\(\int \sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2) \, dx\) [65]

Optimal result

Integrand size = 29, antiderivative size = 330 \[ \int \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\frac {(d e-c f) \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^3}+\frac {\left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) (c+d x)^{3/2} \sqrt {e+f x}}{32 d^3 f^2}-\frac {(5 C d e+11 c C f-8 B d f) (c+d x)^{3/2} (e+f x)^{3/2}}{24 d^2 f^2}+\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {(d e-c f)^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{64 d^{7/2} f^{7/2}} \] Output:

1/64*(-c*f+d*e)*(C*(5*c^2*f^2+6*c*d*e*f+5*d^2*e^2)+8*d*f*(2*A*d*f-B*(c*f+d 
*e)))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/d^3/f^3+1/32*(C*(5*c^2*f^2+6*c*d*e*f+5*d 
^2*e^2)+8*d*f*(2*A*d*f-B*(c*f+d*e)))*(d*x+c)^(3/2)*(f*x+e)^(1/2)/d^3/f^2-1 
/24*(-8*B*d*f+11*C*c*f+5*C*d*e)*(d*x+c)^(3/2)*(f*x+e)^(3/2)/d^2/f^2+1/4*C* 
(d*x+c)^(5/2)*(f*x+e)^(3/2)/d^2/f-1/64*(-c*f+d*e)^2*(C*(5*c^2*f^2+6*c*d*e* 
f+5*d^2*e^2)+8*d*f*(2*A*d*f-B*(c*f+d*e)))*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^ 
(1/2)/(f*x+e)^(1/2))/d^(7/2)/f^(7/2)
 

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.86 \[ \int \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {c+d x} \sqrt {e+f x} \left (C \left (15 c^3 f^3-c^2 d f^2 (7 e+10 f x)+c d^2 f \left (-7 e^2+4 e f x+8 f^2 x^2\right )+d^3 \left (15 e^3-10 e^2 f x+8 e f^2 x^2+48 f^3 x^3\right )\right )+8 d f \left (6 A d f (c f+d (e+2 f x))+B \left (-3 c^2 f^2+2 c d f (e+f x)+d^2 \left (-3 e^2+2 e f x+8 f^2 x^2\right )\right )\right )\right )}{192 d^3 f^3}-\frac {(d e-c f)^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{64 d^{7/2} f^{7/2}} \] Input:

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

Output:

(Sqrt[c + d*x]*Sqrt[e + f*x]*(C*(15*c^3*f^3 - c^2*d*f^2*(7*e + 10*f*x) + c 
*d^2*f*(-7*e^2 + 4*e*f*x + 8*f^2*x^2) + d^3*(15*e^3 - 10*e^2*f*x + 8*e*f^2 
*x^2 + 48*f^3*x^3)) + 8*d*f*(6*A*d*f*(c*f + d*(e + 2*f*x)) + B*(-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)))))/(192*d^3*f^3 
) - ((d*e - c*f)^2*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*d*f 
 - B*(d*e + c*f)))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/(Sqrt[f]*Sqrt[c + d*x]) 
])/(64*d^(7/2)*f^(7/2))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.75, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1194, 27, 90, 60, 60, 66, 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.

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

\(\Big \downarrow \) 1194

\(\displaystyle \frac {\int -\frac {1}{2} \sqrt {c+d x} \sqrt {e+f x} \left (3 C f c^2+5 C d e c-8 A d^2 f+d (5 C d e+11 c C f-8 B d f) x\right )dx}{4 d^2 f}+\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {\int \sqrt {c+d x} \sqrt {e+f x} \left (3 C f c^2+5 C d e c-8 A d^2 f+d (5 C d e+11 c C f-8 B d f) x\right )dx}{8 d^2 f}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {\frac {(c+d x)^{3/2} (e+f x)^{3/2} (-8 B d f+11 c C f+5 C d e)}{3 f}-\frac {\left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right ) \int \sqrt {c+d x} \sqrt {e+f x}dx}{2 f}}{8 d^2 f}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {\frac {(c+d x)^{3/2} (e+f x)^{3/2} (-8 B d f+11 c C f+5 C d e)}{3 f}-\frac {\left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right ) \left (\frac {(d e-c f) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x}}dx}{4 d}+\frac {(c+d x)^{3/2} \sqrt {e+f x}}{2 d}\right )}{2 f}}{8 d^2 f}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {\frac {(c+d x)^{3/2} (e+f x)^{3/2} (-8 B d f+11 c C f+5 C d e)}{3 f}-\frac {\left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right ) \left (\frac {(d e-c f) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{2 f}\right )}{4 d}+\frac {(c+d x)^{3/2} \sqrt {e+f x}}{2 d}\right )}{2 f}}{8 d^2 f}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {\frac {(c+d x)^{3/2} (e+f x)^{3/2} (-8 B d f+11 c C f+5 C d e)}{3 f}-\frac {\left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right ) \left (\frac {(d e-c f) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{f}-\frac {(d e-c f) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{f}\right )}{4 d}+\frac {(c+d x)^{3/2} \sqrt {e+f x}}{2 d}\right )}{2 f}}{8 d^2 f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {\frac {(c+d x)^{3/2} (e+f x)^{3/2} (-8 B d f+11 c C f+5 C d e)}{3 f}-\frac {\left (\frac {(d e-c f) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{\sqrt {d} f^{3/2}}\right )}{4 d}+\frac {(c+d x)^{3/2} \sqrt {e+f x}}{2 d}\right ) \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )}{2 f}}{8 d^2 f}\)

Input:

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

Output:

(C*(c + d*x)^(5/2)*(e + f*x)^(3/2))/(4*d^2*f) - (((5*C*d*e + 11*c*C*f - 8* 
B*d*f)*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*f) - ((C*(5*d^2*e^2 + 6*c*d*e*f 
 + 5*c^2*f^2) + 8*d*f*(2*A*d*f - B*(d*e + c*f)))*(((c + d*x)^(3/2)*Sqrt[e 
+ f*x])/(2*d) + ((d*e - c*f)*((Sqrt[c + d*x]*Sqrt[e + f*x])/f - ((d*e - c* 
f)*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(Sqrt[d]*f^(3 
/2))))/(4*d)))/(2*f))/(8*d^2*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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

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 1194

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x 
)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 
*p + 1))   Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 
2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) 
*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ 
[p, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1206\) vs. \(2(292)=584\).

Time = 0.54 (sec) , antiderivative size = 1207, normalized size of antiderivative = 3.66

method	result	size
default	\(\text {Expression too large to display}\)	\(1207\)

Input:

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

Output:

-1/384*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(-12*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))*c*d^3*e^3*f-6*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))*c^2*d^2*e^2* 
f^2-8*C*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*c*d^2*e*f^2*x+48*B*(d*f)^(1/2) 
*((f*x+e)*(d*x+c))^(1/2)*d^3*e^2*f+48*A*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))*c^2*d^2*f^4-24*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))*d^4*e^3*f-16* 
C*c*d^2*f^3*x^2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+48*A*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))*d^4*e^2*f^2-24* 
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))*c^3*d*f^4-30*C*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*c^3*f^3-96*A*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))*c*d^ 
3*e*f^3-16*C*d^3*e*f^2*x^2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)-96*C*d^3*f^ 
3*x^3*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)-128*B*d^3*f^3*x^2*((f*x+e)*(d*x+ 
c))^(1/2)*(d*f)^(1/2)+15*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))*c^4*f^4+15*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))*d^4*e^4-96*A*(d*f)^(1/2)*((f*x 
+e)*(d*x+c))^(1/2)*c*d^2*f^3-96*A*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*d^3* 
e*f^2+48*B*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*c^2*d*f^3-192*A*(d*f)^(1/2) 
*((f*x+e)*(d*x+c))^(1/2)*d^3*f^3*x+24*B*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x...
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 840, normalized size of antiderivative = 2.55 \[ \int \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

[1/768*(3*(5*C*d^4*e^4 - 4*(C*c*d^3 + 2*B*d^4)*e^3*f - 2*(C*c^2*d^2 - 4*B* 
c*d^3 - 8*A*d^4)*e^2*f^2 - 4*(C*c^3*d - 2*B*c^2*d^2 + 8*A*c*d^3)*e*f^3 + ( 
5*C*c^4 - 8*B*c^3*d + 16*A*c^2*d^2)*f^4)*sqrt(d*f)*log(8*d^2*f^2*x^2 + d^2 
*e^2 + 6*c*d*e*f + c^2*f^2 - 4*(2*d*f*x + d*e + c*f)*sqrt(d*f)*sqrt(d*x + 
c)*sqrt(f*x + e) + 8*(d^2*e*f + c*d*f^2)*x) + 4*(48*C*d^4*f^4*x^3 + 15*C*d 
^4*e^3*f - (7*C*c*d^3 + 24*B*d^4)*e^2*f^2 - (7*C*c^2*d^2 - 16*B*c*d^3 - 48 
*A*d^4)*e*f^3 + 3*(5*C*c^3*d - 8*B*c^2*d^2 + 16*A*c*d^3)*f^4 + 8*(C*d^4*e* 
f^3 + (C*c*d^3 + 8*B*d^4)*f^4)*x^2 - 2*(5*C*d^4*e^2*f^2 - 2*(C*c*d^3 + 4*B 
*d^4)*e*f^3 + (5*C*c^2*d^2 - 8*B*c*d^3 - 48*A*d^4)*f^4)*x)*sqrt(d*x + c)*s 
qrt(f*x + e))/(d^4*f^4), 1/384*(3*(5*C*d^4*e^4 - 4*(C*c*d^3 + 2*B*d^4)*e^3 
*f - 2*(C*c^2*d^2 - 4*B*c*d^3 - 8*A*d^4)*e^2*f^2 - 4*(C*c^3*d - 2*B*c^2*d^ 
2 + 8*A*c*d^3)*e*f^3 + (5*C*c^4 - 8*B*c^3*d + 16*A*c^2*d^2)*f^4)*sqrt(-d*f 
)*arctan(1/2*(2*d*f*x + d*e + c*f)*sqrt(-d*f)*sqrt(d*x + c)*sqrt(f*x + e)/ 
(d^2*f^2*x^2 + c*d*e*f + (d^2*e*f + c*d*f^2)*x)) + 2*(48*C*d^4*f^4*x^3 + 1 
5*C*d^4*e^3*f - (7*C*c*d^3 + 24*B*d^4)*e^2*f^2 - (7*C*c^2*d^2 - 16*B*c*d^3 
 - 48*A*d^4)*e*f^3 + 3*(5*C*c^3*d - 8*B*c^2*d^2 + 16*A*c*d^3)*f^4 + 8*(C*d 
^4*e*f^3 + (C*c*d^3 + 8*B*d^4)*f^4)*x^2 - 2*(5*C*d^4*e^2*f^2 - 2*(C*c*d^3 
+ 4*B*d^4)*e*f^3 + (5*C*c^2*d^2 - 8*B*c*d^3 - 48*A*d^4)*f^4)*x)*sqrt(d*x + 
 c)*sqrt(f*x + e))/(d^4*f^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^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(c*f+d*e>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1035 vs. \(2 (292) = 584\).

Time = 0.27 (sec) , antiderivative size = 1035, normalized size of antiderivative = 3.14 \[ \int \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/192*(192*((d^2*e - c*d*f)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt(d^2*e 
 + (d*x + c)*d*f - c*d*f)))/sqrt(d*f) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f 
)*sqrt(d*x + c))*A*c*abs(d)/d^2 - (sqrt(d^2*e + (d*x + c)*d*f - c*d*f)*(2* 
(d*x + c)*(4*(d*x + c)*(6*(d*x + c)/d^3 + (d^12*e*f^5 - 25*c*d^11*f^6)/(d^ 
14*f^6)) - (5*d^13*e^2*f^4 + 14*c*d^12*e*f^5 - 163*c^2*d^11*f^6)/(d^14*f^6 
)) + 3*(5*d^14*e^3*f^3 + 9*c*d^13*e^2*f^4 + 15*c^2*d^12*e*f^5 - 93*c^3*d^1 
1*f^6)/(d^14*f^6))*sqrt(d*x + c) + 3*(5*d^4*e^4 + 4*c*d^3*e^3*f + 6*c^2*d^ 
2*e^2*f^2 + 20*c^3*d*e*f^3 - 35*c^4*f^4)*log(abs(-sqrt(d*f)*sqrt(d*x + c) 
+ sqrt(d^2*e + (d*x + c)*d*f - c*d*f)))/(sqrt(d*f)*d^2*f^3))*C*abs(d)/d - 
48*(sqrt(d^2*e + (d*x + c)*d*f - c*d*f)*(2*d*x + 2*c + (d*e*f - 5*c*f^2)/f 
^2)*sqrt(d*x + c) + (d^3*e^2 + 2*c*d^2*e*f - 3*c^2*d*f^2)*log(abs(-sqrt(d* 
f)*sqrt(d*x + c) + sqrt(d^2*e + (d*x + c)*d*f - c*d*f)))/(sqrt(d*f)*f))*B* 
c*abs(d)/d^3 - 48*(sqrt(d^2*e + (d*x + c)*d*f - c*d*f)*(2*d*x + 2*c + (d*e 
*f - 5*c*f^2)/f^2)*sqrt(d*x + c) + (d^3*e^2 + 2*c*d^2*e*f - 3*c^2*d*f^2)*l 
og(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt(d^2*e + (d*x + c)*d*f - c*d*f)))/(s 
qrt(d*f)*f))*A*abs(d)/d^2 - 8*(sqrt(d^2*e + (d*x + c)*d*f - c*d*f)*(2*(4*d 
*x + 4*c + (d*e*f^3 - 13*c*f^4)/f^4)*(d*x + c) - 3*(d^2*e^2*f^2 + 2*c*d*e* 
f^3 - 11*c^2*f^4)/f^4)*sqrt(d*x + c) - 3*(d^4*e^3 + c*d^3*e^2*f + 3*c^2*d^ 
2*e*f^2 - 5*c^3*d*f^3)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt(d^2*e + (d* 
x + c)*d*f - c*d*f)))/(sqrt(d*f)*f^2))*C*c*abs(d)/d^4 - 8*(sqrt(d^2*e +...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 2.86 (sec) , antiderivative size = 1024, normalized size of antiderivative = 3.10 \[ \int \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(48*sqrt(e + f*x)*sqrt(c + d*x)*a*c*d**3*f**4 + 48*sqrt(e + f*x)*sqrt(c + 
d*x)*a*d**4*e*f**3 + 96*sqrt(e + f*x)*sqrt(c + d*x)*a*d**4*f**4*x - 24*sqr 
t(e + f*x)*sqrt(c + d*x)*b*c**2*d**2*f**4 + 16*sqrt(e + f*x)*sqrt(c + d*x) 
*b*c*d**3*e*f**3 + 16*sqrt(e + f*x)*sqrt(c + d*x)*b*c*d**3*f**4*x - 24*sqr 
t(e + f*x)*sqrt(c + d*x)*b*d**4*e**2*f**2 + 16*sqrt(e + f*x)*sqrt(c + d*x) 
*b*d**4*e*f**3*x + 64*sqrt(e + f*x)*sqrt(c + d*x)*b*d**4*f**4*x**2 + 15*sq 
rt(e + f*x)*sqrt(c + d*x)*c**4*d*f**4 - 7*sqrt(e + f*x)*sqrt(c + d*x)*c**3 
*d**2*e*f**3 - 10*sqrt(e + f*x)*sqrt(c + d*x)*c**3*d**2*f**4*x - 7*sqrt(e 
+ f*x)*sqrt(c + d*x)*c**2*d**3*e**2*f**2 + 4*sqrt(e + f*x)*sqrt(c + d*x)*c 
**2*d**3*e*f**3*x + 8*sqrt(e + f*x)*sqrt(c + d*x)*c**2*d**3*f**4*x**2 + 15 
*sqrt(e + f*x)*sqrt(c + d*x)*c*d**4*e**3*f - 10*sqrt(e + f*x)*sqrt(c + d*x 
)*c*d**4*e**2*f**2*x + 8*sqrt(e + f*x)*sqrt(c + d*x)*c*d**4*e*f**3*x**2 + 
48*sqrt(e + f*x)*sqrt(c + d*x)*c*d**4*f**4*x**3 - 48*sqrt(f)*sqrt(d)*log(( 
sqrt(f)*sqrt(c + d*x) + sqrt(d)*sqrt(e + f*x))/sqrt(c*f - d*e))*a*c**2*d** 
2*f**4 + 96*sqrt(f)*sqrt(d)*log((sqrt(f)*sqrt(c + d*x) + sqrt(d)*sqrt(e + 
f*x))/sqrt(c*f - d*e))*a*c*d**3*e*f**3 - 48*sqrt(f)*sqrt(d)*log((sqrt(f)*s 
qrt(c + d*x) + sqrt(d)*sqrt(e + f*x))/sqrt(c*f - d*e))*a*d**4*e**2*f**2 + 
24*sqrt(f)*sqrt(d)*log((sqrt(f)*sqrt(c + d*x) + sqrt(d)*sqrt(e + f*x))/sqr 
t(c*f - d*e))*b*c**3*d*f**4 - 24*sqrt(f)*sqrt(d)*log((sqrt(f)*sqrt(c + d*x 
) + sqrt(d)*sqrt(e + f*x))/sqrt(c*f - d*e))*b*c**2*d**2*e*f**3 - 24*sqr...