3.50 \(\int \frac{\sqrt{c+d x} (A+B x+C x^2)}{(a+b x) \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=290 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (2 b d f (4 A b d f-a C (c f+3 d e))+(2 a d f-b c f+b d e) (4 a C d f+b (-4 B d f+c C f+3 C d e)))}{4 b^3 d^{3/2} f^{5/2}}-\frac{2 \sqrt{b c-a d} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right )}{b^3 \sqrt{b e-a f}}-\frac{\sqrt{c+d x} \sqrt{e+f x} (4 a C d f+b (-4 B d f+c C f+3 C d e))}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} \sqrt{e+f x}}{2 b d f} \]

[Out]

-((4*a*C*d*f + b*(3*C*d*e + c*C*f - 4*B*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b^2*d*f^2) + (C*(c + d*x)^(3/2)*
Sqrt[e + f*x])/(2*b*d*f) + ((2*b*d*f*(4*A*b*d*f - a*C*(3*d*e + c*f)) + (b*d*e - b*c*f + 2*a*d*f)*(4*a*C*d*f +
b*(3*C*d*e + c*C*f - 4*B*d*f)))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(4*b^3*d^(3/2)*f^(5/
2)) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt
[e + f*x])])/(b^3*Sqrt[b*e - a*f])

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Rubi [A]  time = 0.672355, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1615, 154, 157, 63, 217, 206, 93, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (2 b d f (4 A b d f-a C (c f+3 d e))+(2 a d f-b c f+b d e) (4 a C d f+b (-4 B d f+c C f+3 C d e)))}{4 b^3 d^{3/2} f^{5/2}}-\frac{2 \sqrt{b c-a d} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right )}{b^3 \sqrt{b e-a f}}-\frac{\sqrt{c+d x} \sqrt{e+f x} (4 a C d f+b (-4 B d f+c C f+3 C d e))}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} \sqrt{e+f x}}{2 b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((4*a*C*d*f + b*(3*C*d*e + c*C*f - 4*B*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b^2*d*f^2) + (C*(c + d*x)^(3/2)*
Sqrt[e + f*x])/(2*b*d*f) + ((2*b*d*f*(4*A*b*d*f - a*C*(3*d*e + c*f)) + (b*d*e - b*c*f + 2*a*d*f)*(4*a*C*d*f +
b*(3*C*d*e + c*C*f - 4*B*d*f)))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(4*b^3*d^(3/2)*f^(5/
2)) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt
[e + f*x])])/(b^3*Sqrt[b*e - a*f])

Rule 1615

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^
(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 3.85997, size = 465, normalized size = 1.6 \[ \frac{-\frac{8 \sqrt{a d-b c} \left (a (a C-b B)+A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{a d-b c}}\right )}{\sqrt{b e-a f}}+\frac{8 \sqrt{d e-c f} \left (a (a C-b B)+A b^2\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sinh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )}{\sqrt{f} \sqrt{e+f x}}+\frac{4 b \sqrt{e+f x} (a C f-b B f+b C e) \left (\sqrt{c+d x} (d e-c f) \sinh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )-\sqrt{f} (c+d x) \sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}}\right )}{f^{5/2} \sqrt{c+d x} \sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}}}+\frac{b^2 C \sqrt{e+f x} \left (\sqrt{f} \sqrt{c+d x} (c f+d (e+2 f x))-\frac{(d e-c f)^{3/2} \sinh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )}{\sqrt{\frac{d (e+f x)}{d e-c f}}}\right )}{d f^{5/2}}}{4 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[d*e - c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])
/Sqrt[d*e - c*f]])/(Sqrt[f]*Sqrt[e + f*x]) + (4*b*(b*C*e - b*B*f + a*C*f)*Sqrt[e + f*x]*(-(Sqrt[f]*Sqrt[d*e -
c*f]*(c + d*x)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (d*e - c*f)*Sqrt[c + d*x]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sq
rt[d*e - c*f]]))/(f^(5/2)*Sqrt[d*e - c*f]*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (b^2*C*Sqrt[e + f*x
]*(Sqrt[f]*Sqrt[c + d*x]*(c*f + d*(e + 2*f*x)) - ((d*e - c*f)^(3/2)*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e -
c*f]])/Sqrt[(d*(e + f*x))/(d*e - c*f)]))/(d*f^(5/2)) - (8*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[-(b*c) + a*d]*ArcTa
n[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/Sqrt[b*e - a*f])/(4*b^3)

________________________________________________________________________________________

Maple [B]  time = 0.033, size = 1822, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(8*A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2
)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a*b^2*d^2*f^2*(d*f)^(1/2)-8*A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b
*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*b^3*c*d*f^2*(d*f)^(1/
2)+8*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^3*d^2*f^2*((a^2*d*f-a*b*c
*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-8*B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1
/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^2*b*d^2*f^2*(d*f)^(1/2)+8*B*ln((-2*a*d*f*x+b*c*f
*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x
+a))*a*b^2*c*d*f^2*(d*f)^(1/2)-8*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))
*a*b^2*d^2*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+4*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*
f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^3*c*d*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-4*B*ln(1/2*(2*d*f*x+2
*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^3*d^2*e*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)
^(1/2)+8*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1
/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^3*d^2*f^2*(d*f)^(1/2)-8*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b
*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^2*b*c*d*f^2*(d*f)^(
1/2)+8*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*b*d^2*f^2*((a^2*d*f-a
*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-4*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1
/2))*a*b^2*c*d*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+4*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)
*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b^2*d^2*e*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-C*ln(1/2*(2*d*f
*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^3*c^2*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/
b^2)^(1/2)-2*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^3*c*d*e*f*((a^2*d
*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+3*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f
)^(1/2))*b^3*d^2*e^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+4*C*x*b^3*d*f*((d*x+c)*(f*x+e))^(1/2)*(d*f)
^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+8*B*b^3*d*f*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)*((a^2*d*f
-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-8*C*a*b^2*d*f*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d
*e+b^2*c*e)/b^2)^(1/2)+2*C*b^3*c*f*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)
^(1/2)-6*C*b^3*d*e*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2))*(f*x+e)^
(1/2)*(d*x+c)^(1/2)/((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)/(d*f)^(1/2)/d/f^2/b^4/((d*x+c)*(f*x+e))^(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((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.80923, size = 797, normalized size = 2.75 \begin{align*} \frac{1}{4} \, \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e} \sqrt{d x + c}{\left (\frac{2 \,{\left (d x + c\right )} C}{b d f{\left | d \right |}} - \frac{C b^{5} c d^{3} f^{2} + 4 \, C a b^{4} d^{4} f^{2} - 4 \, B b^{5} d^{4} f^{2} + 3 \, C b^{5} d^{4} f e}{b^{6} d^{4} f^{3}{\left | d \right |}}\right )} - \frac{2 \,{\left (\sqrt{d f} C a^{2} b c d^{2} - \sqrt{d f} B a b^{2} c d^{2} + \sqrt{d f} A b^{3} c d^{2} - \sqrt{d f} C a^{3} d^{3} + \sqrt{d f} B a^{2} b d^{3} - \sqrt{d f} A a b^{2} d^{3}\right )} \arctan \left (-\frac{b c d f - 2 \, a d^{2} f + b d^{2} e -{\left (\sqrt{d f} \sqrt{d x + c} - \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e}\right )}^{2} b}{2 \, \sqrt{a b c d f^{2} - a^{2} d^{2} f^{2} - b^{2} c d f e + a b d^{2} f e} d}\right )}{\sqrt{a b c d f^{2} - a^{2} d^{2} f^{2} - b^{2} c d f e + a b d^{2} f e} b^{3} d{\left | d \right |}} + \frac{{\left (\sqrt{d f} C b^{2} c^{2} f^{2} + 4 \, \sqrt{d f} C a b c d f^{2} - 4 \, \sqrt{d f} B b^{2} c d f^{2} - 8 \, \sqrt{d f} C a^{2} d^{2} f^{2} + 8 \, \sqrt{d f} B a b d^{2} f^{2} - 8 \, \sqrt{d f} A b^{2} d^{2} f^{2} + 2 \, \sqrt{d f} C b^{2} c d f e - 4 \, \sqrt{d f} C a b d^{2} f e + 4 \, \sqrt{d f} B b^{2} d^{2} f e - 3 \, \sqrt{d f} C b^{2} d^{2} e^{2}\right )} \log \left ({\left (\sqrt{d f} \sqrt{d x + c} - \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e}\right )}^{2}\right )}{8 \, b^{3} d f^{3}{\left | d \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*C/(b*d*f*abs(d)) - (C*b^5*c*d^3*f^2 + 4*C*a
*b^4*d^4*f^2 - 4*B*b^5*d^4*f^2 + 3*C*b^5*d^4*f*e)/(b^6*d^4*f^3*abs(d))) - 2*(sqrt(d*f)*C*a^2*b*c*d^2 - sqrt(d*
f)*B*a*b^2*c*d^2 + sqrt(d*f)*A*b^3*c*d^2 - sqrt(d*f)*C*a^3*d^3 + sqrt(d*f)*B*a^2*b*d^3 - sqrt(d*f)*A*a*b^2*d^3
)*arctan(-1/2*(b*c*d*f - 2*a*d^2*f + b*d^2*e - (sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))
^2*b)/(sqrt(a*b*c*d*f^2 - a^2*d^2*f^2 - b^2*c*d*f*e + a*b*d^2*f*e)*d))/(sqrt(a*b*c*d*f^2 - a^2*d^2*f^2 - b^2*c
*d*f*e + a*b*d^2*f*e)*b^3*d*abs(d)) + 1/8*(sqrt(d*f)*C*b^2*c^2*f^2 + 4*sqrt(d*f)*C*a*b*c*d*f^2 - 4*sqrt(d*f)*B
*b^2*c*d*f^2 - 8*sqrt(d*f)*C*a^2*d^2*f^2 + 8*sqrt(d*f)*B*a*b*d^2*f^2 - 8*sqrt(d*f)*A*b^2*d^2*f^2 + 2*sqrt(d*f)
*C*b^2*c*d*f*e - 4*sqrt(d*f)*C*a*b*d^2*f*e + 4*sqrt(d*f)*B*b^2*d^2*f*e - 3*sqrt(d*f)*C*b^2*d^2*e^2)*log((sqrt(
d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2)/(b^3*d*f^3*abs(d))