\(\int (a+b x) \sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2) \, dx\) [64]

Optimal result

Integrand size = 34, antiderivative size = 715 \[ \int (a+b x) \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\frac {(d e-c f) \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{128 d^4 f^4}+\frac {\left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{64 d^4 f^3}-\frac {\left (2 b d f (5 B d e+11 B c f-8 A d f)+2 a d f (5 C d e+11 c C f-8 B d f)-b C \left (7 d^2 e^2+16 c d e f+25 c^2 f^2\right )\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{48 d^3 f^3}+\frac {(10 a C d f-b (7 C d e+23 c C f-10 B d f)) (c+d x)^{5/2} (e+f x)^{3/2}}{40 d^3 f^2}+\frac {b C (c+d x)^{7/2} (e+f x)^{3/2}}{5 d^3 f}-\frac {(d e-c f)^2 \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{128 d^{9/2} f^{9/2}} \] Output:

1/128*(-c*f+d*e)*(2*a*d*f*(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)))-b*(C*(7*c^3*f^3+9*c^2*d*e*f^2+9*c*d^2*e^2*f+7*d^3*e^3)+2*d 
*f*(8*A*d*f*(c*f+d*e)-B*(5*c^2*f^2+6*c*d*e*f+5*d^2*e^2))))*(d*x+c)^(1/2)*( 
f*x+e)^(1/2)/d^4/f^4+1/64*(2*a*d*f*(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)))-b*(C*(7*c^3*f^3+9*c^2*d*e*f^2+9*c*d^2*e^2*f+7*d^3 
*e^3)+2*d*f*(8*A*d*f*(c*f+d*e)-B*(5*c^2*f^2+6*c*d*e*f+5*d^2*e^2))))*(d*x+c 
)^(3/2)*(f*x+e)^(1/2)/d^4/f^3-1/48*(2*b*d*f*(-8*A*d*f+11*B*c*f+5*B*d*e)+2* 
a*d*f*(-8*B*d*f+11*C*c*f+5*C*d*e)-b*C*(25*c^2*f^2+16*c*d*e*f+7*d^2*e^2))*( 
d*x+c)^(3/2)*(f*x+e)^(3/2)/d^3/f^3+1/40*(10*a*C*d*f-b*(-10*B*d*f+23*C*c*f+ 
7*C*d*e))*(d*x+c)^(5/2)*(f*x+e)^(3/2)/d^3/f^2+1/5*b*C*(d*x+c)^(7/2)*(f*x+e 
)^(3/2)/d^3/f-1/128*(-c*f+d*e)^2*(2*a*d*f*(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)))-b*(C*(7*c^3*f^3+9*c^2*d*e*f^2+9*c*d^2*e^2* 
f+7*d^3*e^3)+2*d*f*(8*A*d*f*(c*f+d*e)-B*(5*c^2*f^2+6*c*d*e*f+5*d^2*e^2)))) 
*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2))/d^(9/2)/f^(9/2)
 

Mathematica [A] (verified)

Time = 2.59 (sec) , antiderivative size = 662, normalized size of antiderivative = 0.93 \[ \int (a+b x) \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {c+d x} \sqrt {e+f x} \left (10 a d f \left (C \left (15 c^3 f^3-c^2 d f^2 (7 e+10 f x)+c d^2 f \left (-7 e^2+4 e f x+8 f^2 x^2\right )+d^3 \left (15 e^3-10 e^2 f x+8 e f^2 x^2+48 f^3 x^3\right )\right )+8 d f \left (6 A d f (c f+d (e+2 f x))+B \left (-3 c^2 f^2+2 c d f (e+f x)+d^2 \left (-3 e^2+2 e f x+8 f^2 x^2\right )\right )\right )\right )+b \left (C \left (-105 c^4 f^4+10 c^3 d f^3 (4 e+7 f x)-2 c^2 d^2 f^2 \left (-17 e^2+11 e f x+28 f^2 x^2\right )+2 c d^3 f \left (20 e^3-11 e^2 f x+8 e f^2 x^2+24 f^3 x^3\right )+d^4 \left (-105 e^4+70 e^3 f x-56 e^2 f^2 x^2+48 e f^3 x^3+384 f^4 x^4\right )\right )+10 d f \left (8 A d f \left (-3 c^2 f^2+2 c d f (e+f x)+d^2 \left (-3 e^2+2 e f x+8 f^2 x^2\right )\right )+B \left (15 c^3 f^3-c^2 d f^2 (7 e+10 f x)+c d^2 f \left (-7 e^2+4 e f x+8 f^2 x^2\right )+d^3 \left (15 e^3-10 e^2 f x+8 e f^2 x^2+48 f^3 x^3\right )\right )\right )\right )\right )}{1920 d^4 f^4}+\frac {(d e-c f)^2 \left (-2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )+b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{128 d^{9/2} f^{9/2}} \] Input:

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

Output:

(Sqrt[c + d*x]*Sqrt[e + f*x]*(10*a*d*f*(C*(15*c^3*f^3 - c^2*d*f^2*(7*e + 1 
0*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)))) + b 
*(C*(-105*c^4*f^4 + 10*c^3*d*f^3*(4*e + 7*f*x) - 2*c^2*d^2*f^2*(-17*e^2 + 
11*e*f*x + 28*f^2*x^2) + 2*c*d^3*f*(20*e^3 - 11*e^2*f*x + 8*e*f^2*x^2 + 24 
*f^3*x^3) + d^4*(-105*e^4 + 70*e^3*f*x - 56*e^2*f^2*x^2 + 48*e*f^3*x^3 + 3 
84*f^4*x^4)) + 10*d*f*(8*A*d*f*(-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)) + B*(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))))))/(1920*d^4*f^4) + ((d*e - c*f)^2*(-2*a*d*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))) + b*(C*(7 
*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d* 
e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2))))*ArcTanh[(Sqrt[d]*Sqrt[ 
e + f*x])/(Sqrt[f]*Sqrt[c + d*x])])/(128*d^(9/2)*f^(9/2))
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.64, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {2118, 27, 164, 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.

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

\(\Big \downarrow \) 2118

\(\displaystyle \frac {\int -\frac {1}{2} b (a+b x) \sqrt {c+d x} \sqrt {e+f x} (4 b c C e+3 a C d e+3 a c C f-10 A b d f-(10 b B d f-6 a C d f-7 b C (d e+c f)) x)dx}{5 b^2 d f}+\frac {C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac {\int (a+b x) \sqrt {c+d x} \sqrt {e+f x} (4 b c C e+3 a C d e+3 a c C f-10 A b d f-(10 b B d f-6 a C d f-7 b C (d e+c f)) x)dx}{10 b d f}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac {\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-6 b d f x (-6 a C d f+10 b B d f-7 b C (c f+d e))-10 a b d f (8 B d f-5 C (c f+d e))-\left (b^2 \left (10 d f (8 A d f-5 B (c f+d e))+C \left (35 c^2 f^2+38 c d e f+35 d^2 e^2\right )\right )\right )\right )}{24 d^2 f^2}-\frac {5 b \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (7 c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right ) \int \sqrt {c+d x} \sqrt {e+f x}dx}{16 d^2 f^2}}{10 b d f}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac {\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-6 b d f x (-6 a C d f+10 b B d f-7 b C (c f+d e))-10 a b d f (8 B d f-5 C (c f+d e))-\left (b^2 \left (10 d f (8 A d f-5 B (c f+d e))+C \left (35 c^2 f^2+38 c d e f+35 d^2 e^2\right )\right )\right )\right )}{24 d^2 f^2}-\frac {5 b \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (7 c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right ) \left (\frac {(d e-c f) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x}}dx}{4 d}+\frac {(c+d x)^{3/2} \sqrt {e+f x}}{2 d}\right )}{16 d^2 f^2}}{10 b d f}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac {\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-6 b d f x (-6 a C d f+10 b B d f-7 b C (c f+d e))-10 a b d f (8 B d f-5 C (c f+d e))-\left (b^2 \left (10 d f (8 A d f-5 B (c f+d e))+C \left (35 c^2 f^2+38 c d e f+35 d^2 e^2\right )\right )\right )\right )}{24 d^2 f^2}-\frac {5 b \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (7 c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right ) \left (\frac {(d e-c f) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{2 f}\right )}{4 d}+\frac {(c+d x)^{3/2} \sqrt {e+f x}}{2 d}\right )}{16 d^2 f^2}}{10 b d f}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac {\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-6 b d f x (-6 a C d f+10 b B d f-7 b C (c f+d e))-10 a b d f (8 B d f-5 C (c f+d e))-\left (b^2 \left (10 d f (8 A d f-5 B (c f+d e))+C \left (35 c^2 f^2+38 c d e f+35 d^2 e^2\right )\right )\right )\right )}{24 d^2 f^2}-\frac {5 b \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (7 c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right ) \left (\frac {(d e-c f) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{f}-\frac {(d e-c f) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{f}\right )}{4 d}+\frac {(c+d x)^{3/2} \sqrt {e+f x}}{2 d}\right )}{16 d^2 f^2}}{10 b d f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac {\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-6 b d f x (-6 a C d f+10 b B d f-7 b C (c f+d e))-10 a b d f (8 B d f-5 C (c f+d e))-\left (b^2 \left (10 d f (8 A d f-5 B (c f+d e))+C \left (35 c^2 f^2+38 c d e f+35 d^2 e^2\right )\right )\right )\right )}{24 d^2 f^2}-\frac {5 b \left (\frac {(d e-c f) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{\sqrt {d} f^{3/2}}\right )}{4 d}+\frac {(c+d x)^{3/2} \sqrt {e+f x}}{2 d}\right ) \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (7 c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right )}{16 d^2 f^2}}{10 b d f}\)

Input:

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

Output:

(C*(a + b*x)^2*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(5*b*d*f) - (((c + d*x)^(3 
/2)*(e + f*x)^(3/2)*(48*a^2*C*d^2*f^2 - 10*a*b*d*f*(8*B*d*f - 5*C*(d*e + c 
*f)) - b^2*(C*(35*d^2*e^2 + 38*c*d*e*f + 35*c^2*f^2) + 10*d*f*(8*A*d*f - 5 
*B*(d*e + c*f))) - 6*b*d*f*(10*b*B*d*f - 6*a*C*d*f - 7*b*C*(d*e + c*f))*x) 
)/(24*d^2*f^2) - (5*b*(2*a*d*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))) - b*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d 
*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e* 
f + 5*c^2*f^2))))*(((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)))/(16*d^2*f^2))/(1 
0*b*d*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 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3024\) vs. \(2(671)=1342\).

Time = 0.57 (sec) , antiderivative size = 3025, normalized size of antiderivative = 4.23

method	result	size
default	\(\text {Expression too large to display}\)	\(3025\)

Input:

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

Output:

-1/3840*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(200*B*((f*x+e)*(d*x+c))^(1/2)*(d*f)^( 
1/2)*b*c^2*d^2*f^4*x-320*A*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)*b*c*d^3*f^4 
*x-320*A*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)*b*d^4*e*f^3*x-320*B*((f*x+e)* 
(d*x+c))^(1/2)*(d*f)^(1/2)*a*c*d^3*f^4*x+200*B*((f*x+e)*(d*x+c))^(1/2)*(d* 
f)^(1/2)*b*d^4*e^2*f^2*x-80*C*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)*b*c^3*d* 
e*f^3+140*C*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)*a*c*d^3*e^2*f^2-320*B*((f* 
x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)*a*d^4*e*f^3*x-68*C*((f*x+e)*(d*x+c))^(1/2) 
*(d*f)^(1/2)*b*c^2*d^2*e^2*f^2-32*C*b*c*d^3*e*f^3*x^2*((f*x+e)*(d*x+c))^(1 
/2)*(d*f)^(1/2)-60*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*c^2*d^3*e^2*f^3-120*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*c*d^4*e^3*f^2-96*C*b*c* 
d^3*f^4*x^3*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)-96*C*b*d^4*e*f^3*x^3*((f*x 
+e)*(d*x+c))^(1/2)*(d*f)^(1/2)-160*B*b*c*d^3*f^4*x^2*((f*x+e)*(d*x+c))^(1/ 
2)*(d*f)^(1/2)-160*B*b*d^4*e*f^3*x^2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)-1 
60*C*a*c*d^3*f^4*x^2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+210*C*((f*x+e)*(d 
*x+c))^(1/2)*(d*f)^(1/2)*b*c^4*f^4-120*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*c^3*d^2*e*f^4-960*A*((f*x+e)* 
(d*x+c))^(1/2)*(d*f)^(1/2)*a*c*d^3*f^4-960*A*((f*x+e)*(d*x+c))^(1/2)*(d*f) 
^(1/2)*a*d^4*e*f^3+150*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*d^5*e^4*f+150*C*ln(1/2*(2*d*f*x+2*((f*x+e)...
 

Fricas [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 1620, normalized size of antiderivative = 2.27 \[ \int (a+b x) \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/7680*(15*(7*C*b*d^5*e^5 - 5*(C*b*c*d^4 + 2*(C*a + B*b)*d^5)*e^4*f - 2* 
(C*b*c^2*d^3 - 4*(C*a + B*b)*c*d^4 - 8*(B*a + A*b)*d^5)*e^3*f^2 - 2*(C*b*c 
^3*d^2 + 16*A*a*d^5 - 2*(C*a + B*b)*c^2*d^3 + 8*(B*a + A*b)*c*d^4)*e^2*f^3 
 - (5*C*b*c^4*d - 64*A*a*c*d^4 - 8*(C*a + B*b)*c^3*d^2 + 16*(B*a + A*b)*c^ 
2*d^3)*e*f^4 + (7*C*b*c^5 - 32*A*a*c^2*d^3 - 10*(C*a + B*b)*c^4*d + 16*(B* 
a + A*b)*c^3*d^2)*f^5)*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*(384*C*b*d^5*f^5*x^4 - 105*C*b*d^5*e^4*f + 1 
0*(4*C*b*c*d^4 + 15*(C*a + B*b)*d^5)*e^3*f^2 + 2*(17*C*b*c^2*d^3 - 35*(C*a 
 + B*b)*c*d^4 - 120*(B*a + A*b)*d^5)*e^2*f^3 + 10*(4*C*b*c^3*d^2 + 48*A*a* 
d^5 - 7*(C*a + B*b)*c^2*d^3 + 16*(B*a + A*b)*c*d^4)*e*f^4 - 15*(7*C*b*c^4* 
d - 32*A*a*c*d^4 - 10*(C*a + B*b)*c^3*d^2 + 16*(B*a + A*b)*c^2*d^3)*f^5 + 
48*(C*b*d^5*e*f^4 + (C*b*c*d^4 + 10*(C*a + B*b)*d^5)*f^5)*x^3 - 8*(7*C*b*d 
^5*e^2*f^3 - 2*(C*b*c*d^4 + 5*(C*a + B*b)*d^5)*e*f^4 + (7*C*b*c^2*d^3 - 10 
*(C*a + B*b)*c*d^4 - 80*(B*a + A*b)*d^5)*f^5)*x^2 + 2*(35*C*b*d^5*e^3*f^2 
- (11*C*b*c*d^4 + 50*(C*a + B*b)*d^5)*e^2*f^3 - (11*C*b*c^2*d^3 - 20*(C*a 
+ B*b)*c*d^4 - 80*(B*a + A*b)*d^5)*e*f^4 + 5*(7*C*b*c^3*d^2 + 96*A*a*d^5 - 
 10*(C*a + B*b)*c^2*d^3 + 16*(B*a + A*b)*c*d^4)*f^5)*x)*sqrt(d*x + c)*sqrt 
(f*x + e))/(d^5*f^5), -1/3840*(15*(7*C*b*d^5*e^5 - 5*(C*b*c*d^4 + 2*(C*a + 
 B*b)*d^5)*e^4*f - 2*(C*b*c^2*d^3 - 4*(C*a + B*b)*c*d^4 - 8*(B*a + A*b)...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)*(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. 2481 vs. \(2 (671) = 1342\).

Time = 0.45 (sec) , antiderivative size = 2481, normalized size of antiderivative = 3.47 \[ \int (a+b x) \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/1920*(1920*((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*a*c*abs(d)/d^2 - 10*(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^11*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*b*c* 
abs(d)/d^2 - 10*(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^11*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*a*abs(d)/d - 10*(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^11*f^6)/(d^14*f^6))*sqrt(d*x + c) + 3*(5*d^4*...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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