\(\int \frac {(d+e x+f x^2)^3}{(a+b x+c x^2)^{3/2}} \, dx\) [153]

Optimal result

Integrand size = 27, antiderivative size = 649 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\left (187 b^3 f^3-4 b c f^2 (114 b e+73 a f)-64 c^3 \left (e^3+6 d e f\right )+16 c^2 f \left (20 a e f+21 b \left (e^2+d f\right )\right )\right ) \sqrt {a+b x+c x^2}}{64 c^5}+\frac {f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x \sqrt {a+b x+c x^2}}{32 c^4}+\frac {f^2 (8 c e-5 b f) x^2 \sqrt {a+b x+c x^2}}{8 c^3}+\frac {f^3 x^3 \sqrt {a+b x+c x^2}}{4 c^2}+\frac {3 \left (105 b^4 f^3-280 b^2 c f^2 (b e+a f)+128 c^4 d \left (e^2+d f\right )+80 c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-64 c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{11/2}} \] Output:

2*(3*a*b^4*c*e*f^2-a*b^5*f^3+a*b^3*c*f*(5*a*f^2-3*c*(d*f+e^2))-b*c^2*(c^3* 
d^3+5*a^3*f^3+3*a*c^2*d*(d*f+e^2)-9*a^2*c*f*(d*f+e^2))-a*b^2*c^2*e*(12*a*f 
^2-c*(6*d*f+e^2))+2*a*c^3*e*(3*c^2*d^2+3*a^2*f^2-a*c*(6*d*f+e^2))-(-2*a*c* 
f+b^2*f-b*c*e+2*c^2*d)*(a^2*c^2*f^2-4*a*b^2*c*f^2+7*a*b*c^2*e*f-2*a*c^3*d* 
f-3*a*c^3*e^2+b^4*f^2-2*b^3*c*e*f+b^2*c^2*d*f+b^2*c^2*e^2-b*c^3*d*e+c^4*d^ 
2)*x)/c^5/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)-1/64*(187*b^3*f^3-4*b*c*f^2*(73 
*a*f+114*b*e)-64*c^3*(6*d*e*f+e^3)+16*c^2*f*(20*a*e*f+21*b*(d*f+e^2)))*(c* 
x^2+b*x+a)^(1/2)/c^5+1/32*f*(41*b^2*f^2-4*c*f*(7*a*f+22*b*e)+48*c^2*(d*f+e 
^2))*x*(c*x^2+b*x+a)^(1/2)/c^4+1/8*f^2*(-5*b*f+8*c*e)*x^2*(c*x^2+b*x+a)^(1 
/2)/c^3+1/4*f^3*x^3*(c*x^2+b*x+a)^(1/2)/c^2+3/128*(105*b^4*f^3-280*b^2*c*f 
^2*(a*f+b*e)+128*c^4*d*(d*f+e^2)+80*c^2*f*(6*a*b*e*f+a^2*f^2+3*b^2*(d*f+e^ 
2))-64*c^3*(3*a*f*(d*f+e^2)+b*(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^(11/2)
 

Mathematica [A] (verified)

Time = 9.51 (sec) , antiderivative size = 771, normalized size of antiderivative = 1.19 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c} \left (315 b^6 f^3 x+105 b^5 f^2 (3 a f+c x (-8 e+f x))-2 b^4 c f \left (105 a f (4 e+9 f x)+c x \left (-360 e^2+140 e f x+3 f \left (-120 d+7 f x^2\right )\right )\right )+8 b^3 c \left (-210 a^2 f^3+a c f \left (90 e^2+530 e f x+f \left (90 d-77 f x^2\right )\right )+c^2 x \left (-24 e^3+30 e^2 f x+3 f^2 x \left (10 d+f x^2\right )+2 e f \left (-72 d+7 f x^2\right )\right )\right )-16 b^2 c^2 \left (-a^2 f^2 (230 e+169 f x)+a c \left (12 e^3+186 e^2 f x+2 e f \left (36 d-43 f x^2\right )+f^2 x \left (186 d-13 f x^2\right )\right )+c^2 x \left (-24 d^2 f+6 d \left (-4 e^2+4 e f x+f^2 x^2\right )+x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )\right )\right )+32 c^3 \left (8 c^3 d^3 x-a^3 f^2 (64 e+15 f x)+a^2 c \left (16 e^3+36 e^2 f x+f^2 x \left (36 d-5 f x^2\right )-32 e f \left (-3 d+f x^2\right )\right )+2 a c^2 \left (-12 d^2 (e+f x)+6 d x \left (-2 e^2+4 e f x+f^2 x^2\right )+x^2 \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )\right )\right )+16 b c^2 \left (113 a^3 f^3+8 c^3 d^2 (d-3 e x)+a^2 c f \left (-156 e^2-244 e f x+f \left (-156 d+49 f x^2\right )\right )+2 a c^2 \left (12 d^2 f+6 d \left (2 e^2+20 e f x-5 f^2 x^2\right )-x \left (-20 e^3+30 e^2 f x+14 e f^2 x^2+3 f^3 x^3\right )\right )\right )\right )-3 \left (b^2-4 a c\right ) \left (105 b^4 f^3-280 b^2 c f^2 (b e+a f)+128 c^4 d \left (e^2+d f\right )+80 c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-64 c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right ) \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{64 c^{11/2} \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}} \] Input:

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

Output:

(Sqrt[c]*(315*b^6*f^3*x + 105*b^5*f^2*(3*a*f + c*x*(-8*e + f*x)) - 2*b^4*c 
*f*(105*a*f*(4*e + 9*f*x) + c*x*(-360*e^2 + 140*e*f*x + 3*f*(-120*d + 7*f* 
x^2))) + 8*b^3*c*(-210*a^2*f^3 + a*c*f*(90*e^2 + 530*e*f*x + f*(90*d - 77* 
f*x^2)) + c^2*x*(-24*e^3 + 30*e^2*f*x + 3*f^2*x*(10*d + f*x^2) + 2*e*f*(-7 
2*d + 7*f*x^2))) - 16*b^2*c^2*(-(a^2*f^2*(230*e + 169*f*x)) + a*c*(12*e^3 
+ 186*e^2*f*x + 2*e*f*(36*d - 43*f*x^2) + f^2*x*(186*d - 13*f*x^2)) + c^2* 
x*(-24*d^2*f + 6*d*(-4*e^2 + 4*e*f*x + f^2*x^2) + x*(4*e^3 + 6*e^2*f*x + 4 
*e*f^2*x^2 + f^3*x^3))) + 32*c^3*(8*c^3*d^3*x - a^3*f^2*(64*e + 15*f*x) + 
a^2*c*(16*e^3 + 36*e^2*f*x + f^2*x*(36*d - 5*f*x^2) - 32*e*f*(-3*d + f*x^2 
)) + 2*a*c^2*(-12*d^2*(e + f*x) + 6*d*x*(-2*e^2 + 4*e*f*x + f^2*x^2) + x^2 
*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))) + 16*b*c^2*(113*a^3*f^3 + 8 
*c^3*d^2*(d - 3*e*x) + a^2*c*f*(-156*e^2 - 244*e*f*x + f*(-156*d + 49*f*x^ 
2)) + 2*a*c^2*(12*d^2*f + 6*d*(2*e^2 + 20*e*f*x - 5*f^2*x^2) - x*(-20*e^3 
+ 30*e^2*f*x + 14*e*f^2*x^2 + 3*f^3*x^3)))) - 3*(b^2 - 4*a*c)*(105*b^4*f^3 
 - 280*b^2*c*f^2*(b*e + a*f) + 128*c^4*d*(e^2 + d*f) + 80*c^2*f*(6*a*b*e*f 
 + a^2*f^2 + 3*b^2*(e^2 + d*f)) - 64*c^3*(3*a*f*(e^2 + d*f) + b*(e^3 + 6*d 
*e*f)))*Sqrt[a + x*(b + c*x)]*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*( 
b + c*x)])])/(64*c^(11/2)*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 2.95 (sec) , antiderivative size = 709, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2191, 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.

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

\(\Big \downarrow \) 2191

\(\displaystyle \frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {\frac {\left (b^2-4 a c\right ) f^3 x^4}{c}+\frac {\left (b^2-4 a c\right ) f^2 (3 c e-b f) x^3}{c^2}+\frac {\left (b^2-4 a c\right ) f \left (3 \left (e^2+d f\right ) c^2-f (3 b e+a f) c+b^2 f^2\right ) x^2}{c^3}-\frac {\left (b^2-4 a c\right ) \left (-\left (\left (e^3+6 d f e\right ) c^3\right )+3 f \left (a e f+b \left (e^2+d f\right )\right ) c^2-b f^2 (3 b e+2 a f) c+b^3 f^3\right ) x}{c^4}+\frac {\left (b^2-4 a c\right ) \left (f^3 b^4-3 c f^2 (b e+a f) b^2+3 c^4 d \left (e^2+d f\right )+c^2 f \left (3 \left (e^2+d f\right ) b^2+6 a e f b+a^2 f^2\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d f e\right )\right )\right )}{c^5}}{2 \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\frac {\left (b^2-4 a c\right ) f^3 x^4}{c}+\frac {\left (b^2-4 a c\right ) f^2 (3 c e-b f) x^3}{c^2}+\frac {\left (b^2-4 a c\right ) f \left (3 \left (e^2+d f\right ) c^2-f (3 b e+a f) c+b^2 f^2\right ) x^2}{c^3}-\frac {\left (b^2-4 a c\right ) \left (-\left (\left (e^3+6 d f e\right ) c^3\right )+3 f \left (a e f+b \left (e^2+d f\right )\right ) c^2-b f^2 (3 b e+2 a f) c+b^3 f^3\right ) x}{c^4}+\frac {\left (b^2-4 a c\right ) \left (f^3 b^4-3 c f^2 (b e+a f) b^2+3 c^4 d \left (e^2+d f\right )+c^2 f \left (3 \left (e^2+d f\right ) b^2+6 a e f b+a^2 f^2\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d f e\right )\right )\right )}{c^5}}{\sqrt {c x^2+b x+a}}dx}{b^2-4 a c}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\int \frac {\frac {3 \left (b^2-4 a c\right ) f^2 (8 c e-5 b f) x^3}{c}+\frac {2 \left (b^2-4 a c\right ) f \left (12 \left (e^2+d f\right ) c^2-f (12 b e+7 a f) c+4 b^2 f^2\right ) x^2}{c^2}-\frac {8 \left (b^2-4 a c\right ) \left (-\left (\left (e^3+6 d f e\right ) c^3\right )+3 f \left (a e f+b \left (e^2+d f\right )\right ) c^2-b f^2 (3 b e+2 a f) c+b^3 f^3\right ) x}{c^3}+\frac {8 \left (b^2-4 a c\right ) \left (f^3 b^4-3 c f^2 (b e+a f) b^2+3 c^4 d \left (e^2+d f\right )+c^2 f \left (3 \left (e^2+d f\right ) b^2+6 a e f b+a^2 f^2\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d f e\right )\right )\right )}{c^4}}{2 \sqrt {c x^2+b x+a}}dx}{4 c}+\frac {f^3 x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{4 c^2}}{b^2-4 a c}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\frac {3 \left (b^2-4 a c\right ) f^2 (8 c e-5 b f) x^3}{c}+\frac {2 \left (b^2-4 a c\right ) f \left (12 \left (e^2+d f\right ) c^2-f (12 b e+7 a f) c+4 b^2 f^2\right ) x^2}{c^2}-\frac {8 \left (b^2-4 a c\right ) \left (-\left (\left (e^3+6 d f e\right ) c^3\right )+3 f \left (a e f+b \left (e^2+d f\right )\right ) c^2-b f^2 (3 b e+2 a f) c+b^3 f^3\right ) x}{c^3}+\frac {8 \left (b^2-4 a c\right ) \left (f^3 b^4-3 c f^2 (b e+a f) b^2+3 c^4 d \left (e^2+d f\right )+c^2 f \left (3 \left (e^2+d f\right ) b^2+6 a e f b+a^2 f^2\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d f e\right )\right )\right )}{c^4}}{\sqrt {c x^2+b x+a}}dx}{8 c}+\frac {f^3 x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{4 c^2}}{b^2-4 a c}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\int \frac {3 \left (\frac {\left (b^2-4 a c\right ) f \left (48 \left (e^2+d f\right ) c^2-4 f (22 b e+7 a f) c+41 b^2 f^2\right ) x^2}{c}-\frac {4 \left (b^2-4 a c\right ) \left (-4 \left (e^3+6 d f e\right ) c^3+4 f \left (5 a e f+3 b \left (e^2+d f\right )\right ) c^2-b f^2 (12 b e+13 a f) c+4 b^3 f^3\right ) x}{c^2}+\frac {16 \left (b^2-4 a c\right ) \left (f^3 b^4-3 c f^2 (b e+a f) b^2+3 c^4 d \left (e^2+d f\right )+c^2 f \left (3 \left (e^2+d f\right ) b^2+6 a e f b+a^2 f^2\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d f e\right )\right )\right )}{c^3}\right )}{2 \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {f^2 x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (8 c e-5 b f)}{c^2}}{8 c}+\frac {f^3 x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{4 c^2}}{b^2-4 a c}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {\frac {\left (b^2-4 a c\right ) f \left (48 \left (e^2+d f\right ) c^2-4 f (22 b e+7 a f) c+41 b^2 f^2\right ) x^2}{c}-\frac {4 \left (b^2-4 a c\right ) \left (-4 \left (e^3+6 d f e\right ) c^3+4 f \left (5 a e f+3 b \left (e^2+d f\right )\right ) c^2-b f^2 (12 b e+13 a f) c+4 b^3 f^3\right ) x}{c^2}+\frac {16 \left (b^2-4 a c\right ) \left (f^3 b^4-3 c f^2 (b e+a f) b^2+3 c^4 d \left (e^2+d f\right )+c^2 f \left (3 \left (e^2+d f\right ) b^2+6 a e f b+a^2 f^2\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d f e\right )\right )\right )}{c^3}}{\sqrt {c x^2+b x+a}}dx}{2 c}+\frac {f^2 x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (8 c e-5 b f)}{c^2}}{8 c}+\frac {f^3 x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{4 c^2}}{b^2-4 a c}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (b^2-4 a c\right ) \left (2 \left (32 f^3 b^4-c f^2 (96 b e+137 a f) b^2+96 c^4 d \left (e^2+d f\right )+4 c^2 f \left (24 \left (e^2+d f\right ) b^2+70 a e f b+15 a^2 f^2\right )-16 c^3 \left (9 a f \left (e^2+d f\right )+2 b \left (e^3+6 d f e\right )\right )\right )-c \left (-64 \left (e^3+6 d f e\right ) c^3+16 f \left (20 a e f+21 b \left (e^2+d f\right )\right ) c^2-4 b f^2 (114 b e+73 a f) c+187 b^3 f^3\right ) x\right )}{2 c^2 \sqrt {c x^2+b x+a}}dx}{2 c}+\frac {f x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{2 c^2}}{2 c}+\frac {f^2 x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (8 c e-5 b f)}{c^2}}{8 c}+\frac {f^3 x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{4 c^2}}{b^2-4 a c}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\left (b^2-4 a c\right ) \int \frac {2 \left (32 f^3 b^4-c f^2 (96 b e+137 a f) b^2+96 c^4 d \left (e^2+d f\right )+4 c^2 f \left (24 \left (e^2+d f\right ) b^2+70 a e f b+15 a^2 f^2\right )-16 c^3 \left (9 a f \left (e^2+d f\right )+2 b \left (e^3+6 d f e\right )\right )\right )-c \left (-64 \left (e^3+6 d f e\right ) c^3+16 f \left (20 a e f+21 b \left (e^2+d f\right )\right ) c^2-4 b f^2 (114 b e+73 a f) c+187 b^3 f^3\right ) x}{\sqrt {c x^2+b x+a}}dx}{4 c^3}+\frac {f x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{2 c^2}}{2 c}+\frac {f^2 x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (8 c e-5 b f)}{c^2}}{8 c}+\frac {f^3 x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{4 c^2}}{b^2-4 a c}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\frac {\frac {\left (b^2-4 a c\right ) \left (\frac {3}{2} \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx-\sqrt {a+b x+c x^2} \left (16 c^2 f \left (20 a e f+21 b \left (d f+e^2\right )\right )-4 b c f^2 (73 a f+114 b e)+187 b^3 f^3-64 c^3 \left (6 d e f+e^3\right )\right )\right )}{4 c^3}+\frac {f x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{2 c^2}}{2 c}+\frac {f^2 x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (8 c e-5 b f)}{c^2}}{8 c}+\frac {f^3 x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{4 c^2}}{b^2-4 a c}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {\frac {\left (b^2-4 a c\right ) \left (3 \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}-\sqrt {a+b x+c x^2} \left (16 c^2 f \left (20 a e f+21 b \left (d f+e^2\right )\right )-4 b c f^2 (73 a f+114 b e)+187 b^3 f^3-64 c^3 \left (6 d e f+e^3\right )\right )\right )}{4 c^3}+\frac {f x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{2 c^2}}{2 c}+\frac {f^2 x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (8 c e-5 b f)}{c^2}}{8 c}+\frac {f^3 x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{4 c^2}}{b^2-4 a c}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {\left (b^2-4 a c\right ) \left (\frac {3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right )}{2 \sqrt {c}}-\sqrt {a+b x+c x^2} \left (16 c^2 f \left (20 a e f+21 b \left (d f+e^2\right )\right )-4 b c f^2 (73 a f+114 b e)+187 b^3 f^3-64 c^3 \left (6 d e f+e^3\right )\right )\right )}{4 c^3}+\frac {f x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{2 c^2}}{2 c}+\frac {f^2 x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (8 c e-5 b f)}{c^2}}{8 c}+\frac {f^3 x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{4 c^2}}{b^2-4 a c}+\frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

Input:

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

Output:

(2*(3*a*b^4*c*e*f^2 - a*b^5*f^3 + a*b^3*c*f*(5*a*f^2 - 3*c*(e^2 + d*f)) - 
b*c^2*(c^3*d^3 + 5*a^3*f^3 + 3*a*c^2*d*(e^2 + d*f) - 9*a^2*c*f*(e^2 + d*f) 
) - a*b^2*c^2*e*(12*a*f^2 - c*(e^2 + 6*d*f)) + 2*a*c^3*e*(3*c^2*d^2 + 3*a^ 
2*f^2 - a*c*(e^2 + 6*d*f)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*(c^4*d^2 
- b*c^3*d*e + b^2*c^2*e^2 - 3*a*c^3*e^2 + b^2*c^2*d*f - 2*a*c^3*d*f - 2*b^ 
3*c*e*f + 7*a*b*c^2*e*f + b^4*f^2 - 4*a*b^2*c*f^2 + a^2*c^2*f^2)*x))/(c^5* 
(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (((b^2 - 4*a*c)*f^3*x^3*Sqrt[a + b* 
x + c*x^2])/(4*c^2) + (((b^2 - 4*a*c)*f^2*(8*c*e - 5*b*f)*x^2*Sqrt[a + b*x 
 + c*x^2])/c^2 + (((b^2 - 4*a*c)*f*(41*b^2*f^2 - 4*c*f*(22*b*e + 7*a*f) + 
48*c^2*(e^2 + d*f))*x*Sqrt[a + b*x + c*x^2])/(2*c^2) + ((b^2 - 4*a*c)*(-(( 
187*b^3*f^3 - 4*b*c*f^2*(114*b*e + 73*a*f) - 64*c^3*(e^3 + 6*d*e*f) + 16*c 
^2*f*(20*a*e*f + 21*b*(e^2 + d*f)))*Sqrt[a + b*x + c*x^2]) + (3*(105*b^4*f 
^3 - 280*b^2*c*f^2*(b*e + a*f) + 128*c^4*d*(e^2 + d*f) + 80*c^2*f*(6*a*b*e 
*f + a^2*f^2 + 3*b^2*(e^2 + d*f)) - 64*c^3*(3*a*f*(e^2 + d*f) + b*(e^3 + 6 
*d*e*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[ 
c])))/(4*c^3))/(2*c))/(8*c))/(b^2 - 4*a*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 2191

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = 
PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P 
q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + 
c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ 
(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int 
[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* 
(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 
2 - 4*a*c, 0] && LtQ[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 = 3.07 (sec) , antiderivative size = 1136, normalized size of antiderivative = 1.75

method	result	size
risch	\(\text {Expression too large to display}\)	\(1136\)
default	\(\text {Expression too large to display}\)	\(2266\)

Input:

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

Output:

1/64*(16*c^3*f^3*x^3-40*b*c^2*f^3*x^2+64*c^3*e*f^2*x^2-56*a*c^2*f^3*x+82*b 
^2*c*f^3*x-176*b*c^2*e*f^2*x+96*c^3*d*f^2*x+96*c^3*e^2*f*x+292*a*b*c*f^3-3 
20*a*c^2*e*f^2-187*b^3*f^3+456*b^2*c*e*f^2-336*b*c^2*d*f^2-336*b*c^2*e^2*f 
+384*c^3*d*e*f+64*c^3*e^3)/c^5*(c*x^2+b*x+a)^(1/2)+1/128/c^5*(3*c*(80*a^2* 
c^2*f^3-280*a*b^2*c*f^3+480*a*b*c^2*e*f^2-192*a*c^3*d*f^2-192*a*c^3*e^2*f+ 
105*b^4*f^3-280*b^3*c*e*f^2+240*b^2*c^2*d*f^2+240*b^2*c^2*e^2*f-384*b*c^3* 
d*e*f-64*b*c^3*e^3+128*c^4*d^2*f+128*c^4*d*e^2)*(-x/c/(c*x^2+b*x+a)^(1/2)- 
1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^ 
(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+(-144*a^2*b* 
c^2*f^3+384*a^2*c^3*e*f^2-328*a*b^3*c*f^3+288*a*b^2*c^2*e*f^2+192*a*b*c^3* 
d*f^2+192*a*b*c^3*e^2*f-768*a*c^4*d*e*f-128*a*c^4*e^3+187*b^5*f^3-456*b^4* 
c*e*f^2+336*b^3*c^2*d*f^2+336*b^3*c^2*e^2*f-384*b^2*c^3*d*e*f-64*b^2*c^3*e 
^3+384*c^5*d^2*e)*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x 
^2+b*x+a)^(1/2))+256*c^5*d^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+374 
*a*b^4*f^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+224*a^3*c^2*f^3*(2*c* 
x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-128*a*b*c^3*e^3*(2*c*x+b)/(4*a*c-b^2) 
/(c*x^2+b*x+a)^(1/2)-912*a^2*b^2*c*f^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a) 
^(1/2)-384*a^2*c^3*d*f^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-384*a^2 
*c^3*e^2*f*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+672*a*b^2*c^2*d*f^2*( 
2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+672*a*b^2*c^2*e^2*f*(2*c*x+b)/...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1570 vs. \(2 (621) = 1242\).

Time = 0.97 (sec) , antiderivative size = 3143, normalized size of antiderivative = 4.84 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((f*x^2+e*x+d)^3/(c*x^2+b*x+a)^(3/2),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 = 0.17 (sec) , antiderivative size = 1093, normalized size of antiderivative = 1.68 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/64*(((2*(4*(2*(b^2*c^4*f^3 - 4*a*c^5*f^3)*x/(b^2*c^5 - 4*a*c^6) + (8*b^2 
*c^4*e*f^2 - 32*a*c^5*e*f^2 - 3*b^3*c^3*f^3 + 12*a*b*c^4*f^3)/(b^2*c^5 - 4 
*a*c^6))*x + (48*b^2*c^4*e^2*f - 192*a*c^5*e^2*f + 48*b^2*c^4*d*f^2 - 192* 
a*c^5*d*f^2 - 56*b^3*c^3*e*f^2 + 224*a*b*c^4*e*f^2 + 21*b^4*c^2*f^3 - 104* 
a*b^2*c^3*f^3 + 80*a^2*c^4*f^3)/(b^2*c^5 - 4*a*c^6))*x + (64*b^2*c^4*e^3 - 
 256*a*c^5*e^3 + 384*b^2*c^4*d*e*f - 1536*a*c^5*d*e*f - 240*b^3*c^3*e^2*f 
+ 960*a*b*c^4*e^2*f - 240*b^3*c^3*d*f^2 + 960*a*b*c^4*d*f^2 + 280*b^4*c^2* 
e*f^2 - 1376*a*b^2*c^3*e*f^2 + 1024*a^2*c^4*e*f^2 - 105*b^5*c*f^3 + 616*a* 
b^3*c^2*f^3 - 784*a^2*b*c^3*f^3)/(b^2*c^5 - 4*a*c^6))*x - (256*c^6*d^3 - 3 
84*b*c^5*d^2*e + 384*b^2*c^4*d*e^2 - 768*a*c^5*d*e^2 - 192*b^3*c^3*e^3 + 6 
40*a*b*c^4*e^3 + 384*b^2*c^4*d^2*f - 768*a*c^5*d^2*f - 1152*b^3*c^3*d*e*f 
+ 3840*a*b*c^4*d*e*f + 720*b^4*c^2*e^2*f - 2976*a*b^2*c^3*e^2*f + 1152*a^2 
*c^4*e^2*f + 720*b^4*c^2*d*f^2 - 2976*a*b^2*c^3*d*f^2 + 1152*a^2*c^4*d*f^2 
 - 840*b^5*c*e*f^2 + 4240*a*b^3*c^2*e*f^2 - 3904*a^2*b*c^3*e*f^2 + 315*b^6 
*f^3 - 1890*a*b^4*c*f^3 + 2704*a^2*b^2*c^2*f^3 - 480*a^3*c^3*f^3)/(b^2*c^5 
 - 4*a*c^6))*x - (128*b*c^5*d^3 - 768*a*c^5*d^2*e + 384*a*b*c^4*d*e^2 - 19 
2*a*b^2*c^3*e^3 + 512*a^2*c^4*e^3 + 384*a*b*c^4*d^2*f - 1152*a*b^2*c^3*d*e 
*f + 3072*a^2*c^4*d*e*f + 720*a*b^3*c^2*e^2*f - 2496*a^2*b*c^3*e^2*f + 720 
*a*b^3*c^2*d*f^2 - 2496*a^2*b*c^3*d*f^2 - 840*a*b^4*c*e*f^2 + 3680*a^2*b^2 
*c^2*e*f^2 - 2048*a^3*c^3*e*f^2 + 315*a*b^5*f^3 - 1680*a^2*b^3*c*f^3 + ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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