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

Optimal. Leaf size=519 \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} (b e g-10 c d g+8 c e f) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 g \sqrt{b^2-4 a c}}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}\right )}{15 c^2 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c g (-6 a e g-5 b d g+3 b e f)+2 b^2 e g^2-2 c^2 f (4 e f-5 d g)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} (-b e g-5 c d g+4 c e f-3 c e g x)}{15 c g^2} \]

[Out]

(-2*Sqrt[f + g*x]*(4*c*e*f - 5*c*d*g - b*e*g - 3*c*e*g*x)*Sqrt[a + b*x + c*x^2])/(15*c*g^2) - (Sqrt[2]*Sqrt[b^
2 - 4*a*c]*(2*b^2*e*g^2 - 2*c^2*f*(4*e*f - 5*d*g) + c*g*(3*b*e*f - 5*b*d*g - 6*a*e*g))*Sqrt[f + g*x]*Sqrt[-((c
*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/S
qrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(15*c^2*g^3*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]*(8*c*e*f - 10*c*d*g + b*e*
g)*(c*f^2 - b*f*g + a*g^2)*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)])/(15*c^2*g^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.535008, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {814, 843, 718, 424, 419} \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} (b e g-10 c d g+8 c e f) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c g (-6 a e g-5 b d g+3 b e f)+2 b^2 e g^2-2 c^2 f (4 e f-5 d g)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} (-b e g-5 c d g+4 c e f-3 c e g x)}{15 c g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[f + g*x]*(4*c*e*f - 5*c*d*g - b*e*g - 3*c*e*g*x)*Sqrt[a + b*x + c*x^2])/(15*c*g^2) - (Sqrt[2]*Sqrt[b^
2 - 4*a*c]*(2*b^2*e*g^2 - 2*c^2*f*(4*e*f - 5*d*g) + c*g*(3*b*e*f - 5*b*d*g - 6*a*e*g))*Sqrt[f + g*x]*Sqrt[-((c
*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/S
qrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(15*c^2*g^3*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]*(8*c*e*f - 10*c*d*g + b*e*
g)*(c*f^2 - b*f*g + a*g^2)*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)])/(15*c^2*g^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

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 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 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 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)])

Rubi steps

\begin{align*} \int \frac{(d+e x) \sqrt{a+b x+c x^2}}{\sqrt{f+g x}} \, dx &=-\frac{2 \sqrt{f+g x} (4 c e f-5 c d g-b e g-3 c e g x) \sqrt{a+b x+c x^2}}{15 c g^2}-\frac{2 \int \frac{\frac{1}{2} \left (5 c d g (b f-2 a g)-b e f (4 c f-b g)+2 a e g \left (c f+\frac{b g}{2}\right )\right )+\frac{1}{2} \left (2 b^2 e g^2-2 c^2 f (4 e f-5 d g)+c g (3 b e f-5 b d g-6 a e g)\right ) x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{15 c g^2}\\ &=-\frac{2 \sqrt{f+g x} (4 c e f-5 c d g-b e g-3 c e g x) \sqrt{a+b x+c x^2}}{15 c g^2}-\frac{\left ((8 c e f-10 c d g+b e g) \left (c f^2-b f g+a g^2\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{15 c g^3}-\frac{\left (2 b^2 e g^2-2 c^2 f (4 e f-5 d g)+c g (3 b e f-5 b d g-6 a e g)\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{15 c g^3}\\ &=-\frac{2 \sqrt{f+g x} (4 c e f-5 c d g-b e g-3 c e g x) \sqrt{a+b x+c x^2}}{15 c g^2}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (2 b^2 e g^2-2 c^2 f (4 e f-5 d g)+c g (3 b e f-5 b d g-6 a e g)\right ) \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 )}{15 c^2 g^3 \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} (8 c e f-10 c d g+b e g) \left (c f^2-b f g+a g^2\right ) \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 )}{15 c^2 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \sqrt{f+g x} (4 c e f-5 c d g-b e g-3 c e g x) \sqrt{a+b x+c x^2}}{15 c g^2}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (2 b^2 e g^2-2 c^2 f (4 e f-5 d g)+c g (3 b e f-5 b d g-6 a e g)\right ) \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 )}{15 c^2 g^3 \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} (8 c e f-10 c d g+b e g) \left (c f^2-b f g+a g^2\right ) \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 )}{15 c^2 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 10.6082, size = 911, normalized size = 1.76 \[ \frac{2 \sqrt{a+x (b+c x)} \left (\left (2 f (4 e f-5 d g) c^2+g (-3 b e f+5 b d g+6 a e g) c-2 b^2 e g^2\right ) \left (c \left (\frac{f}{f+g x}-1\right )^2+\frac{g \left (-\frac{f b}{f+g x}+b+\frac{a g}{f+g x}\right )}{f+g x}\right )+\frac{i \sqrt{1-\frac{2 \left (c f^2+g (a g-b f)\right )}{\left (2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt{\frac{2 \left (c f^2+g (a g-b f)\right )}{\left (-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}+1} \left (\left (2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}\right ) \left (2 f (5 d g-4 e f) c^2+g (3 b e f-5 b d g-6 a e g) c+2 b^2 e g^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right )|-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )+\left (2 b^3 e g^3-b^2 \left (-c f e+2 \sqrt{\left (b^2-4 a c\right ) g^2} e+5 c d g\right ) g^2+b c \left (\sqrt{\left (b^2-4 a c\right ) g^2} (5 d g-3 e f)-8 a e g^2\right ) g+2 c \left (a \left (-2 c f e+3 \sqrt{\left (b^2-4 a c\right ) g^2} e+10 c d g\right ) g^2+c f \sqrt{\left (b^2-4 a c\right ) g^2} (4 e f-5 d g)\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right ),-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )\right )}{2 \sqrt{2} \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}} \sqrt{f+g x}}\right ) (f+g x)^{3/2}}{15 c^2 g^4 \sqrt{c x^2+b x+a} \sqrt{\frac{(f+g x)^2 \left (c \left (\frac{f}{f+g x}-1\right )^2+\frac{g \left (-\frac{f b}{f+g x}+b+\frac{a g}{f+g x}\right )}{f+g x}\right )}{g^2}}}+\left (\frac{2 (-4 c e f+5 c d g+b e g)}{15 c g^2}+\frac{2 e x}{5 g}\right ) \sqrt{a+x (b+c x)} \sqrt{f+g x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(-4*c*e*f + 5*c*d*g + b*e*g))/(15*c*g^2) + (2*e*x)/(5*g))*Sqrt[f + g*x]*Sqrt[a + x*(b + c*x)] + (2*(f + g*
x)^(3/2)*Sqrt[a + x*(b + c*x)]*((-2*b^2*e*g^2 + 2*c^2*f*(4*e*f - 5*d*g) + c*g*(-3*b*e*f + 5*b*d*g + 6*a*e*g))*
(c*(-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)) + ((I/2)*Sqrt[1 - (2*(c*f^2 +
g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[1 + (2*(c*f^2 + g*(-(b*f) + a*g)
))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(2*b^2*e*g^2
+ 2*c^2*f*(-4*e*f + 5*d*g) + c*g*(3*b*e*f - 5*b*d*g - 6*a*e*g))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*
f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g
^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] + (2*b^3*e*g^3 - b^2*g^2*(-(c*e*f) + 5*c*d*g + 2*e*Sqrt[(b^2 -
4*a*c)*g^2]) + b*c*g*(-8*a*e*g^2 + Sqrt[(b^2 - 4*a*c)*g^2]*(-3*e*f + 5*d*g)) + 2*c*(c*f*Sqrt[(b^2 - 4*a*c)*g^2
]*(4*e*f - 5*d*g) + a*g^2*(-2*c*e*f + 10*c*d*g + 3*e*Sqrt[(b^2 - 4*a*c)*g^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*S
qrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[
(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))]))/(Sqrt[2]*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c
*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[f + g*x])))/(15*c^2*g^4*Sqrt[a + b*x + c*x^2]*Sqrt[((f + g*x)^2*(c*(
-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)))/g^2])

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Maple [B]  time = 0.37, size = 6207, normalized size = 12. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}}{\sqrt{g x + f}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}}{\sqrt{g x + f}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right ) \sqrt{a + b x + c x^{2}}}{\sqrt{f + g x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out