\(\int \frac {\sqrt {c+d x} (A+B x+C x^2)}{(a+b x)^2 \sqrt {e+f x}} \, dx\) [51]

Optimal result

Integrand size = 36, antiderivative size = 364 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^2 \sqrt {e+f x}} \, dx=\frac {\left (2 a^2 C d f+b^2 (c C e+A d f)-a b (C d e+c C f+B d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{b^2 (b c-a d) f (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{b (b c-a d) (b e-a f) (a+b x)}-\frac {(4 a C d f+b (C d e-c C f-2 B d f)) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{b^3 \sqrt {d} f^{3/2}}+\frac {\left (4 a^3 C d f-b^3 (2 B c e+A d e-A c f)+a b^2 (4 c C e+3 B d e+B c f)-a^2 b (5 C d e+3 c C f+2 B d f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{b^3 \sqrt {b c-a d} (b e-a f)^{3/2}} \]

[Out]

-(4*a*C*d*f+b*(-2*B*d*f-C*c*f+C*d*e))*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2))/b^3/f^(3/2)/d^(1/2)
+(4*a^3*C*d*f-b^3*(-A*c*f+A*d*e+2*B*c*e)+a*b^2*(B*c*f+3*B*d*e+4*C*c*e)-a^2*b*(2*B*d*f+3*C*c*f+5*C*d*e))*arctan
h((-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*f+b*e)^(3/2)/(-a*d+b*c)^(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*a^2*C*d*f+b^2*(A*d*f+C*c*e)-a*b*(B
*d*f+C*c*f+C*d*e))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^2/(-a*d+b*c)/f/(-a*f+b*e)

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 364, 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, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^2 \sqrt {e+f x}} \, dx=\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right )}{b^2 f (b c-a d) (b e-a f)}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (4 a^3 C d f-a^2 b (2 B d f+3 c C f+5 C d e)+a b^2 (B c f+3 B d e+4 c C e)-b^3 (-A c f+A d e+2 B c e)\right )}{b^3 \sqrt {b c-a d} (b e-a f)^{3/2}}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}-\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) (4 a C d f+b (-2 B d f-c C f+C d e))}{b^3 \sqrt {d} f^{3/2}} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 11.28 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^2 \sqrt {e+f x}} \, dx=\frac {-\frac {2 b \left (A b^2+a (-b B+a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b e-a f) (a+b x)}+\frac {4 (b B-2 a C) \sqrt {d e-c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \text {arcsinh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} \sqrt {e+f x}}+\frac {2 b C \sqrt {e+f x} \left (\sqrt {f} \sqrt {c+d x}-\frac {\sqrt {d e-c f} \text {arcsinh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {\frac {d (e+f x)}{d e-c f}}}\right )}{f^{3/2}}-\frac {4 (b B-2 a C) \sqrt {-b c+a d} \text {arctanh}\left (\frac {\sqrt {-b e+a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )}{\sqrt {-b e+a f}}+\frac {2 b \left (A b^2+a (-b B+a C)\right ) (-d e+c f) \text {arctanh}\left (\frac {\sqrt {-b e+a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )}{\sqrt {-b c+a d} (-b e+a f)^{3/2}}}{2 b^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3669\) vs. \(2(332)=664\).

Time = 1.70 (sec) , antiderivative size = 3670, normalized size of antiderivative = 10.08

method	result	size
default	\(\text {Expression too large to display}\)	\(3670\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^2/(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. 1354 vs. \(2 (331) = 662\).

Time = 1.40 (sec) , antiderivative size = 1354, normalized size of antiderivative = 3.72 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^2 \sqrt {e+f x}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}