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

3.1.44.1 Optimal result

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

output

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))*arctanh((-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
 

3.1.44.2 Mathematica [A] (verified)

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

input

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

output

((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)
 

3.1.44.3 Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2118, 27, 171, 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.

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

\(\Big \downarrow \) 2118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

\(\displaystyle \frac {\frac {\frac {\int \frac {2 b d e (4 b c f (2 A b d f-a C (d e+c f))+a (d e+3 c f) (2 a C d f+b (C d e+c C f-2 B d f)))-a (d e+c f) (4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f)))-\left (-\left (\left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right ) b^3\right )-2 a d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right ) b^2-8 a^2 d^2 f^2 (C d e+c C f+2 B d f) b+16 a^3 C d^3 f^3\right ) x}{2 (a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (4 b d f (2 A b d f-a C (c f+d e))+(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e)))}{b d}}{4 b f}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{2 b f}}{2 b d f}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {2 b d e (4 b c f (2 A b d f-a C (d e+c f))+a (d e+3 c f) (2 a C d f+b (C d e+c C f-2 B d f)))-a (d e+c f) (4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f)))-\left (-\left (\left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right ) b^3\right )-2 a d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right ) b^2-8 a^2 d^2 f^2 (C d e+c C f+2 B d f) b+16 a^3 C d^3 f^3\right ) x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (4 b d f (2 A b d f-a C (c f+d e))+(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e)))}{b d}}{4 b f}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{2 b f}}{2 b d f}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {\frac {\frac {\frac {16 d^2 f^2 (b c-a d) (b e-a f) \left (A b^2-a (b B-a C)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}-\frac {\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )-\left (b^3 \left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{b}}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (4 b d f (2 A b d f-a C (c f+d e))+(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e)))}{b d}}{4 b f}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{2 b f}}{2 b d f}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {\frac {\frac {16 d^2 f^2 (b c-a d) (b e-a f) \left (A b^2-a (b B-a C)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}-\frac {2 \left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )-\left (b^3 \left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right ) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (4 b d f (2 A b d f-a C (c f+d e))+(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e)))}{b d}}{4 b f}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{2 b f}}{2 b d f}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {\frac {\frac {32 d^2 f^2 (b c-a d) (b e-a f) \left (A b^2-a (b B-a C)\right ) \int \frac {1}{-b c+a d+\frac {(b e-a f) (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}-\frac {2 \left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )-\left (b^3 \left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right ) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (4 b d f (2 A b d f-a C (c f+d e))+(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e)))}{b d}}{4 b f}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{2 b f}}{2 b d f}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {-\frac {2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )-\left (b^3 \left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right )}{b \sqrt {d} \sqrt {f}}-\frac {32 d^2 f^2 \sqrt {b c-a d} \sqrt {b e-a f} \left (A b^2-a (b B-a C)\right ) \text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right )}{b}}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (4 b d f (2 A b d f-a C (c f+d e))+(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e)))}{b d}}{4 b f}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{2 b f}}{2 b d f}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}\)

input

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

output

(C*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*b*d*f) + (-1/2*((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))/(b*f) + (((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])/(b*d) + ((-2*(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]*Sqrt[e + f*x])])/(b*Sqrt[d]*Sqrt[f]) - (32*(A*b^ 
2 - a*(b*B - a*C))*d^2*Sqrt[b*c - a*d]*f^2*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)/(2*b*d))/(4* 
b*f))/(2*b*d*f)
 

3.1.44.3.1 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 2118

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, Expo 
n[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] + Simp[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]] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] && PolyQ[Px, x]
 

3.1.44.4 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	\(\text {Expression too large to display}\)	\(3898\)

input

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

output

-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*...
 

3.1.44.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.1.44.6 Sympy [F]

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

input

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

output

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

3.1.44.7 Maxima [F(-2)]

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

input

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

3.1.44.8 Giac [F(-2)]

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

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

3.1.44.9 Mupad [F(-1)]

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

input

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

output

\text{Hanged}