\(\int \frac {\sqrt {c+d x} (A+B x+C x^2)}{(a+b x)^3 \sqrt {e+f x}} \, dx\) [52]

Optimal result

Integrand size = 36, antiderivative size = 484 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^3 \sqrt {e+f x}} \, dx=\frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {2 C \sqrt {d} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{b^3 \sqrt {f}}-\frac {\left (8 a^4 C d^2 f^2-4 a^3 b C d f (5 d e+3 c f)+3 a^2 b^2 C \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )-a b^3 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (12 C e^2-B e f+2 A f^2\right )\right )-b^4 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{4 b^3 (b c-a d)^{3/2} (b e-a f)^{5/2}} \]

[Out]

-1/4*(8*a^4*C*d^2*f^2-4*a^3*b*C*d*f*(3*c*f+5*d*e)+3*a^2*b^2*C*(c^2*f^2+10*c*d*e*f+5*d^2*e^2)-a*b^3*(d^2*e*(-4*
A*f+3*B*e)+c^2*f*(-B*f+8*C*e)+2*c*d*(2*A*f^2-B*e*f+12*C*e^2))-b^4*(A*d^2*e^2-2*c*d*e*(-A*f+2*B*e)-c^2*(3*A*f^2
-4*B*e*f+8*C*e^2)))*arctanh((-a*f+b*e)^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2)/(f*x+e)^(1/2))/b^3/(-a*d+b*c)^(3/2
)/(-a*f+b*e)^(5/2)+2*C*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2))*d^(1/2)/b^3/f^(1/2)-1/2*(A*b^2-a*(
B*b-C*a))*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^2+1/4*(4*a^3*C*d*f-a^2*b*C*(5*c*f+7*d*e)
-b^3*(-3*A*c*f-A*d*e+4*B*c*e)+a*b^2*(-4*A*d*f+B*c*f+3*B*d*e+8*C*c*e))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^2/(-a*d+b*
c)/(-a*f+b*e)^2/(b*x+a)

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1627, 154, 163, 65, 223, 212, 95, 214} \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^3 \sqrt {e+f x}} \, dx=\frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b C (5 c f+7 d e)+a b^2 (-4 A d f+B c f+3 B d e+8 c C e)-b^3 (-3 A c f-A d e+4 B c e)\right )}{4 b^2 (a+b x) (b c-a d) (b e-a f)^2}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (8 a^4 C d^2 f^2-4 a^3 b C d f (3 c f+5 d e)+3 a^2 b^2 C \left (c^2 f^2+10 c d e f+5 d^2 e^2\right )-a b^3 \left (2 c d \left (2 A f^2-B e f+12 C e^2\right )+d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)\right )-b^4 \left (-\left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )\right )-2 c d e (2 B e-A f)+A d^2 e^2\right )\right )}{4 b^3 (b c-a d)^{3/2} (b e-a f)^{5/2}}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}+\frac {2 C \sqrt {d} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{b^3 \sqrt {f}} \]

[In]

Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^3*Sqrt[e + f*x]),x]

[Out]

((4*a^3*C*d*f - a^2*b*C*(7*d*e + 5*c*f) - b^3*(4*B*c*e - A*d*e - 3*A*c*f) + a*b^2*(8*c*C*e + 3*B*d*e + B*c*f -
4*A*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)) - ((A*b^2 - a*(b*B - a*C))
*(c + d*x)^(3/2)*Sqrt[e + f*x])/(2*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2) + (2*C*Sqrt[d]*ArcTanh[(Sqrt[f]*Sqrt
[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b^3*Sqrt[f]) - ((8*a^4*C*d^2*f^2 - 4*a^3*b*C*d*f*(5*d*e + 3*c*f) + 3*a^2
*b^2*C*(5*d^2*e^2 + 10*c*d*e*f + c^2*f^2) - a*b^3*(d^2*e*(3*B*e - 4*A*f) + c^2*f*(8*C*e - B*f) + 2*c*d*(12*C*e
^2 - B*e*f + 2*A*f^2)) - b^4*(A*d^2*e^2 - 2*c*d*e*(2*B*e - A*f) - c^2*(8*C*e^2 - 4*B*e*f + 3*A*f^2)))*ArcTanh[
(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(4*b^3*(b*c - a*d)^(3/2)*(b*e - a*f)^(5/2))

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 154

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

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} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac {\int \frac {\sqrt {c+d x} \left (-\frac {a^2 C (3 d e+c f)+b^2 (4 B c e-A d e-3 A c f)-a b (4 c C e+3 B d e+B c f-4 A d f)}{2 b}-\frac {2 C (b c-a d) (b e-a f) x}{b}\right )}{(a+b x)^2 \sqrt {e+f x}} \, dx}{2 (b c-a d) (b e-a f)} \\ & = \frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac {\int \frac {\frac {4 a^3 C d f (d e+c f)-a^2 b C \left (7 d^2 e^2+14 c d e f+3 c^2 f^2\right )+a b^2 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (8 C e^2-B e f+2 A f^2\right )\right )+b^3 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )}{4 b}-\frac {2 C d (b c-a d) (b e-a f)^2 x}{b}}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{2 b (b c-a d) (b e-a f)^2} \\ & = \frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {(C d) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{b^3}+\frac {\left (8 a^4 C d^2 f^2-4 a^3 b C d f (5 d e+3 c f)+3 a^2 b^2 C \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )-a b^3 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (12 C e^2-B e f+2 A f^2\right )\right )-b^4 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{8 b^3 (b c-a d) (b e-a f)^2} \\ & = \frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {(2 C) \text {Subst}\left (\int \frac {1}{\sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{b^3}+\frac {\left (8 a^4 C d^2 f^2-4 a^3 b C d f (5 d e+3 c f)+3 a^2 b^2 C \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )-a b^3 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (12 C e^2-B e f+2 A f^2\right )\right )-b^4 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )\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 )}{4 b^3 (b c-a d) (b e-a f)^2} \\ & = \frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac {\left (8 a^4 C d^2 f^2-4 a^3 b C d f (5 d e+3 c f)+3 a^2 b^2 C \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )-a b^3 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (12 C e^2-B e f+2 A f^2\right )\right )-b^4 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{4 b^3 (b c-a d)^{3/2} (b e-a f)^{5/2}}+\frac {(2 C) \text {Subst}\left (\int \frac {1}{1-\frac {f x^2}{d}} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{b^3} \\ & = \frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {2 C \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{b^3 \sqrt {f}}-\frac {\left (8 a^4 C d^2 f^2-4 a^3 b C d f (5 d e+3 c f)+3 a^2 b^2 C \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )-a b^3 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (12 C e^2-B e f+2 A f^2\right )\right )-b^4 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{4 b^3 (b c-a d)^{3/2} (b e-a f)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 13.01 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^3 \sqrt {e+f x}} \, dx=-\frac {\frac {4 b (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x}}{(b e-a f) (a+b x)}+\frac {2 b^2 \left (A b^2+a (-b B+a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x)^2}-\frac {8 C \sqrt {d e-c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \text {arcsinh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} \sqrt {e+f x}}+\frac {8 C \sqrt {-b c+a d} \text {arctanh}\left (\frac {\sqrt {-b e+a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )}{\sqrt {-b e+a f}}-\frac {4 b (b B-2 a C) (-d e+c f) \text {arctanh}\left (\frac {\sqrt {-b e+a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )}{\sqrt {-b c+a d} (-b e+a f)^{3/2}}+\frac {b \left (A b^2+a (-b B+a C)\right ) (b d e+3 b c f-4 a d f) \left (\sqrt {-b c+a d} \sqrt {-b e+a f} \sqrt {c+d x} \sqrt {e+f x}-(d e-c f) (a+b x) \text {arctanh}\left (\frac {\sqrt {-b e+a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )\right )}{(-b c+a d)^{3/2} (-b e+a f)^{5/2} (a+b x)}}{4 b^3} \]

[In]

Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^3*Sqrt[e + f*x]),x]

[Out]

-1/4*((4*b*(b*B - 2*a*C)*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*e - a*f)*(a + b*x)) + (2*b^2*(A*b^2 + a*(-(b*B) + a*
C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - (8*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]) + (8*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])])/Sqrt[-(b*e) + a*f] - (
4*b*(b*B - 2*a*C)*(-(d*e) + c*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])
])/(Sqrt[-(b*c) + a*d]*(-(b*e) + a*f)^(3/2)) + (b*(A*b^2 + a*(-(b*B) + a*C))*(b*d*e + 3*b*c*f - 4*a*d*f)*(Sqrt
[-(b*c) + a*d]*Sqrt[-(b*e) + a*f]*Sqrt[c + d*x]*Sqrt[e + f*x] - (d*e - c*f)*(a + b*x)*ArcTanh[(Sqrt[-(b*e) + a
*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])]))/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(5/2)*(a + b*x))
)/b^3

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(9099\) vs. \(2(446)=892\).

Time = 1.70 (sec) , antiderivative size = 9100, normalized size of antiderivative = 18.80

method	result	size
default	\(\text {Expression too large to display}\)	\(9100\)

[In]

int((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^3/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^3 \sqrt {e+f x}} \, dx=\text {Timed out} \]

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^3/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^3 \sqrt {e+f x}} \, dx=\int \frac {\sqrt {c + d x} \left (A + B x + C x^{2}\right )}{\left (a + b x\right )^{3} \sqrt {e + f x}}\, dx \]

[In]

integrate((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**3/(f*x+e)**(1/2),x)

[Out]

Integral(sqrt(c + d*x)*(A + B*x + C*x**2)/((a + b*x)**3*sqrt(e + f*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^3 \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^3/(f*x+e)^(1/2),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^2)>0)', see `as
sume?` for m

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7922 vs. \(2 (445) = 890\).

Time = 23.15 (sec) , antiderivative size = 7922, normalized size of antiderivative = 16.37 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \]

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^3/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

-1/4*(8*sqrt(d*f)*C*b^4*c^2*d^2*e^2 - 24*sqrt(d*f)*C*a*b^3*c*d^3*e^2 + 4*sqrt(d*f)*B*b^4*c*d^3*e^2 + 15*sqrt(d
*f)*C*a^2*b^2*d^4*e^2 - 3*sqrt(d*f)*B*a*b^3*d^4*e^2 - sqrt(d*f)*A*b^4*d^4*e^2 - 8*sqrt(d*f)*C*a*b^3*c^2*d^2*e*
f - 4*sqrt(d*f)*B*b^4*c^2*d^2*e*f + 30*sqrt(d*f)*C*a^2*b^2*c*d^3*e*f + 2*sqrt(d*f)*B*a*b^3*c*d^3*e*f - 2*sqrt(
d*f)*A*b^4*c*d^3*e*f - 20*sqrt(d*f)*C*a^3*b*d^4*e*f + 4*sqrt(d*f)*A*a*b^3*d^4*e*f + 3*sqrt(d*f)*C*a^2*b^2*c^2*
d^2*f^2 + sqrt(d*f)*B*a*b^3*c^2*d^2*f^2 + 3*sqrt(d*f)*A*b^4*c^2*d^2*f^2 - 12*sqrt(d*f)*C*a^3*b*c*d^3*f^2 - 4*s
qrt(d*f)*A*a*b^3*c*d^3*f^2 + 8*sqrt(d*f)*C*a^4*d^4*f^2)*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))/((b^6*c*e^2*abs(d) - a*b^5*d*e^2*abs(d) - 2*a*b^5*c*e*f*abs(d) + 2*a^2*b^4*d*e*f*abs(d) + a^2*
b^4*c*f^2*abs(d) - a^3*b^3*d*f^2*abs(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)*d) - sqr
t(d*f)*C*d*log((sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2)/(b^3*f*abs(d)) + 1/2*(8*sqrt
(d*f)*C*a*b^4*c*d^9*e^5 - 4*sqrt(d*f)*B*b^5*c*d^9*e^5 - 9*sqrt(d*f)*C*a^2*b^3*d^10*e^5 + 5*sqrt(d*f)*B*a*b^4*d
^10*e^5 - sqrt(d*f)*A*b^5*d^10*e^5 - 32*sqrt(d*f)*C*a*b^4*c^2*d^8*e^4*f + 16*sqrt(d*f)*B*b^5*c^2*d^8*e^4*f + 3
1*sqrt(d*f)*C*a^2*b^3*c*d^9*e^4*f - 19*sqrt(d*f)*B*a*b^4*c*d^9*e^4*f + 7*sqrt(d*f)*A*b^5*c*d^9*e^4*f + 6*sqrt(
d*f)*C*a^3*b^2*d^10*e^4*f - 2*sqrt(d*f)*B*a^2*b^3*d^10*e^4*f - 2*sqrt(d*f)*A*a*b^4*d^10*e^4*f + 48*sqrt(d*f)*C
*a*b^4*c^3*d^7*e^3*f^2 - 24*sqrt(d*f)*B*b^5*c^3*d^7*e^3*f^2 - 34*sqrt(d*f)*C*a^2*b^3*c^2*d^8*e^3*f^2 + 26*sqrt
(d*f)*B*a*b^4*c^2*d^8*e^3*f^2 - 18*sqrt(d*f)*A*b^5*c^2*d^8*e^3*f^2 - 24*sqrt(d*f)*C*a^3*b^2*c*d^9*e^3*f^2 + 8*
sqrt(d*f)*B*a^2*b^3*c*d^9*e^3*f^2 + 8*sqrt(d*f)*A*a*b^4*c*d^9*e^3*f^2 - 32*sqrt(d*f)*C*a*b^4*c^4*d^6*e^2*f^3 +
16*sqrt(d*f)*B*b^5*c^4*d^6*e^2*f^3 + 6*sqrt(d*f)*C*a^2*b^3*c^3*d^7*e^2*f^3 - 14*sqrt(d*f)*B*a*b^4*c^3*d^7*e^2
*f^3 + 22*sqrt(d*f)*A*b^5*c^3*d^7*e^2*f^3 + 36*sqrt(d*f)*C*a^3*b^2*c^2*d^8*e^2*f^3 - 12*sqrt(d*f)*B*a^2*b^3*c^
2*d^8*e^2*f^3 - 12*sqrt(d*f)*A*a*b^4*c^2*d^8*e^2*f^3 + 8*sqrt(d*f)*C*a*b^4*c^5*d^5*e*f^4 - 4*sqrt(d*f)*B*b^5*c
^5*d^5*e*f^4 + 11*sqrt(d*f)*C*a^2*b^3*c^4*d^6*e*f^4 + sqrt(d*f)*B*a*b^4*c^4*d^6*e*f^4 - 13*sqrt(d*f)*A*b^5*c^4
*d^6*e*f^4 - 24*sqrt(d*f)*C*a^3*b^2*c^3*d^7*e*f^4 + 8*sqrt(d*f)*B*a^2*b^3*c^3*d^7*e*f^4 + 8*sqrt(d*f)*A*a*b^4*
c^3*d^7*e*f^4 - 5*sqrt(d*f)*C*a^2*b^3*c^5*d^5*f^5 + sqrt(d*f)*B*a*b^4*c^5*d^5*f^5 + 3*sqrt(d*f)*A*b^5*c^5*d^5*
f^5 + 6*sqrt(d*f)*C*a^3*b^2*c^4*d^6*f^5 - 2*sqrt(d*f)*B*a^2*b^3*c^4*d^6*f^5 - 2*sqrt(d*f)*A*a*b^4*c^4*d^6*f^5
- 24*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*b^4*c*d^7*e^4 + 12*sqrt(d
*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*b^5*c*d^7*e^4 + 27*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^3*d^8*e^4 - 15*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^4*d^8*e^4 + 3*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^5*d^8*e^4 + 24*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*b^4*c^2*d^6*e^3*f - 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f -
c*d*f))^2*B*b^5*c^2*d^6*e^3*f + 32*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^3*c*d^7*e^3*f - 8*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^4
*c*d^7*e^3*f - 16*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^5*c*d^7*e^3*
f - 68*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*b^2*d^8*e^3*f + 32*sq
rt(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^3*d^8*e^3*f + 4*sqrt(d*f)*(s
qrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*A*a*b^4*d^8*e^3*f + 24*sqrt(d*f)*(sqrt(d*f)*sq
rt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a*b^4*c^3*d^5*e^2*f^2 - 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*
x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*b^5*c^3*d^5*e^2*f^2 - 130*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^3*c^2*d^6*e^2*f^2 + 58*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^4*c^2*d^6*e^2*f^2 + 14*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^5*c^2*d^6*e^2*f^2 + 92*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*b^2*c*d^7*e^2*f^2 - 56*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^3*c*d^7*e^2*f^2 + 20*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^4*c*d^7*e^2*f^2 + 32*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^4*b*d^8*e^2*f^2 - 8*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^3*b^2*d^8*e^2*f^2 - 16*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^2*b
^3*d^8*e^2*f^2 - 24*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*b^4*c^4*d^
4*e*f^3 + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*b^5*c^4*d^4*e*f^3 +
56*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^3*c^3*d^5*e*f^3 - 32*s
qrt(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^4*c^3*d^5*e*f^3 + 8*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^5*c^3*d^5*e*f^3 + 20*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*b^2*c^2*d^6*e*f^3 + 16*sqrt(d*f)*(sqrt(d*f)*sq
rt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a^2*b^3*c^2*d^6*e*f^3 - 52*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^4*c^2*d^6*e*f^3 - 64*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^4*b*c*d^7*e*f^3 + 16*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^3*b^2*c*d^7*e*f^3 + 32*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^2*b^3*c*d^7*e*f^3 + 15*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^3*c^4*d^4*f^4 - 3*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^4*c^4*d^4*f^4 - 9*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^5*c^4*d^4*f^4 - 44*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*b^2*c
^3*d^5*f^4 + 8*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^3*c^3*d^5*f
^4 + 28*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^4*c^3*d^5*f^4 + 32*s
qrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^4*b*c^2*d^6*f^4 - 8*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^3*b^2*c^2*d^6*f^4 - 16*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^2*b^3*c^2*d^6*f^4 + 24*sqrt(d*f)*(sqrt(d*f)*sqrt
(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a*b^4*c*d^5*e^3 - 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*b^5*c*d^5*e^3 - 27*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e +
(d*x + c)*d*f - c*d*f))^4*C*a^2*b^3*d^6*e^3 + 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*
d*f - c*d*f))^4*B*a*b^4*d^6*e^3 - 3*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^
4*A*b^5*d^6*e^3 + 16*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a*b^4*c^2*d
^4*e^2*f - 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*b^5*c^2*d^4*e^2*f -
97*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^2*b^3*c*d^5*e^2*f + 45*sqr
t(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a*b^4*c*d^5*e^2*f + 7*sqrt(d*f)*(sq
rt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*b^5*c*d^5*e^2*f + 90*sqrt(d*f)*(sqrt(d*f)*sqr
t(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^3*b^2*d^6*e^2*f - 46*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c
) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^2*b^3*d^6*e^2*f + 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d
^2*e + (d*x + c)*d*f - c*d*f))^4*A*a*b^4*d^6*e^2*f + 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x
+ c)*d*f - c*d*f))^4*C*a*b^4*c^3*d^3*e*f^2 - 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d
*f - c*d*f))^4*B*b^5*c^3*d^3*e*f^2 - 101*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d
*f))^4*C*a^2*b^3*c^2*d^4*e*f^2 + 49*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^
4*B*a*b^4*c^2*d^4*e*f^2 + 3*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*b^5*
c^2*d^4*e*f^2 + 188*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^3*b^2*c*d^
5*e*f^2 - 84*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^2*b^3*c*d^5*e*f^2
- 20*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a*b^4*c*d^5*e*f^2 - 120*sq
rt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^4*b*d^6*e*f^2 + 56*sqrt(d*f)*(sq
rt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^3*b^2*d^6*e*f^2 + 8*sqrt(d*f)*(sqrt(d*f)*sq
rt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a^2*b^3*d^6*e*f^2 - 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x +
c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^2*b^3*c^3*d^3*f^3 + 3*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqr
t(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a*b^4*c^3*d^3*f^3 + 9*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e +
(d*x + c)*d*f - c*d*f))^4*A*b^5*c^3*d^3*f^3 + 58*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d
*f - c*d*f))^4*C*a^3*b^2*c^2*d^4*f^3 - 14*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*
d*f))^4*B*a^2*b^3*c^2*d^4*f^3 - 30*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4
*A*a*b^4*c^2*d^4*f^3 - 88*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^4*b*
c*d^5*f^3 + 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^3*b^2*c*d^5*f^3
+ 40*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a^2*b^3*c*d^5*f^3 + 48*sqr
t(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*C*a^5*d^6*f^3 - 16*sqrt(d*f)*(sqrt(d*
f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*B*a^4*b*d^6*f^3 - 16*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x +
c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^4*A*a^3*b^2*d^6*f^3 - 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d
^2*e + (d*x + c)*d*f - c*d*f))^6*C*a*b^4*c*d^3*e^2 + 4*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x
+ c)*d*f - c*d*f))^6*B*b^5*c*d^3*e^2 + 9*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d
*f))^6*C*a^2*b^3*d^4*e^2 - 5*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a*b
^4*d^4*e^2 + sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*b^5*d^4*e^2 - 8*sqr
t(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a*b^4*c^2*d^2*e*f + 4*sqrt(d*f)*(sq
rt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*b^5*c^2*d^2*e*f + 34*sqrt(d*f)*(sqrt(d*f)*sqr
t(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^2*b^3*c*d^3*e*f - 18*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c
) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a*b^4*c*d^3*e*f + 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2
*e + (d*x + c)*d*f - c*d*f))^6*A*b^5*c*d^3*e*f - 28*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c
)*d*f - c*d*f))^6*C*a^3*b^2*d^4*e*f + 16*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d
*f))^6*B*a^2*b^3*d^4*e*f - 4*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a*b
^4*d^4*e*f + 5*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^2*b^3*c^2*d^2*f
^2 - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a*b^4*c^2*d^2*f^2 - 3*sqrt(
d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*b^5*c^2*d^2*f^2 - 20*sqrt(d*f)*(sqrt(
d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*C*a^3*b^2*c*d^3*f^2 + 8*sqrt(d*f)*(sqrt(d*f)*sqrt(
d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*B*a^2*b^3*c*d^3*f^2 + 4*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^6*A*a*b^4*c*d^3*f^2 + 16*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e
+ (d*x + c)*d*f - c*d*f))^6*C*a^4*b*d^4*f^2 - 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d
*f - c*d*f))^6*B*a^3*b^2*d^4*f^2)/((b^6*c*e^2*abs(d) - a*b^5*d*e^2*abs(d) - 2*a*b^5*c*e*f*abs(d) + 2*a^2*b^4*d
*e*f*abs(d) + a^2*b^4*c*f^2*abs(d) - a^3*b^3*d*f^2*abs(d))*(b*d^4*e^2 - 2*b*c*d^3*e*f + b*c^2*d^2*f^2 - 2*(sqr
t(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)^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^3 \sqrt {e+f x}} \, dx=\text {Hanged} \]

[In]

int(((c + d*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(a + b*x)^3),x)

[Out]

\text{Hanged}