\(\int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 (A+B x+C x^2) \, dx\) [45]

Optimal result

Integrand size = 40, antiderivative size = 554 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx=\frac {\left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) x \sqrt {a+b x} \sqrt {a c-b c x}}{16 b^4}-\frac {\left (424 a^4 C f^3+280 b^4 e^2 (B e+3 A f)-105 a^3 b f^2 (3 C e+B f)-210 a b^3 e \left (C e^2+3 f (B e+A f)\right )+112 a^2 b^2 f \left (3 C e^2+f (3 B e+A f)\right )\right ) (a+b x)^{3/2} (a c-b c x)^{3/2}}{840 b^6 c}+\frac {f \left (26 a^2 C f^2-7 b^2 \left (3 C e^2+f (3 B e+A f)\right )\right ) x^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}{35 b^4 c}-\frac {f^2 (21 b C e+7 b B f-24 a C f) x^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}{42 b^3 c}+\frac {\left (32 a^3 C f^3-7 a^2 b f^2 (3 C e+B f)-14 b^3 \left (C e^3+3 e f (B e+A f)\right )\right ) (a+b x)^{5/2} (a c-b c x)^{3/2}}{56 b^6 c}-\frac {C f^3 (a+b x)^{11/2} (a c-b c x)^{3/2}}{7 b^6 c}+\frac {a^2 \sqrt {c} \left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{8 b^5} \] Output:

1/16*(a^4*f^2*(B*f+3*C*e)+2*a^2*b^2*e^2*(3*B*f+C*e)+A*(6*a^2*b^2*e*f^2+8*b 
^4*e^3))*x*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^4-1/840*(424*a^4*C*f^3+280*b 
^4*e^2*(3*A*f+B*e)-105*a^3*b*f^2*(B*f+3*C*e)-210*a*b^3*e*(C*e^2+3*f*(A*f+B 
*e))+112*a^2*b^2*f*(3*C*e^2+f*(A*f+3*B*e)))*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/ 
2)/b^6/c+1/35*f*(26*a^2*C*f^2-7*b^2*(3*C*e^2+f*(A*f+3*B*e)))*x^2*(b*x+a)^( 
3/2)*(-b*c*x+a*c)^(3/2)/b^4/c-1/42*f^2*(7*B*b*f-24*C*a*f+21*C*b*e)*x^3*(b* 
x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/b^3/c+1/56*(32*a^3*C*f^3-7*a^2*b*f^2*(B*f+3* 
C*e)-14*b^3*(C*e^3+3*e*f*(A*f+B*e)))*(b*x+a)^(5/2)*(-b*c*x+a*c)^(3/2)/b^6/ 
c-1/7*C*f^3*(b*x+a)^(11/2)*(-b*c*x+a*c)^(3/2)/b^6/c+1/8*a^2*c^(1/2)*(a^4*f 
^2*(B*f+3*C*e)+2*a^2*b^2*e^2*(3*B*f+C*e)+A*(6*a^2*b^2*e*f^2+8*b^4*e^3))*ar 
ctan(c^(1/2)*(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2))/b^5
 

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.73 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {c (a-b x)} \left (-\sqrt {a-b x} \sqrt {a+b x} \left (128 a^6 C f^3+a^4 b^2 f \left (7 f (96 B e+32 A f+15 B f x)+C \left (672 e^2+315 e f x+64 f^2 x^2\right )\right )+2 a^2 b^4 \left (7 A f \left (120 e^2+45 e f x+8 f^2 x^2\right )+7 B \left (40 e^3+45 e^2 f x+24 e f^2 x^2+5 f^3 x^3\right )+3 C x \left (35 e^3+56 e^2 f x+35 e f^2 x^2+8 f^3 x^3\right )\right )-4 b^6 x \left (21 A \left (10 e^3+20 e^2 f x+15 e f^2 x^2+4 f^3 x^3\right )+x \left (7 B \left (20 e^3+45 e^2 f x+36 e f^2 x^2+10 f^3 x^3\right )+3 C x \left (35 e^3+84 e^2 f x+70 e f^2 x^2+20 f^3 x^3\right )\right )\right )\right )+210 a^2 b \left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{1680 b^6 \sqrt {a-b x}} \] Input:

Integrate[Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^3*(A + B*x + C*x^2),x]
 

Output:

(Sqrt[c*(a - b*x)]*(-(Sqrt[a - b*x]*Sqrt[a + b*x]*(128*a^6*C*f^3 + a^4*b^2 
*f*(7*f*(96*B*e + 32*A*f + 15*B*f*x) + C*(672*e^2 + 315*e*f*x + 64*f^2*x^2 
)) + 2*a^2*b^4*(7*A*f*(120*e^2 + 45*e*f*x + 8*f^2*x^2) + 7*B*(40*e^3 + 45* 
e^2*f*x + 24*e*f^2*x^2 + 5*f^3*x^3) + 3*C*x*(35*e^3 + 56*e^2*f*x + 35*e*f^ 
2*x^2 + 8*f^3*x^3)) - 4*b^6*x*(21*A*(10*e^3 + 20*e^2*f*x + 15*e*f^2*x^2 + 
4*f^3*x^3) + x*(7*B*(20*e^3 + 45*e^2*f*x + 36*e*f^2*x^2 + 10*f^3*x^3) + 3* 
C*x*(35*e^3 + 84*e^2*f*x + 70*e*f^2*x^2 + 20*f^3*x^3))))) + 210*a^2*b*(a^4 
*f^2*(3*C*e + B*f) + 2*a^2*b^2*e^2*(C*e + 3*B*f) + A*(8*b^4*e^3 + 6*a^2*b^ 
2*e*f^2))*ArcTan[Sqrt[a + b*x]/Sqrt[a - b*x]]))/(1680*b^6*Sqrt[a - b*x])
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 512, normalized size of antiderivative = 0.92, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2113, 2185, 25, 27, 687, 27, 687, 25, 27, 676, 211, 224, 216}

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.

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

\(\Big \downarrow \) 2113

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 687

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {(e+f x)^3 \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-7 B f)}{6 c}-\frac {\int -3 b^2 c (e+f x)^2 \left (f \left ((5 C e+7 B f) a^2+14 A b^2 e\right )+\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) x\right ) \sqrt {a^2 c-b^2 c x^2}dx}{6 b^2 c}}{7 b^2 f}-\frac {C (e+f x)^4 \left (a^2 c-b^2 c x^2\right )^{3/2}}{7 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {1}{2} \int (e+f x)^2 \left (f \left ((5 C e+7 B f) a^2+14 A b^2 e\right )+\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) x\right ) \sqrt {a^2 c-b^2 c x^2}dx+\frac {(e+f x)^3 \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-7 B f)}{6 c}}{7 b^2 f}-\frac {C (e+f x)^4 \left (a^2 c-b^2 c x^2\right )^{3/2}}{7 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {1}{2} \left (-\frac {\int -c (e+f x) \left (\left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x b^2+f \left (16 C f^2 a^4+b^2 e (19 C e+49 B f) a^2+14 A \left (5 e^2 b^4+2 a^2 f^2 b^2\right )\right )\right ) \sqrt {a^2 c-b^2 c x^2}dx}{5 b^2 c}-\frac {(e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (2 A f+B e)\right )\right )}{5 b^2 c}\right )+\frac {(e+f x)^3 \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-7 B f)}{6 c}}{7 b^2 f}-\frac {C (e+f x)^4 \left (a^2 c-b^2 c x^2\right )^{3/2}}{7 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {1}{2} \left (\frac {\int c (e+f x) \left (\left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x b^2+f \left (16 C f^2 a^4+b^2 e (19 C e+49 B f) a^2+14 A \left (5 e^2 b^4+2 a^2 f^2 b^2\right )\right )\right ) \sqrt {a^2 c-b^2 c x^2}dx}{5 b^2 c}-\frac {(e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (2 A f+B e)\right )\right )}{5 b^2 c}\right )+\frac {(e+f x)^3 \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-7 B f)}{6 c}}{7 b^2 f}-\frac {C (e+f x)^4 \left (a^2 c-b^2 c x^2\right )^{3/2}}{7 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {1}{2} \left (\frac {\int (e+f x) \left (\left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x b^2+f \left (16 C f^2 a^4+b^2 e (19 C e+49 B f) a^2+14 A \left (5 e^2 b^4+2 a^2 f^2 b^2\right )\right )\right ) \sqrt {a^2 c-b^2 c x^2}dx}{5 b^2}-\frac {(e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (2 A f+B e)\right )\right )}{5 b^2 c}\right )+\frac {(e+f x)^3 \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-7 B f)}{6 c}}{7 b^2 f}-\frac {C (e+f x)^4 \left (a^2 c-b^2 c x^2\right )^{3/2}}{7 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {1}{2} \left (\frac {\frac {35}{4} f \left (a^4 f^2 (B f+3 C e)+A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+2 a^2 b^2 e^2 (3 B f+C e)\right ) \int \sqrt {a^2 c-b^2 c x^2}dx-\frac {f x \left (a^2 c-b^2 c x^2\right )^{3/2} \left (a^2 f^2 (35 B f+41 C e)-b^2 \left (6 C e^3-14 e f (7 A f+B e)\right )\right )}{4 c}-\frac {2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (7 f (A f+3 B e)+15 C e^2\right )-\left (b^4 \left (3 C e^4-7 e^2 f (12 A f+B e)\right )\right )\right )}{3 b^2 c}}{5 b^2}-\frac {(e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (2 A f+B e)\right )\right )}{5 b^2 c}\right )+\frac {(e+f x)^3 \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-7 B f)}{6 c}}{7 b^2 f}-\frac {C (e+f x)^4 \left (a^2 c-b^2 c x^2\right )^{3/2}}{7 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {1}{2} \left (\frac {\frac {35}{4} f \left (a^4 f^2 (B f+3 C e)+A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+2 a^2 b^2 e^2 (3 B f+C e)\right ) \left (\frac {1}{2} a^2 c \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}}dx+\frac {1}{2} x \sqrt {a^2 c-b^2 c x^2}\right )-\frac {f x \left (a^2 c-b^2 c x^2\right )^{3/2} \left (a^2 f^2 (35 B f+41 C e)-b^2 \left (6 C e^3-14 e f (7 A f+B e)\right )\right )}{4 c}-\frac {2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (7 f (A f+3 B e)+15 C e^2\right )-\left (b^4 \left (3 C e^4-7 e^2 f (12 A f+B e)\right )\right )\right )}{3 b^2 c}}{5 b^2}-\frac {(e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (2 A f+B e)\right )\right )}{5 b^2 c}\right )+\frac {(e+f x)^3 \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-7 B f)}{6 c}}{7 b^2 f}-\frac {C (e+f x)^4 \left (a^2 c-b^2 c x^2\right )^{3/2}}{7 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {1}{2} \left (\frac {\frac {35}{4} f \left (a^4 f^2 (B f+3 C e)+A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+2 a^2 b^2 e^2 (3 B f+C e)\right ) \left (\frac {1}{2} a^2 c \int \frac {1}{\frac {b^2 c x^2}{a^2 c-b^2 c x^2}+1}d\frac {x}{\sqrt {a^2 c-b^2 c x^2}}+\frac {1}{2} x \sqrt {a^2 c-b^2 c x^2}\right )-\frac {f x \left (a^2 c-b^2 c x^2\right )^{3/2} \left (a^2 f^2 (35 B f+41 C e)-b^2 \left (6 C e^3-14 e f (7 A f+B e)\right )\right )}{4 c}-\frac {2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (7 f (A f+3 B e)+15 C e^2\right )-\left (b^4 \left (3 C e^4-7 e^2 f (12 A f+B e)\right )\right )\right )}{3 b^2 c}}{5 b^2}-\frac {(e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (2 A f+B e)\right )\right )}{5 b^2 c}\right )+\frac {(e+f x)^3 \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-7 B f)}{6 c}}{7 b^2 f}-\frac {C (e+f x)^4 \left (a^2 c-b^2 c x^2\right )^{3/2}}{7 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {(e+f x)^3 \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-7 B f)}{6 c}+\frac {1}{2} \left (\frac {-\frac {f x \left (a^2 c-b^2 c x^2\right )^{3/2} \left (a^2 f^2 (35 B f+41 C e)-b^2 \left (6 C e^3-14 e f (7 A f+B e)\right )\right )}{4 c}+\frac {35}{4} f \left (\frac {a^2 \sqrt {c} \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{2 b}+\frac {1}{2} x \sqrt {a^2 c-b^2 c x^2}\right ) \left (a^4 f^2 (B f+3 C e)+A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+2 a^2 b^2 e^2 (3 B f+C e)\right )-\frac {2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (7 f (A f+3 B e)+15 C e^2\right )-\left (b^4 \left (3 C e^4-7 e^2 f (12 A f+B e)\right )\right )\right )}{3 b^2 c}}{5 b^2}-\frac {(e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2} \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (2 A f+B e)\right )\right )}{5 b^2 c}\right )}{7 b^2 f}-\frac {C (e+f x)^4 \left (a^2 c-b^2 c x^2\right )^{3/2}}{7 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

Input:

Int[Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^3*(A + B*x + C*x^2),x]
 

Output:

(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(-1/7*(C*(e + f*x)^4*(a^2*c - b^2*c*x^2)^ 
(3/2))/(b^2*c*f) + (((3*C*e - 7*B*f)*(e + f*x)^3*(a^2*c - b^2*c*x^2)^(3/2) 
)/(6*c) + (-1/5*((8*a^2*C*f^2 - b^2*(3*C*e^2 - 7*f*(B*e + 2*A*f)))*(e + f* 
x)^2*(a^2*c - b^2*c*x^2)^(3/2))/(b^2*c) + ((-2*(8*a^4*C*f^4 + 2*a^2*b^2*f^ 
2*(15*C*e^2 + 7*f*(3*B*e + A*f)) - b^4*(3*C*e^4 - 7*e^2*f*(B*e + 12*A*f))) 
*(a^2*c - b^2*c*x^2)^(3/2))/(3*b^2*c) - (f*(a^2*f^2*(41*C*e + 35*B*f) - b^ 
2*(6*C*e^3 - 14*e*f*(B*e + 7*A*f)))*x*(a^2*c - b^2*c*x^2)^(3/2))/(4*c) + ( 
35*f*(a^4*f^2*(3*C*e + B*f) + 2*a^2*b^2*e^2*(C*e + 3*B*f) + A*(8*b^4*e^3 + 
 6*a^2*b^2*e*f^2))*((x*Sqrt[a^2*c - b^2*c*x^2])/2 + (a^2*Sqrt[c]*ArcTan[(b 
*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(2*b)))/4)/(5*b^2))/2)/(7*b^2*f)))/S 
qrt[a^2*c - b^2*c*x^2]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

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 676

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim 
p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p 
+ 3))/(c*(2*p + 3))   Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g 
, p}, x] &&  !LeQ[p, -1]
 

rule 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 0])
 

rule 2113

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. 
)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ 
m]/(a*c + b*d*x^2)^FracPart[m])   Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a 
*d, 0] && EqQ[m, n] &&  !IntegerQ[m]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {\left (-240 C \,f^{3} x^{6} b^{6}-280 B \,b^{6} f^{3} x^{5}-840 C \,b^{6} e \,f^{2} x^{5}-336 A \,b^{6} f^{3} x^{4}-1008 B \,b^{6} e \,f^{2} x^{4}+48 C \,a^{2} b^{4} f^{3} x^{4}-1008 C \,b^{6} e^{2} f \,x^{4}-1260 A \,b^{6} e \,f^{2} x^{3}+70 B \,a^{2} b^{4} f^{3} x^{3}-1260 B \,b^{6} e^{2} f \,x^{3}+210 C \,a^{2} b^{4} e \,f^{2} x^{3}-420 C \,b^{6} e^{3} x^{3}+112 A \,a^{2} b^{4} f^{3} x^{2}-1680 A \,b^{6} e^{2} f \,x^{2}+336 B \,a^{2} b^{4} e \,f^{2} x^{2}-560 B \,b^{6} e^{3} x^{2}+64 C \,a^{4} b^{2} f^{3} x^{2}+336 C \,a^{2} b^{4} e^{2} f \,x^{2}+630 A \,a^{2} b^{4} e \,f^{2} x -840 A \,b^{6} e^{3} x +105 B \,a^{4} b^{2} f^{3} x +630 B \,a^{2} b^{4} e^{2} f x +315 C \,a^{4} b^{2} e \,f^{2} x +210 C \,a^{2} b^{4} e^{3} x +224 A \,a^{4} b^{2} f^{3}+1680 A \,a^{2} e^{2} f \,b^{4}+672 B \,a^{4} b^{2} e \,f^{2}+560 B \,a^{2} e^{3} b^{4}+128 C \,a^{6} f^{3}+672 C \,a^{4} b^{2} e^{2} f \right ) \left (-b x +a \right ) \sqrt {b x +a}\, c}{1680 b^{6} \sqrt {-c \left (b x -a \right )}}+\frac {a^{2} \left (6 A \,a^{2} b^{2} e \,f^{2}+8 A \,b^{4} e^{3}+B \,a^{4} f^{3}+6 B \,a^{2} e^{2} f \,b^{2}+3 C \,a^{4} e \,f^{2}+2 C \,a^{2} e^{3} b^{2}\right ) \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c}{16 b^{4} \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(575\)
default	\(\text {Expression too large to display}\)	\(1370\)

Input:

int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(f*x+e)^3*(C*x^2+B*x+A),x,method=_RET 
URNVERBOSE)
 

Output:

-1/1680*(-240*C*b^6*f^3*x^6-280*B*b^6*f^3*x^5-840*C*b^6*e*f^2*x^5-336*A*b^ 
6*f^3*x^4-1008*B*b^6*e*f^2*x^4+48*C*a^2*b^4*f^3*x^4-1008*C*b^6*e^2*f*x^4-1 
260*A*b^6*e*f^2*x^3+70*B*a^2*b^4*f^3*x^3-1260*B*b^6*e^2*f*x^3+210*C*a^2*b^ 
4*e*f^2*x^3-420*C*b^6*e^3*x^3+112*A*a^2*b^4*f^3*x^2-1680*A*b^6*e^2*f*x^2+3 
36*B*a^2*b^4*e*f^2*x^2-560*B*b^6*e^3*x^2+64*C*a^4*b^2*f^3*x^2+336*C*a^2*b^ 
4*e^2*f*x^2+630*A*a^2*b^4*e*f^2*x-840*A*b^6*e^3*x+105*B*a^4*b^2*f^3*x+630* 
B*a^2*b^4*e^2*f*x+315*C*a^4*b^2*e*f^2*x+210*C*a^2*b^4*e^3*x+224*A*a^4*b^2* 
f^3+1680*A*a^2*b^4*e^2*f+672*B*a^4*b^2*e*f^2+560*B*a^2*b^4*e^3+128*C*a^6*f 
^3+672*C*a^4*b^2*e^2*f)/b^6*(-b*x+a)*(b*x+a)^(1/2)/(-c*(b*x-a))^(1/2)*c+1/ 
16*a^2/b^4*(6*A*a^2*b^2*e*f^2+8*A*b^4*e^3+B*a^4*f^3+6*B*a^2*b^2*e^2*f+3*C* 
a^4*e*f^2+2*C*a^2*b^2*e^3)/(b^2*c)^(1/2)*arctan((b^2*c)^(1/2)*x/(-b^2*c*x^ 
2+a^2*c)^(1/2))*(-(b*x+a)*c*(b*x-a))^(1/2)/(b*x+a)^(1/2)/(-c*(b*x-a))^(1/2 
)*c
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 1001, normalized size of antiderivative = 1.81 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(f*x+e)^3*(C*x^2+B*x+A),x, algo 
rithm="fricas")
 

Output:

[1/3360*(105*(6*B*a^4*b^3*e^2*f + B*a^6*b*f^3 + 2*(C*a^4*b^3 + 4*A*a^2*b^5 
)*e^3 + 3*(C*a^6*b + 2*A*a^4*b^3)*e*f^2)*sqrt(-c)*log(2*b^2*c*x^2 + 2*sqrt 
(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(-c)*x - a^2*c) + 2*(240*C*b^6*f^3*x^6 
- 560*B*a^2*b^4*e^3 - 672*B*a^4*b^2*e*f^2 + 280*(3*C*b^6*e*f^2 + B*b^6*f^3 
)*x^5 + 48*(21*C*b^6*e^2*f + 21*B*b^6*e*f^2 - (C*a^2*b^4 - 7*A*b^6)*f^3)*x 
^4 - 336*(2*C*a^4*b^2 + 5*A*a^2*b^4)*e^2*f - 32*(4*C*a^6 + 7*A*a^4*b^2)*f^ 
3 + 70*(6*C*b^6*e^3 + 18*B*b^6*e^2*f - B*a^2*b^4*f^3 - 3*(C*a^2*b^4 - 6*A* 
b^6)*e*f^2)*x^3 + 16*(35*B*b^6*e^3 - 21*B*a^2*b^4*e*f^2 - 21*(C*a^2*b^4 - 
5*A*b^6)*e^2*f - (4*C*a^4*b^2 + 7*A*a^2*b^4)*f^3)*x^2 - 105*(6*B*a^2*b^4*e 
^2*f + B*a^4*b^2*f^3 + 2*(C*a^2*b^4 - 4*A*b^6)*e^3 + 3*(C*a^4*b^2 + 2*A*a^ 
2*b^4)*e*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^6, -1/1680*(105*(6*B* 
a^4*b^3*e^2*f + B*a^6*b*f^3 + 2*(C*a^4*b^3 + 4*A*a^2*b^5)*e^3 + 3*(C*a^6*b 
 + 2*A*a^4*b^3)*e*f^2)*sqrt(c)*arctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*s 
qrt(c)*x/(b^2*c*x^2 - a^2*c)) - (240*C*b^6*f^3*x^6 - 560*B*a^2*b^4*e^3 - 6 
72*B*a^4*b^2*e*f^2 + 280*(3*C*b^6*e*f^2 + B*b^6*f^3)*x^5 + 48*(21*C*b^6*e^ 
2*f + 21*B*b^6*e*f^2 - (C*a^2*b^4 - 7*A*b^6)*f^3)*x^4 - 336*(2*C*a^4*b^2 + 
 5*A*a^2*b^4)*e^2*f - 32*(4*C*a^6 + 7*A*a^4*b^2)*f^3 + 70*(6*C*b^6*e^3 + 1 
8*B*b^6*e^2*f - B*a^2*b^4*f^3 - 3*(C*a^2*b^4 - 6*A*b^6)*e*f^2)*x^3 + 16*(3 
5*B*b^6*e^3 - 21*B*a^2*b^4*e*f^2 - 21*(C*a^2*b^4 - 5*A*b^6)*e^2*f - (4*C*a 
^4*b^2 + 7*A*a^2*b^4)*f^3)*x^2 - 105*(6*B*a^2*b^4*e^2*f + B*a^4*b^2*f^3...
 

Sympy [F]

\[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx=\int \sqrt {- c \left (- a + b x\right )} \sqrt {a + b x} \left (e + f x\right )^{3} \left (A + B x + C x^{2}\right )\, dx \] Input:

integrate((b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2)*(f*x+e)**3*(C*x**2+B*x+A),x)
 

Output:

Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x)*(e + f*x)**3*(A + B*x + C*x**2) 
, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.05 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx=-\frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C f^{3} x^{4}}{7 \, b^{2} c} + \frac {A a^{2} \sqrt {c} e^{3} \arcsin \left (\frac {b x}{a}\right )}{2 \, b} + \frac {1}{2} \, \sqrt {-b^{2} c x^{2} + a^{2} c} A e^{3} x - \frac {4 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C a^{2} f^{3} x^{2}}{35 \, b^{4} c} + \frac {{\left (3 \, C e f^{2} + B f^{3}\right )} a^{6} \sqrt {c} \arcsin \left (\frac {b x}{a}\right )}{16 \, b^{5}} + \frac {{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} a^{4} \sqrt {c} \arcsin \left (\frac {b x}{a}\right )}{8 \, b^{3}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} B e^{3}}{3 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} A e^{2} f}{b^{2} c} - \frac {8 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C a^{4} f^{3}}{105 \, b^{6} c} + \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e f^{2} + B f^{3}\right )} a^{4} x}{16 \, b^{4}} + \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} a^{2} x}{8 \, b^{2}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (3 \, C e f^{2} + B f^{3}\right )} x^{3}}{6 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} x^{2}}{5 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (3 \, C e f^{2} + B f^{3}\right )} a^{2} x}{8 \, b^{4} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} x}{4 \, b^{2} c} - \frac {2 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} a^{2}}{15 \, b^{4} c} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(f*x+e)^3*(C*x^2+B*x+A),x, algo 
rithm="maxima")
 

Output:

-1/7*(-b^2*c*x^2 + a^2*c)^(3/2)*C*f^3*x^4/(b^2*c) + 1/2*A*a^2*sqrt(c)*e^3* 
arcsin(b*x/a)/b + 1/2*sqrt(-b^2*c*x^2 + a^2*c)*A*e^3*x - 4/35*(-b^2*c*x^2 
+ a^2*c)^(3/2)*C*a^2*f^3*x^2/(b^4*c) + 1/16*(3*C*e*f^2 + B*f^3)*a^6*sqrt(c 
)*arcsin(b*x/a)/b^5 + 1/8*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*a^4*sqrt(c)*arcs 
in(b*x/a)/b^3 - 1/3*(-b^2*c*x^2 + a^2*c)^(3/2)*B*e^3/(b^2*c) - (-b^2*c*x^2 
 + a^2*c)^(3/2)*A*e^2*f/(b^2*c) - 8/105*(-b^2*c*x^2 + a^2*c)^(3/2)*C*a^4*f 
^3/(b^6*c) + 1/16*sqrt(-b^2*c*x^2 + a^2*c)*(3*C*e*f^2 + B*f^3)*a^4*x/b^4 + 
 1/8*sqrt(-b^2*c*x^2 + a^2*c)*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*a^2*x/b^2 - 
1/6*(-b^2*c*x^2 + a^2*c)^(3/2)*(3*C*e*f^2 + B*f^3)*x^3/(b^2*c) - 1/5*(-b^2 
*c*x^2 + a^2*c)^(3/2)*(3*C*e^2*f + 3*B*e*f^2 + A*f^3)*x^2/(b^2*c) - 1/8*(- 
b^2*c*x^2 + a^2*c)^(3/2)*(3*C*e*f^2 + B*f^3)*a^2*x/(b^4*c) - 1/4*(-b^2*c*x 
^2 + a^2*c)^(3/2)*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*x/(b^2*c) - 2/15*(-b^2*c 
*x^2 + a^2*c)^(3/2)*(3*C*e^2*f + 3*B*e*f^2 + A*f^3)*a^2/(b^4*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2671 vs. \(2 (510) = 1020\).

Time = 2.19 (sec) , antiderivative size = 2671, normalized size of antiderivative = 4.82 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(f*x+e)^3*(C*x^2+B*x+A),x, algo 
rithm="giac")
 

Output:

-1/1680*(1680*(2*a*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 
 2*a*c)))/sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*A*a*b^5*e^3 
 - 840*(2*a^2*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a* 
c)))/sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*(b*x - 2*a))*B*a* 
b^4*e^3 - 840*(2*a^2*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c 
 + 2*a*c)))/sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*(b*x - 2*a 
))*A*b^5*e^3 - 2520*(2*a^2*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x 
+ a)*c + 2*a*c)))/sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*(b*x 
 - 2*a))*A*a*b^4*e^2*f + 280*(6*a^3*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sq 
rt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - ((2*b*x - 5*a)*(b*x + a) + 9*a^2)*sq 
rt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*C*a*b^3*e^3 + 280*(6*a^3*c*log(abs 
(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - ((2*b*x 
 - 5*a)*(b*x + a) + 9*a^2)*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*B*b^4 
*e^3 + 840*(6*a^3*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 
2*a*c)))/sqrt(-c) - ((2*b*x - 5*a)*(b*x + a) + 9*a^2)*sqrt(-(b*x + a)*c + 
2*a*c)*sqrt(b*x + a))*B*a*b^3*e^2*f + 840*(6*a^3*c*log(abs(-sqrt(b*x + a)* 
sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - ((2*b*x - 5*a)*(b*x + a 
) + 9*a^2)*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*A*b^4*e^2*f + 840*(6* 
a^3*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt( 
-c) - ((2*b*x - 5*a)*(b*x + a) + 9*a^2)*sqrt(-(b*x + a)*c + 2*a*c)*sqrt...
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx=\text {Hanged} \] Input:

int((e + f*x)^3*(a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)*(A + B*x + C*x^2),x)
 

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 951, normalized size of antiderivative = 1.72 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(f*x+e)^3*(C*x^2+B*x+A),x)
 

Output:

(sqrt(c)*( - 210*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**6*b**2*f**3 - 63 
0*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**6*b*c*e*f**2 - 1260*asin(sqrt(a 
 - b*x)/(sqrt(a)*sqrt(2)))*a**5*b**3*e*f**2 - 1260*asin(sqrt(a - b*x)/(sqr 
t(a)*sqrt(2)))*a**4*b**4*e**2*f - 420*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)) 
)*a**4*b**3*c*e**3 - 1680*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b**5* 
e**3 - 128*sqrt(a + b*x)*sqrt(a - b*x)*a**6*c*f**3 - 224*sqrt(a + b*x)*sqr 
t(a - b*x)*a**5*b**2*f**3 - 672*sqrt(a + b*x)*sqrt(a - b*x)*a**4*b**3*e*f* 
*2 - 105*sqrt(a + b*x)*sqrt(a - b*x)*a**4*b**3*f**3*x - 672*sqrt(a + b*x)* 
sqrt(a - b*x)*a**4*b**2*c*e**2*f - 315*sqrt(a + b*x)*sqrt(a - b*x)*a**4*b* 
*2*c*e*f**2*x - 64*sqrt(a + b*x)*sqrt(a - b*x)*a**4*b**2*c*f**3*x**2 - 168 
0*sqrt(a + b*x)*sqrt(a - b*x)*a**3*b**4*e**2*f - 630*sqrt(a + b*x)*sqrt(a 
- b*x)*a**3*b**4*e*f**2*x - 112*sqrt(a + b*x)*sqrt(a - b*x)*a**3*b**4*f**3 
*x**2 - 560*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b**5*e**3 - 630*sqrt(a + b*x) 
*sqrt(a - b*x)*a**2*b**5*e**2*f*x - 336*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b 
**5*e*f**2*x**2 - 70*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b**5*f**3*x**3 - 210 
*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b**4*c*e**3*x - 336*sqrt(a + b*x)*sqrt(a 
 - b*x)*a**2*b**4*c*e**2*f*x**2 - 210*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b** 
4*c*e*f**2*x**3 - 48*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b**4*c*f**3*x**4 + 8 
40*sqrt(a + b*x)*sqrt(a - b*x)*a*b**6*e**3*x + 1680*sqrt(a + b*x)*sqrt(a - 
 b*x)*a*b**6*e**2*f*x**2 + 1260*sqrt(a + b*x)*sqrt(a - b*x)*a*b**6*e*f*...