\(\int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx\) [907]

Optimal result

Integrand size = 31, antiderivative size = 675 \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {2} \sqrt {b^2-4 a c} g \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c e \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (e f-d g) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} e^2 \sqrt {a+b x+c x^2}} \]

[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] (verified)

Time = 0.97 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {971, 732, 430, 948, 175, 552, 551, 435} \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {2} g \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} (e f-d g) \sqrt {2 c f-g \left (b-\sqrt {b^2-4 a c}\right )} \sqrt {1-\frac {2 c (f+g x)}{2 c f-g \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} e^2 \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} g \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}} \]

[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 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 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 971

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*} \text {integral}& = \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 ) \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 )}{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 ) \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 )}{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 ) \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^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 ) \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^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 ) \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^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*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1385\) vs. \(2(675)=1350\).

Time = 13.60 (sec) , antiderivative size = 1385, normalized size of antiderivative = 2.05 \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {2} \sqrt {\frac {c (f+g x)}{2 c f+\left (-b+\sqrt {b^2-4 a c}\right ) g}} \left (\frac {2 f g \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {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-b g+\sqrt {b^2-4 a c} g}\right )}{c e \sqrt {\frac {-b+\sqrt {b^2-4 a c}-2 c x}{\sqrt {b^2-4 a c}}}}-\frac {d g^2 \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {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-b g+\sqrt {b^2-4 a c} g}\right )}{c e^2 \sqrt {\frac {-b+\sqrt {b^2-4 a c}-2 c x}{\sqrt {b^2-4 a c}}}}+\frac {g \left (-b+\sqrt {b^2-4 a c}-2 c x\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}} \left (\left (-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g\right ) E\left (\arcsin \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 (-b+\sqrt {b^2-4 a c}\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 \operatorname {EllipticF}\left (\arcsin \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 (-b+\sqrt {b^2-4 a c}\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+\sqrt {b^2-4 a c}-2 c x\right )}{2 c f+\left (-b+\sqrt {b^2-4 a c}\right ) g}}}-\frac {4 \sqrt {b^2-4 a c} f^2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c} e}{2 c d-b e+\sqrt {b^2-4 a c} e},\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-b g+\sqrt {b^2-4 a c} g}\right )}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}+\frac {8 \sqrt {b^2-4 a c} d f g \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c} e}{2 c d-b e+\sqrt {b^2-4 a c} e},\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-b g+\sqrt {b^2-4 a c} g}\right )}{e \left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right )}-\frac {4 \sqrt {b^2-4 a c} d^2 g^2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c} e}{2 c d-b e+\sqrt {b^2-4 a c} e},\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-b g+\sqrt {b^2-4 a c} g}\right )}{e^2 \left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right )}\right )}{\sqrt {f+g x} \sqrt {a+x (b+c x)}} \]

[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)])

Maple [A] (verified)

Time = 2.00 (sec) , antiderivative size = 1122, normalized size of antiderivative = 1.66

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1122\)
default	\(\text {Expression too large to display}\)	\(1879\)

[In]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)