3.907 \(\int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx\)

Optimal. Leaf size=675 \[ \frac {2 \sqrt {2} g \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} (e f-d g) \sqrt {2 c f-g \left (b-\sqrt {b^2-4 a c}\right )} \sqrt {1-\frac {2 c (f+g x)}{2 c f-g \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} e^2 \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} g \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.65, antiderivative size = 675, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {957, 718, 419, 934, 169, 538, 537, 424} \[ \frac {2 \sqrt {2} g \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} (e f-d g) \sqrt {2 c f-g \left (b-\sqrt {b^2-4 a c}\right )} \sqrt {1-\frac {2 c (f+g x)}{2 c f-g \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} e^2 \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} g \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*Sqrt[b^2 - 4*a*c]*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)])/(c*e*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]
*Sqrt[b^2 - 4*a*c]*g*(e*f - d*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)])/(c*e^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]) - (Sqr
t[2]*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]*(e*f - d*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[c]*e^2*Sqrt[a + b*x + c*x^2]
)

Rule 169

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 419

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

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 538

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

Rule 718

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 934

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 957

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Int
[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*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[
n + 1/2]

Rubi steps

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

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Mathematica [B]  time = 4.55, size = 1385, normalized size = 2.05 \[ \frac {\sqrt {2} \sqrt {\frac {c (f+g x)}{2 c f+\left (\sqrt {b^2-4 a c}-b\right ) g}} \left (-\frac {4 \sqrt {b^2-4 a c} \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \Pi \left (\frac {2 \sqrt {b^2-4 a c} e}{2 c d-b e+\sqrt {b^2-4 a c} e};\sin ^{-1}\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-b g+\sqrt {b^2-4 a c} g}\right ) f^2}{2 c d+\left (\sqrt {b^2-4 a c}-b\right ) e}+\frac {2 g \left (b+2 c x-\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}} F\left (\sin ^{-1}\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-b g+\sqrt {b^2-4 a c} g}\right ) f}{c e \sqrt {\frac {-b-2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}+\frac {8 \sqrt {b^2-4 a c} d g \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \Pi \left (\frac {2 \sqrt {b^2-4 a c} e}{2 c d-b e+\sqrt {b^2-4 a c} e};\sin ^{-1}\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-b g+\sqrt {b^2-4 a c} g}\right ) f}{e \left (2 c d+\left (\sqrt {b^2-4 a c}-b\right ) e\right )}-\frac {d g^2 \left (b+2 c x-\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}} F\left (\sin ^{-1}\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-b g+\sqrt {b^2-4 a c} g}\right )}{c e^2 \sqrt {\frac {-b-2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}+\frac {g \left (-b-2 c x+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) g-2 c f}} \left (\left (\left (b+\sqrt {b^2-4 a c}\right ) g-2 c f\right ) E\left (\sin ^{-1}\left (\sqrt {2} \sqrt {\frac {c (f+g x)}{2 c f-b g+\sqrt {b^2-4 a c} g}}\right )|\frac {2 c f+\left (\sqrt {b^2-4 a c}-b\right ) g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )-\left (b+\sqrt {b^2-4 a c}\right ) g F\left (\sin ^{-1}\left (\sqrt {2} \sqrt {\frac {c (f+g x)}{2 c f-b g+\sqrt {b^2-4 a c} g}}\right )|\frac {2 c f+\left (\sqrt {b^2-4 a c}-b\right ) g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )\right )}{2 c^2 e \sqrt {\frac {g \left (-b-2 c x+\sqrt {b^2-4 a c}\right )}{2 c f+\left (\sqrt {b^2-4 a c}-b\right ) g}}}-\frac {4 \sqrt {b^2-4 a c} d^2 g^2 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \Pi \left (\frac {2 \sqrt {b^2-4 a c} e}{2 c d-b e+\sqrt {b^2-4 a c} e};\sin ^{-1}\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-b g+\sqrt {b^2-4 a c} g}\right )}{e^2 \left (2 c d+\left (\sqrt {b^2-4 a c}-b\right ) e\right )}\right )}{\sqrt {f+g x} \sqrt {a+x (b+c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[2]*Sqrt[(c*(f + g*x))/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]*((2*f*g*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*Sqrt
[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]*EllipticF[ArcSin[Sqrt[(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/Sqr
t[b^2 - 4*a*c]]/Sqrt[2]], (2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g)])/(c*e*Sqrt[(-b + Sqrt[b
^2 - 4*a*c] - 2*c*x)/Sqrt[b^2 - 4*a*c]]) - (d*g^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*Sqrt[(b + Sqrt[b^2 - 4*a*c]
+ 2*c*x)/Sqrt[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*g + Sqrt[b^2 - 4*a*c]*g)])/(c*e^2*Sqrt[(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/S
qrt[b^2 - 4*a*c]]) + (g*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*Sqrt[(g*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*f + (b
+ Sqrt[b^2 - 4*a*c])*g)]*((-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)*EllipticE[ArcSin[Sqrt[2]*Sqrt[(c*(f + g*x))/(2
*c*f - b*g + Sqrt[b^2 - 4*a*c]*g)]], (2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]
- (b + Sqrt[b^2 - 4*a*c])*g*EllipticF[ArcSin[Sqrt[2]*Sqrt[(c*(f + g*x))/(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g)]]
, (2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]))/(2*c^2*e*Sqrt[(g*(-b + Sqrt[b^2
- 4*a*c] - 2*c*x))/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]) - (4*Sqrt[b^2 - 4*a*c]*f^2*Sqrt[(c*(a + x*(b + c*x))
)/(-b^2 + 4*a*c)]*EllipticPi[(2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e), ArcSin[Sqrt[(-b + Sq
rt[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*g + Sqrt[b^2 - 4*a*c]
*g)])/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e) + (8*Sqrt[b^2 - 4*a*c]*d*f*g*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a
*c)]*EllipticPi[(2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e), 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*g + Sqrt[b^2 - 4*a*c]*g)])/(e*(2*c
*d + (-b + Sqrt[b^2 - 4*a*c])*e)) - (4*Sqrt[b^2 - 4*a*c]*d^2*g^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*El
lipticPi[(2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e), 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*g + Sqrt[b^2 - 4*a*c]*g)])/(e^2*(2*c*d +
(-b + Sqrt[b^2 - 4*a*c])*e))))/(Sqrt[f + g*x]*Sqrt[a + x*(b + c*x)])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 1879, normalized size = 2.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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