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

Optimal. Leaf size=246 \[ \frac{\sqrt{d+e x} \sqrt{f+g x} \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{8 e^2 g^3}-\frac{(e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right ) \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{8 e^{5/2} g^{7/2}}-\frac{(d+e x)^{3/2} \sqrt{f+g x} (-6 b e g+7 c d g+5 c e f)}{12 e^2 g^2}+\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g} \]

[Out]

((c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*Sqrt[d + e*x]*Sqrt[f + g*x])/(8*e^2
*g^3) - ((5*c*e*f + 7*c*d*g - 6*b*e*g)*(d + e*x)^(3/2)*Sqrt[f + g*x])/(12*e^2*g^2) + (c*(d + e*x)^(5/2)*Sqrt[f
 + g*x])/(3*e^2*g) - ((e*f - d*g)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*Ar
cTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(8*e^(5/2)*g^(7/2))

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Rubi [A]  time = 0.255505, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {951, 80, 50, 63, 217, 206} \[ \frac{\sqrt{d+e x} \sqrt{f+g x} \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{8 e^2 g^3}-\frac{(e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right ) \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{8 e^{5/2} g^{7/2}}-\frac{(d+e x)^{3/2} \sqrt{f+g x} (-6 b e g+7 c d g+5 c e f)}{12 e^2 g^2}+\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*Sqrt[d + e*x]*Sqrt[f + g*x])/(8*e^2
*g^3) - ((5*c*e*f + 7*c*d*g - 6*b*e*g)*(d + e*x)^(3/2)*Sqrt[f + g*x])/(12*e^2*g^2) + (c*(d + e*x)^(5/2)*Sqrt[f
 + g*x])/(3*e^2*g) - ((e*f - d*g)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*Ar
cTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(8*e^(5/2)*g^(7/2))

Rule 951

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Simp[(c^p*(d + e*x)^(m + 2*p)*(f + g*x)^(n + 1))/(g*e^(2*p)*(m + n + 2*p + 1)), x] + Dist[1/(g*e^(2*p)*(m +
n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*
(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), 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] && IGtQ[p, 0] && NeQ[m + n + 2*
p + 1, 0] && (IntegerQ[n] ||  !IntegerQ[m])

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x} \left (a+b x+c x^2\right )}{\sqrt{f+g x}} \, dx &=\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g}+\frac{\int \frac{\sqrt{d+e x} \left (\frac{1}{2} \left (6 a e^2 g-c d (5 e f+d g)\right )-\frac{1}{2} e (5 c e f+7 c d g-6 b e g) x\right )}{\sqrt{f+g x}} \, dx}{3 e^2 g}\\ &=-\frac{(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt{f+g x}}{12 e^2 g^2}+\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g}+\frac{\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \int \frac{\sqrt{d+e x}}{\sqrt{f+g x}} \, dx}{8 e^2 g^2}\\ &=\frac{\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \sqrt{d+e x} \sqrt{f+g x}}{8 e^2 g^3}-\frac{(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt{f+g x}}{12 e^2 g^2}+\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g}-\frac{\left ((e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{f+g x}} \, dx}{16 e^2 g^3}\\ &=\frac{\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \sqrt{d+e x} \sqrt{f+g x}}{8 e^2 g^3}-\frac{(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt{f+g x}}{12 e^2 g^2}+\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g}-\frac{\left ((e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{f-\frac{d g}{e}+\frac{g x^2}{e}}} \, dx,x,\sqrt{d+e x}\right )}{8 e^3 g^3}\\ &=\frac{\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \sqrt{d+e x} \sqrt{f+g x}}{8 e^2 g^3}-\frac{(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt{f+g x}}{12 e^2 g^2}+\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g}-\frac{\left ((e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{g x^2}{e}} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{8 e^3 g^3}\\ &=\frac{\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \sqrt{d+e x} \sqrt{f+g x}}{8 e^2 g^3}-\frac{(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt{f+g x}}{12 e^2 g^2}+\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g}-\frac{(e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{8 e^{5/2} g^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.08633, size = 225, normalized size = 0.91 \[ \frac{-e \sqrt{g} \sqrt{d+e x} (f+g x) \left (c \left (3 d^2 g^2-2 d e g (g x-2 f)+e^2 \left (-15 f^2+10 f g x-8 g^2 x^2\right )\right )-6 e g (4 a e g+b (d g-3 e f+2 e g x))\right )-3 (e f-d g)^{3/2} \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right ) \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{24 e^3 g^{7/2} \sqrt{f+g x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(e*Sqrt[g]*Sqrt[d + e*x]*(f + g*x)*(-6*e*g*(4*a*e*g + b*(-3*e*f + d*g + 2*e*g*x)) + c*(3*d^2*g^2 - 2*d*e*g*(
-2*f + g*x) + e^2*(-15*f^2 + 10*f*g*x - 8*g^2*x^2)))) - 3*(e*f - d*g)^(3/2)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^
2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e
*f - d*g]])/(24*e^3*g^(7/2)*Sqrt[f + g*x])

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Maple [B]  time = 0.301, size = 763, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(16*x^2*c*e^2*g^2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+24*ln(1/2*(2*e*g*x+2*((
g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*a*d*e^2*g^3-24*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/
2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*a*e^3*f*g^2-6*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e
*f)/(e*g)^(1/2))*b*d^2*e*g^3-12*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*b*
d*e^2*f*g^2+18*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*b*e^3*f^2*g+3*ln(1/
2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^3*g^3+3*ln(1/2*(2*e*g*x+2*((g*x+f)*
(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e*f*g^2+9*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*
g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d*e^2*f^2*g-15*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)
/(e*g)^(1/2))*c*e^3*f^3+24*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*x*b*e^2*g^2+4*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/
2)*x*c*d*e*g^2-20*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*x*c*e^2*f*g+48*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*a*e^2
*g^2+12*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*b*d*e*g^2-36*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*b*e^2*f*g-6*(e*g)
^(1/2)*((g*x+f)*(e*x+d))^(1/2)*c*d^2*g^2-8*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*c*d*e*f*g+30*(e*g)^(1/2)*((g*x+
f)*(e*x+d))^(1/2)*c*e^2*f^2)/g^3/((g*x+f)*(e*x+d))^(1/2)/e^2/(e*g)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.12624, size = 1277, normalized size = 5.19 \begin{align*} \left [-\frac{3 \,{\left (5 \, c e^{3} f^{3} - 3 \,{\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2} g -{\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f g^{2} -{\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} g^{3}\right )} \sqrt{e g} \log \left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 4 \,{\left (2 \, e g x + e f + d g\right )} \sqrt{e g} \sqrt{e x + d} \sqrt{g x + f} + 8 \,{\left (e^{2} f g + d e g^{2}\right )} x\right ) - 4 \,{\left (8 \, c e^{3} g^{3} x^{2} + 15 \, c e^{3} f^{2} g - 2 \,{\left (2 \, c d e^{2} + 9 \, b e^{3}\right )} f g^{2} - 3 \,{\left (c d^{2} e - 2 \, b d e^{2} - 8 \, a e^{3}\right )} g^{3} - 2 \,{\left (5 \, c e^{3} f g^{2} -{\left (c d e^{2} + 6 \, b e^{3}\right )} g^{3}\right )} x\right )} \sqrt{e x + d} \sqrt{g x + f}}{96 \, e^{3} g^{4}}, \frac{3 \,{\left (5 \, c e^{3} f^{3} - 3 \,{\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2} g -{\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f g^{2} -{\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} g^{3}\right )} \sqrt{-e g} \arctan \left (\frac{{\left (2 \, e g x + e f + d g\right )} \sqrt{-e g} \sqrt{e x + d} \sqrt{g x + f}}{2 \,{\left (e^{2} g^{2} x^{2} + d e f g +{\left (e^{2} f g + d e g^{2}\right )} x\right )}}\right ) + 2 \,{\left (8 \, c e^{3} g^{3} x^{2} + 15 \, c e^{3} f^{2} g - 2 \,{\left (2 \, c d e^{2} + 9 \, b e^{3}\right )} f g^{2} - 3 \,{\left (c d^{2} e - 2 \, b d e^{2} - 8 \, a e^{3}\right )} g^{3} - 2 \,{\left (5 \, c e^{3} f g^{2} -{\left (c d e^{2} + 6 \, b e^{3}\right )} g^{3}\right )} x\right )} \sqrt{e x + d} \sqrt{g x + f}}{48 \, e^{3} g^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/96*(3*(5*c*e^3*f^3 - 3*(c*d*e^2 + 2*b*e^3)*f^2*g - (c*d^2*e - 4*b*d*e^2 - 8*a*e^3)*f*g^2 - (c*d^3 - 2*b*d^
2*e + 8*a*d*e^2)*g^3)*sqrt(e*g)*log(8*e^2*g^2*x^2 + e^2*f^2 + 6*d*e*f*g + d^2*g^2 + 4*(2*e*g*x + e*f + d*g)*sq
rt(e*g)*sqrt(e*x + d)*sqrt(g*x + f) + 8*(e^2*f*g + d*e*g^2)*x) - 4*(8*c*e^3*g^3*x^2 + 15*c*e^3*f^2*g - 2*(2*c*
d*e^2 + 9*b*e^3)*f*g^2 - 3*(c*d^2*e - 2*b*d*e^2 - 8*a*e^3)*g^3 - 2*(5*c*e^3*f*g^2 - (c*d*e^2 + 6*b*e^3)*g^3)*x
)*sqrt(e*x + d)*sqrt(g*x + f))/(e^3*g^4), 1/48*(3*(5*c*e^3*f^3 - 3*(c*d*e^2 + 2*b*e^3)*f^2*g - (c*d^2*e - 4*b*
d*e^2 - 8*a*e^3)*f*g^2 - (c*d^3 - 2*b*d^2*e + 8*a*d*e^2)*g^3)*sqrt(-e*g)*arctan(1/2*(2*e*g*x + e*f + d*g)*sqrt
(-e*g)*sqrt(e*x + d)*sqrt(g*x + f)/(e^2*g^2*x^2 + d*e*f*g + (e^2*f*g + d*e*g^2)*x)) + 2*(8*c*e^3*g^3*x^2 + 15*
c*e^3*f^2*g - 2*(2*c*d*e^2 + 9*b*e^3)*f*g^2 - 3*(c*d^2*e - 2*b*d*e^2 - 8*a*e^3)*g^3 - 2*(5*c*e^3*f*g^2 - (c*d*
e^2 + 6*b*e^3)*g^3)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(e^3*g^4)]

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Sympy [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)**(1/2)*(c*x**2+b*x+a)/(g*x+f)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.23661, size = 393, normalized size = 1.6 \begin{align*} \frac{1}{24} \, \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}}{\left (2 \,{\left (x e + d\right )}{\left (\frac{4 \,{\left (x e + d\right )} c e^{\left (-3\right )}}{g} - \frac{{\left (7 \, c d g^{4} e^{6} + 5 \, c f g^{3} e^{7} - 6 \, b g^{4} e^{7}\right )} e^{\left (-9\right )}}{g^{5}}\right )} + \frac{3 \,{\left (c d^{2} g^{4} e^{6} + 2 \, c d f g^{3} e^{7} - 2 \, b d g^{4} e^{7} + 5 \, c f^{2} g^{2} e^{8} - 6 \, b f g^{3} e^{8} + 8 \, a g^{4} e^{8}\right )} e^{\left (-9\right )}}{g^{5}}\right )} \sqrt{x e + d} - \frac{{\left (c d^{3} g^{3} + c d^{2} f g^{2} e - 2 \, b d^{2} g^{3} e + 3 \, c d f^{2} g e^{2} - 4 \, b d f g^{2} e^{2} + 8 \, a d g^{3} e^{2} - 5 \, c f^{3} e^{3} + 6 \, b f^{2} g e^{3} - 8 \, a f g^{2} e^{3}\right )} e^{\left (-\frac{5}{2}\right )} \log \left ({\left | -\sqrt{x e + d} \sqrt{g} e^{\frac{1}{2}} + \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}} \right |}\right )}{8 \, g^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt((x*e + d)*g*e - d*g*e + f*e^2)*(2*(x*e + d)*(4*(x*e + d)*c*e^(-3)/g - (7*c*d*g^4*e^6 + 5*c*f*g^3*e^7
 - 6*b*g^4*e^7)*e^(-9)/g^5) + 3*(c*d^2*g^4*e^6 + 2*c*d*f*g^3*e^7 - 2*b*d*g^4*e^7 + 5*c*f^2*g^2*e^8 - 6*b*f*g^3
*e^8 + 8*a*g^4*e^8)*e^(-9)/g^5)*sqrt(x*e + d) - 1/8*(c*d^3*g^3 + c*d^2*f*g^2*e - 2*b*d^2*g^3*e + 3*c*d*f^2*g*e
^2 - 4*b*d*f*g^2*e^2 + 8*a*d*g^3*e^2 - 5*c*f^3*e^3 + 6*b*f^2*g*e^3 - 8*a*f*g^2*e^3)*e^(-5/2)*log(abs(-sqrt(x*e
 + d)*sqrt(g)*e^(1/2) + sqrt((x*e + d)*g*e - d*g*e + f*e^2)))/g^(7/2)