\(\int (a+b x+c x^2)^{3/2} (d+e x+f x^2)^2 \, dx\) [140]

Optimal result

Integrand size = 27, antiderivative size = 564 \[ \int \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )^2 \, dx=-\frac {\left (b^2-4 a c\right ) \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^6}+\frac {\left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}+\frac {\left (5376 c^3 d e-693 b^3 f^2+36 b c f (56 b e+31 a f)-32 c^2 \left (48 a e f+49 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4}+\frac {\left (99 b^2 f^2-12 c f (24 b e+7 a f)+224 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{5/2}}{1344 c^3}+\frac {f (32 c e-11 b f) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\left (b^2-4 a c\right )^2 \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2}} \] Output:

-1/16384*(-4*a*c+b^2)*(768*c^4*d^2+99*b^4*f^2-72*b^2*c*f*(3*a*f+4*b*e)-128 
*c^3*(6*b*d*e+a*(2*d*f+e^2))+16*c^2*(24*a*b*e*f+3*a^2*f^2+14*b^2*(2*d*f+e^ 
2)))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^6+1/6144*(768*c^4*d^2+99*b^4*f^2-72*b 
^2*c*f*(3*a*f+4*b*e)-128*c^3*(6*b*d*e+a*(2*d*f+e^2))+16*c^2*(24*a*b*e*f+3* 
a^2*f^2+14*b^2*(2*d*f+e^2)))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^5+1/13440*(53 
76*c^3*d*e-693*b^3*f^2+36*b*c*f*(31*a*f+56*b*e)-32*c^2*(48*a*e*f+49*b*(2*d 
*f+e^2)))*(c*x^2+b*x+a)^(5/2)/c^4+1/1344*(99*b^2*f^2-12*c*f*(7*a*f+24*b*e) 
+224*c^2*(2*d*f+e^2))*x*(c*x^2+b*x+a)^(5/2)/c^3+1/112*f*(-11*b*f+32*c*e)*x 
^2*(c*x^2+b*x+a)^(5/2)/c^2+1/8*f^2*x^3*(c*x^2+b*x+a)^(5/2)/c+1/32768*(-4*a 
*c+b^2)^2*(768*c^4*d^2+99*b^4*f^2-72*b^2*c*f*(3*a*f+4*b*e)-128*c^3*(6*b*d* 
e+a*(2*d*f+e^2))+16*c^2*(24*a*b*e*f+3*a^2*f^2+14*b^2*(2*d*f+e^2)))*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 = 12.01 (sec) , antiderivative size = 829, normalized size of antiderivative = 1.47 \[ \int \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )^2 \, dx=\frac {430080 d^2 (b+2 c x) (a+x (b+c x))^{3/2}+1376256 d e (a+x (b+c x))^{5/2}+573440 \left (e^2+2 d f\right ) x (a+x (b+c x))^{5/2}+983040 e f x^2 (a+x (b+c x))^{5/2}+430080 f^2 x^3 (a+x (b+c x))^{5/2}+\frac {80640 \left (b^2-4 a c\right ) d^2 \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{3/2}}-\frac {26880 b d e \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )}{c^{5/2}}+\frac {96 e f \left (-256 c^{5/2} \left (-21 b^2+16 a c+30 b c x\right ) (a+x (b+c x))^{5/2}-35 b \left (3 b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{c^{9/2}}-\frac {224 \left (e^2+2 d f\right ) \left (1792 b c^{5/2} (a+x (b+c x))^{5/2}-5 \left (7 b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{c^{7/2}}-\frac {3 f^2 \left (112640 b c^{9/2} x^2 (a+x (b+c x))^{5/2}+256 c^{5/2} \left (231 b^3-372 a b c-330 b^2 c x+280 a c^2 x\right ) (a+x (b+c x))^{5/2}-35 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{c^{11/2}}}{3440640 c} \] Input:

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

Output:

(430080*d^2*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) + 1376256*d*e*(a + x*(b + 
c*x))^(5/2) + 573440*(e^2 + 2*d*f)*x*(a + x*(b + c*x))^(5/2) + 983040*e*f* 
x^2*(a + x*(b + c*x))^(5/2) + 430080*f^2*x^3*(a + x*(b + c*x))^(5/2) + (80 
640*(b^2 - 4*a*c)*d^2*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 
 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(3/2) 
 - (26880*b*d*e*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 
 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTa 
nh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])))/c^(5/2) + (96*e*f*(-2 
56*c^(5/2)*(-21*b^2 + 16*a*c + 30*b*c*x)*(a + x*(b + c*x))^(5/2) - 35*b*(3 
*b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4 
*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh 
[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))))/c^(9/2) - (224*(e^2 + 
2*d*f)*(1792*b*c^(5/2)*(a + x*(b + c*x))^(5/2) - 5*(7*b^2 - 4*a*c)*(16*c^( 
3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c]*(b + 
 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[ 
c]*Sqrt[a + x*(b + c*x)])]))))/c^(7/2) - (3*f^2*(112640*b*c^(9/2)*x^2*(a + 
 x*(b + c*x))^(5/2) + 256*c^(5/2)*(231*b^3 - 372*a*b*c - 330*b^2*c*x + 280 
*a*c^2*x)*(a + x*(b + c*x))^(5/2) - 35*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)* 
(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[ 
c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x...
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.76, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2192, 27, 2192, 27, 2192, 27, 1160, 1087, 1087, 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.

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\int \frac {1}{2} \left (c x^2+b x+a\right )^{3/2} \left (f (32 c e-11 b f) x^3-2 \left (3 a f^2-8 c \left (e^2+2 d f\right )\right ) x^2+32 c d e x+16 c d^2\right )dx}{8 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (c x^2+b x+a\right )^{3/2} \left (f (32 c e-11 b f) x^3-2 \left (3 a f^2-8 c \left (e^2+2 d f\right )\right ) x^2+32 c d e x+16 c d^2\right )dx}{16 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\int \frac {1}{2} \left (c x^2+b x+a\right )^{3/2} \left (224 c^2 d^2+\left (224 \left (e^2+2 d f\right ) c^2-12 f (24 b e+7 a f) c+99 b^2 f^2\right ) x^2+4 \left (112 d e c^2-32 a e f c+11 a b f^2\right ) x\right )dx}{7 c}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{7 c}}{16 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \left (c x^2+b x+a\right )^{3/2} \left (224 c^2 d^2+\left (224 \left (e^2+2 d f\right ) c^2-12 f (24 b e+7 a f) c+99 b^2 f^2\right ) x^2+4 \left (112 d e c^2-32 a e f c+11 a b f^2\right ) x\right )dx}{14 c}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{7 c}}{16 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\int \frac {1}{2} \left (2688 d^2 c^3-448 a \left (e^2+2 d f\right ) c^2+24 a f (24 b e+7 a f) c-198 a b^2 f^2+\left (-693 f^2 b^3+36 c f (56 b e+31 a f) b+5376 c^3 d e-32 c^2 \left (48 a e f+49 b \left (e^2+2 d f\right )\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{6 c}+\frac {x \left (a+b x+c x^2\right )^{5/2} \left (-12 c f (7 a f+24 b e)+99 b^2 f^2+224 c^2 \left (2 d f+e^2\right )\right )}{6 c}}{14 c}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{7 c}}{16 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \left (2 \left (1344 d^2 c^3-224 a \left (e^2+2 d f\right ) c^2+12 a f (24 b e+7 a f) c-99 a b^2 f^2\right )+\left (-693 f^2 b^3+36 c f (56 b e+31 a f) b+5376 c^3 d e-32 c^2 \left (48 a e f+49 b \left (e^2+2 d f\right )\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{12 c}+\frac {x \left (a+b x+c x^2\right )^{5/2} \left (-12 c f (7 a f+24 b e)+99 b^2 f^2+224 c^2 \left (2 d f+e^2\right )\right )}{6 c}}{14 c}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{7 c}}{16 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\frac {\frac {7 \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{2 c}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-32 c^2 \left (48 a e f+49 b \left (2 d f+e^2\right )\right )+36 b c f (31 a f+56 b e)-693 b^3 f^2+5376 c^3 d e\right )}{5 c}}{12 c}+\frac {x \left (a+b x+c x^2\right )^{5/2} \left (-12 c f (7 a f+24 b e)+99 b^2 f^2+224 c^2 \left (2 d f+e^2\right )\right )}{6 c}}{14 c}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{7 c}}{16 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\frac {\frac {7 \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{2 c}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-32 c^2 \left (48 a e f+49 b \left (2 d f+e^2\right )\right )+36 b c f (31 a f+56 b e)-693 b^3 f^2+5376 c^3 d e\right )}{5 c}}{12 c}+\frac {x \left (a+b x+c x^2\right )^{5/2} \left (-12 c f (7 a f+24 b e)+99 b^2 f^2+224 c^2 \left (2 d f+e^2\right )\right )}{6 c}}{14 c}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{7 c}}{16 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\frac {\frac {7 \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{2 c}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-32 c^2 \left (48 a e f+49 b \left (2 d f+e^2\right )\right )+36 b c f (31 a f+56 b e)-693 b^3 f^2+5376 c^3 d e\right )}{5 c}}{12 c}+\frac {x \left (a+b x+c x^2\right )^{5/2} \left (-12 c f (7 a f+24 b e)+99 b^2 f^2+224 c^2 \left (2 d f+e^2\right )\right )}{6 c}}{14 c}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{7 c}}{16 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {\frac {7 \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{2 c}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-32 c^2 \left (48 a e f+49 b \left (2 d f+e^2\right )\right )+36 b c f (31 a f+56 b e)-693 b^3 f^2+5376 c^3 d e\right )}{5 c}}{12 c}+\frac {x \left (a+b x+c x^2\right )^{5/2} \left (-12 c f (7 a f+24 b e)+99 b^2 f^2+224 c^2 \left (2 d f+e^2\right )\right )}{6 c}}{14 c}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{7 c}}{16 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {7 \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right ) \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )}{2 c}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-32 c^2 \left (48 a e f+49 b \left (2 d f+e^2\right )\right )+36 b c f (31 a f+56 b e)-693 b^3 f^2+5376 c^3 d e\right )}{5 c}}{12 c}+\frac {x \left (a+b x+c x^2\right )^{5/2} \left (-12 c f (7 a f+24 b e)+99 b^2 f^2+224 c^2 \left (2 d f+e^2\right )\right )}{6 c}}{14 c}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{7 c}}{16 c}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

Input:

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

Output:

(f^2*x^3*(a + b*x + c*x^2)^(5/2))/(8*c) + ((f*(32*c*e - 11*b*f)*x^2*(a + b 
*x + c*x^2)^(5/2))/(7*c) + (((99*b^2*f^2 - 12*c*f*(24*b*e + 7*a*f) + 224*c 
^2*(e^2 + 2*d*f))*x*(a + b*x + c*x^2)^(5/2))/(6*c) + (((5376*c^3*d*e - 693 
*b^3*f^2 + 36*b*c*f*(56*b*e + 31*a*f) - 32*c^2*(48*a*e*f + 49*b*(e^2 + 2*d 
*f)))*(a + b*x + c*x^2)^(5/2))/(5*c) + (7*(768*c^4*d^2 + 99*b^4*f^2 - 72*b 
^2*c*f*(4*b*e + 3*a*f) - 128*c^3*(6*b*d*e + a*(e^2 + 2*d*f)) + 16*c^2*(24* 
a*b*e*f + 3*a^2*f^2 + 14*b^2*(e^2 + 2*d*f)))*(((b + 2*c*x)*(a + b*x + c*x^ 
2)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4 
*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2]) 
])/(8*c^(3/2))))/(16*c)))/(2*c))/(12*c))/(14*c))/(16*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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1199\) vs. \(2(530)=1060\).

Time = 2.43 (sec) , antiderivative size = 1200, normalized size of antiderivative = 2.13

method	result	size
risch	\(\text {Expression too large to display}\)	\(1200\)
default	\(\text {Expression too large to display}\)	\(1654\)

Input:

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

Output:

1/1720320/c^6*(215040*c^7*f^2*x^7+261120*b*c^6*f^2*x^6+491520*c^7*e*f*x^6+ 
322560*a*c^6*f^2*x^5+3840*b^2*c^5*f^2*x^5+614400*b*c^6*e*f*x^5+573440*c^7* 
d*f*x^5+286720*c^7*e^2*x^5+19968*a*b*c^5*f^2*x^4+786432*a*c^6*e*f*x^4-4224 
*b^3*c^4*f^2*x^4+12288*b^2*c^5*e*f*x^4+745472*b*c^6*d*f*x^4+372736*b*c^6*e 
^2*x^4+688128*c^7*d*e*x^4+26880*a^2*c^5*f^2*x^3-27264*a*b^2*c^4*f^2*x^3+67 
584*a*b*c^5*e*f*x^3+1003520*a*c^6*d*f*x^3+501760*a*c^6*e^2*x^3+4752*b^4*c^ 
3*f^2*x^3-13824*b^3*c^4*e*f*x^3+21504*b^2*c^5*d*f*x^3+10752*b^2*c^5*e^2*x^ 
3+946176*b*c^6*d*e*x^3+430080*c^7*d^2*x^3-57984*a^2*b*c^4*f^2*x^2+98304*a^ 
2*c^5*e*f*x^2+37440*a*b^3*c^3*f^2*x^2-95232*a*b^2*c^4*e*f*x^2+129024*a*b*c 
^5*d*f*x^2+64512*a*b*c^5*e^2*x^2+1376256*a*c^6*d*e*x^2-5544*b^5*c^2*f^2*x^ 
2+16128*b^4*c^3*e*f*x^2-25088*b^3*c^4*d*f*x^2-12544*b^3*c^4*e^2*x^2+43008* 
b^2*c^5*d*e*x^2+645120*b*c^6*d^2*x^2-40320*a^3*c^4*f^2*x+113376*a^2*b^2*c^ 
3*f^2*x-224256*a^2*b*c^4*e*f*x+215040*a^2*c^5*d*f*x+107520*a^2*c^5*e^2*x-5 
3928*a*b^4*c^2*f^2*x+139776*a*b^3*c^3*e*f*x-193536*a*b^2*c^4*d*f*x-96768*a 
*b^2*c^4*e^2*x+301056*a*b*c^5*d*e*x+1075200*a*c^6*d^2*x+6930*b^6*c*f^2*x-2 
0160*b^5*c^2*e*f*x+31360*b^4*c^3*d*f*x+15680*b^4*c^3*e^2*x-53760*b^3*c^4*d 
*e*x+53760*b^2*c^5*d^2*x+176448*a^3*b*c^3*f^2-196608*a^3*c^4*e*f-244944*a^ 
2*b^3*c^2*f^2+526848*a^2*b^2*c^3*e*f-580608*a^2*b*c^4*d*f-290304*a^2*b*c^4 
*e^2+688128*a^2*c^5*d*e+91980*a*b^5*c*f^2-241920*a*b^4*c^2*e*f+340480*a*b^ 
3*c^3*d*f+170240*a*b^3*c^3*e^2-537600*a*b^2*c^4*d*e+537600*a*b*c^5*d^2-...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1088 vs. \(2 (530) = 1060\).

Time = 0.62 (sec) , antiderivative size = 2179, normalized size of antiderivative = 3.86 \[ \int \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

[1/6881280*(105*(768*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^2 - 768*(b^5*c 
^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d*e + 32*(7*b^6*c^2 - 60*a*b^4*c^3 + 144* 
a^2*b^2*c^4 - 64*a^3*c^5)*e^2 + 3*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 
 - 1280*a^3*b^2*c^3 + 256*a^4*c^4)*f^2 + 32*(2*(7*b^6*c^2 - 60*a*b^4*c^3 + 
 144*a^2*b^2*c^4 - 64*a^3*c^5)*d - 3*(3*b^7*c - 28*a*b^5*c^2 + 80*a^2*b^3* 
c^3 - 64*a^3*b*c^4)*e)*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*(215040*c^8*f^2*x^7 + 15 
360*(32*c^8*e*f + 17*b*c^7*f^2)*x^6 + 1280*(224*c^8*e^2 + 3*(b^2*c^6 + 84* 
a*c^7)*f^2 + 32*(14*c^8*d + 15*b*c^7*e)*f)*x^5 + 128*(5376*c^8*d*e + 2912* 
b*c^7*e^2 - 3*(11*b^3*c^5 - 52*a*b*c^6)*f^2 + 32*(182*b*c^7*d + 3*(b^2*c^6 
 + 64*a*c^7)*e)*f)*x^4 + 16*(26880*c^8*d^2 + 59136*b*c^7*d*e + 224*(3*b^2* 
c^6 + 140*a*c^7)*e^2 + 3*(99*b^4*c^4 - 568*a*b^2*c^5 + 560*a^2*c^6)*f^2 + 
32*(14*(3*b^2*c^6 + 140*a*c^7)*d - 3*(9*b^3*c^5 - 44*a*b*c^6)*e)*f)*x^3 - 
26880*(3*b^3*c^5 - 20*a*b*c^6)*d^2 + 5376*(15*b^4*c^4 - 100*a*b^2*c^5 + 12 
8*a^2*c^6)*d*e - 224*(105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*e^2 - 
3*(3465*b^7*c - 30660*a*b^5*c^2 + 81648*a^2*b^3*c^3 - 58816*a^3*b*c^4)*f^2 
 + 8*(80640*b*c^7*d^2 + 5376*(b^2*c^6 + 32*a*c^7)*d*e - 224*(7*b^3*c^5 - 3 
6*a*b*c^6)*e^2 - 3*(231*b^5*c^3 - 1560*a*b^3*c^4 + 2416*a^2*b*c^5)*f^2 - 3 
2*(14*(7*b^3*c^5 - 36*a*b*c^6)*d - 3*(21*b^4*c^4 - 124*a*b^2*c^5 + 128*a^2 
*c^6)*e)*f)*x^2 - 32*(14*(105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5942 vs. \(2 (580) = 1160\).

Time = 0.89 (sec) , antiderivative size = 5942, normalized size of antiderivative = 10.54 \[ \int \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(c*f**2*x**7/8 + x**6*(17*b*c*f**2/16 + 
2*c**2*e*f)/(7*c) + x**5*(9*a*c*f**2/8 + b**2*f**2 + 4*b*c*e*f - 13*b*(17* 
b*c*f**2/16 + 2*c**2*e*f)/(14*c) + 2*c**2*d*f + c**2*e**2)/(6*c) + x**4*(2 
*a*b*f**2 + 4*a*c*e*f - 6*a*(17*b*c*f**2/16 + 2*c**2*e*f)/(7*c) + 2*b**2*e 
*f + 4*b*c*d*f + 2*b*c*e**2 - 11*b*(9*a*c*f**2/8 + b**2*f**2 + 4*b*c*e*f - 
 13*b*(17*b*c*f**2/16 + 2*c**2*e*f)/(14*c) + 2*c**2*d*f + c**2*e**2)/(12*c 
) + 2*c**2*d*e)/(5*c) + x**3*(a**2*f**2 + 4*a*b*e*f + 4*a*c*d*f + 2*a*c*e* 
*2 - 5*a*(9*a*c*f**2/8 + b**2*f**2 + 4*b*c*e*f - 13*b*(17*b*c*f**2/16 + 2* 
c**2*e*f)/(14*c) + 2*c**2*d*f + c**2*e**2)/(6*c) + 2*b**2*d*f + b**2*e**2 
+ 4*b*c*d*e - 9*b*(2*a*b*f**2 + 4*a*c*e*f - 6*a*(17*b*c*f**2/16 + 2*c**2*e 
*f)/(7*c) + 2*b**2*e*f + 4*b*c*d*f + 2*b*c*e**2 - 11*b*(9*a*c*f**2/8 + b** 
2*f**2 + 4*b*c*e*f - 13*b*(17*b*c*f**2/16 + 2*c**2*e*f)/(14*c) + 2*c**2*d* 
f + c**2*e**2)/(12*c) + 2*c**2*d*e)/(10*c) + c**2*d**2)/(4*c) + x**2*(2*a* 
*2*e*f + 4*a*b*d*f + 2*a*b*e**2 + 4*a*c*d*e - 4*a*(2*a*b*f**2 + 4*a*c*e*f 
- 6*a*(17*b*c*f**2/16 + 2*c**2*e*f)/(7*c) + 2*b**2*e*f + 4*b*c*d*f + 2*b*c 
*e**2 - 11*b*(9*a*c*f**2/8 + b**2*f**2 + 4*b*c*e*f - 13*b*(17*b*c*f**2/16 
+ 2*c**2*e*f)/(14*c) + 2*c**2*d*f + c**2*e**2)/(12*c) + 2*c**2*d*e)/(5*c) 
+ 2*b**2*d*e + 2*b*c*d**2 - 7*b*(a**2*f**2 + 4*a*b*e*f + 4*a*c*d*f + 2*a*c 
*e**2 - 5*a*(9*a*c*f**2/8 + b**2*f**2 + 4*b*c*e*f - 13*b*(17*b*c*f**2/16 + 
 2*c**2*e*f)/(14*c) + 2*c**2*d*f + c**2*e**2)/(6*c) + 2*b**2*d*f + b**2...
 

Maxima [F(-2)]

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

integrate((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)^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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1132 vs. \(2 (530) = 1060\).

Time = 0.35 (sec) , antiderivative size = 1132, normalized size of antiderivative = 2.01 \[ \int \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/1720320*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*(14*c*f^2*x + (32*c^8* 
e*f + 17*b*c^7*f^2)/c^7)*x + (224*c^8*e^2 + 448*c^8*d*f + 480*b*c^7*e*f + 
3*b^2*c^6*f^2 + 252*a*c^7*f^2)/c^7)*x + (5376*c^8*d*e + 2912*b*c^7*e^2 + 5 
824*b*c^7*d*f + 96*b^2*c^6*e*f + 6144*a*c^7*e*f - 33*b^3*c^5*f^2 + 156*a*b 
*c^6*f^2)/c^7)*x + (26880*c^8*d^2 + 59136*b*c^7*d*e + 672*b^2*c^6*e^2 + 31 
360*a*c^7*e^2 + 1344*b^2*c^6*d*f + 62720*a*c^7*d*f - 864*b^3*c^5*e*f + 422 
4*a*b*c^6*e*f + 297*b^4*c^4*f^2 - 1704*a*b^2*c^5*f^2 + 1680*a^2*c^6*f^2)/c 
^7)*x + (80640*b*c^7*d^2 + 5376*b^2*c^6*d*e + 172032*a*c^7*d*e - 1568*b^3* 
c^5*e^2 + 8064*a*b*c^6*e^2 - 3136*b^3*c^5*d*f + 16128*a*b*c^6*d*f + 2016*b 
^4*c^4*e*f - 11904*a*b^2*c^5*e*f + 12288*a^2*c^6*e*f - 693*b^5*c^3*f^2 + 4 
680*a*b^3*c^4*f^2 - 7248*a^2*b*c^5*f^2)/c^7)*x + (26880*b^2*c^6*d^2 + 5376 
00*a*c^7*d^2 - 26880*b^3*c^5*d*e + 150528*a*b*c^6*d*e + 7840*b^4*c^4*e^2 - 
 48384*a*b^2*c^5*e^2 + 53760*a^2*c^6*e^2 + 15680*b^4*c^4*d*f - 96768*a*b^2 
*c^5*d*f + 107520*a^2*c^6*d*f - 10080*b^5*c^3*e*f + 69888*a*b^3*c^4*e*f - 
112128*a^2*b*c^5*e*f + 3465*b^6*c^2*f^2 - 26964*a*b^4*c^3*f^2 + 56688*a^2* 
b^2*c^4*f^2 - 20160*a^3*c^5*f^2)/c^7)*x - (80640*b^3*c^5*d^2 - 537600*a*b* 
c^6*d^2 - 80640*b^4*c^4*d*e + 537600*a*b^2*c^5*d*e - 688128*a^2*c^6*d*e + 
23520*b^5*c^3*e^2 - 170240*a*b^3*c^4*e^2 + 290304*a^2*b*c^5*e^2 + 47040*b^ 
5*c^3*d*f - 340480*a*b^3*c^4*d*f + 580608*a^2*b*c^5*d*f - 30240*b^6*c^2*e* 
f + 241920*a*b^4*c^3*e*f - 526848*a^2*b^2*c^4*e*f + 196608*a^3*c^5*e*f ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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