3.10.3 \(\int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx\) [903]

Optimal. Leaf size=188 \[ \frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}} \]

[Out]

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/(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)

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Rubi [A]
time = 0.05, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {732, 435} \begin {gather*} \frac {\sqrt {2} \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 (\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 \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 )}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[f + g*x]/Sqrt[a + b*x + c*x^2],x]

[Out]

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

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 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]

Rubi steps

\begin {align*} \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx &=\frac {\left (\sqrt {2} \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}}\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 \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 {\sqrt {2} \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+\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 \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}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 20.56, size = 365, normalized size = 1.94 \begin {gather*} \frac {i \left (2 c f+\left (-b+\sqrt {b^2-4 a c}\right ) g\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 {1-\frac {2 c (f+g x)}{2 c f+\left (-b+\sqrt {b^2-4 a c}\right ) g}} \left (E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {f+g x}\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 )-F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {f+g x}\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 )}{\sqrt {2} c g \sqrt {\frac {c}{-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[f + g*x]/Sqrt[a + b*x + c*x^2],x]

[Out]

(I*(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)*Sqrt[(g*(b + Sqrt[b^2 - 4*a*c] + 2*c*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)]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(-
2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[f + g*x]], (2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f + (-b + Sqrt[b^
2 - 4*a*c])*g)] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[f + g*x]], (2*
c*f - (b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]))/(Sqrt[2]*c*g*Sqrt[c/(-2*c*f + (b + Sq
rt[b^2 - 4*a*c])*g)]*Sqrt[a + x*(b + c*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(746\) vs. \(2(164)=328\).
time = 0.12, size = 747, normalized size = 3.97

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 f \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}+\frac {2 g \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}\)	\(746\)
default	\(\frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}\, \left (g \sqrt {-4 a c +b^{2}}+b g -2 c f \right ) \sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-4 a c +b^{2}}+b g -2 c f}}\, \sqrt {\frac {\left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) g}{2 c f -b g +g \sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {\left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) g}{g \sqrt {-4 a c +b^{2}}+b g -2 c f}}\, \left (\EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-4 a c +b^{2}}+b g -2 c f}}, \sqrt {-\frac {g \sqrt {-4 a c +b^{2}}+b g -2 c f}{2 c f -b g +g \sqrt {-4 a c +b^{2}}}}\right ) g b -2 f \EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-4 a c +b^{2}}+b g -2 c f}}, \sqrt {-\frac {g \sqrt {-4 a c +b^{2}}+b g -2 c f}{2 c f -b g +g \sqrt {-4 a c +b^{2}}}}\right ) c -\EllipticF \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-4 a c +b^{2}}+b g -2 c f}}, \sqrt {-\frac {g \sqrt {-4 a c +b^{2}}+b g -2 c f}{2 c f -b g +g \sqrt {-4 a c +b^{2}}}}\right ) g \sqrt {-4 a c +b^{2}}-\EllipticE \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-4 a c +b^{2}}+b g -2 c f}}, \sqrt {-\frac {g \sqrt {-4 a c +b^{2}}+b g -2 c f}{2 c f -b g +g \sqrt {-4 a c +b^{2}}}}\right ) b g +2 \EllipticE \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-4 a c +b^{2}}+b g -2 c f}}, \sqrt {-\frac {g \sqrt {-4 a c +b^{2}}+b g -2 c f}{2 c f -b g +g \sqrt {-4 a c +b^{2}}}}\right ) c f +\sqrt {-4 a c +b^{2}}\, \EllipticE \left (\sqrt {2}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-4 a c +b^{2}}+b g -2 c f}}, \sqrt {-\frac {g \sqrt {-4 a c +b^{2}}+b g -2 c f}{2 c f -b g +g \sqrt {-4 a c +b^{2}}}}\right ) g \right )}{2 g \left (c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a \right ) c^{2}}\)	\(747\)

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),x,method=_RETURNVERBOSE)

[Out]

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

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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),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.39, size = 359, normalized size = 1.91 \begin {gather*} -\frac {2 \, {\left (3 \, \sqrt {c g} c g {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right ) - {\left (2 \, c f - b g\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right )}}{3 \, c^{2} g} \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),x, algorithm="fricas")

[Out]

-2/3*(3*sqrt(c*g)*c*g*weierstrassZeta(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3*f^3
- 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), weierstrassPInverse(4/3*(c^2*f
^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2
*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), 1/3*(3*c*g*x + c*f + b*g)/(c*g))) - (2*c*f - b*g)*sqrt(c*g)*weierstrassPInvers
e(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2
)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), 1/3*(3*c*g*x + c*f + b*g)/(c*g)))/(c^2*g)

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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}}}\, 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),x)

[Out]

Integral(sqrt(f + g*x)/sqrt(a + b*x + c*x**2), 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),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {f+g\,x}}{\sqrt {c\,x^2+b\,x+a}} \,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),x)

[Out]

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

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