\(\int (g+h x)^3 (a+b x+c x^2)^{3/2} (d+e x+f x^2) \, dx\) [33]

Optimal result

Integrand size = 32, antiderivative size = 1153 \[ \int (g+h x)^3 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx =\text {Too large to display} \] Output:

-1/32768*(-4*a*c+b^2)*(1536*c^5*d*g^3-143*b^5*f*h^3-256*c^4*g*(a*f*g^2+3*a 
*h*(d*h+e*g)+3*b*g*(3*d*h+e*g))+22*b^3*c*h^2*(20*a*f*h+9*b*(e*h+3*f*g))-48 
*b*c^2*h*(5*a^2*f*h^2+9*a*b*h*(e*h+3*f*g)+6*b^2*(d*h^2+3*e*g*h+3*f*g^2))+3 
2*c^3*(3*a^2*h^2*(e*h+3*f*g)+14*b^2*g*(f*g^2+3*h*(d*h+e*g))+12*a*b*h*(3*f* 
g^2+h*(d*h+3*e*g))))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^7+1/12288*(1536*c^5*d 
*g^3-143*b^5*f*h^3-256*c^4*g*(a*f*g^2+3*a*h*(d*h+e*g)+3*b*g*(3*d*h+e*g))+2 
2*b^3*c*h^2*(20*a*f*h+9*b*(e*h+3*f*g))-48*b*c^2*h*(5*a^2*f*h^2+9*a*b*h*(e* 
h+3*f*g)+6*b^2*(d*h^2+3*e*g*h+3*f*g^2))+32*c^3*(3*a^2*h^2*(e*h+3*f*g)+14*b 
^2*g*(f*g^2+3*h*(d*h+e*g))+12*a*b*h*(3*f*g^2+h*(d*h+3*e*g))))*(2*c*x+b)*(c 
*x^2+b*x+a)^(3/2)/c^6+1/2016*(143*b^2*f*h/c+12*c*(9*e*g-5*f*g^2/h+24*d*h)- 
128*a*f*h-198*b*e*h-48*b*f*g)*(h*x+g)^2*(c*x^2+b*x+a)^(5/2)/c^2+1/144*(18* 
c*e-13*b*f-10*c*f*g/h)*(h*x+g)^3*(c*x^2+b*x+a)^(5/2)/c^2+1/9*f*(h*x+g)^4*( 
c*x^2+b*x+a)^(5/2)/c/h+1/80640*(3003*b^4*f*h^3-192*c^4*(5*f*g^4-3*g^2*h*(6 
4*d*h+3*e*g))/h-198*b^2*c*h^2*(38*a*f*h+21*b*(e*h+3*f*g))+8*c^2*h*(256*a^2 
*f*h^2+837*a*b*h*(e*h+3*f*g)+b^2*(1553*f*g^2+756*h*(d*h+3*e*g)))-16*c^3*(3 
2*a*h*(17*f*g^2+9*h*(d*h+3*e*g))+b*g*(13*f*g^2+9*h*(196*d*h+141*e*g)))-10* 
c*(429*b^3*f*h^3-22*b*c*h^2*(34*a*f*h+27*b*e*h+29*b*f*g)+16*c^3*(5*f*g^3-9 
*g*h*(12*d*h+e*g))+8*c^2*h*(a*h*(63*e*h+61*f*g)+3*b*(f*g^2+6*h*(6*d*h+7*e* 
g))))*x)*(c*x^2+b*x+a)^(5/2)/c^5+1/65536*(-4*a*c+b^2)^2*(1536*c^5*d*g^3-14 
3*b^5*f*h^3-256*c^4*g*(a*f*g^2+3*a*h*(d*h+e*g)+3*b*g*(3*d*h+e*g))+22*b^...
 

Mathematica [A] (verified)

Time = 16.29 (sec) , antiderivative size = 1683, normalized size of antiderivative = 1.46 \[ \int (g+h x)^3 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(72*h*(f*g^2 + h*(-(e*g) + d*h))*(g + h*x)^2*(a + x*(b + c*x))^(5/2) + 63* 
h*(-2*f*g + e*h)*(g + h*x)^3*(a + x*(b + c*x))^(5/2) + 56*f*h*(g + h*x)^4* 
(a + x*(b + c*x))^(5/2) - (f*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-45045*b^8* 
h^5 + 2310*b^7*c*h^4*(135*g + 13*h*x) - 84*b^6*c*h^3*(-5225*a*h^2 + c*(108 
00*g^2 + 2475*g*h*x + 286*h^2*x^2)) + 72*b^5*c^2*h^2*(-7*a*h^2*(5475*g + 5 
17*h*x) + 2*c*(9800*g^3 + 4200*g^2*h*x + 1155*g*h^2*x^2 + 143*h^3*x^3)) - 
16*b^4*c^2*h*(86499*a^2*h^4 - 9*a*c*h^2*(50400*g^2 + 11235*g*h*x + 1276*h^ 
2*x^2) + 2*c^2*(37800*g^4 + 29400*g^3*h*x + 15120*g^2*h^2*x^2 + 4455*g*h^3 
*x^3 + 572*h^4*x^4)) - 128*b*c^4*(a^3*h^4*(41355*g + 3701*h*x) - 6*a^2*c*h 
^2*(22680*g^3 + 8760*g^2*h*x + 2265*g*h^2*x^2 + 269*h^3*x^3) + 40*a*c^2*(6 
30*g^5 + 434*g^4*h*x - 1036*g^3*h^2*x^2 - 2292*g^2*h^3*x^3 - 1675*g*h^4*x^ 
4 - 433*h^5*x^5) + 80*c^3*x^2*(378*g^5 + 1162*g^4*h*x + 1288*g^3*h^2*x^2 + 
 456*g^2*h^3*x^3 - 131*g*h^4*x^4 - 91*h^5*x^5)) - 256*c^4*(1024*a^4*h^5 - 
a^3*c*h^3*(23040*g^2 + 4725*g*h*x + 512*h^2*x^2) + 80*c^4*g*x^3*(126*g^4 + 
 448*g^3*h*x + 616*g^2*h^2*x^2 + 384*g*h^3*x^3 + 91*h^4*x^4) - 2*a^2*c^2*h 
*(-17920*g^4 - 3640*g^3*h*x + 7680*g^2*h^2*x^2 + 7385*g*h^3*x^3 + 2048*h^4 
*x^4) + 40*a*c^3*x*(630*g^5 + 1792*g^4*h*x + 2044*g^3*h^2*x^2 + 960*g^2*h^ 
3*x^3 + 49*g*h^4*x^4 - 64*h^5*x^5)) + 32*b^3*c^3*(9*a^2*h^4*(25515*g + 235 
3*h*x) - 4*a*c*h^2*(79800*g^3 + 32760*g^2*h*x + 8775*g*h^2*x^2 + 1067*h^3* 
x^3) + 40*c^2*(378*g^5 + 630*g^4*h*x + 588*g^3*h^2*x^2 + 324*g^2*h^3*x^...
 

Rubi [A] (verified)

Time = 2.04 (sec) , antiderivative size = 801, normalized size of antiderivative = 0.69, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2184, 27, 1236, 27, 1236, 27, 1225, 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.

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

\(\Big \downarrow \) 2184

\(\displaystyle \frac {\int -\frac {1}{2} h (g+h x)^3 (5 b f g-18 c d h+8 a f h+(10 c f g-18 c e h+13 b f h) x) \left (c x^2+b x+a\right )^{3/2}dx}{9 c h^2}+\frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{5/2}}{9 c h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{5/2}}{9 c h}-\frac {\int (g+h x)^3 (5 b f g-18 c d h+8 a f h+(10 c f g-18 c e h+13 b f h) x) \left (c x^2+b x+a\right )^{3/2}dx}{18 c h}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{5/2}}{9 c h}-\frac {\frac {\int -\frac {1}{2} (g+h x)^2 \left (65 f g h b^2+78 a f h^2 b-30 c g (f g+3 e h) b+4 c h (72 c d g-17 a f g-27 a e h)+\left (-12 \left (5 f g^2-3 h (3 e g+8 d h)\right ) c^2-2 h (24 b f g+99 b e h+64 a f h) c+143 b^2 f h^2\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{8 c}+\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{5/2} (13 b f h-18 c e h+10 c f g)}{8 c}}{18 c h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{5/2}}{9 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{5/2} (13 b f h-18 c e h+10 c f g)}{8 c}-\frac {\int (g+h x)^2 \left (65 f g h b^2+78 a f h^2 b-30 c g (f g+3 e h) b+4 c h (72 c d g-17 a f g-27 a e h)+\left (-12 \left (5 f g^2-3 h (3 e g+8 d h)\right ) c^2-2 h (24 b f g+99 b e h+64 a f h) c+143 b^2 f h^2\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{16 c}}{18 c h}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{5/2}}{9 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{5/2} (13 b f h-18 c e h+10 c f g)}{8 c}-\frac {\frac {\int -\frac {1}{2} (g+h x) \left (715 f g h^2 b^3+2 \left (286 a f h^3-5 c g h (115 f g+99 e h)\right ) b^2-4 c \left (a h^2 (481 f g+198 e h)-30 c g \left (f g^2+3 h (5 e g+4 d h)\right )\right ) b-8 c h \left (504 c^2 d g^2+64 a^2 f h^2-a c \left (89 f g^2+9 h (27 e g+16 d h)\right )\right )+3 \left (16 \left (5 f g^3-9 g h (e g+12 d h)\right ) c^3+8 h \left (a h (61 f g+63 e h)+3 b \left (f g^2+6 h (7 e g+6 d h)\right )\right ) c^2-22 b h^2 (29 b f g+27 b e h+34 a f h) c+429 b^3 f h^3\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{7 c}+\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} \left (-2 c h (64 a f h+99 b e h+24 b f g)+143 b^2 f h^2-12 c^2 \left (5 f g^2-3 h (8 d h+3 e g)\right )\right )}{7 c}}{16 c}}{18 c h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{5/2}}{9 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{5/2} (13 b f h-18 c e h+10 c f g)}{8 c}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} \left (-2 c h (64 a f h+99 b e h+24 b f g)+143 b^2 f h^2-12 c^2 \left (5 f g^2-3 h (8 d h+3 e g)\right )\right )}{7 c}-\frac {\int (g+h x) \left (715 f g h^2 b^3+2 \left (286 a f h^3-5 c g h (115 f g+99 e h)\right ) b^2-4 c \left (a h^2 (481 f g+198 e h)-30 c \left (f g^3+3 h (5 e g+4 d h) g\right )\right ) b-8 c h \left (504 c^2 d g^2+64 a^2 f h^2-a c \left (89 f g^2+9 h (27 e g+16 d h)\right )\right )+3 \left (16 \left (5 f g^3-9 g h (e g+12 d h)\right ) c^3+8 h \left (a h (61 f g+63 e h)+3 b \left (f g^2+6 h (7 e g+6 d h)\right )\right ) c^2-22 b h^2 (29 b f g+27 b e h+34 a f h) c+429 b^3 f h^3\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{14 c}}{16 c}}{18 c h}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{5/2}}{9 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{5/2} (13 b f h-18 c e h+10 c f g)}{8 c}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} \left (-2 c h (64 a f h+99 b e h+24 b f g)+143 b^2 f h^2-12 c^2 \left (5 f g^2-3 h (8 d h+3 e g)\right )\right )}{7 c}-\frac {-\frac {21 h \left (32 c^3 \left (3 a^2 h^2 (e h+3 f g)+12 a b h \left (h (d h+3 e g)+3 f g^2\right )+14 b^2 g \left (3 h (d h+e g)+f g^2\right )\right )-48 b c^2 h \left (5 a^2 f h^2+9 a b h (e h+3 f g)+6 b^2 \left (d h^2+3 e g h+3 f g^2\right )\right )+22 b^3 c h^2 (20 a f h+9 b (e h+3 f g))-256 c^4 g \left (3 a h (d h+e g)+a f g^2+3 b g (3 d h+e g)\right )-143 b^5 f h^3+1536 c^5 d g^3\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (8 c^2 h^2 \left (256 a^2 f h^2+837 a b h (e h+3 f g)+b^2 \left (756 h (d h+3 e g)+1553 f g^2\right )\right )-10 c h x \left (8 c^2 h \left (a h (63 e h+61 f g)+3 b \left (6 h (6 d h+7 e g)+f g^2\right )\right )-22 b c h^2 (34 a f h+27 b e h+29 b f g)+429 b^3 f h^3+16 c^3 \left (5 f g^3-9 g h (12 d h+e g)\right )\right )-198 b^2 c h^3 (38 a f h+21 b (e h+3 f g))-16 c^3 h \left (32 a h \left (9 h (d h+3 e g)+17 f g^2\right )+b g \left (9 h (196 d h+141 e g)+13 f g^2\right )\right )+3003 b^4 f h^4-192 c^4 \left (5 f g^4-3 g^2 h (64 d h+3 e g)\right )\right )}{20 c^2}}{14 c}}{16 c}}{18 c h}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{5/2}}{9 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{5/2} (13 b f h-18 c e h+10 c f g)}{8 c}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} \left (-2 c h (64 a f h+99 b e h+24 b f g)+143 b^2 f h^2-12 c^2 \left (5 f g^2-3 h (8 d h+3 e g)\right )\right )}{7 c}-\frac {-\frac {21 h \left (32 c^3 \left (3 a^2 h^2 (e h+3 f g)+12 a b h \left (h (d h+3 e g)+3 f g^2\right )+14 b^2 g \left (3 h (d h+e g)+f g^2\right )\right )-48 b c^2 h \left (5 a^2 f h^2+9 a b h (e h+3 f g)+6 b^2 \left (d h^2+3 e g h+3 f g^2\right )\right )+22 b^3 c h^2 (20 a f h+9 b (e h+3 f g))-256 c^4 g \left (3 a h (d h+e g)+a f g^2+3 b g (3 d h+e g)\right )-143 b^5 f h^3+1536 c^5 d g^3\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (8 c^2 h^2 \left (256 a^2 f h^2+837 a b h (e h+3 f g)+b^2 \left (756 h (d h+3 e g)+1553 f g^2\right )\right )-10 c h x \left (8 c^2 h \left (a h (63 e h+61 f g)+3 b \left (6 h (6 d h+7 e g)+f g^2\right )\right )-22 b c h^2 (34 a f h+27 b e h+29 b f g)+429 b^3 f h^3+16 c^3 \left (5 f g^3-9 g h (12 d h+e g)\right )\right )-198 b^2 c h^3 (38 a f h+21 b (e h+3 f g))-16 c^3 h \left (32 a h \left (9 h (d h+3 e g)+17 f g^2\right )+b g \left (9 h (196 d h+141 e g)+13 f g^2\right )\right )+3003 b^4 f h^4-192 c^4 \left (5 f g^4-3 g^2 h (64 d h+3 e g)\right )\right )}{20 c^2}}{14 c}}{16 c}}{18 c h}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{5/2}}{9 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{5/2} (13 b f h-18 c e h+10 c f g)}{8 c}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} \left (-2 c h (64 a f h+99 b e h+24 b f g)+143 b^2 f h^2-12 c^2 \left (5 f g^2-3 h (8 d h+3 e g)\right )\right )}{7 c}-\frac {-\frac {21 h \left (32 c^3 \left (3 a^2 h^2 (e h+3 f g)+12 a b h \left (h (d h+3 e g)+3 f g^2\right )+14 b^2 g \left (3 h (d h+e g)+f g^2\right )\right )-48 b c^2 h \left (5 a^2 f h^2+9 a b h (e h+3 f g)+6 b^2 \left (d h^2+3 e g h+3 f g^2\right )\right )+22 b^3 c h^2 (20 a f h+9 b (e h+3 f g))-256 c^4 g \left (3 a h (d h+e g)+a f g^2+3 b g (3 d h+e g)\right )-143 b^5 f h^3+1536 c^5 d g^3\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (8 c^2 h^2 \left (256 a^2 f h^2+837 a b h (e h+3 f g)+b^2 \left (756 h (d h+3 e g)+1553 f g^2\right )\right )-10 c h x \left (8 c^2 h \left (a h (63 e h+61 f g)+3 b \left (6 h (6 d h+7 e g)+f g^2\right )\right )-22 b c h^2 (34 a f h+27 b e h+29 b f g)+429 b^3 f h^3+16 c^3 \left (5 f g^3-9 g h (12 d h+e g)\right )\right )-198 b^2 c h^3 (38 a f h+21 b (e h+3 f g))-16 c^3 h \left (32 a h \left (9 h (d h+3 e g)+17 f g^2\right )+b g \left (9 h (196 d h+141 e g)+13 f g^2\right )\right )+3003 b^4 f h^4-192 c^4 \left (5 f g^4-3 g^2 h (64 d h+3 e g)\right )\right )}{20 c^2}}{14 c}}{16 c}}{18 c h}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {f (g+h x)^4 \left (c x^2+b x+a\right )^{5/2}}{9 c h}-\frac {\frac {(10 c f g-18 c e h+13 b f h) (g+h x)^3 \left (c x^2+b x+a\right )^{5/2}}{8 c}-\frac {\frac {\left (-12 \left (5 f g^2-3 h (3 e g+8 d h)\right ) c^2-2 h (24 b f g+99 b e h+64 a f h) c+143 b^2 f h^2\right ) (g+h x)^2 \left (c x^2+b x+a\right )^{5/2}}{7 c}-\frac {-\frac {\left (-192 \left (5 f g^4-3 g^2 h (3 e g+64 d h)\right ) c^4-16 h \left (32 a h \left (17 f g^2+9 h (3 e g+d h)\right )+b g \left (13 f g^2+9 h (141 e g+196 d h)\right )\right ) c^3+8 h^2 \left (\left (1553 f g^2+756 h (3 e g+d h)\right ) b^2+837 a h (3 f g+e h) b+256 a^2 f h^2\right ) c^2-198 b^2 h^3 (38 a f h+21 b (3 f g+e h)) c-10 h \left (16 \left (5 f g^3-9 g h (e g+12 d h)\right ) c^3+8 h \left (a h (61 f g+63 e h)+3 b \left (f g^2+6 h (7 e g+6 d h)\right )\right ) c^2-22 b h^2 (29 b f g+27 b e h+34 a f h) c+429 b^3 f h^3\right ) x c+3003 b^4 f h^4\right ) \left (c x^2+b x+a\right )^{5/2}}{20 c^2}-\frac {21 h \left (-143 f h^3 b^5+22 c h^2 (20 a f h+9 b (3 f g+e h)) b^3-48 c^2 h \left (6 \left (3 f g^2+3 e h g+d h^2\right ) b^2+9 a h (3 f g+e h) b+5 a^2 f h^2\right ) b+1536 c^5 d g^3-256 c^4 g \left (a f g^2+3 b (e g+3 d h) g+3 a h (e g+d h)\right )+32 c^3 \left (14 g \left (f g^2+3 h (e g+d h)\right ) b^2+12 a h \left (3 f g^2+h (3 e g+d h)\right ) b+3 a^2 h^2 (3 f g+e h)\right )\right ) \left (\frac {(b+2 c x) \left (c x^2+b x+a\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {c x^2+b x+a}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{8 c^2}}{14 c}}{16 c}}{18 c h}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {f (g+h x)^4 \left (c x^2+b x+a\right )^{5/2}}{9 c h}-\frac {\frac {(10 c f g-18 c e h+13 b f h) (g+h x)^3 \left (c x^2+b x+a\right )^{5/2}}{8 c}-\frac {\frac {\left (-12 \left (5 f g^2-3 h (3 e g+8 d h)\right ) c^2-2 h (24 b f g+99 b e h+64 a f h) c+143 b^2 f h^2\right ) (g+h x)^2 \left (c x^2+b x+a\right )^{5/2}}{7 c}-\frac {-\frac {\left (-192 \left (5 f g^4-3 g^2 h (3 e g+64 d h)\right ) c^4-16 h \left (32 a h \left (17 f g^2+9 h (3 e g+d h)\right )+b g \left (13 f g^2+9 h (141 e g+196 d h)\right )\right ) c^3+8 h^2 \left (\left (1553 f g^2+756 h (3 e g+d h)\right ) b^2+837 a h (3 f g+e h) b+256 a^2 f h^2\right ) c^2-198 b^2 h^3 (38 a f h+21 b (3 f g+e h)) c-10 h \left (16 \left (5 f g^3-9 g h (e g+12 d h)\right ) c^3+8 h \left (a h (61 f g+63 e h)+3 b \left (f g^2+6 h (7 e g+6 d h)\right )\right ) c^2-22 b h^2 (29 b f g+27 b e h+34 a f h) c+429 b^3 f h^3\right ) x c+3003 b^4 f h^4\right ) \left (c x^2+b x+a\right )^{5/2}}{20 c^2}-\frac {21 h \left (-143 f h^3 b^5+22 c h^2 (20 a f h+9 b (3 f g+e h)) b^3-48 c^2 h \left (6 \left (3 f g^2+3 e h g+d h^2\right ) b^2+9 a h (3 f g+e h) b+5 a^2 f h^2\right ) b+1536 c^5 d g^3-256 c^4 g \left (a f g^2+3 b (e g+3 d h) g+3 a h (e g+d h)\right )+32 c^3 \left (14 g \left (f g^2+3 h (e g+d h)\right ) b^2+12 a h \left (3 f g^2+h (3 e g+d h)\right ) b+3 a^2 h^2 (3 f g+e h)\right )\right ) \left (\frac {(b+2 c x) \left (c x^2+b x+a\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {c x^2+b x+a}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{8 c^2}}{14 c}}{16 c}}{18 c h}\)

Input:

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

Output:

(f*(g + h*x)^4*(a + b*x + c*x^2)^(5/2))/(9*c*h) - (((10*c*f*g - 18*c*e*h + 
 13*b*f*h)*(g + h*x)^3*(a + b*x + c*x^2)^(5/2))/(8*c) - (((143*b^2*f*h^2 - 
 2*c*h*(24*b*f*g + 99*b*e*h + 64*a*f*h) - 12*c^2*(5*f*g^2 - 3*h*(3*e*g + 8 
*d*h)))*(g + h*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c) - (-1/20*((3003*b^4*f*h 
^4 - 192*c^4*(5*f*g^4 - 3*g^2*h*(3*e*g + 64*d*h)) - 198*b^2*c*h^3*(38*a*f* 
h + 21*b*(3*f*g + e*h)) + 8*c^2*h^2*(256*a^2*f*h^2 + 837*a*b*h*(3*f*g + e* 
h) + b^2*(1553*f*g^2 + 756*h*(3*e*g + d*h))) - 16*c^3*h*(32*a*h*(17*f*g^2 
+ 9*h*(3*e*g + d*h)) + b*g*(13*f*g^2 + 9*h*(141*e*g + 196*d*h))) - 10*c*h* 
(429*b^3*f*h^3 - 22*b*c*h^2*(29*b*f*g + 27*b*e*h + 34*a*f*h) + 16*c^3*(5*f 
*g^3 - 9*g*h*(e*g + 12*d*h)) + 8*c^2*h*(a*h*(61*f*g + 63*e*h) + 3*b*(f*g^2 
 + 6*h*(7*e*g + 6*d*h))))*x)*(a + b*x + c*x^2)^(5/2))/c^2 - (21*h*(1536*c^ 
5*d*g^3 - 143*b^5*f*h^3 - 256*c^4*g*(a*f*g^2 + 3*a*h*(e*g + d*h) + 3*b*g*( 
e*g + 3*d*h)) + 22*b^3*c*h^2*(20*a*f*h + 9*b*(3*f*g + e*h)) - 48*b*c^2*h*( 
5*a^2*f*h^2 + 9*a*b*h*(3*f*g + e*h) + 6*b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) + 
 32*c^3*(3*a^2*h^2*(3*f*g + e*h) + 14*b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 12 
*a*b*h*(3*f*g^2 + h*(3*e*g + d*h))))*(((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)))/(8*c^2))/(14*c))/(16*c))/(18*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 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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2869\) vs. \(2(1116)=2232\).

Time = 0.45 (sec) , antiderivative size = 2870, normalized size of antiderivative = 2.49

method	result	size
default	\(\text {Expression too large to display}\)	\(2870\)
risch	\(\text {Expression too large to display}\)	\(3146\)

Input:

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

Output:

d*g^3*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+ 
b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c 
*x^2+b*x+a)^(1/2))))+g^2*(3*d*h+e*g)*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1 
/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c* 
x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x 
+a)^(1/2)))))+h^2*(e*h+3*f*g)*(1/8*x^3*(c*x^2+b*x+a)^(5/2)/c-11/16*b/c*(1/ 
7*x^2*(c*x^2+b*x+a)^(5/2)/c-9/14*b/c*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c 
*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3 
/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^( 
3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b) 
/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^( 
1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))) 
)-2/7*a/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a 
)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c 
-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))-3/8*a/c*(1/6 
*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8* 
(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2 
+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a) 
^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c 
*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2374 vs. \(2 (1116) = 2232\).

Time = 2.49 (sec) , antiderivative size = 4751, normalized size of antiderivative = 4.12 \[ \int (g+h x)^3 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19122 vs. \(2 (1210) = 2420\).

Time = 1.52 (sec) , antiderivative size = 19122, normalized size of antiderivative = 16.58 \[ \int (g+h x)^3 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(c*f*h**3*x**8/9 + x**7*(19*b*c*f*h**3/1 
8 + c**2*e*h**3 + 3*c**2*f*g*h**2)/(8*c) + x**6*(10*a*c*f*h**3/9 + b**2*f* 
h**3 + 2*b*c*e*h**3 + 6*b*c*f*g*h**2 - 15*b*(19*b*c*f*h**3/18 + c**2*e*h** 
3 + 3*c**2*f*g*h**2)/(16*c) + c**2*d*h**3 + 3*c**2*e*g*h**2 + 3*c**2*f*g** 
2*h)/(7*c) + x**5*(2*a*b*f*h**3 + 2*a*c*e*h**3 + 6*a*c*f*g*h**2 - 7*a*(19* 
b*c*f*h**3/18 + c**2*e*h**3 + 3*c**2*f*g*h**2)/(8*c) + b**2*e*h**3 + 3*b** 
2*f*g*h**2 + 2*b*c*d*h**3 + 6*b*c*e*g*h**2 + 6*b*c*f*g**2*h - 13*b*(10*a*c 
*f*h**3/9 + b**2*f*h**3 + 2*b*c*e*h**3 + 6*b*c*f*g*h**2 - 15*b*(19*b*c*f*h 
**3/18 + c**2*e*h**3 + 3*c**2*f*g*h**2)/(16*c) + c**2*d*h**3 + 3*c**2*e*g* 
h**2 + 3*c**2*f*g**2*h)/(14*c) + 3*c**2*d*g*h**2 + 3*c**2*e*g**2*h + c**2* 
f*g**3)/(6*c) + x**4*(a**2*f*h**3 + 2*a*b*e*h**3 + 6*a*b*f*g*h**2 + 2*a*c* 
d*h**3 + 6*a*c*e*g*h**2 + 6*a*c*f*g**2*h - 6*a*(10*a*c*f*h**3/9 + b**2*f*h 
**3 + 2*b*c*e*h**3 + 6*b*c*f*g*h**2 - 15*b*(19*b*c*f*h**3/18 + c**2*e*h**3 
 + 3*c**2*f*g*h**2)/(16*c) + c**2*d*h**3 + 3*c**2*e*g*h**2 + 3*c**2*f*g**2 
*h)/(7*c) + b**2*d*h**3 + 3*b**2*e*g*h**2 + 3*b**2*f*g**2*h + 6*b*c*d*g*h* 
*2 + 6*b*c*e*g**2*h + 2*b*c*f*g**3 - 11*b*(2*a*b*f*h**3 + 2*a*c*e*h**3 + 6 
*a*c*f*g*h**2 - 7*a*(19*b*c*f*h**3/18 + c**2*e*h**3 + 3*c**2*f*g*h**2)/(8* 
c) + b**2*e*h**3 + 3*b**2*f*g*h**2 + 2*b*c*d*h**3 + 6*b*c*e*g*h**2 + 6*b*c 
*f*g**2*h - 13*b*(10*a*c*f*h**3/9 + b**2*f*h**3 + 2*b*c*e*h**3 + 6*b*c*f*g 
*h**2 - 15*b*(19*b*c*f*h**3/18 + c**2*e*h**3 + 3*c**2*f*g*h**2)/(16*c) ...
 

Maxima [F(-2)]

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

integrate((h*x+g)^3*(c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),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. 2902 vs. \(2 (1116) = 2232\).

Time = 0.26 (sec) , antiderivative size = 2902, normalized size of antiderivative = 2.52 \[ \int (g+h x)^3 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/10321920*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*(14*(16*c*f*h^3*x + (5 
4*c^9*f*g*h^2 + 18*c^9*e*h^3 + 19*b*c^8*f*h^3)/c^8)*x + (864*c^9*f*g^2*h + 
 864*c^9*e*g*h^2 + 918*b*c^8*f*g*h^2 + 288*c^9*d*h^3 + 306*b*c^8*e*h^3 + 3 
*b^2*c^7*f*h^3 + 320*a*c^8*f*h^3)/c^8)*x + (1344*c^9*f*g^3 + 4032*c^9*e*g^ 
2*h + 4320*b*c^8*f*g^2*h + 4032*c^9*d*g*h^2 + 4320*b*c^8*e*g*h^2 + 54*b^2* 
c^7*f*g*h^2 + 4536*a*c^8*f*g*h^2 + 1440*b*c^8*d*h^3 + 18*b^2*c^7*e*h^3 + 1 
512*a*c^8*e*h^3 - 13*b^3*c^6*f*h^3 + 60*a*b*c^7*f*h^3)/c^8)*x + (16128*c^9 
*e*g^3 + 17472*b*c^8*f*g^3 + 48384*c^9*d*g^2*h + 52416*b*c^8*e*g^2*h + 864 
*b^2*c^7*f*g^2*h + 55296*a*c^8*f*g^2*h + 52416*b*c^8*d*g*h^2 + 864*b^2*c^7 
*e*g*h^2 + 55296*a*c^8*e*g*h^2 - 594*b^3*c^6*f*g*h^2 + 2808*a*b*c^7*f*g*h^ 
2 + 288*b^2*c^7*d*h^3 + 18432*a*c^8*d*h^3 - 198*b^3*c^6*e*h^3 + 936*a*b*c^ 
7*e*h^3 + 143*b^4*c^5*f*h^3 - 804*a*b^2*c^6*f*h^3 + 768*a^2*c^7*f*h^3)/c^8 
)*x + (161280*c^9*d*g^3 + 177408*b*c^8*e*g^3 + 4032*b^2*c^7*f*g^3 + 188160 
*a*c^8*f*g^3 + 532224*b*c^8*d*g^2*h + 12096*b^2*c^7*e*g^2*h + 564480*a*c^8 
*e*g^2*h - 7776*b^3*c^6*f*g^2*h + 38016*a*b*c^7*f*g^2*h + 12096*b^2*c^7*d* 
g*h^2 + 564480*a*c^8*d*g*h^2 - 7776*b^3*c^6*e*g*h^2 + 38016*a*b*c^7*e*g*h^ 
2 + 5346*b^4*c^5*f*g*h^2 - 30672*a*b^2*c^6*f*g*h^2 + 30240*a^2*c^7*f*g*h^2 
 - 2592*b^3*c^6*d*h^3 + 12672*a*b*c^7*d*h^3 + 1782*b^4*c^5*e*h^3 - 10224*a 
*b^2*c^6*e*h^3 + 10080*a^2*c^7*e*h^3 - 1287*b^5*c^4*f*h^3 + 8536*a*b^3*c^5 
*f*h^3 - 12912*a^2*b*c^6*f*h^3)/c^8)*x + (483840*b*c^8*d*g^3 + 16128*b^...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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