\(\int \frac {\sqrt {c+d x} (A+B x+C x^2)}{(a+b x)^2 \sqrt {e+f x}} \, dx\) [73]

Optimal result

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

(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)-(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)-(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/d^(1/2)/f^ 
(3/2)+(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))*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)^(1/2)/(-a*f+b*e)^(3/2)
 

Mathematica [A] (verified)

Time = 11.35 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^2 \sqrt {e+f x}} \, dx=\frac {-\frac {2 b \left (A b^2+a (-b B+a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b e-a f) (a+b x)}+\frac {4 (b B-2 a C) \sqrt {d e-c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \text {arcsinh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} \sqrt {e+f x}}+\frac {2 b C \sqrt {e+f x} \left (\sqrt {f} \sqrt {c+d x}-\frac {\sqrt {d e-c f} \text {arcsinh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {\frac {d (e+f x)}{d e-c f}}}\right )}{f^{3/2}}-\frac {4 (b B-2 a C) \sqrt {-b c+a d} \text {arctanh}\left (\frac {\sqrt {-b e+a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )}{\sqrt {-b e+a f}}+\frac {2 b \left (A b^2+a (-b B+a C)\right ) (-d e+c f) \text {arctanh}\left (\frac {\sqrt {-b e+a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )}{\sqrt {-b c+a d} (-b e+a f)^{3/2}}}{2 b^3} \] Input:

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

Output:

((-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]*Arc 
Sinh[(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)
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2116, 27, 171, 27, 175, 66, 104, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2116

\(\displaystyle -\frac {\int -\frac {\sqrt {c+d x} \left (C (3 d e+c f) a^2-b (2 c C e+3 B d e+B c f-2 A d f) a+b^2 (2 B c e+A d e-A c f)+2 b \left (\frac {2 C d f a^2}{b}-(C d e+c C f+B d f) a+b (c C e+A d f)\right ) x\right )}{2 b (a+b x) \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c+d x} \left (C (3 d e+c f) a^2-b (2 c C e+3 B d e+B c f-2 A d f) a+b^2 (2 B c e+A d e-A c f)+2 b \left (\frac {2 C d f a^2}{b}-(C d e+c C f+B d f) a+b (c C e+A d f)\right ) x\right )}{(a+b x) \sqrt {e+f x}}dx}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {\frac {\int \frac {(b c-a d) \left (2 C f (d e+c f) a^2-b (B f (d e+c f)+C e (d e+3 c f)) a+b^2 f (2 B c e+A d e-A c f)-(b e-a f) (4 a C d f+b (C d e-c C f-2 B d f)) x\right )}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b f}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (\frac {2 a^2 C d f}{b}-a (B d f+c C f+C d e)+b (A d f+c C e)\right )}{f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {(b c-a d) \int \frac {2 C f (d e+c f) a^2-b (B f (d e+c f)+C e (d e+3 c f)) a+b^2 f (2 B c e+A d e-A c f)-(b e-a f) (4 a C d f+b (C d e-c C f-2 B d f)) x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b f}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (\frac {2 a^2 C d f}{b}-a (B d f+c C f+C d e)+b (A d f+c C e)\right )}{f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {\frac {(b c-a d) \left (-\frac {f \left (4 a^3 C d f-a^2 b (2 B d f+3 c C f+5 C d e)+a b^2 (B c f+3 B d e+4 c C e)-b^3 (-A c f+A d e+2 B c e)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}-\frac {(b e-a f) (4 a C d f+b (-2 B d f-c C f+C d e)) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{b}\right )}{b f}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (\frac {2 a^2 C d f}{b}-a (B d f+c C f+C d e)+b (A d f+c C e)\right )}{f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {(b c-a d) \left (-\frac {f \left (4 a^3 C d f-a^2 b (2 B d f+3 c C f+5 C d e)+a b^2 (B c f+3 B d e+4 c C e)-b^3 (-A c f+A d e+2 B c e)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}-\frac {2 (b e-a f) (4 a C d f+b (-2 B d f-c C f+C d e)) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}\right )}{b f}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (\frac {2 a^2 C d f}{b}-a (B d f+c C f+C d e)+b (A d f+c C e)\right )}{f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {(b c-a d) \left (-\frac {2 f \left (4 a^3 C d f-a^2 b (2 B d f+3 c C f+5 C d e)+a b^2 (B c f+3 B d e+4 c C e)-b^3 (-A c f+A d e+2 B c e)\right ) \int \frac {1}{-b c+a d+\frac {(b e-a f) (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}-\frac {2 (b e-a f) (4 a C d f+b (-2 B d f-c C f+C d e)) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}\right )}{b f}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (\frac {2 a^2 C d f}{b}-a (B d f+c C f+C d e)+b (A d f+c C e)\right )}{f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (\frac {2 a^2 C d f}{b}-a (B d f+c C f+C d e)+b (A d f+c C e)\right )}{f}+\frac {(b c-a d) \left (\frac {2 f \text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (4 a^3 C d f-a^2 b (2 B d f+3 c C f+5 C d e)+a b^2 (B c f+3 B d e+4 c C e)-b^3 (-A c f+A d e+2 B c e)\right )}{b \sqrt {b c-a d} \sqrt {b e-a f}}-\frac {2 (b e-a f) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) (4 a C d f+b (-2 B d f-c C f+C d e))}{b \sqrt {d} \sqrt {f}}\right )}{b f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[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] /; Fre 
eQ[{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 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(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 221

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 2116

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] + Si 
mp[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]
 

Maple [B] (verified)

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

Time = 0.98 (sec) , antiderivative size = 3670, normalized size of antiderivative = 10.08

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(((-(2*a*d*f)/b^2)>0)', see `assu 
me?` 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 = 1.23 (sec) , antiderivative size = 1354, normalized size of antiderivative = 3.72 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^2 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

(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)*sqr 
t(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)*(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*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)*s 
qrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*C*a^3*d^3*f - 2*s...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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