\(\int \frac {\sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2)}{(a+b x)^2} \, dx\) [45]

Optimal result

Integrand size = 36, antiderivative size = 521 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^2} \, dx=\frac {\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^3 d f (b e-a f)}+\frac {\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt {c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}+\frac {\left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{4 b^4 d^{3/2} f^{3/2}}+\frac {\left (6 a^3 C d f-b^3 (2 B c e+A d e+A c f)+a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)-a^2 b (4 B d f+5 C (d e+c f))\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{b^4 \sqrt {b c-a d} \sqrt {b e-a f}} \]

[Out]

-(A*b^2-a*(B*b-C*a))*(d*x+c)^(3/2)*(f*x+e)^(3/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)+1/4*(24*a^2*C*d^2*f^2-8*a*b*d
*f*(2*B*d*f+C*c*f+C*d*e)-b^2*(C*(-c*f+d*e)^2-4*d*f*(2*A*d*f+B*c*f+B*d*e)))*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/
2)/(f*x+e)^(1/2))/b^4/d^(3/2)/f^(3/2)+(6*a^3*C*d*f-b^3*(A*c*f+A*d*e+2*B*c*e)+a*b^2*(2*A*d*f+3*B*c*f+3*B*d*e+4*
C*c*e)-a^2*b*(4*B*d*f+5*C*(c*f+d*e)))*arctanh((-a*f+b*e)^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2)/(f*x+e)^(1/2))/b
^4/(-a*d+b*c)^(1/2)/(-a*f+b*e)^(1/2)+1/2*(3*a^2*C*d*f+b^2*(2*A*d*f+C*c*e)-a*b*(2*B*d*f+C*c*f+C*d*e))*(f*x+e)^(
3/2)*(d*x+c)^(1/2)/b^2/(-a*d+b*c)/f/(-a*f+b*e)+1/4*(12*a^2*C*d*f^2-a*b*f*(8*B*d*f+C*c*f+7*C*d*e)+b^2*(4*d*f*(A
*f+B*e)-C*e*(-c*f+d*e)))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^3/d/f/(-a*f+b*e)

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1627, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^2} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (24 a^2 C d^2 f^2-8 a b d f (2 B d f+c C f+C d e)-\left (b^2 \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )\right )\right )}{4 b^4 d^{3/2} f^{3/2}}+\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (3 a^2 C d f-a b (2 B d f+c C f+C d e)+b^2 (2 A d f+c C e)\right )}{2 b^2 f (b c-a d) (b e-a f)}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (12 a^2 C d f^2-a b f (8 B d f+c C f+7 C d e)+b^2 (4 d f (A f+B e)-C e (d e-c f))\right )}{4 b^3 d f (b e-a f)}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (6 a^3 C d f-a^2 b (4 B d f+5 C (c f+d e))+a b^2 (2 A d f+3 B c f+3 B d e+4 c C e)-b^3 (A c f+A d e+2 B c e)\right )}{b^4 \sqrt {b c-a d} \sqrt {b e-a f}}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)} \]

[In]

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

[Out]

((12*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 8*B*d*f) + b^2*(4*d*f*(B*e + A*f) - C*e*(d*e - c*f)))*Sqrt[c + d*x
]*Sqrt[e + f*x])/(4*b^3*d*f*(b*e - a*f)) + ((3*a^2*C*d*f + b^2*(c*C*e + 2*A*d*f) - a*b*(C*d*e + c*C*f + 2*B*d*
f))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(2*b^2*(b*c - a*d)*f*(b*e - a*f)) - ((A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2
)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(b*e - a*f)*(a + b*x)) + ((24*a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 2*B
*d*f) - b^2*(C*(d*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt
[e + f*x])])/(4*b^4*d^(3/2)*f^(3/2)) + ((6*a^3*C*d*f - b^3*(2*B*c*e + A*d*e + A*c*f) + a*b^2*(4*c*C*e + 3*B*d*
e + 3*B*c*f + 2*A*d*f) - a^2*b*(4*B*d*f + 5*C*(d*e + c*f)))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c
- a*d]*Sqrt[e + f*x])])/(b^4*Sqrt[b*c - a*d]*Sqrt[b*e - a*f])

Rule 65

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 95

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 159

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 163

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 212

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

Rule 214

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

Rule 223

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 1627

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(a + b*x)^(m + 1)*
(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}-\frac {\int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (-\frac {3 a^2 C (d e+c f)+b^2 (2 B c e+A d e+A c f)-a b (2 c C e+3 B d e+3 B c f-2 A d f)}{2 b}+\left (-\frac {3 a^2 C d f}{b}-b (c C e+2 A d f)+a (C d e+c C f+2 B d f)\right ) x\right )}{a+b x} \, dx}{(b c-a d) (b e-a f)} \\ & = \frac {\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt {c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}-\frac {\int \frac {\sqrt {e+f x} \left (-\frac {(b c-a d) \left (3 a^2 C f (d e+3 c f)+2 b^2 f (2 B c e+A d e+A c f)-a b (2 B f (d e+3 c f)+C e (d e+7 c f))\right )}{2 b}-\frac {(b c-a d) \left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) x}{2 b}\right )}{(a+b x) \sqrt {c+d x}} \, dx}{2 b (b c-a d) f (b e-a f)} \\ & = \frac {\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^3 d f (b e-a f)}+\frac {\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt {c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}-\frac {\int \frac {-\frac {(b c-a d) (b e-a f) \left (12 a^2 C d f (d e+c f)+4 b^2 d f (2 B c e+A d e+A c f)-a b \left (8 B d f (d e+c f)+C \left (d^2 e^2+14 c d e f+c^2 f^2\right )\right )\right )}{4 b}-\frac {(b c-a d) (b e-a f) \left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) x}{4 b}}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{2 b^2 d (b c-a d) f (b e-a f)} \\ & = \frac {\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^3 d f (b e-a f)}+\frac {\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt {c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}-\frac {\left (6 a^3 C d f-b^3 (2 B c e+A d e+A c f)+a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)-a^2 b (4 B d f+5 C (d e+c f))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{2 b^4}+\frac {\left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{8 b^4 d f} \\ & = \frac {\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^3 d f (b e-a f)}+\frac {\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt {c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}-\frac {\left (6 a^3 C d f-b^3 (2 B c e+A d e+A c f)+a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)-a^2 b (4 B d f+5 C (d e+c f))\right ) \text {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^4}+\frac {\left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \text {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^4 d^2 f} \\ & = \frac {\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^3 d f (b e-a f)}+\frac {\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt {c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}+\frac {\left (6 a^3 C d f-b^3 (2 B c e+A d e+A c f)+a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)-a^2 b (4 B d f+5 C (d e+c f))\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{b^4 \sqrt {b c-a d} \sqrt {b e-a f}}+\frac {\left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \text {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^4 d^2 f} \\ & = \frac {\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^3 d f (b e-a f)}+\frac {\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt {c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}+\frac {\left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{4 b^4 d^{3/2} f^{3/2}}+\frac {\left (6 a^3 C d f-b^3 (2 B c e+A d e+A c f)+a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)-a^2 b (4 B d f+5 C (d e+c f))\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{b^4 \sqrt {b c-a d} \sqrt {b e-a f}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.13 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^2} \, dx=\frac {\frac {b \sqrt {c+d x} \sqrt {e+f x} \left (-12 a^2 C d f+a b (c C f+8 B d f+C d (e-6 f x))+b^2 (-4 A d f+x (c C f+4 B d f+C d (e+2 f x)))\right )}{d f (a+b x)}+\frac {4 \left (-6 a^3 C d f+b^3 (2 B c e+A d e+A c f)-a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)+a^2 b (4 B d f+5 C (d e+c f))\right ) \arctan \left (\frac {\sqrt {b c-a d} \sqrt {e+f x}}{\sqrt {-b e+a f} \sqrt {c+d x}}\right )}{\sqrt {b c-a d} \sqrt {-b e+a f}}-\frac {\left (-24 a^2 C d^2 f^2+8 a b d f (C d e+c C f+2 B d f)+b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{d^{3/2} f^{3/2}}}{4 b^4} \]

[In]

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

[Out]

((b*Sqrt[c + d*x]*Sqrt[e + f*x]*(-12*a^2*C*d*f + a*b*(c*C*f + 8*B*d*f + C*d*(e - 6*f*x)) + b^2*(-4*A*d*f + x*(
c*C*f + 4*B*d*f + C*d*(e + 2*f*x)))))/(d*f*(a + b*x)) + (4*(-6*a^3*C*d*f + b^3*(2*B*c*e + A*d*e + A*c*f) - a*b
^2*(4*c*C*e + 3*B*d*e + 3*B*c*f + 2*A*d*f) + a^2*b*(4*B*d*f + 5*C*(d*e + c*f)))*ArcTan[(Sqrt[b*c - a*d]*Sqrt[e
+ f*x])/(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])])/(Sqrt[b*c - a*d]*Sqrt[-(b*e) + a*f]) - ((-24*a^2*C*d^2*f^2 + 8*a
*b*d*f*(C*d*e + c*C*f + 2*B*d*f) + b^2*(C*(d*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)))*ArcTanh[(Sqrt[d]*S
qrt[e + f*x])/(Sqrt[f]*Sqrt[c + d*x])])/(d^(3/2)*f^(3/2)))/(4*b^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4679\) vs. \(2(479)=958\).

Time = 1.69 (sec) , antiderivative size = 4680, normalized size of antiderivative = 8.98

method	result	size
default	\(\text {Expression too large to display}\)	\(4680\)

[In]

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

[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^2*b^2*d^2*f^2*(d*f)^(1/2)-16*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^3*d^2*f^2*x*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-8*A*b^4*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*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^4*d^2*f^2*x*((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^4*c^2*f^2*x*((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^4*d
^2*e^2*x*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+4*C*b^4*d*f*x^2*((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)+12*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^2*d^2*e*f*(d*f)^(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))*a*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)+8*B*b^4*d*f*x*((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^4*c*f*x*((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^4
*d*e*x*((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)+16*B*a*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)-24*C*a^2*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*a*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)+2*C*a*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)+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))*a*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)-16*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^2*d^2*f
^2*x*(d*f)^(1/2)+24*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^4*d^2*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))*a*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)-16*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^3*b*d^2*f^2*(d*f)^(1/2)-16*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^2*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)+24*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^3*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)
-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^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)-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
^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)+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^3*d^2*f^2*x*(d*f)^(1
/2)-4*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^4*c*d*f^2*x*(d*f)^(1/2)-4*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^4*d^2*e*f*x*(d*f)^(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^4*c*d*f^2*x*((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^4*d^2*e*f*x*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-20*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*b*c*d*f^2*(d
*f)^(1/2)-20*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*b*d^2*e*f*(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^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)-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^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)+24*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*b*d^2*f^2*x*(d*f)^(1/2)+24*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^2*d^2*f^2*x*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/
2)-4*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^3*c*d*f^2*(d*f)^(1/2)-4*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^3*d^2*e*f*(d*f)^(1/
2)+12*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^2*c*d*f^2*(d*f)^(1/2)-12*C*a*b^3*d*f*x*((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)+12*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^3*c*d*f^2*x*(d*f)^(1/2)+12
*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^3*d^2*e*f*x*(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))*b^4*c*d*e*f*x*(d*f)^(1/2)
-20*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^2*c*d*f^2*x*(d*f)^(1/2)-20*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^2*d^2*e*f*x*(
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*b^3*c*d*f^2*x*((a
^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(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*b^3*d^2*e*f*x*((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^4*c*d*e*f*x*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-8*B*l
n((-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^3*c*d*e*f*(d*f)^(1/2)+16*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^2*c*d*e*f*(d*f)^(1/2)+16
*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*b^3*c*d*e*f*x*(d*f)^(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))*a*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))*(f*x+e)^(1/2)*(d*x+c)^(
1/2)/f/d/((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)/(d*f)^(1/2)/(b*x+a)/((d*x+c)*(f*x+e))^(1/2)/b^5

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(2*a*d*f-b*c*f>0)', see `assume
?` for more

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1507 vs. \(2 (478) = 956\).

Time = 1.54 (sec) , antiderivative size = 1507, normalized size of antiderivative = 2.89 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*sqrt(d^2*e + (d*x + c)*d*f - c*d*f)*sqrt(d*x + c)*(2*(d*x + c)*C*abs(d)/(b^2*d^3) + (C*b^7*d^4*e*f*abs(d)
- C*b^7*c*d^3*f^2*abs(d) - 8*C*a*b^6*d^4*f^2*abs(d) + 4*B*b^7*d^4*f^2*abs(d))/(b^9*d^6*f^2)) + (4*sqrt(d*f)*C*
a*b^2*c*e*abs(d) - 2*sqrt(d*f)*B*b^3*c*e*abs(d) - 5*sqrt(d*f)*C*a^2*b*d*e*abs(d) + 3*sqrt(d*f)*B*a*b^2*d*e*abs
(d) - sqrt(d*f)*A*b^3*d*e*abs(d) - 5*sqrt(d*f)*C*a^2*b*c*f*abs(d) + 3*sqrt(d*f)*B*a*b^2*c*f*abs(d) - sqrt(d*f)
*A*b^3*c*f*abs(d) + 6*sqrt(d*f)*C*a^3*d*f*abs(d) - 4*sqrt(d*f)*B*a^2*b*d*f*abs(d) + 2*sqrt(d*f)*A*a*b^2*d*f*ab
s(d))*arctan(-1/2*(b*d^2*e + b*c*d*f - 2*a*d^2*f - (sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d
*f))^2*b)/(sqrt(-b^2*c*d*e*f + a*b*d^2*e*f + a*b*c*d*f^2 - a^2*d^2*f^2)*d))/(sqrt(-b^2*c*d*e*f + a*b*d^2*e*f +
a*b*c*d*f^2 - a^2*d^2*f^2)*b^4*d) - 2*(sqrt(d*f)*C*a^2*b*d^3*e^2*abs(d) - sqrt(d*f)*B*a*b^2*d^3*e^2*abs(d) +
sqrt(d*f)*A*b^3*d^3*e^2*abs(d) - 2*sqrt(d*f)*C*a^2*b*c*d^2*e*f*abs(d) + 2*sqrt(d*f)*B*a*b^2*c*d^2*e*f*abs(d) -
2*sqrt(d*f)*A*b^3*c*d^2*e*f*abs(d) + sqrt(d*f)*C*a^2*b*c^2*d*f^2*abs(d) - sqrt(d*f)*B*a*b^2*c^2*d*f^2*abs(d)
+ sqrt(d*f)*A*b^3*c^2*d*f^2*abs(d) - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))
^2*C*a^2*b*d*e*abs(d) + sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a*b^2*d*
e*abs(d) - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*b^3*d*e*abs(d) - sqrt
(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^2*b*c*f*abs(d) + sqrt(d*f)*(sqrt(d
*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a*b^2*c*f*abs(d) - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*b^3*c*f*abs(d) + 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d
^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^3*d*f*abs(d) - 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x +
c)*d*f - c*d*f))^2*B*a^2*b*d*f*abs(d) + 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c
*d*f))^2*A*a*b^2*d*f*abs(d))/((b*d^4*e^2 - 2*b*c*d^3*e*f + b*c^2*d^2*f^2 - 2*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d
^2*e + (d*x + c)*d*f - c*d*f))^2*b*d^2*e - 2*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2
*b*c*d*f + 4*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*a*d^2*f + (sqrt(d*f)*sqrt(d*x +
c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*b)*b^4) + 1/8*(C*b^2*d^2*e^2*abs(d) - 2*C*b^2*c*d*e*f*abs(d) + 8*
C*a*b*d^2*e*f*abs(d) - 4*B*b^2*d^2*e*f*abs(d) + C*b^2*c^2*f^2*abs(d) + 8*C*a*b*c*d*f^2*abs(d) - 4*B*b^2*c*d*f^
2*abs(d) - 24*C*a^2*d^2*f^2*abs(d) + 16*B*a*b*d^2*f^2*abs(d) - 8*A*b^2*d^2*f^2*abs(d))*log((sqrt(d*f)*sqrt(d*x
+ c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2)/(sqrt(d*f)*b^4*d^2*f)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^2} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}