\(\int \frac {(g+h x)^3 (d+e x+f x^2)}{\sqrt {a+b x+c x^2}} \, dx\) [63]

Optimal result

Integrand size = 32, antiderivative size = 674 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\sqrt {a+b x+c x^2}} \, dx=\frac {\left (60 c e g-48 b f g-\frac {12 c f g^2}{h}+80 c d h-70 b e h-64 a f h+\frac {63 b^2 f h}{c}\right ) (g+h x)^2 \sqrt {a+b x+c x^2}}{240 c^2}+\frac {\left (10 c e-9 b f-\frac {2 c f g}{h}\right ) (g+h x)^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {f (g+h x)^4 \sqrt {a+b x+c x^2}}{5 c h}+\frac {\left (945 b^4 f h^3-\frac {64 c^4 \left (3 f g^4-5 g^2 h (3 e g+16 d h)\right )}{h}-210 b^2 c h^2 (14 a f h+5 b (3 f g+e h))+8 c^2 h \left (128 a^2 f h^2+275 a b h (3 f g+e h)+3 b^2 \left (129 f g^2+50 h (3 e g+d h)\right )\right )-16 c^3 \left (16 a h \left (13 f g^2+5 h (3 e g+d h)\right )+b g \left (39 f g^2+5 h (47 e g+54 d h)\right )\right )-2 c \left (315 b^3 f h^3-14 b c h^2 (39 b f g+25 b e h+46 a f h)+16 c^3 \left (3 f g^3-5 g h (3 e g+10 d h)\right )+8 c^2 h \left (21 b f g^2+10 b h (8 e g+5 d h)+a h (71 f g+45 e h)\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}+\frac {\left (256 c^5 d g^3-63 b^5 f h^3+70 b^3 c h^2 (3 b f g+b e h+4 a f h)-128 c^4 g \left (a f g^2+3 a h (e g+d h)+b g (e g+3 d h)\right )-80 b c^2 h \left (3 a^2 f h^2+3 a b h (3 f g+e h)+b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )+96 c^3 \left (a^2 h^2 (3 f g+e h)+b^2 g \left (f g^2+3 h (e g+d h)\right )+2 a b h \left (3 f g^2+h (3 e g+d h)\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}} \] Output:

1/240*(60*c*e*g-48*b*f*g-12*c*f*g^2/h+80*c*d*h-70*b*e*h-64*a*f*h+63*b^2*f* 
h/c)*(h*x+g)^2*(c*x^2+b*x+a)^(1/2)/c^2+1/40*(10*c*e-9*b*f-2*c*f*g/h)*(h*x+ 
g)^3*(c*x^2+b*x+a)^(1/2)/c^2+1/5*f*(h*x+g)^4*(c*x^2+b*x+a)^(1/2)/c/h+1/192 
0*(945*b^4*f*h^3-64*c^4*(3*f*g^4-5*g^2*h*(16*d*h+3*e*g))/h-210*b^2*c*h^2*( 
14*a*f*h+5*b*(e*h+3*f*g))+8*c^2*h*(128*a^2*f*h^2+275*a*b*h*(e*h+3*f*g)+3*b 
^2*(129*f*g^2+50*h*(d*h+3*e*g)))-16*c^3*(16*a*h*(13*f*g^2+5*h*(d*h+3*e*g)) 
+b*g*(39*f*g^2+5*h*(54*d*h+47*e*g)))-2*c*(315*b^3*f*h^3-14*b*c*h^2*(46*a*f 
*h+25*b*e*h+39*b*f*g)+16*c^3*(3*f*g^3-5*g*h*(10*d*h+3*e*g))+8*c^2*h*(21*b* 
f*g^2+10*b*h*(5*d*h+8*e*g)+a*h*(45*e*h+71*f*g)))*x)*(c*x^2+b*x+a)^(1/2)/c^ 
5+1/256*(256*c^5*d*g^3-63*b^5*f*h^3+70*b^3*c*h^2*(4*a*f*h+b*e*h+3*b*f*g)-1 
28*c^4*g*(a*f*g^2+3*a*h*(d*h+e*g)+b*g*(3*d*h+e*g))-80*b*c^2*h*(3*a^2*f*h^2 
+3*a*b*h*(e*h+3*f*g)+b^2*(d*h^2+3*e*g*h+3*f*g^2))+96*c^3*(a^2*h^2*(e*h+3*f 
*g)+b^2*g*(f*g^2+3*h*(d*h+e*g))+2*a*b*h*(3*f*g^2+h*(d*h+3*e*g))))*arctanh( 
1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)
 

Mathematica [A] (verified)

Time = 4.38 (sec) , antiderivative size = 588, normalized size of antiderivative = 0.87 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^4 f h^3-210 b^2 c h^2 (5 b e h+14 a f h+3 b f (5 g+h x))+32 c^4 \left (10 d h \left (18 g^2+9 g h x+2 h^2 x^2\right )+15 e \left (4 g^3+6 g^2 h x+4 g h^2 x^2+h^3 x^3\right )+3 f x \left (10 g^3+20 g^2 h x+15 g h^2 x^2+4 h^3 x^3\right )\right )+4 c^2 h \left (256 a^2 f h^2+2 a b h (825 f g+275 e h+161 f h x)+b^2 \left (25 h (36 e g+12 d h+7 e h x)+3 f \left (300 g^2+175 g h x+42 h^2 x^2\right )\right )\right )-16 c^3 \left (a h \left (5 h (48 e g+16 d h+9 e h x)+f \left (240 g^2+135 g h x+32 h^2 x^2\right )\right )+b \left (3 f \left (30 g^3+50 g^2 h x+35 g h^2 x^2+9 h^3 x^3\right )+5 h \left (2 d h (27 g+5 h x)+e \left (54 g^2+30 g h x+7 h^2 x^2\right )\right )\right )\right )\right )+15 \left (-256 c^5 d g^3+63 b^5 f h^3-70 b^3 c h^2 (3 b f g+b e h+4 a f h)+128 c^4 g \left (a f g^2+3 a h (e g+d h)+b g (e g+3 d h)\right )+80 b c^2 h \left (3 a^2 f h^2+3 a b h (3 f g+e h)+b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )-96 c^3 \left (a^2 h^2 (3 f g+e h)+b^2 g \left (f g^2+3 h (e g+d h)\right )+2 a b h \left (3 f g^2+h (3 e g+d h)\right )\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{3840 c^{11/2}} \] Input:

Integrate[((g + h*x)^3*(d + e*x + f*x^2))/Sqrt[a + b*x + c*x^2],x]
 

Output:

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^4*f*h^3 - 210*b^2*c*h^2*(5*b*e*h + 
 14*a*f*h + 3*b*f*(5*g + h*x)) + 32*c^4*(10*d*h*(18*g^2 + 9*g*h*x + 2*h^2* 
x^2) + 15*e*(4*g^3 + 6*g^2*h*x + 4*g*h^2*x^2 + h^3*x^3) + 3*f*x*(10*g^3 + 
20*g^2*h*x + 15*g*h^2*x^2 + 4*h^3*x^3)) + 4*c^2*h*(256*a^2*f*h^2 + 2*a*b*h 
*(825*f*g + 275*e*h + 161*f*h*x) + b^2*(25*h*(36*e*g + 12*d*h + 7*e*h*x) + 
 3*f*(300*g^2 + 175*g*h*x + 42*h^2*x^2))) - 16*c^3*(a*h*(5*h*(48*e*g + 16* 
d*h + 9*e*h*x) + f*(240*g^2 + 135*g*h*x + 32*h^2*x^2)) + b*(3*f*(30*g^3 + 
50*g^2*h*x + 35*g*h^2*x^2 + 9*h^3*x^3) + 5*h*(2*d*h*(27*g + 5*h*x) + e*(54 
*g^2 + 30*g*h*x + 7*h^2*x^2))))) + 15*(-256*c^5*d*g^3 + 63*b^5*f*h^3 - 70* 
b^3*c*h^2*(3*b*f*g + b*e*h + 4*a*f*h) + 128*c^4*g*(a*f*g^2 + 3*a*h*(e*g + 
d*h) + b*g*(e*g + 3*d*h)) + 80*b*c^2*h*(3*a^2*f*h^2 + 3*a*b*h*(3*f*g + e*h 
) + b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) - 96*c^3*(a^2*h^2*(3*f*g + e*h) + b^2 
*g*(f*g^2 + 3*h*(e*g + d*h)) + 2*a*b*h*(3*f*g^2 + h*(3*e*g + d*h))))*Log[b 
 + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(3840*c^(11/2))
 

Rubi [A] (verified)

Time = 2.93 (sec) , antiderivative size = 709, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2184, 27, 1236, 27, 1236, 27, 1225, 1092, 219}

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.

\(\displaystyle \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {\int -\frac {h (g+h x)^3 (b f g-10 c d h+8 a f h+(2 c f g-10 c e h+9 b f h) x)}{2 \sqrt {c x^2+b x+a}}dx}{5 c h^2}+\frac {f (g+h x)^4 \sqrt {a+b x+c x^2}}{5 c h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {f (g+h x)^4 \sqrt {a+b x+c x^2}}{5 c h}-\frac {\int \frac {(g+h x)^3 (b f g-10 c d h+8 a f h+(2 c f g-10 c e h+9 b f h) x)}{\sqrt {c x^2+b x+a}}dx}{10 c h}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {f (g+h x)^4 \sqrt {a+b x+c x^2}}{5 c h}-\frac {\frac {\int -\frac {(g+h x)^2 \left (9 f g h b^2+54 a f h^2 b-2 c g (3 f g+5 e h) b+4 c h (20 c d g-13 a f g-15 a e h)+\left (-4 \left (3 f g^2-5 h (3 e g+4 d h)\right ) c^2-2 h (24 b f g+35 b e h+32 a f h) c+63 b^2 f h^2\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{4 c}+\frac {(g+h x)^3 \sqrt {a+b x+c x^2} (9 b f h-10 c e h+2 c f g)}{4 c}}{10 c h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {f (g+h x)^4 \sqrt {a+b x+c x^2}}{5 c h}-\frac {\frac {(g+h x)^3 \sqrt {a+b x+c x^2} (9 b f h-10 c e h+2 c f g)}{4 c}-\frac {\int \frac {(g+h x)^2 \left (9 f g h b^2+54 a f h^2 b-2 c g (3 f g+5 e h) b+4 c h (20 c d g-13 a f g-15 a e h)+\left (-\left (\left (12 f g^2-20 h (3 e g+4 d h)\right ) c^2\right )-2 h (24 b f g+35 b e h+32 a f h) c+63 b^2 f h^2\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{8 c}}{10 c h}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {f (g+h x)^4 \sqrt {a+b x+c x^2}}{5 c h}-\frac {\frac {(g+h x)^3 \sqrt {a+b x+c x^2} (9 b f h-10 c e h+2 c f g)}{4 c}-\frac {\frac {\int -\frac {(g+h x) \left (63 f g h^2 b^3+2 \left (126 a f h^3-c g h (51 f g+35 e h)\right ) b^2+4 c \left (6 c f g^3+10 c h (3 e g+2 d h) g-5 a h^2 (29 f g+14 e h)\right ) b-8 c h \left (60 c^2 d g^2+32 a^2 f h^2-a c \left (33 f g^2+75 e h g+40 d h^2\right )\right )+\left (\left (48 f g^3-80 g h (3 e g+10 d h)\right ) c^3+8 h \left (21 b f g^2+10 b h (8 e g+5 d h)+a h (71 f g+45 e h)\right ) c^2-14 b h^2 (39 b f g+25 b e h+46 a f h) c+315 b^3 f h^3\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {(g+h x)^2 \sqrt {a+b x+c x^2} \left (-2 c h (32 a f h+35 b e h+24 b f g)+63 b^2 f h^2-4 c^2 \left (3 f g^2-5 h (4 d h+3 e g)\right )\right )}{3 c}}{8 c}}{10 c h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {f (g+h x)^4 \sqrt {a+b x+c x^2}}{5 c h}-\frac {\frac {(g+h x)^3 \sqrt {a+b x+c x^2} (9 b f h-10 c e h+2 c f g)}{4 c}-\frac {\frac {(g+h x)^2 \sqrt {a+b x+c x^2} \left (-2 c h (32 a f h+35 b e h+24 b f g)+63 b^2 f h^2-4 c^2 \left (3 f g^2-5 h (4 d h+3 e g)\right )\right )}{3 c}-\frac {\int \frac {(g+h x) \left (63 f g h^2 b^3+2 \left (126 a f h^3-c g h (51 f g+35 e h)\right ) b^2+4 c \left (6 c f g^3+10 c h (3 e g+2 d h) g-5 a h^2 (29 f g+14 e h)\right ) b-8 c h \left (60 c^2 d g^2+32 a^2 f h^2-a c \left (33 f g^2+75 e h g+40 d h^2\right )\right )+\left (\left (48 f g^3-80 g h (3 e g+10 d h)\right ) c^3+8 h \left (21 b f g^2+10 b h (8 e g+5 d h)+a h (71 f g+45 e h)\right ) c^2-14 b h^2 (39 b f g+25 b e h+46 a f h) c+315 b^3 f h^3\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{6 c}}{8 c}}{10 c h}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {f (g+h x)^4 \sqrt {a+b x+c x^2}}{5 c h}-\frac {\frac {(g+h x)^3 \sqrt {a+b x+c x^2} (9 b f h-10 c e h+2 c f g)}{4 c}-\frac {\frac {(g+h x)^2 \sqrt {a+b x+c x^2} \left (-2 c h (32 a f h+35 b e h+24 b f g)+63 b^2 f h^2-4 c^2 \left (3 f g^2-5 h (4 d h+3 e g)\right )\right )}{3 c}-\frac {-\frac {15 h \left (96 c^3 \left (a^2 h^2 (e h+3 f g)+2 a b h \left (h (d h+3 e g)+3 f g^2\right )+b^2 g \left (3 h (d h+e g)+f g^2\right )\right )-80 b c^2 h \left (3 a^2 f h^2+3 a b h (e h+3 f g)+b^2 \left (d h^2+3 e g h+3 f g^2\right )\right )+70 b^3 c h^2 (4 a f h+b e h+3 b f g)-128 c^4 g \left (3 a h (d h+e g)+a f g^2+b g (3 d h+e g)\right )-63 b^5 f h^3+256 c^5 d g^3\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}-\frac {\sqrt {a+b x+c x^2} \left (8 c^2 h^2 \left (128 a^2 f h^2+275 a b h (e h+3 f g)+3 b^2 \left (50 h (d h+3 e g)+129 f g^2\right )\right )-2 c h x \left (8 c^2 h \left (a h (45 e h+71 f g)+10 b h (5 d h+8 e g)+21 b f g^2\right )-14 b c h^2 (46 a f h+25 b e h+39 b f g)+315 b^3 f h^3+c^3 \left (48 f g^3-80 g h (10 d h+3 e g)\right )\right )-210 b^2 c h^3 (14 a f h+5 b (e h+3 f g))-16 c^3 h \left (16 a h \left (5 h (d h+3 e g)+13 f g^2\right )+b g \left (5 h (54 d h+47 e g)+39 f g^2\right )\right )+945 b^4 f h^4-64 c^4 \left (3 f g^4-5 g^2 h (16 d h+3 e g)\right )\right )}{4 c^2}}{6 c}}{8 c}}{10 c h}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {f (g+h x)^4 \sqrt {a+b x+c x^2}}{5 c h}-\frac {\frac {(g+h x)^3 \sqrt {a+b x+c x^2} (9 b f h-10 c e h+2 c f g)}{4 c}-\frac {\frac {(g+h x)^2 \sqrt {a+b x+c x^2} \left (-2 c h (32 a f h+35 b e h+24 b f g)+63 b^2 f h^2-4 c^2 \left (3 f g^2-5 h (4 d h+3 e g)\right )\right )}{3 c}-\frac {-\frac {15 h \left (96 c^3 \left (a^2 h^2 (e h+3 f g)+2 a b h \left (h (d h+3 e g)+3 f g^2\right )+b^2 g \left (3 h (d h+e g)+f g^2\right )\right )-80 b c^2 h \left (3 a^2 f h^2+3 a b h (e h+3 f g)+b^2 \left (d h^2+3 e g h+3 f g^2\right )\right )+70 b^3 c h^2 (4 a f h+b e h+3 b f g)-128 c^4 g \left (3 a h (d h+e g)+a f g^2+b g (3 d h+e g)\right )-63 b^5 f h^3+256 c^5 d g^3\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}-\frac {\sqrt {a+b x+c x^2} \left (8 c^2 h^2 \left (128 a^2 f h^2+275 a b h (e h+3 f g)+3 b^2 \left (50 h (d h+3 e g)+129 f g^2\right )\right )-2 c h x \left (8 c^2 h \left (a h (45 e h+71 f g)+10 b h (5 d h+8 e g)+21 b f g^2\right )-14 b c h^2 (46 a f h+25 b e h+39 b f g)+315 b^3 f h^3+c^3 \left (48 f g^3-80 g h (10 d h+3 e g)\right )\right )-210 b^2 c h^3 (14 a f h+5 b (e h+3 f g))-16 c^3 h \left (16 a h \left (5 h (d h+3 e g)+13 f g^2\right )+b g \left (5 h (54 d h+47 e g)+39 f g^2\right )\right )+945 b^4 f h^4-64 c^4 \left (3 f g^4-5 g^2 h (16 d h+3 e g)\right )\right )}{4 c^2}}{6 c}}{8 c}}{10 c h}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {f (g+h x)^4 \sqrt {a+b x+c x^2}}{5 c h}-\frac {\frac {(g+h x)^3 \sqrt {a+b x+c x^2} (9 b f h-10 c e h+2 c f g)}{4 c}-\frac {\frac {(g+h x)^2 \sqrt {a+b x+c x^2} \left (-2 c h (32 a f h+35 b e h+24 b f g)+63 b^2 f h^2-4 c^2 \left (3 f g^2-5 h (4 d h+3 e g)\right )\right )}{3 c}-\frac {-\frac {15 h \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (96 c^3 \left (a^2 h^2 (e h+3 f g)+2 a b h \left (h (d h+3 e g)+3 f g^2\right )+b^2 g \left (3 h (d h+e g)+f g^2\right )\right )-80 b c^2 h \left (3 a^2 f h^2+3 a b h (e h+3 f g)+b^2 \left (d h^2+3 e g h+3 f g^2\right )\right )+70 b^3 c h^2 (4 a f h+b e h+3 b f g)-128 c^4 g \left (3 a h (d h+e g)+a f g^2+b g (3 d h+e g)\right )-63 b^5 f h^3+256 c^5 d g^3\right )}{8 c^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (8 c^2 h^2 \left (128 a^2 f h^2+275 a b h (e h+3 f g)+3 b^2 \left (50 h (d h+3 e g)+129 f g^2\right )\right )-2 c h x \left (8 c^2 h \left (a h (45 e h+71 f g)+10 b h (5 d h+8 e g)+21 b f g^2\right )-14 b c h^2 (46 a f h+25 b e h+39 b f g)+315 b^3 f h^3+c^3 \left (48 f g^3-80 g h (10 d h+3 e g)\right )\right )-210 b^2 c h^3 (14 a f h+5 b (e h+3 f g))-16 c^3 h \left (16 a h \left (5 h (d h+3 e g)+13 f g^2\right )+b g \left (5 h (54 d h+47 e g)+39 f g^2\right )\right )+945 b^4 f h^4-64 c^4 \left (3 f g^4-5 g^2 h (16 d h+3 e g)\right )\right )}{4 c^2}}{6 c}}{8 c}}{10 c h}\)

Input:

Int[((g + h*x)^3*(d + e*x + f*x^2))/Sqrt[a + b*x + c*x^2],x]
 

Output:

(f*(g + h*x)^4*Sqrt[a + b*x + c*x^2])/(5*c*h) - (((2*c*f*g - 10*c*e*h + 9* 
b*f*h)*(g + h*x)^3*Sqrt[a + b*x + c*x^2])/(4*c) - (((63*b^2*f*h^2 - 2*c*h* 
(24*b*f*g + 35*b*e*h + 32*a*f*h) - 4*c^2*(3*f*g^2 - 5*h*(3*e*g + 4*d*h)))* 
(g + h*x)^2*Sqrt[a + b*x + c*x^2])/(3*c) - (-1/4*((945*b^4*f*h^4 - 64*c^4* 
(3*f*g^4 - 5*g^2*h*(3*e*g + 16*d*h)) - 210*b^2*c*h^3*(14*a*f*h + 5*b*(3*f* 
g + e*h)) + 8*c^2*h^2*(128*a^2*f*h^2 + 275*a*b*h*(3*f*g + e*h) + 3*b^2*(12 
9*f*g^2 + 50*h*(3*e*g + d*h))) - 16*c^3*h*(16*a*h*(13*f*g^2 + 5*h*(3*e*g + 
 d*h)) + b*g*(39*f*g^2 + 5*h*(47*e*g + 54*d*h))) - 2*c*h*(315*b^3*f*h^3 - 
14*b*c*h^2*(39*b*f*g + 25*b*e*h + 46*a*f*h) + c^3*(48*f*g^3 - 80*g*h*(3*e* 
g + 10*d*h)) + 8*c^2*h*(21*b*f*g^2 + 10*b*h*(8*e*g + 5*d*h) + a*h*(71*f*g 
+ 45*e*h)))*x)*Sqrt[a + b*x + c*x^2])/c^2 - (15*h*(256*c^5*d*g^3 - 63*b^5* 
f*h^3 + 70*b^3*c*h^2*(3*b*f*g + b*e*h + 4*a*f*h) - 128*c^4*g*(a*f*g^2 + 3* 
a*h*(e*g + d*h) + b*g*(e*g + 3*d*h)) - 80*b*c^2*h*(3*a^2*f*h^2 + 3*a*b*h*( 
3*f*g + e*h) + b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) + 96*c^3*(a^2*h^2*(3*f*g + 
 e*h) + b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 2*a*b*h*(3*f*g^2 + h*(3*e*g + d* 
h))))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2))) 
/(6*c))/(8*c))/(10*c*h)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1225

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

rule 1236

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

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 810, normalized size of antiderivative = 1.20

method	result	size
risch	\(\frac {\left (384 f \,h^{3} c^{4} x^{4}-432 b \,c^{3} f \,h^{3} x^{3}+480 c^{4} e \,h^{3} x^{3}+1440 c^{4} f g \,h^{2} x^{3}-512 a \,c^{3} f \,h^{3} x^{2}+504 b^{2} c^{2} f \,h^{3} x^{2}-560 b \,c^{3} e \,h^{3} x^{2}-1680 b \,c^{3} f g \,h^{2} x^{2}+640 c^{4} d \,h^{3} x^{2}+1920 c^{4} e g \,h^{2} x^{2}+1920 c^{4} f \,g^{2} h \,x^{2}+1288 a b \,c^{2} f \,h^{3} x -720 a \,c^{3} e \,h^{3} x -2160 a \,c^{3} f g \,h^{2} x -630 b^{3} c f \,h^{3} x +700 b^{2} c^{2} e \,h^{3} x +2100 b^{2} c^{2} f g \,h^{2} x -800 b \,c^{3} d \,h^{3} x -2400 b \,c^{3} e g \,h^{2} x -2400 b \,c^{3} f \,g^{2} h x +2880 c^{4} d g \,h^{2} x +2880 c^{4} e \,g^{2} h x +960 c^{4} f \,g^{3} x +1024 a^{2} c^{2} f \,h^{3}-2940 a \,b^{2} c f \,h^{3}+2200 a b \,c^{2} e \,h^{3}+6600 a b \,c^{2} f g \,h^{2}-1280 a \,c^{3} d \,h^{3}-3840 a \,c^{3} e g \,h^{2}-3840 a \,c^{3} f \,g^{2} h +945 b^{4} f \,h^{3}-1050 b^{3} c e \,h^{3}-3150 b^{3} c f g \,h^{2}+1200 b^{2} c^{2} d \,h^{3}+3600 b^{2} c^{2} e g \,h^{2}+3600 b^{2} c^{2} f \,g^{2} h -4320 b \,c^{3} d g \,h^{2}-4320 b \,c^{3} e \,g^{2} h -1440 b \,c^{3} f \,g^{3}+5760 c^{4} d \,g^{2} h +1920 c^{4} e \,g^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{5}}-\frac {\left (240 a^{2} b \,c^{2} f \,h^{3}-96 a^{2} c^{3} e \,h^{3}-288 a^{2} c^{3} f g \,h^{2}-280 a \,b^{3} c f \,h^{3}+240 a \,b^{2} c^{2} e \,h^{3}+720 a \,b^{2} c^{2} f g \,h^{2}-192 a b \,c^{3} d \,h^{3}-576 a b \,c^{3} e g \,h^{2}-576 a b \,c^{3} f \,g^{2} h +384 a \,c^{4} d g \,h^{2}+384 a \,c^{4} e \,g^{2} h +128 a \,c^{4} f \,g^{3}+63 b^{5} f \,h^{3}-70 b^{4} c e \,h^{3}-210 b^{4} c f g \,h^{2}+80 b^{3} c^{2} d \,h^{3}+240 b^{3} c^{2} e g \,h^{2}+240 b^{3} c^{2} f \,g^{2} h -288 b^{2} c^{3} d g \,h^{2}-288 b^{2} c^{3} e \,g^{2} h -96 b^{2} c^{3} f \,g^{3}+384 b \,c^{4} d \,g^{2} h +128 b \,c^{4} e \,g^{3}-256 c^{5} d \,g^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {11}{2}}}\)	\(810\)
default	\(\text {Expression too large to display}\)	\(1301\)

Input:

int((h*x+g)^3*(f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/1920*(384*c^4*f*h^3*x^4-432*b*c^3*f*h^3*x^3+480*c^4*e*h^3*x^3+1440*c^4*f 
*g*h^2*x^3-512*a*c^3*f*h^3*x^2+504*b^2*c^2*f*h^3*x^2-560*b*c^3*e*h^3*x^2-1 
680*b*c^3*f*g*h^2*x^2+640*c^4*d*h^3*x^2+1920*c^4*e*g*h^2*x^2+1920*c^4*f*g^ 
2*h*x^2+1288*a*b*c^2*f*h^3*x-720*a*c^3*e*h^3*x-2160*a*c^3*f*g*h^2*x-630*b^ 
3*c*f*h^3*x+700*b^2*c^2*e*h^3*x+2100*b^2*c^2*f*g*h^2*x-800*b*c^3*d*h^3*x-2 
400*b*c^3*e*g*h^2*x-2400*b*c^3*f*g^2*h*x+2880*c^4*d*g*h^2*x+2880*c^4*e*g^2 
*h*x+960*c^4*f*g^3*x+1024*a^2*c^2*f*h^3-2940*a*b^2*c*f*h^3+2200*a*b*c^2*e* 
h^3+6600*a*b*c^2*f*g*h^2-1280*a*c^3*d*h^3-3840*a*c^3*e*g*h^2-3840*a*c^3*f* 
g^2*h+945*b^4*f*h^3-1050*b^3*c*e*h^3-3150*b^3*c*f*g*h^2+1200*b^2*c^2*d*h^3 
+3600*b^2*c^2*e*g*h^2+3600*b^2*c^2*f*g^2*h-4320*b*c^3*d*g*h^2-4320*b*c^3*e 
*g^2*h-1440*b*c^3*f*g^3+5760*c^4*d*g^2*h+1920*c^4*e*g^3)*(c*x^2+b*x+a)^(1/ 
2)/c^5-1/256*(240*a^2*b*c^2*f*h^3-96*a^2*c^3*e*h^3-288*a^2*c^3*f*g*h^2-280 
*a*b^3*c*f*h^3+240*a*b^2*c^2*e*h^3+720*a*b^2*c^2*f*g*h^2-192*a*b*c^3*d*h^3 
-576*a*b*c^3*e*g*h^2-576*a*b*c^3*f*g^2*h+384*a*c^4*d*g*h^2+384*a*c^4*e*g^2 
*h+128*a*c^4*f*g^3+63*b^5*f*h^3-70*b^4*c*e*h^3-210*b^4*c*f*g*h^2+80*b^3*c^ 
2*d*h^3+240*b^3*c^2*e*g*h^2+240*b^3*c^2*f*g^2*h-288*b^2*c^3*d*g*h^2-288*b^ 
2*c^3*e*g^2*h-96*b^2*c^3*f*g^3+384*b*c^4*d*g^2*h+128*b*c^4*e*g^3-256*c^5*d 
*g^3)/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 1435, normalized size of antiderivative = 2.13 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)^3*(f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas 
")
 

Output:

[-1/7680*(15*(32*(8*c^5*d - 4*b*c^4*e + (3*b^2*c^3 - 4*a*c^4)*f)*g^3 - 48* 
(8*b*c^4*d - 2*(3*b^2*c^3 - 4*a*c^4)*e + (5*b^3*c^2 - 12*a*b*c^3)*f)*g^2*h 
 + 6*(16*(3*b^2*c^3 - 4*a*c^4)*d - 8*(5*b^3*c^2 - 12*a*b*c^3)*e + (35*b^4* 
c - 120*a*b^2*c^2 + 48*a^2*c^3)*f)*g*h^2 - (16*(5*b^3*c^2 - 12*a*b*c^3)*d 
- 2*(35*b^4*c - 120*a*b^2*c^2 + 48*a^2*c^3)*e + (63*b^5 - 280*a*b^3*c + 24 
0*a^2*b*c^2)*f)*h^3)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 
 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(384*c^5*f*h^3*x^4 + 480*(4*c 
^5*e - 3*b*c^4*f)*g^3 + 240*(24*c^5*d - 18*b*c^4*e + (15*b^2*c^3 - 16*a*c^ 
4)*f)*g^2*h - 30*(144*b*c^4*d - 8*(15*b^2*c^3 - 16*a*c^4)*e + 5*(21*b^3*c^ 
2 - 44*a*b*c^3)*f)*g*h^2 + (80*(15*b^2*c^3 - 16*a*c^4)*d - 50*(21*b^3*c^2 
- 44*a*b*c^3)*e + (945*b^4*c - 2940*a*b^2*c^2 + 1024*a^2*c^3)*f)*h^3 + 48* 
(30*c^5*f*g*h^2 + (10*c^5*e - 9*b*c^4*f)*h^3)*x^3 + 8*(240*c^5*f*g^2*h + 3 
0*(8*c^5*e - 7*b*c^4*f)*g*h^2 + (80*c^5*d - 70*b*c^4*e + (63*b^2*c^3 - 64* 
a*c^4)*f)*h^3)*x^2 + 2*(480*c^5*f*g^3 + 240*(6*c^5*e - 5*b*c^4*f)*g^2*h + 
30*(48*c^5*d - 40*b*c^4*e + (35*b^2*c^3 - 36*a*c^4)*f)*g*h^2 - (400*b*c^4* 
d - 10*(35*b^2*c^3 - 36*a*c^4)*e + 7*(45*b^3*c^2 - 92*a*b*c^3)*f)*h^3)*x)* 
sqrt(c*x^2 + b*x + a))/c^6, -1/3840*(15*(32*(8*c^5*d - 4*b*c^4*e + (3*b^2* 
c^3 - 4*a*c^4)*f)*g^3 - 48*(8*b*c^4*d - 2*(3*b^2*c^3 - 4*a*c^4)*e + (5*b^3 
*c^2 - 12*a*b*c^3)*f)*g^2*h + 6*(16*(3*b^2*c^3 - 4*a*c^4)*d - 8*(5*b^3*c^2 
 - 12*a*b*c^3)*e + (35*b^4*c - 120*a*b^2*c^2 + 48*a^2*c^3)*f)*g*h^2 - (...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1613 vs. \(2 (712) = 1424\).

Time = 1.36 (sec) , antiderivative size = 1613, normalized size of antiderivative = 2.39 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)**3*(f*x**2+e*x+d)/(c*x**2+b*x+a)**(1/2),x)
 

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(f*h**3*x**4/(5*c) + x**3*(-9*b*f*h**3/( 
10*c) + e*h**3 + 3*f*g*h**2)/(4*c) + x**2*(-4*a*f*h**3/(5*c) - 7*b*(-9*b*f 
*h**3/(10*c) + e*h**3 + 3*f*g*h**2)/(8*c) + d*h**3 + 3*e*g*h**2 + 3*f*g**2 
*h)/(3*c) + x*(-3*a*(-9*b*f*h**3/(10*c) + e*h**3 + 3*f*g*h**2)/(4*c) - 5*b 
*(-4*a*f*h**3/(5*c) - 7*b*(-9*b*f*h**3/(10*c) + e*h**3 + 3*f*g*h**2)/(8*c) 
 + d*h**3 + 3*e*g*h**2 + 3*f*g**2*h)/(6*c) + 3*d*g*h**2 + 3*e*g**2*h + f*g 
**3)/(2*c) + (-2*a*(-4*a*f*h**3/(5*c) - 7*b*(-9*b*f*h**3/(10*c) + e*h**3 + 
 3*f*g*h**2)/(8*c) + d*h**3 + 3*e*g*h**2 + 3*f*g**2*h)/(3*c) - 3*b*(-3*a*( 
-9*b*f*h**3/(10*c) + e*h**3 + 3*f*g*h**2)/(4*c) - 5*b*(-4*a*f*h**3/(5*c) - 
 7*b*(-9*b*f*h**3/(10*c) + e*h**3 + 3*f*g*h**2)/(8*c) + d*h**3 + 3*e*g*h** 
2 + 3*f*g**2*h)/(6*c) + 3*d*g*h**2 + 3*e*g**2*h + f*g**3)/(4*c) + 3*d*g**2 
*h + e*g**3)/c) + (-a*(-3*a*(-9*b*f*h**3/(10*c) + e*h**3 + 3*f*g*h**2)/(4* 
c) - 5*b*(-4*a*f*h**3/(5*c) - 7*b*(-9*b*f*h**3/(10*c) + e*h**3 + 3*f*g*h** 
2)/(8*c) + d*h**3 + 3*e*g*h**2 + 3*f*g**2*h)/(6*c) + 3*d*g*h**2 + 3*e*g**2 
*h + f*g**3)/(2*c) - b*(-2*a*(-4*a*f*h**3/(5*c) - 7*b*(-9*b*f*h**3/(10*c) 
+ e*h**3 + 3*f*g*h**2)/(8*c) + d*h**3 + 3*e*g*h**2 + 3*f*g**2*h)/(3*c) - 3 
*b*(-3*a*(-9*b*f*h**3/(10*c) + e*h**3 + 3*f*g*h**2)/(4*c) - 5*b*(-4*a*f*h* 
*3/(5*c) - 7*b*(-9*b*f*h**3/(10*c) + e*h**3 + 3*f*g*h**2)/(8*c) + d*h**3 + 
 3*e*g*h**2 + 3*f*g**2*h)/(6*c) + 3*d*g*h**2 + 3*e*g**2*h + f*g**3)/(4*c) 
+ 3*d*g**2*h + e*g**3)/(2*c) + d*g**3)*Piecewise((log(b + 2*sqrt(c)*sqr...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((h*x+g)^3*(f*x^2+e*x+d)/(c*x^2+b*x+a)^(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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 798, normalized size of antiderivative = 1.18 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

integrate((h*x+g)^3*(f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*f*h^3*x/c + (30*c^4*f*g*h^2 + 10* 
c^4*e*h^3 - 9*b*c^3*f*h^3)/c^5)*x + (240*c^4*f*g^2*h + 240*c^4*e*g*h^2 - 2 
10*b*c^3*f*g*h^2 + 80*c^4*d*h^3 - 70*b*c^3*e*h^3 + 63*b^2*c^2*f*h^3 - 64*a 
*c^3*f*h^3)/c^5)*x + (480*c^4*f*g^3 + 1440*c^4*e*g^2*h - 1200*b*c^3*f*g^2* 
h + 1440*c^4*d*g*h^2 - 1200*b*c^3*e*g*h^2 + 1050*b^2*c^2*f*g*h^2 - 1080*a* 
c^3*f*g*h^2 - 400*b*c^3*d*h^3 + 350*b^2*c^2*e*h^3 - 360*a*c^3*e*h^3 - 315* 
b^3*c*f*h^3 + 644*a*b*c^2*f*h^3)/c^5)*x + (1920*c^4*e*g^3 - 1440*b*c^3*f*g 
^3 + 5760*c^4*d*g^2*h - 4320*b*c^3*e*g^2*h + 3600*b^2*c^2*f*g^2*h - 3840*a 
*c^3*f*g^2*h - 4320*b*c^3*d*g*h^2 + 3600*b^2*c^2*e*g*h^2 - 3840*a*c^3*e*g* 
h^2 - 3150*b^3*c*f*g*h^2 + 6600*a*b*c^2*f*g*h^2 + 1200*b^2*c^2*d*h^3 - 128 
0*a*c^3*d*h^3 - 1050*b^3*c*e*h^3 + 2200*a*b*c^2*e*h^3 + 945*b^4*f*h^3 - 29 
40*a*b^2*c*f*h^3 + 1024*a^2*c^2*f*h^3)/c^5) - 1/256*(256*c^5*d*g^3 - 128*b 
*c^4*e*g^3 + 96*b^2*c^3*f*g^3 - 128*a*c^4*f*g^3 - 384*b*c^4*d*g^2*h + 288* 
b^2*c^3*e*g^2*h - 384*a*c^4*e*g^2*h - 240*b^3*c^2*f*g^2*h + 576*a*b*c^3*f* 
g^2*h + 288*b^2*c^3*d*g*h^2 - 384*a*c^4*d*g*h^2 - 240*b^3*c^2*e*g*h^2 + 57 
6*a*b*c^3*e*g*h^2 + 210*b^4*c*f*g*h^2 - 720*a*b^2*c^2*f*g*h^2 + 288*a^2*c^ 
3*f*g*h^2 - 80*b^3*c^2*d*h^3 + 192*a*b*c^3*d*h^3 + 70*b^4*c*e*h^3 - 240*a* 
b^2*c^2*e*h^3 + 96*a^2*c^3*e*h^3 - 63*b^5*f*h^3 + 280*a*b^3*c*f*h^3 - 240* 
a^2*b*c^2*f*h^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b 
))/c^(11/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (g+h\,x\right )}^3\,\left (f\,x^2+e\,x+d\right )}{\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int(((g + h*x)^3*(d + e*x + f*x^2))/(a + b*x + c*x^2)^(1/2),x)
 

Output:

int(((g + h*x)^3*(d + e*x + f*x^2))/(a + b*x + c*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (h x +g \right )^{3} \left (f \,x^{2}+e x +d \right )}{\sqrt {c \,x^{2}+b x +a}}d x \] Input:

int((h*x+g)^3*(f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x)
 

Output:

int((h*x+g)^3*(f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x)