-d*g+e*f)/(e*x+d)^2-3/4*e^2*(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)/(a*e^2-b*d*e+c*d^2)^2/(-d*g+e*f)^2/(e*x+d)+3/8*e*
(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)/(a*e^2-b*d*e+c*d^2)^2/(-d*g+e*f)^2*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*(7*d*g-6*e*f)+e*(a*e*g-4*b*d*g+3*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)/(c*d^2+e*(a*e-b*d))^2/(-d*g+e*f)*2^(1/2)/(g*x+f)^(1/2)/(a+x*(c*x+b))^(1/2)+1/8*(c^2*d^2*(1
5*d^2*g^2-20*d*e*f*g+8*e^2*f^2)+2*c*e*(b*d*(-10*d^2*g^2+11*d*e*f*g-4*e^2*f^2)+a*e*(3*d^2*g^2+2*d*e*f*g-2*e^2*f
^2))+e^2*(3*a^2*e^2*g^2+2*a*b*e*g*(-4*d*g+e*f)+b^2*(8*d^2*g^2-8*d*e*f*g+3*e^2*f^2)))*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)/(c*d^2+e*(a*e-b*d))^2/(d*g-e*f)^3*2^(1/2)/c^(1/2)/(a+x*(c*x+b))^(
1/2)

Rubi [A] (verified)

Time = 4.76 (sec) , antiderivative size = 1762, normalized size of antiderivative = 1.58, number of steps used = 25, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {953, 6874, 732, 430, 857, 435, 948, 175, 552, 551} \[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {3 (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 x^2+b x+a} e^2}{4 \left (c d^2-b e d+a e^2\right )^2 (e f-d g)^2 (d+e x)}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a} e^2}{2 \left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)^2}+\frac {3 \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 (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\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 ) e}{4 \sqrt {2} \left (c d^2-b e d+a e^2\right )^2 (e f-d g)^2 \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 {3 \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 (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\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 ) e}{2 \sqrt {2} \left (c d^2-b e d+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {3 \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 (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\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 )}{2 \sqrt {2} \left (c d^2-b e d+a e^2\right )^2 (e f-d g)^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {\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}} \operatorname {EllipticF}\left (\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} \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 {3 \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}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\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 )}{4 \sqrt {2} \sqrt {c} \left (c d^2-b e d+a e^2\right )^2 (e f-d g)^3 \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}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\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} \left (c d^2-b e d+a e^2\right ) (e f-d g)^2 \sqrt {c x^2+b x+a}} \]

[In]

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

[Out]

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 37.30 (sec) , antiderivative size = 40396, normalized size of antiderivative = 36.26 \[ \int \frac {1}{(d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [A] (verified)

Time = 4.55 (sec) , antiderivative size = 1686, normalized size of antiderivative = 1.51

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1686\)
default	\(\text {Expression too large to display}\)	\(64947\)

[In]

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

[Out]

((g*x+f)*(c*x^2+b*x+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)*(1/2*e^2/(a*d*e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*
f+c*d^3*g-c*d^2*e*f)*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)/(e*x+d)^2+3/4*e^2*(a*e^2*g-2*b*d*e*g+b*e^
2*f+3*c*d^2*g-2*c*d*e*f)/(a*d*e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)^2*(c*g*x^3+b*g*x^2+c*f*x^2+
a*g*x+b*f*x+a*f)^(1/2)/(e*x+d)-1/4*c*g*(a*d*e^2*g+2*a*e^3*f-4*b*d^2*e*g+b*d*e^2*f+7*c*d^3*g-4*c*d^2*e*f)/(a*d*
e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)^2*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-1/2*(b
+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*
((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*
f*x+a*f)^(1/2)*EllipticF(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/
c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))-3/4*e*c*g*(a*e^2*g-2*b*d*e*g+b*e^2*f+3*c*d^2*g-2*c*d*e*f)/(a*d
*e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)^2*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-1/2*(
b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)
*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b
*f*x+a*f)^(1/2)*((-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(
1/2),((-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2)^
(1/2))*EllipticF(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g
-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)))+1/4*(3*a^2*e^4*g^2-8*a*b*d*e^3*g^2+2*a*b*e^4*f*g+6*a*c*d^2*e^2*g^2+4*
a*c*d*e^3*f*g-4*a*c*e^4*f^2+8*b^2*d^2*e^2*g^2-8*b^2*d*e^3*f*g+3*b^2*e^4*f^2-20*b*c*d^3*e*g^2+22*b*c*d^2*e^2*f*
g-8*b*c*d*e^3*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-b*d^2*e*g+b*d*e^2*f+c*
d^3*g-c*d^2*e*f)^2/e*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x
-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/
(-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)/(-f/g+d/e)*Elliptic
Pi(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),(-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g+d/e),((-f/g+1/
2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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