\(\int \frac {(d+e x+f x^2)^3}{\sqrt {a+b x+c x^2}} \, dx\) [147]

Optimal result

Integrand size = 27, antiderivative size = 717 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\sqrt {a+b x+c x^2}} \, dx=\frac {\left (23040 c^5 d^2 e-3465 b^5 f^3+420 b^3 c f^2 (27 b e+34 a f)-504 b c^2 f \left (70 a b e f+22 a^2 f^2+25 b^2 \left (e^2+d f\right )\right )-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (e^2+d f\right )+50 b^2 \left (e^3+6 d e f\right )\right )\right ) \sqrt {a+b x+c x^2}}{7680 c^6}+\frac {\left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x \sqrt {a+b x+c x^2}}{3840 c^5}-\frac {\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt {a+b x+c x^2}}{960 c^4}+\frac {f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}+\frac {f^2 (36 c e-11 b f) x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\left (1024 c^6 d^3+231 b^6 f^3-252 b^4 c f^2 (3 b e+5 a f)-1536 c^5 d \left (b d e+a \left (e^2+d f\right )\right )+840 b^2 c^2 f \left (4 a b e f+2 a^2 f^2+b^2 \left (e^2+d f\right )\right )+384 c^4 \left (3 b^2 d \left (e^2+d f\right )+3 a^2 f \left (e^2+d f\right )+2 a b e \left (e^2+6 d f\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{13/2}} \] Output:

1/7680*(23040*c^5*d^2*e-3465*b^5*f^3+420*b^3*c*f^2*(34*a*f+27*b*e)-504*b*c 
^2*f*(70*a*b*e*f+22*a^2*f^2+25*b^2*(d*f+e^2))-640*c^4*(27*b*d*(d*f+e^2)+8* 
a*e*(6*d*f+e^2))+96*c^3*(128*a^2*e*f^2+275*a*b*f*(d*f+e^2)+50*b^2*(6*d*e*f 
+e^3)))*(c*x^2+b*x+a)^(1/2)/c^6+1/3840*(1155*b^4*f^3-252*b^2*c*f^2*(14*a*f 
+15*b*e)+5760*c^4*d*(d*f+e^2)+24*c^2*f*(322*a*b*e*f+50*a^2*f^2+175*b^2*(d* 
f+e^2))-160*c^3*(27*a*f*(d*f+e^2)+10*b*(6*d*e*f+e^3)))*x*(c*x^2+b*x+a)^(1/ 
2)/c^5-1/960*(231*b^3*f^3-36*b*c*f^2*(13*a*f+21*b*e)-320*c^3*(6*d*e*f+e^3) 
+24*c^2*f*(32*a*e*f+35*b*(d*f+e^2)))*x^2*(c*x^2+b*x+a)^(1/2)/c^4+1/480*f*( 
99*b^2*f^2-4*c*f*(25*a*f+81*b*e)+360*c^2*(d*f+e^2))*x^3*(c*x^2+b*x+a)^(1/2 
)/c^3+1/60*f^2*(-11*b*f+36*c*e)*x^4*(c*x^2+b*x+a)^(1/2)/c^2+1/6*f^3*x^5*(c 
*x^2+b*x+a)^(1/2)/c+1/1024*(1024*c^6*d^3+231*b^6*f^3-252*b^4*c*f^2*(5*a*f+ 
3*b*e)-1536*c^5*d*(b*d*e+a*(d*f+e^2))+840*b^2*c^2*f*(4*a*b*e*f+2*a^2*f^2+b 
^2*(d*f+e^2))+384*c^4*(3*b^2*d*(d*f+e^2)+3*a^2*f*(d*f+e^2)+2*a*b*e*(6*d*f+ 
e^2))-320*c^3*(9*a^2*b*e*f^2+a^3*f^3+9*a*b^2*f*(d*f+e^2)+b^3*(6*d*e*f+e^3) 
))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(13/2)
 

Mathematica [A] (verified)

Time = 7.78 (sec) , antiderivative size = 618, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-3465 b^5 f^3+210 b^3 c f^2 (54 b e+68 a f+11 b f x)-168 b c^2 f \left (66 a^2 f^2+42 a b f (5 e+f x)+b^2 \left (75 e^2+75 d f+45 e f x+11 f^2 x^2\right )\right )+128 c^5 \left (90 d^2 (2 e+f x)+15 d x \left (6 e^2+8 e f x+3 f^2 x^2\right )+x^2 \left (20 e^3+45 e^2 f x+36 e f^2 x^2+10 f^3 x^3\right )\right )+48 c^3 \left (2 a^2 f^2 (128 e+25 f x)+b^2 \left (100 e^3+175 e^2 f x+6 e f \left (100 d+21 f x^2\right )+f^2 x \left (175 d+33 f x^2\right )\right )+2 a b f \left (275 e^2+161 e f x+f \left (275 d+39 f x^2\right )\right )\right )-64 c^4 \left (a \left (80 e^3+135 e^2 f x+96 e f \left (5 d+f x^2\right )+5 f^2 x \left (27 d+5 f x^2\right )\right )+b \left (270 d^2 f+15 d \left (18 e^2+20 e f x+7 f^2 x^2\right )+x \left (50 e^3+105 e^2 f x+81 e f^2 x^2+22 f^3 x^3\right )\right )\right )\right )+15 \left (1024 c^6 d^3+231 b^6 f^3-252 b^4 c f^2 (3 b e+5 a f)-1536 c^5 d \left (b d e+a \left (e^2+d f\right )\right )+840 b^2 c^2 f \left (4 a b e f+2 a^2 f^2+b^2 \left (e^2+d f\right )\right )+384 c^4 \left (3 b^2 d \left (e^2+d f\right )+3 a^2 f \left (e^2+d f\right )+2 a b e \left (e^2+6 d f\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{7680 c^{13/2}} \] Input:

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

Output:

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^5*f^3 + 210*b^3*c*f^2*(54*b*e + 68 
*a*f + 11*b*f*x) - 168*b*c^2*f*(66*a^2*f^2 + 42*a*b*f*(5*e + f*x) + b^2*(7 
5*e^2 + 75*d*f + 45*e*f*x + 11*f^2*x^2)) + 128*c^5*(90*d^2*(2*e + f*x) + 1 
5*d*x*(6*e^2 + 8*e*f*x + 3*f^2*x^2) + x^2*(20*e^3 + 45*e^2*f*x + 36*e*f^2* 
x^2 + 10*f^3*x^3)) + 48*c^3*(2*a^2*f^2*(128*e + 25*f*x) + b^2*(100*e^3 + 1 
75*e^2*f*x + 6*e*f*(100*d + 21*f*x^2) + f^2*x*(175*d + 33*f*x^2)) + 2*a*b* 
f*(275*e^2 + 161*e*f*x + f*(275*d + 39*f*x^2))) - 64*c^4*(a*(80*e^3 + 135* 
e^2*f*x + 96*e*f*(5*d + f*x^2) + 5*f^2*x*(27*d + 5*f*x^2)) + b*(270*d^2*f 
+ 15*d*(18*e^2 + 20*e*f*x + 7*f^2*x^2) + x*(50*e^3 + 105*e^2*f*x + 81*e*f^ 
2*x^2 + 22*f^3*x^3)))) + 15*(1024*c^6*d^3 + 231*b^6*f^3 - 252*b^4*c*f^2*(3 
*b*e + 5*a*f) - 1536*c^5*d*(b*d*e + a*(e^2 + d*f)) + 840*b^2*c^2*f*(4*a*b* 
e*f + 2*a^2*f^2 + b^2*(e^2 + d*f)) + 384*c^4*(3*b^2*d*(e^2 + d*f) + 3*a^2* 
f*(e^2 + d*f) + 2*a*b*e*(e^2 + 6*d*f)) - 320*c^3*(9*a^2*b*e*f^2 + a^3*f^3 
+ 9*a*b^2*f*(e^2 + d*f) + b^3*(e^3 + 6*d*e*f)))*ArcTanh[(Sqrt[c]*x)/(-Sqrt 
[a] + Sqrt[a + x*(b + c*x)])])/(7680*c^(13/2))
 

Rubi [A] (verified)

Time = 2.44 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 1160, 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.

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\int \frac {f^2 (36 c e-11 b f) x^5-2 f \left (5 a f^2-18 c \left (e^2+d f\right )\right ) x^4+12 c e \left (e^2+6 d f\right ) x^3+36 c d \left (e^2+d f\right ) x^2+36 c d^2 e x+12 c d^3}{2 \sqrt {c x^2+b x+a}}dx}{6 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {f^2 (36 c e-11 b f) x^5-2 f \left (5 a f^2-18 c \left (e^2+d f\right )\right ) x^4+12 c e \left (e^2+6 d f\right ) x^3+36 c d \left (e^2+d f\right ) x^2+36 c d^2 e x+12 c d^3}{\sqrt {c x^2+b x+a}}dx}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\int \frac {f \left (360 \left (e^2+d f\right ) c^2-4 f (81 b e+25 a f) c+99 b^2 f^2\right ) x^4-8 \left (-11 a b f^3+36 a c e f^2-15 c^2 \left (e^3+6 d f e\right )\right ) x^3+360 c^2 d \left (e^2+d f\right ) x^2+360 c^2 d^2 e x+120 c^2 d^3}{2 \sqrt {c x^2+b x+a}}dx}{5 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {f \left (360 \left (e^2+d f\right ) c^2-4 f (81 b e+25 a f) c+99 b^2 f^2\right ) x^4-8 \left (-11 a b f^3+36 a c e f^2-15 c^2 \left (e^3+6 d f e\right )\right ) x^3+360 c^2 d \left (e^2+d f\right ) x^2+360 c^2 d^2 e x+120 c^2 d^3}{\sqrt {c x^2+b x+a}}dx}{10 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\int \frac {3 \left (320 d^3 c^3+960 d^2 e x c^3-\left (-320 \left (e^3+6 d f e\right ) c^3+24 f \left (32 a e f+35 b \left (e^2+d f\right )\right ) c^2-36 b f^2 (21 b e+13 a f) c+231 b^3 f^3\right ) x^3-2 \left (-480 d \left (e^2+d f\right ) c^3+360 a f \left (e^2+d f\right ) c^2-4 a f^2 (81 b e+25 a f) c+99 a b^2 f^3\right ) x^2\right )}{2 \sqrt {c x^2+b x+a}}dx}{4 c}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{4 c}}{10 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {320 d^3 c^3+960 d^2 e x c^3-\left (-320 \left (e^3+6 d f e\right ) c^3+24 f \left (32 a e f+35 b \left (e^2+d f\right )\right ) c^2-36 b f^2 (21 b e+13 a f) c+231 b^3 f^3\right ) x^3-2 \left (-480 d \left (e^2+d f\right ) c^3+360 a f \left (e^2+d f\right ) c^2-4 a f^2 (81 b e+25 a f) c+99 a b^2 f^3\right ) x^2}{\sqrt {c x^2+b x+a}}dx}{8 c}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{4 c}}{10 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {1920 d^3 c^4+\left (1155 f^3 b^4-252 c f^2 (15 b e+14 a f) b^2+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (175 \left (e^2+d f\right ) b^2+322 a e f b+50 a^2 f^2\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d f e\right )\right )\right ) x^2+4 \left (1440 d^2 e c^4-320 a e \left (e^2+6 d f\right ) c^3+24 a f \left (32 a e f+35 b \left (e^2+d f\right )\right ) c^2-36 a b f^2 (21 b e+13 a f) c+231 a b^3 f^3\right ) x}{2 \sqrt {c x^2+b x+a}}dx}{3 c}-\frac {x^2 \sqrt {a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{3 c}\right )}{8 c}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{4 c}}{10 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {1920 d^3 c^4+\left (1155 f^3 b^4-252 c f^2 (15 b e+14 a f) b^2+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (175 \left (e^2+d f\right ) b^2+322 a e f b+50 a^2 f^2\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d f e\right )\right )\right ) x^2+4 \left (1440 d^2 e c^4-320 a e \left (e^2+6 d f\right ) c^3+24 a f \left (32 a e f+35 b \left (e^2+d f\right )\right ) c^2-36 a b f^2 (21 b e+13 a f) c+231 a b^3 f^3\right ) x}{\sqrt {c x^2+b x+a}}dx}{6 c}-\frac {x^2 \sqrt {a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{3 c}\right )}{8 c}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{4 c}}{10 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {\int \frac {7680 d^3 c^5-11520 a d \left (e^2+d f\right ) c^4+320 a \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d f e\right )\right ) c^3-48 a f \left (175 \left (e^2+d f\right ) b^2+322 a e f b+50 a^2 f^2\right ) c^2+504 a b^2 f^2 (15 b e+14 a f) c-2310 a b^4 f^3+\left (-3465 f^3 b^5+420 c f^2 (27 b e+34 a f) b^3-504 c^2 f \left (25 \left (e^2+d f\right ) b^2+70 a e f b+22 a^2 f^2\right ) b+23040 c^5 d^2 e-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (50 \left (e^3+6 d f e\right ) b^2+275 a f \left (e^2+d f\right ) b+128 a^2 e f^2\right )\right ) x}{2 \sqrt {c x^2+b x+a}}dx}{2 c}+\frac {x \sqrt {a+b x+c x^2} \left (24 c^2 f \left (50 a^2 f^2+322 a b e f+175 b^2 \left (d f+e^2\right )\right )-252 b^2 c f^2 (14 a f+15 b e)-160 c^3 \left (27 a f \left (d f+e^2\right )+10 b \left (6 d e f+e^3\right )\right )+1155 b^4 f^3+5760 c^4 d \left (d f+e^2\right )\right )}{2 c}}{6 c}-\frac {x^2 \sqrt {a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{3 c}\right )}{8 c}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{4 c}}{10 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {\int \frac {2 \left (3840 d^3 c^5-5760 a d \left (e^2+d f\right ) c^4+160 a \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d f e\right )\right ) c^3-24 a f \left (175 \left (e^2+d f\right ) b^2+322 a e f b+50 a^2 f^2\right ) c^2+252 a b^2 f^2 (15 b e+14 a f) c-1155 a b^4 f^3\right )+\left (-3465 f^3 b^5+420 c f^2 (27 b e+34 a f) b^3-504 c^2 f \left (25 \left (e^2+d f\right ) b^2+70 a e f b+22 a^2 f^2\right ) b+23040 c^5 d^2 e-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (50 \left (e^3+6 d f e\right ) b^2+275 a f \left (e^2+d f\right ) b+128 a^2 e f^2\right )\right ) x}{\sqrt {c x^2+b x+a}}dx}{4 c}+\frac {x \sqrt {a+b x+c x^2} \left (24 c^2 f \left (50 a^2 f^2+322 a b e f+175 b^2 \left (d f+e^2\right )\right )-252 b^2 c f^2 (14 a f+15 b e)-160 c^3 \left (27 a f \left (d f+e^2\right )+10 b \left (6 d e f+e^3\right )\right )+1155 b^4 f^3+5760 c^4 d \left (d f+e^2\right )\right )}{2 c}}{6 c}-\frac {x^2 \sqrt {a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{3 c}\right )}{8 c}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{4 c}}{10 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {\frac {15 \left (384 c^4 \left (3 a^2 f \left (d f+e^2\right )+2 a b e \left (6 d f+e^2\right )+3 b^2 d \left (d f+e^2\right )\right )+840 b^2 c^2 f \left (2 a^2 f^2+4 a b e f+b^2 \left (d f+e^2\right )\right )-320 c^3 \left (a^3 f^3+9 a^2 b e f^2+9 a b^2 f \left (d f+e^2\right )+b^3 \left (6 d e f+e^3\right )\right )-252 b^4 c f^2 (5 a f+3 b e)-1536 c^5 d \left (a \left (d f+e^2\right )+b d e\right )+231 b^6 f^3+1024 c^6 d^3\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 c}+\frac {\sqrt {a+b x+c x^2} \left (96 c^3 \left (128 a^2 e f^2+275 a b f \left (d f+e^2\right )+50 b^2 \left (6 d e f+e^3\right )\right )-504 b c^2 f \left (22 a^2 f^2+70 a b e f+25 b^2 \left (d f+e^2\right )\right )+420 b^3 c f^2 (34 a f+27 b e)-640 c^4 \left (8 a e \left (6 d f+e^2\right )+27 b d \left (d f+e^2\right )\right )-3465 b^5 f^3+23040 c^5 d^2 e\right )}{c}}{4 c}+\frac {x \sqrt {a+b x+c x^2} \left (24 c^2 f \left (50 a^2 f^2+322 a b e f+175 b^2 \left (d f+e^2\right )\right )-252 b^2 c f^2 (14 a f+15 b e)-160 c^3 \left (27 a f \left (d f+e^2\right )+10 b \left (6 d e f+e^3\right )\right )+1155 b^4 f^3+5760 c^4 d \left (d f+e^2\right )\right )}{2 c}}{6 c}-\frac {x^2 \sqrt {a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{3 c}\right )}{8 c}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{4 c}}{10 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {\frac {15 \left (384 c^4 \left (3 a^2 f \left (d f+e^2\right )+2 a b e \left (6 d f+e^2\right )+3 b^2 d \left (d f+e^2\right )\right )+840 b^2 c^2 f \left (2 a^2 f^2+4 a b e f+b^2 \left (d f+e^2\right )\right )-320 c^3 \left (a^3 f^3+9 a^2 b e f^2+9 a b^2 f \left (d f+e^2\right )+b^3 \left (6 d e f+e^3\right )\right )-252 b^4 c f^2 (5 a f+3 b e)-1536 c^5 d \left (a \left (d f+e^2\right )+b d e\right )+231 b^6 f^3+1024 c^6 d^3\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}+\frac {\sqrt {a+b x+c x^2} \left (96 c^3 \left (128 a^2 e f^2+275 a b f \left (d f+e^2\right )+50 b^2 \left (6 d e f+e^3\right )\right )-504 b c^2 f \left (22 a^2 f^2+70 a b e f+25 b^2 \left (d f+e^2\right )\right )+420 b^3 c f^2 (34 a f+27 b e)-640 c^4 \left (8 a e \left (6 d f+e^2\right )+27 b d \left (d f+e^2\right )\right )-3465 b^5 f^3+23040 c^5 d^2 e\right )}{c}}{4 c}+\frac {x \sqrt {a+b x+c x^2} \left (24 c^2 f \left (50 a^2 f^2+322 a b e f+175 b^2 \left (d f+e^2\right )\right )-252 b^2 c f^2 (14 a f+15 b e)-160 c^3 \left (27 a f \left (d f+e^2\right )+10 b \left (6 d e f+e^3\right )\right )+1155 b^4 f^3+5760 c^4 d \left (d f+e^2\right )\right )}{2 c}}{6 c}-\frac {x^2 \sqrt {a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{3 c}\right )}{8 c}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{4 c}}{10 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {x \sqrt {a+b x+c x^2} \left (24 c^2 f \left (50 a^2 f^2+322 a b e f+175 b^2 \left (d f+e^2\right )\right )-252 b^2 c f^2 (14 a f+15 b e)-160 c^3 \left (27 a f \left (d f+e^2\right )+10 b \left (6 d e f+e^3\right )\right )+1155 b^4 f^3+5760 c^4 d \left (d f+e^2\right )\right )}{2 c}+\frac {\frac {\sqrt {a+b x+c x^2} \left (96 c^3 \left (128 a^2 e f^2+275 a b f \left (d f+e^2\right )+50 b^2 \left (6 d e f+e^3\right )\right )-504 b c^2 f \left (22 a^2 f^2+70 a b e f+25 b^2 \left (d f+e^2\right )\right )+420 b^3 c f^2 (34 a f+27 b e)-640 c^4 \left (8 a e \left (6 d f+e^2\right )+27 b d \left (d f+e^2\right )\right )-3465 b^5 f^3+23040 c^5 d^2 e\right )}{c}+\frac {15 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (384 c^4 \left (3 a^2 f \left (d f+e^2\right )+2 a b e \left (6 d f+e^2\right )+3 b^2 d \left (d f+e^2\right )\right )+840 b^2 c^2 f \left (2 a^2 f^2+4 a b e f+b^2 \left (d f+e^2\right )\right )-320 c^3 \left (a^3 f^3+9 a^2 b e f^2+9 a b^2 f \left (d f+e^2\right )+b^3 \left (6 d e f+e^3\right )\right )-252 b^4 c f^2 (5 a f+3 b e)-1536 c^5 d \left (a \left (d f+e^2\right )+b d e\right )+231 b^6 f^3+1024 c^6 d^3\right )}{2 c^{3/2}}}{4 c}}{6 c}-\frac {x^2 \sqrt {a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{3 c}\right )}{8 c}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{4 c}}{10 c}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{5 c}}{12 c}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}\)

Input:

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

Output:

(f^3*x^5*Sqrt[a + b*x + c*x^2])/(6*c) + ((f^2*(36*c*e - 11*b*f)*x^4*Sqrt[a 
 + b*x + c*x^2])/(5*c) + ((f*(99*b^2*f^2 - 4*c*f*(81*b*e + 25*a*f) + 360*c 
^2*(e^2 + d*f))*x^3*Sqrt[a + b*x + c*x^2])/(4*c) + (3*(-1/3*((231*b^3*f^3 
- 36*b*c*f^2*(21*b*e + 13*a*f) - 320*c^3*(e^3 + 6*d*e*f) + 24*c^2*f*(32*a* 
e*f + 35*b*(e^2 + d*f)))*x^2*Sqrt[a + b*x + c*x^2])/c + (((1155*b^4*f^3 - 
252*b^2*c*f^2*(15*b*e + 14*a*f) + 5760*c^4*d*(e^2 + d*f) + 24*c^2*f*(322*a 
*b*e*f + 50*a^2*f^2 + 175*b^2*(e^2 + d*f)) - 160*c^3*(27*a*f*(e^2 + d*f) + 
 10*b*(e^3 + 6*d*e*f)))*x*Sqrt[a + b*x + c*x^2])/(2*c) + (((23040*c^5*d^2* 
e - 3465*b^5*f^3 + 420*b^3*c*f^2*(27*b*e + 34*a*f) - 504*b*c^2*f*(70*a*b*e 
*f + 22*a^2*f^2 + 25*b^2*(e^2 + d*f)) - 640*c^4*(27*b*d*(e^2 + d*f) + 8*a* 
e*(e^2 + 6*d*f)) + 96*c^3*(128*a^2*e*f^2 + 275*a*b*f*(e^2 + d*f) + 50*b^2* 
(e^3 + 6*d*e*f)))*Sqrt[a + b*x + c*x^2])/c + (15*(1024*c^6*d^3 + 231*b^6*f 
^3 - 252*b^4*c*f^2*(3*b*e + 5*a*f) - 1536*c^5*d*(b*d*e + a*(e^2 + d*f)) + 
840*b^2*c^2*f*(4*a*b*e*f + 2*a^2*f^2 + b^2*(e^2 + d*f)) + 384*c^4*(3*b^2*d 
*(e^2 + d*f) + 3*a^2*f*(e^2 + d*f) + 2*a*b*e*(e^2 + 6*d*f)) - 320*c^3*(9*a 
^2*b*e*f^2 + a^3*f^3 + 9*a*b^2*f*(e^2 + d*f) + b^3*(e^3 + 6*d*e*f)))*ArcTa 
nh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(3/2)))/(4*c))/(6* 
c)))/(8*c))/(10*c))/(12*c)
 

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 1160

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

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 2.77 (sec) , antiderivative size = 837, normalized size of antiderivative = 1.17

method	result	size
risch	\(-\frac {\left (-1280 f^{3} c^{5} x^{5}+1408 b \,c^{4} f^{3} x^{4}-4608 c^{5} e \,f^{2} x^{4}+1600 a \,c^{4} f^{3} x^{3}-1584 b^{2} c^{3} f^{3} x^{3}+5184 b \,c^{4} e \,f^{2} x^{3}-5760 c^{5} d \,f^{2} x^{3}-5760 c^{5} e^{2} f \,x^{3}-3744 a b \,c^{3} f^{3} x^{2}+6144 a \,c^{4} e \,f^{2} x^{2}+1848 b^{3} c^{2} f^{3} x^{2}-6048 b^{2} c^{3} e \,f^{2} x^{2}+6720 b \,c^{4} d \,f^{2} x^{2}+6720 b \,c^{4} e^{2} f \,x^{2}-15360 c^{5} d e f \,x^{2}-2560 c^{5} e^{3} x^{2}-2400 a^{2} c^{3} f^{3} x +7056 a \,b^{2} c^{2} f^{3} x -15456 a b \,c^{3} e \,f^{2} x +8640 a \,c^{4} d \,f^{2} x +8640 a \,c^{4} e^{2} f x -2310 b^{4} c \,f^{3} x +7560 b^{3} c^{2} e \,f^{2} x -8400 b^{2} c^{3} d \,f^{2} x -8400 b^{2} c^{3} e^{2} f x +19200 b \,c^{4} d e f x +3200 b \,c^{4} e^{3} x -11520 c^{5} d^{2} f x -11520 c^{5} d \,e^{2} x +11088 a^{2} b \,c^{2} f^{3}-12288 a^{2} c^{3} e \,f^{2}-14280 a \,b^{3} c \,f^{3}+35280 a \,b^{2} c^{2} e \,f^{2}-26400 a b \,c^{3} d \,f^{2}-26400 a b \,c^{3} e^{2} f +30720 a \,c^{4} d e f +5120 a \,c^{4} e^{3}+3465 b^{5} f^{3}-11340 b^{4} c e \,f^{2}+12600 b^{3} c^{2} d \,f^{2}+12600 b^{3} c^{2} e^{2} f -28800 b^{2} c^{3} d e f -4800 b^{2} c^{3} e^{3}+17280 b \,c^{4} d^{2} f +17280 b \,c^{4} d \,e^{2}-23040 c^{5} d^{2} e \right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{6}}-\frac {\left (320 a^{3} c^{3} f^{3}-1680 a^{2} b^{2} c^{2} f^{3}+2880 a^{2} b \,c^{3} e \,f^{2}-1152 a^{2} c^{4} d \,f^{2}-1152 a^{2} c^{4} e^{2} f +1260 a \,b^{4} c \,f^{3}-3360 a \,b^{3} c^{2} e \,f^{2}+2880 a \,b^{2} c^{3} d \,f^{2}+2880 a \,b^{2} c^{3} e^{2} f -4608 a b \,c^{4} d e f -768 a b \,c^{4} e^{3}+1536 a \,c^{5} d^{2} f +1536 a \,c^{5} d \,e^{2}-231 b^{6} f^{3}+756 b^{5} c e \,f^{2}-840 b^{4} c^{2} d \,f^{2}-840 b^{4} c^{2} e^{2} f +1920 b^{3} c^{3} d e f +320 b^{3} c^{3} e^{3}-1152 b^{2} c^{4} d^{2} f -1152 b^{2} c^{4} d \,e^{2}+1536 b \,c^{5} d^{2} e -1024 c^{6} d^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {13}{2}}}\)	\(837\)
default	\(\text {Expression too large to display}\)	\(2174\)

Input:

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

Output:

-1/7680*(-1280*c^5*f^3*x^5+1408*b*c^4*f^3*x^4-4608*c^5*e*f^2*x^4+1600*a*c^ 
4*f^3*x^3-1584*b^2*c^3*f^3*x^3+5184*b*c^4*e*f^2*x^3-5760*c^5*d*f^2*x^3-576 
0*c^5*e^2*f*x^3-3744*a*b*c^3*f^3*x^2+6144*a*c^4*e*f^2*x^2+1848*b^3*c^2*f^3 
*x^2-6048*b^2*c^3*e*f^2*x^2+6720*b*c^4*d*f^2*x^2+6720*b*c^4*e^2*f*x^2-1536 
0*c^5*d*e*f*x^2-2560*c^5*e^3*x^2-2400*a^2*c^3*f^3*x+7056*a*b^2*c^2*f^3*x-1 
5456*a*b*c^3*e*f^2*x+8640*a*c^4*d*f^2*x+8640*a*c^4*e^2*f*x-2310*b^4*c*f^3* 
x+7560*b^3*c^2*e*f^2*x-8400*b^2*c^3*d*f^2*x-8400*b^2*c^3*e^2*f*x+19200*b*c 
^4*d*e*f*x+3200*b*c^4*e^3*x-11520*c^5*d^2*f*x-11520*c^5*d*e^2*x+11088*a^2* 
b*c^2*f^3-12288*a^2*c^3*e*f^2-14280*a*b^3*c*f^3+35280*a*b^2*c^2*e*f^2-2640 
0*a*b*c^3*d*f^2-26400*a*b*c^3*e^2*f+30720*a*c^4*d*e*f+5120*a*c^4*e^3+3465* 
b^5*f^3-11340*b^4*c*e*f^2+12600*b^3*c^2*d*f^2+12600*b^3*c^2*e^2*f-28800*b^ 
2*c^3*d*e*f-4800*b^2*c^3*e^3+17280*b*c^4*d^2*f+17280*b*c^4*d*e^2-23040*c^5 
*d^2*e)/c^6*(c*x^2+b*x+a)^(1/2)-1/1024*(320*a^3*c^3*f^3-1680*a^2*b^2*c^2*f 
^3+2880*a^2*b*c^3*e*f^2-1152*a^2*c^4*d*f^2-1152*a^2*c^4*e^2*f+1260*a*b^4*c 
*f^3-3360*a*b^3*c^2*e*f^2+2880*a*b^2*c^3*d*f^2+2880*a*b^2*c^3*e^2*f-4608*a 
*b*c^4*d*e*f-768*a*b*c^4*e^3+1536*a*c^5*d^2*f+1536*a*c^5*d*e^2-231*b^6*f^3 
+756*b^5*c*e*f^2-840*b^4*c^2*d*f^2-840*b^4*c^2*e^2*f+1920*b^3*c^3*d*e*f+32 
0*b^3*c^3*e^3-1152*b^2*c^4*d^2*f-1152*b^2*c^4*d*e^2+1536*b*c^5*d^2*e-1024* 
c^6*d^3)/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 1583, normalized size of antiderivative = 2.21 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/30720*(15*(1024*c^6*d^3 - 1536*b*c^5*d^2*e + 384*(3*b^2*c^4 - 4*a*c^5) 
*d*e^2 - 64*(5*b^3*c^3 - 12*a*b*c^4)*e^3 + (231*b^6 - 1260*a*b^4*c + 1680* 
a^2*b^2*c^2 - 320*a^3*c^3)*f^3 + 12*(2*(35*b^4*c^2 - 120*a*b^2*c^3 + 48*a^ 
2*c^4)*d - (63*b^5*c - 280*a*b^3*c^2 + 240*a^2*b*c^3)*e)*f^2 + 24*(16*(3*b 
^2*c^4 - 4*a*c^5)*d^2 - 16*(5*b^3*c^3 - 12*a*b*c^4)*d*e + (35*b^4*c^2 - 12 
0*a*b^2*c^3 + 48*a^2*c^4)*e^2)*f)*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*(1280*c^6*f^3*x^ 
5 + 23040*c^6*d^2*e - 17280*b*c^5*d*e^2 + 128*(36*c^6*e*f^2 - 11*b*c^5*f^3 
)*x^4 + 320*(15*b^2*c^4 - 16*a*c^5)*e^3 - 21*(165*b^5*c - 680*a*b^3*c^2 + 
528*a^2*b*c^3)*f^3 + 16*(360*c^6*e^2*f + (99*b^2*c^4 - 100*a*c^5)*f^3 + 36 
*(10*c^6*d - 9*b*c^5*e)*f^2)*x^3 - 12*(50*(21*b^3*c^3 - 44*a*b*c^4)*d - (9 
45*b^4*c^2 - 2940*a*b^2*c^3 + 1024*a^2*c^4)*e)*f^2 + 8*(320*c^6*e^3 - 3*(7 
7*b^3*c^3 - 156*a*b*c^4)*f^3 - 12*(70*b*c^5*d - (63*b^2*c^4 - 64*a*c^5)*e) 
*f^2 + 120*(16*c^6*d*e - 7*b*c^5*e^2)*f)*x^2 - 120*(144*b*c^5*d^2 - 16*(15 
*b^2*c^4 - 16*a*c^5)*d*e + 5*(21*b^3*c^3 - 44*a*b*c^4)*e^2)*f + 2*(5760*c^ 
6*d*e^2 - 1600*b*c^5*e^3 + 3*(385*b^4*c^2 - 1176*a*b^2*c^3 + 400*a^2*c^4)* 
f^3 + 12*(10*(35*b^2*c^4 - 36*a*c^5)*d - 7*(45*b^3*c^3 - 92*a*b*c^4)*e)*f^ 
2 + 120*(48*c^6*d^2 - 80*b*c^5*d*e + (35*b^2*c^4 - 36*a*c^5)*e^2)*f)*x)*sq 
rt(c*x^2 + b*x + a))/c^7, -1/15360*(15*(1024*c^6*d^3 - 1536*b*c^5*d^2*e + 
384*(3*b^2*c^4 - 4*a*c^5)*d*e^2 - 64*(5*b^3*c^3 - 12*a*b*c^4)*e^3 + (23...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1787 vs. \(2 (750) = 1500\).

Time = 0.98 (sec) , antiderivative size = 1787, normalized size of antiderivative = 2.49 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(f**3*x**5/(6*c) + x**4*(-11*b*f**3/(12* 
c) + 3*e*f**2)/(5*c) + x**3*(-5*a*f**3/(6*c) - 9*b*(-11*b*f**3/(12*c) + 3* 
e*f**2)/(10*c) + 3*d*f**2 + 3*e**2*f)/(4*c) + x**2*(-4*a*(-11*b*f**3/(12*c 
) + 3*e*f**2)/(5*c) - 7*b*(-5*a*f**3/(6*c) - 9*b*(-11*b*f**3/(12*c) + 3*e* 
f**2)/(10*c) + 3*d*f**2 + 3*e**2*f)/(8*c) + 6*d*e*f + e**3)/(3*c) + x*(-3* 
a*(-5*a*f**3/(6*c) - 9*b*(-11*b*f**3/(12*c) + 3*e*f**2)/(10*c) + 3*d*f**2 
+ 3*e**2*f)/(4*c) - 5*b*(-4*a*(-11*b*f**3/(12*c) + 3*e*f**2)/(5*c) - 7*b*( 
-5*a*f**3/(6*c) - 9*b*(-11*b*f**3/(12*c) + 3*e*f**2)/(10*c) + 3*d*f**2 + 3 
*e**2*f)/(8*c) + 6*d*e*f + e**3)/(6*c) + 3*d**2*f + 3*d*e**2)/(2*c) + (-2* 
a*(-4*a*(-11*b*f**3/(12*c) + 3*e*f**2)/(5*c) - 7*b*(-5*a*f**3/(6*c) - 9*b* 
(-11*b*f**3/(12*c) + 3*e*f**2)/(10*c) + 3*d*f**2 + 3*e**2*f)/(8*c) + 6*d*e 
*f + e**3)/(3*c) - 3*b*(-3*a*(-5*a*f**3/(6*c) - 9*b*(-11*b*f**3/(12*c) + 3 
*e*f**2)/(10*c) + 3*d*f**2 + 3*e**2*f)/(4*c) - 5*b*(-4*a*(-11*b*f**3/(12*c 
) + 3*e*f**2)/(5*c) - 7*b*(-5*a*f**3/(6*c) - 9*b*(-11*b*f**3/(12*c) + 3*e* 
f**2)/(10*c) + 3*d*f**2 + 3*e**2*f)/(8*c) + 6*d*e*f + e**3)/(6*c) + 3*d**2 
*f + 3*d*e**2)/(4*c) + 3*d**2*e)/c) + (-a*(-3*a*(-5*a*f**3/(6*c) - 9*b*(-1 
1*b*f**3/(12*c) + 3*e*f**2)/(10*c) + 3*d*f**2 + 3*e**2*f)/(4*c) - 5*b*(-4* 
a*(-11*b*f**3/(12*c) + 3*e*f**2)/(5*c) - 7*b*(-5*a*f**3/(6*c) - 9*b*(-11*b 
*f**3/(12*c) + 3*e*f**2)/(10*c) + 3*d*f**2 + 3*e**2*f)/(8*c) + 6*d*e*f + e 
**3)/(6*c) + 3*d**2*f + 3*d*e**2)/(2*c) - b*(-2*a*(-4*a*(-11*b*f**3/(12...
 

Maxima [F(-2)]

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

integrate((f*x^2+e*x+d)^3/(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 = 0.21 (sec) , antiderivative size = 818, normalized size of antiderivative = 1.14 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*f^3*x/c + (36*c^5*e*f^2 - 11* 
b*c^4*f^3)/c^6)*x + (360*c^5*e^2*f + 360*c^5*d*f^2 - 324*b*c^4*e*f^2 + 99* 
b^2*c^3*f^3 - 100*a*c^4*f^3)/c^6)*x + (320*c^5*e^3 + 1920*c^5*d*e*f - 840* 
b*c^4*e^2*f - 840*b*c^4*d*f^2 + 756*b^2*c^3*e*f^2 - 768*a*c^4*e*f^2 - 231* 
b^3*c^2*f^3 + 468*a*b*c^3*f^3)/c^6)*x + (5760*c^5*d*e^2 - 1600*b*c^4*e^3 + 
 5760*c^5*d^2*f - 9600*b*c^4*d*e*f + 4200*b^2*c^3*e^2*f - 4320*a*c^4*e^2*f 
 + 4200*b^2*c^3*d*f^2 - 4320*a*c^4*d*f^2 - 3780*b^3*c^2*e*f^2 + 7728*a*b*c 
^3*e*f^2 + 1155*b^4*c*f^3 - 3528*a*b^2*c^2*f^3 + 1200*a^2*c^3*f^3)/c^6)*x 
+ (23040*c^5*d^2*e - 17280*b*c^4*d*e^2 + 4800*b^2*c^3*e^3 - 5120*a*c^4*e^3 
 - 17280*b*c^4*d^2*f + 28800*b^2*c^3*d*e*f - 30720*a*c^4*d*e*f - 12600*b^3 
*c^2*e^2*f + 26400*a*b*c^3*e^2*f - 12600*b^3*c^2*d*f^2 + 26400*a*b*c^3*d*f 
^2 + 11340*b^4*c*e*f^2 - 35280*a*b^2*c^2*e*f^2 + 12288*a^2*c^3*e*f^2 - 346 
5*b^5*f^3 + 14280*a*b^3*c*f^3 - 11088*a^2*b*c^2*f^3)/c^6) - 1/1024*(1024*c 
^6*d^3 - 1536*b*c^5*d^2*e + 1152*b^2*c^4*d*e^2 - 1536*a*c^5*d*e^2 - 320*b^ 
3*c^3*e^3 + 768*a*b*c^4*e^3 + 1152*b^2*c^4*d^2*f - 1536*a*c^5*d^2*f - 1920 
*b^3*c^3*d*e*f + 4608*a*b*c^4*d*e*f + 840*b^4*c^2*e^2*f - 2880*a*b^2*c^3*e 
^2*f + 1152*a^2*c^4*e^2*f + 840*b^4*c^2*d*f^2 - 2880*a*b^2*c^3*d*f^2 + 115 
2*a^2*c^4*d*f^2 - 756*b^5*c*e*f^2 + 3360*a*b^3*c^2*e*f^2 - 2880*a^2*b*c^3* 
e*f^2 + 231*b^6*f^3 - 1260*a*b^4*c*f^3 + 1680*a^2*b^2*c^2*f^3 - 320*a^3*c^ 
3*f^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(1...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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