\(\int \frac {\sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2)}{a+b x} \, dx\) [44]

Optimal result

Integrand size = 36, antiderivative size = 450 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{a+b x} \, dx=\frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}-\frac {\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{8 b^4 d^{5/2} f^{5/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {b c-a d} \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{b^4} \]

[Out]

1/3*C*(d*x+c)^(3/2)*(f*x+e)^(3/2)/b/d/f-1/8*(16*a^3*C*d^3*f^3-8*a^2*b*d^2*f^2*(2*B*d*f+C*c*f+C*d*e)-2*a*b^2*d*
f*(C*(-c*f+d*e)^2-4*d*f*(2*A*d*f+B*c*f+B*d*e))-b^3*(C*(-c*f+d*e)^2*(c*f+d*e)-2*d*f*(B*(-c*f+d*e)^2-4*A*d*f*(c*
f+d*e))))*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2))/b^4/d^(5/2)/f^(5/2)-2*(A*b^2-a*(B*b-C*a))*arcta
nh((-a*f+b*e)^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2)/(f*x+e)^(1/2))*(-a*d+b*c)^(1/2)*(-a*f+b*e)^(1/2)/b^4-1/4*(2
*a*C*d*f+b*(-2*B*d*f+C*c*f+C*d*e))*(f*x+e)^(3/2)*(d*x+c)^(1/2)/b^2/d/f^2+1/8*(4*b*d*f*(2*A*b*d*f-a*C*(c*f+d*e)
)+(4*a*d*f-b*c*f+b*d*e)*(2*a*C*d*f+b*(-2*B*d*f+C*c*f+C*d*e)))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^3/d^2/f^2

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1629, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{a+b x} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )-\left (b^3 \left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right )}{8 b^4 d^{5/2} f^{5/2}}-\frac {2 \sqrt {b c-a d} \sqrt {b e-a f} \left (A b^2-a (b B-a C)\right ) \text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right )}{b^4}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (\frac {(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e))}{b d f}-4 a C (c f+d e)+8 A b d f\right )}{8 b^2 d f}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f} \]

[In]

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

[Out]

((8*A*b*d*f - 4*a*C*(d*e + c*f) + ((b*d*e - b*c*f + 4*a*d*f)*(2*a*C*d*f + b*(C*d*e + c*C*f - 2*B*d*f)))/(b*d*f
))*Sqrt[c + d*x]*Sqrt[e + f*x])/(8*b^2*d*f) - ((2*a*C*d*f + b*(C*d*e + c*C*f - 2*B*d*f))*Sqrt[c + d*x]*(e + f*
x)^(3/2))/(4*b^2*d*f^2) + (C*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*b*d*f) - ((16*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2
*(C*d*e + c*C*f + 2*B*d*f) - 2*a*b^2*d*f*(C*(d*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)) - b^3*(C*(d*e - c
*f)^2*(d*e + c*f) - 2*d*f*(B*(d*e - c*f)^2 - 4*A*d*f*(d*e + c*f))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*S
qrt[e + f*x])])/(8*b^4*d^(5/2)*f^(5/2)) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*Sqrt[b*e - a*f]*ArcTanh[(
Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/b^4

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 1629

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]

Rubi steps \begin{align*} \text {integral}& = \frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac {\int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (\frac {3}{2} b (2 A b d f-a C (d e+c f))-\frac {3}{2} b (2 a C d f+b (C d e+c C f-2 B d f)) x\right )}{a+b x} \, dx}{3 b^2 d f} \\ & = -\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac {\int \frac {\sqrt {e+f x} \left (\frac {3}{4} b (4 b c f (2 A b d f-a C (d e+c f))+a (d e+3 c f) (2 a C d f+b (C d e+c C f-2 B d f)))+\frac {3}{4} b (4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) x\right )}{(a+b x) \sqrt {c+d x}} \, dx}{6 b^3 d f^2} \\ & = \frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac {\int \frac {\frac {3}{8} b \left (16 A b^3 c d^2 e f^2-8 a^3 C d^2 f^2 (d e+c f)+2 a^2 b d f \left (4 B d f (d e+c f)+C \left (d^2 e^2+6 c d e f+c^2 f^2\right )\right )+a b^2 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (4 A d f (d e+c f)+B \left (d^2 e^2+6 c d e f+c^2 f^2\right )\right )\right )\right )-\frac {3}{8} b \left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{6 b^4 d^2 f^2} \\ & = \frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac {\left (\left (A b^2-a (b B-a C)\right ) (b c-a d) (b e-a f)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{b^4}-\frac {\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{16 b^4 d^2 f^2} \\ & = \frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac {\left (2 \left (A b^2-a (b B-a C)\right ) (b c-a d) (b e-a 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 (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\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 )}{8 b^4 d^3 f^2} \\ & = \frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {b c-a d} \sqrt {b e-a f} \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}-\frac {\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\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 )}{8 b^4 d^3 f^2} \\ & = \frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}-\frac {\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{8 b^4 d^{5/2} f^{5/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {b c-a d} \sqrt {b e-a f} \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{a+b x} \, dx=\frac {\frac {b \sqrt {c+d x} \sqrt {e+f x} \left (24 a^2 C d^2 f^2-6 a b d f (c C f+4 B d f+C d (e+2 f x))+b^2 \left (6 d f (B c f+4 A d f+B d (e+2 f x))+C \left (-3 c^2 f^2+2 c d f (e+f x)+d^2 \left (-3 e^2+2 e f x+8 f^2 x^2\right )\right )\right )\right )}{d^2 f^2}-48 \left (A b^2+a (-b B+a C)\right ) \sqrt {b c-a d} \sqrt {-b e+a f} \arctan \left (\frac {\sqrt {b c-a d} \sqrt {e+f x}}{\sqrt {-b e+a f} \sqrt {c+d x}}\right )+\frac {3 \left (-16 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)+2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )+b^3 \left (C (d e-c f)^2 (d e+c f)+2 d f \left (-B (d e-c f)^2+4 A d f (d e+c f)\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{d^{5/2} f^{5/2}}}{24 b^4} \]

[In]

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

[Out]

((b*Sqrt[c + d*x]*Sqrt[e + f*x]*(24*a^2*C*d^2*f^2 - 6*a*b*d*f*(c*C*f + 4*B*d*f + C*d*(e + 2*f*x)) + b^2*(6*d*f
*(B*c*f + 4*A*d*f + B*d*(e + 2*f*x)) + C*(-3*c^2*f^2 + 2*c*d*f*(e + f*x) + d^2*(-3*e^2 + 2*e*f*x + 8*f^2*x^2))
)))/(d^2*f^2) - 48*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[b*c - a*d]*Sqrt[-(b*e) + a*f]*ArcTan[(Sqrt[b*c - a*d]*Sqrt[
e + f*x])/(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])] + (3*(-16*a^3*C*d^3*f^3 + 8*a^2*b*d^2*f^2*(C*d*e + c*C*f + 2*B*d
*f) + 2*a*b^2*d*f*(C*(d*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)) + b^3*(C*(d*e - c*f)^2*(d*e + c*f) + 2*d
*f*(-(B*(d*e - c*f)^2) + 4*A*d*f*(d*e + c*f))))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/(Sqrt[f]*Sqrt[c + d*x])])/(d^(
5/2)*f^(5/2)))/(24*b^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3897\) vs. \(2(406)=812\).

Time = 5.72 (sec) , antiderivative size = 3898, normalized size of antiderivative = 8.66

method	result	size
default	\(\text {Expression too large to display}\)	\(3898\)

[In]

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

[Out]

-1/48*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(-3*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^3*e^3+24*B*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(
1/2)*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^3*e*f^2+24*B*((a^2*d*
f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^2*f^3-3*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^3*f^3-6*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^3*e^2*f+48*C*((a^2*d*f-a*b*c*f-a*b*d*e
+b^2*c*e)/b^2)^(1/2)*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^3*f^3
-24*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^3*e*f^2-12*B*((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)
*(d*f)^(1/2)*b^4*d^2*e*f+48*C*(d*f)^(1/2)*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^3*f^3-48*A*(d*f)^(1/2)*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^2*f^3-6*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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*d*f^3-24*B*((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)*(d*f)^(1/2)*b^4*d^2*f^2*x+48*B*(d*f)^(1/2)*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^3*e*f^2-
24*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^2*f^3-24*A*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^2*f^3-24*A*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)
^(1/2)*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^3*e*f^2+48*A*(d*f)^(1
/2)*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^3*f^3-48*B*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^3*f^3+6*B*((a^2*d*f-a*b*c*f-a*b*d*e+b^
2*c*e)/b^2)^(1/2)*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*d*f^3+6*
B*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^3*e^2*f-48*B*(d*f)^(1/2)*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^3*f^3+48*A*((a^2*d*f-a*b*c*f-a*b
*d*e+b^2*c*e)/b^2)^(1/2)*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^3
*f^3-48*A*((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)*(d*f)^(1/2)*b^4*d^2*f^2+12*C*(
(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)*(d*f)^(1/2)*a*b^3*c*d*f^2+12*C*((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)*(d*f)^(1/2)*a*b^3*d^2*e*f-4*C*((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)*(d*f)^(1/2)*b^4*c*d*e*f-48*B*(d*f)^(1/2)*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^2*e*f^2+24*C*((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)*(d*f)^(1/2)
*a*b^3*d^2*f^2*x-4*C*((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)*(d*f)^(1/2)*b^4*c*d
*f^2*x-4*C*((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)*(d*f)^(1/2)*b^4*d^2*e*f*x+12*
C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^2*e*f^2+48*C*(d*f)^(1/2)*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^2*e*f^2+6*C*((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)*(d*f)^(1/2)*b^4*c^2*f^2+6*C*((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)*(d*f)^(1/2)*b^4*d^2*e^2+48*B*(d*f)^(1/2)*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^2*f^3+3*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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*d*e*f^2-12*B*((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)*(d*f)^(1/2)*b^4*c*d*f^2-48*C*(d*f)^(1/2)*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^3*e*f^2+48*A*(d*f)^
(1/2)*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^2*e*f^2+3*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*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^2*e^2*f-48*C*(d*f)^(1/2)*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^2*f^3-48*C*((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)*(d*f)^(1
/2)*a^2*b^2*d^2*f^2-48*A*(d*f)^(1/2)*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^3*e*f^2+48*B*((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)*(d*f)^(1/2)*a*b^3*d^2*f^2-12*B*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b
^2)^(1/2)*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^2*e*f^2-16*C*b^4
*d^2*f^2*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)*(d*f)^(1/2))/((d*x+c)*(f*x+
e))^(1/2)/b^5/d^2/f^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)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a),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 [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}