Integrand size = 32, antiderivative size = 930 \[ \int (g+h x)^3 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {\left (256 c^5 d g^3-33 b^5 f h^3+6 b^3 c h^2 (20 a f h+7 b (3 f g+e h))-8 b c^2 h \left (10 a^2 f h^2+14 a b h (3 f g+e h)+7 b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )-64 c^4 g \left (2 b g (e g+3 d h)+a \left (f g^2+3 h (e g+d h)\right )\right )+16 c^3 \left (2 a^2 h^2 (3 f g+e h)+5 b^2 g \left (f g^2+3 h (e g+d h)\right )+6 a b h \left (3 f g^2+h (3 e g+d h)\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^6}+\frac {\left (33 b^2 f h^2-2 c h (8 b f g+21 b e h+16 a f h)-4 c^2 \left (3 f g^2-7 h (e g+2 d h)\right )\right ) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3 h}-\frac {(6 c f g-14 c e h+11 b f h) (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2 h}+\frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}+\frac {\left (1155 b^4 f h^4-128 c^4 g^2 \left (3 f g^2-7 h (e g+12 d h)\right )-42 b^2 c h^3 (78 a f h+35 b (3 f g+e h))+8 c^2 h^2 \left (128 a^2 f h^2+343 a b h (3 f g+e h)+b^2 \left (537 f g^2+245 h (3 e g+d h)\right )\right )-16 c^3 h \left (16 a h \left (15 f g^2+7 h (3 e g+d h)\right )+b g \left (17 f g^2+21 h (19 e g+25 d h)\right )\right )-6 c h \left (231 b^3 f h^3-6 b c h^2 (59 b f g+49 b e h+74 a f h)+16 c^3 g \left (3 f g^2-7 h (e g+7 d h)\right )+8 c^2 h \left (a h (41 f g+35 e h)+b \left (5 f g^2+7 h (9 e g+7 d h)\right )\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5 h}-\frac {\left (b^2-4 a c\right ) \left (256 c^5 d g^3-33 b^5 f h^3+6 b^3 c h^2 (20 a f h+7 b (3 f g+e h))-8 b c^2 h \left (10 a^2 f h^2+14 a b h (3 f g+e h)+7 b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )-64 c^4 g \left (2 b g (e g+3 d h)+a \left (f g^2+3 h (e g+d h)\right )\right )+16 c^3 \left (2 a^2 h^2 (3 f g+e h)+5 b^2 g \left (f g^2+3 h (e g+d h)\right )+6 a b h \left (3 f g^2+h (3 e g+d h)\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{13/2}} \]
1/280*(33*b^2*f*h^2-2*c*h*(16*a*f*h+21*b*e*h+8*b*f*g)-4*c^2*(3*f*g^2-7*h*( 2*d*h+e*g)))*(h*x+g)^2*(c*x^2+b*x+a)^(3/2)/c^3/h-1/84*(11*b*f*h-14*c*e*h+6 *c*f*g)*(h*x+g)^3*(c*x^2+b*x+a)^(3/2)/c^2/h+1/7*f*(h*x+g)^4*(c*x^2+b*x+a)^ (3/2)/c/h+1/13440*(1155*b^4*f*h^4-128*c^4*g^2*(3*f*g^2-7*h*(12*d*h+e*g))-4 2*b^2*c*h^3*(78*a*f*h+35*b*(e*h+3*f*g))+8*c^2*h^2*(128*a^2*f*h^2+343*a*b*h *(e*h+3*f*g)+b^2*(537*f*g^2+245*h*(d*h+3*e*g)))-16*c^3*h*(16*a*h*(15*f*g^2 +7*h*(d*h+3*e*g))+b*g*(17*f*g^2+21*h*(25*d*h+19*e*g)))-6*c*h*(231*b^3*f*h^ 3-6*b*c*h^2*(74*a*f*h+49*b*e*h+59*b*f*g)+16*c^3*g*(3*f*g^2-7*h*(7*d*h+e*g) )+8*c^2*h*(a*h*(35*e*h+41*f*g)+b*(5*f*g^2+7*h*(7*d*h+9*e*g))))*x)*(c*x^2+b *x+a)^(3/2)/c^5/h-1/2048*(-4*a*c+b^2)*(256*c^5*d*g^3-33*b^5*f*h^3+6*b^3*c* h^2*(20*a*f*h+7*b*(e*h+3*f*g))-8*b*c^2*h*(10*a^2*f*h^2+14*a*b*h*(e*h+3*f*g )+7*b^2*(d*h^2+3*e*g*h+3*f*g^2))-64*c^4*g*(2*b*g*(3*d*h+e*g)+a*(f*g^2+3*h* (d*h+e*g)))+16*c^3*(2*a^2*h^2*(e*h+3*f*g)+5*b^2*g*(f*g^2+3*h*(d*h+e*g))+6* a*b*h*(3*f*g^2+h*(d*h+3*e*g))))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a )^(1/2))/c^(13/2)+1/1024*(256*c^5*d*g^3-33*b^5*f*h^3+6*b^3*c*h^2*(20*a*f*h +7*b*(e*h+3*f*g))-8*b*c^2*h*(10*a^2*f*h^2+14*a*b*h*(e*h+3*f*g)+7*b^2*(d*h^ 2+3*e*g*h+3*f*g^2))-64*c^4*g*(2*b*g*(3*d*h+e*g)+a*(f*g^2+3*h*(d*h+e*g)))+1 6*c^3*(2*a^2*h^2*(e*h+3*f*g)+5*b^2*g*(f*g^2+3*h*(d*h+e*g))+6*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^6
Time = 11.64 (sec) , antiderivative size = 1093, normalized size of antiderivative = 1.18 \[ \int (g+h x)^3 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-3465 b^6 f h^3+210 b^5 c h^2 (63 f g+21 e h+11 f h x)-84 b^4 c h \left (-260 a f h^2+35 c h (6 e g+2 d h+e h x)+c f \left (210 g^2+105 g h x+22 h^2 x^2\right )\right )-16 b^2 c^2 \left (2163 a^2 f h^3-2 a c h \left (7 h (345 e g+115 d h+56 e h x)+3 f \left (805 g^2+392 g h x+81 h^2 x^2\right )\right )+2 c^2 \left (7 d h \left (180 g^2+75 g h x+14 h^2 x^2\right )+21 e \left (20 g^3+25 g^2 h x+14 g h^2 x^2+3 h^3 x^3\right )+f x \left (175 g^3+294 g^2 h x+189 g h^2 x^2+44 h^3 x^3\right )\right )\right )+16 b^3 c^2 \left (-42 a h^2 (35 e h+3 f (35 g+6 h x))+c \left (f \left (525 g^3+735 g^2 h x+441 g h^2 x^2+99 h^3 x^3\right )+7 h \left (5 d h (45 g+7 h x)+3 e \left (75 g^2+35 g h x+7 h^2 x^2\right )\right )\right )\right )+32 b c^3 \left (a^2 h^2 (2373 f g+791 e h+397 f h x)-2 a c \left (f \left (455 g^3+609 g^2 h x+357 g h^2 x^2+79 h^3 x^3\right )+7 h \left (d h (195 g+29 h x)+e \left (195 g^2+87 g h x+17 h^2 x^2\right )\right )\right )+4 c^2 \left (21 d \left (10 g^3+10 g^2 h x+5 g h^2 x^2+h^3 x^3\right )+x \left (7 e \left (10 g^3+15 g^2 h x+9 g h^2 x^2+2 h^3 x^3\right )+f x \left (35 g^3+63 g^2 h x+42 g h^2 x^2+10 h^3 x^3\right )\right )\right )\right )+64 c^3 \left (128 a^3 f h^3-a^2 c h \left (7 h (96 e g+32 d h+15 e h x)+f \left (672 g^2+315 g h x+64 h^2 x^2\right )\right )+2 a c^2 \left (7 d h \left (120 g^2+45 g h x+8 h^2 x^2\right )+7 e \left (40 g^3+45 g^2 h x+24 g h^2 x^2+5 h^3 x^3\right )+3 f x \left (35 g^3+56 g^2 h x+35 g h^2 x^2+8 h^3 x^3\right )\right )+4 c^3 x \left (21 d \left (10 g^3+20 g^2 h x+15 g h^2 x^2+4 h^3 x^3\right )+x \left (7 e \left (20 g^3+45 g^2 h x+36 g h^2 x^2+10 h^3 x^3\right )+3 f x \left (35 g^3+84 g^2 h x+70 g h^2 x^2+20 h^3 x^3\right )\right )\right )\right )\right )+105 \left (b^2-4 a c\right ) \left (-256 c^5 d g^3+33 b^5 f h^3+64 c^4 g \left (a f g^2+3 a h (e g+d h)+2 b g (e g+3 d h)\right )-6 b^3 c h^2 (20 a f h+7 b (3 f g+e h))+8 b c^2 h \left (10 a^2 f h^2+14 a b h (3 f g+e h)+7 b^2 \left (3 f g^2+3 e g h+d h^2\right )\right )-16 c^3 \left (2 a^2 h^2 (3 f g+e h)+5 b^2 g \left (f g^2+3 h (e g+d h)\right )+6 a b h \left (3 f g^2+h (3 e g+d h)\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{215040 c^{13/2}} \]
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^6*f*h^3 + 210*b^5*c*h^2*(63*f*g + 21*e*h + 11*f*h*x) - 84*b^4*c*h*(-260*a*f*h^2 + 35*c*h*(6*e*g + 2*d*h + e*h*x) + c*f*(210*g^2 + 105*g*h*x + 22*h^2*x^2)) - 16*b^2*c^2*(2163*a^2*f* h^3 - 2*a*c*h*(7*h*(345*e*g + 115*d*h + 56*e*h*x) + 3*f*(805*g^2 + 392*g*h *x + 81*h^2*x^2)) + 2*c^2*(7*d*h*(180*g^2 + 75*g*h*x + 14*h^2*x^2) + 21*e* (20*g^3 + 25*g^2*h*x + 14*g*h^2*x^2 + 3*h^3*x^3) + f*x*(175*g^3 + 294*g^2* h*x + 189*g*h^2*x^2 + 44*h^3*x^3))) + 16*b^3*c^2*(-42*a*h^2*(35*e*h + 3*f* (35*g + 6*h*x)) + c*(f*(525*g^3 + 735*g^2*h*x + 441*g*h^2*x^2 + 99*h^3*x^3 ) + 7*h*(5*d*h*(45*g + 7*h*x) + 3*e*(75*g^2 + 35*g*h*x + 7*h^2*x^2)))) + 3 2*b*c^3*(a^2*h^2*(2373*f*g + 791*e*h + 397*f*h*x) - 2*a*c*(f*(455*g^3 + 60 9*g^2*h*x + 357*g*h^2*x^2 + 79*h^3*x^3) + 7*h*(d*h*(195*g + 29*h*x) + e*(1 95*g^2 + 87*g*h*x + 17*h^2*x^2))) + 4*c^2*(21*d*(10*g^3 + 10*g^2*h*x + 5*g *h^2*x^2 + h^3*x^3) + x*(7*e*(10*g^3 + 15*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^ 3) + f*x*(35*g^3 + 63*g^2*h*x + 42*g*h^2*x^2 + 10*h^3*x^3)))) + 64*c^3*(12 8*a^3*f*h^3 - a^2*c*h*(7*h*(96*e*g + 32*d*h + 15*e*h*x) + f*(672*g^2 + 315 *g*h*x + 64*h^2*x^2)) + 2*a*c^2*(7*d*h*(120*g^2 + 45*g*h*x + 8*h^2*x^2) + 7*e*(40*g^3 + 45*g^2*h*x + 24*g*h^2*x^2 + 5*h^3*x^3) + 3*f*x*(35*g^3 + 56* g^2*h*x + 35*g*h^2*x^2 + 8*h^3*x^3)) + 4*c^3*x*(21*d*(10*g^3 + 20*g^2*h*x + 15*g*h^2*x^2 + 4*h^3*x^3) + x*(7*e*(20*g^3 + 45*g^2*h*x + 36*g*h^2*x^2 + 10*h^3*x^3) + 3*f*x*(35*g^3 + 84*g^2*h*x + 70*g*h^2*x^2 + 20*h^3*x^3))...
Time = 1.72 (sec) , antiderivative size = 755, normalized size of antiderivative = 0.81, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2184, 27, 1236, 27, 1236, 27, 1225, 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.
| \(\displaystyle \int (g+h x)^3 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {\int -\frac {1}{2} h (g+h x)^3 (3 b f g-14 c d h+8 a f h+(6 c f g-14 c e h+11 b f h) x) \sqrt {c x^2+b x+a}dx}{7 c h^2}+\frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}-\frac {\int (g+h x)^3 (3 b f g-14 c d h+8 a f h+(6 c f g-14 c e h+11 b f h) x) \sqrt {c x^2+b x+a}dx}{14 c h}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}-\frac {\frac {\int -\frac {3}{2} (g+h x)^2 \left (11 f g h b^2+22 a f h^2 b-2 c g (3 f g+7 e h) b+4 c h (14 c d g-5 a f g-7 a e h)+\left (-4 \left (3 f g^2-7 h (e g+2 d h)\right ) c^2-2 h (8 b f g+21 b e h+16 a f h) c+33 b^2 f h^2\right ) x\right ) \sqrt {c x^2+b x+a}dx}{6 c}+\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{3/2} (11 b f h-14 c e h+6 c f g)}{6 c}}{14 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{3/2} (11 b f h-14 c e h+6 c f g)}{6 c}-\frac {\int (g+h x)^2 \left (11 f g h b^2+22 a f h^2 b-2 c g (3 f g+7 e h) b+4 c h (14 c d g-5 a f g-7 a e h)+\left (-4 \left (3 f g^2-7 h (e g+2 d h)\right ) c^2-2 h (8 b f g+21 b e h+16 a f h) c+33 b^2 f h^2\right ) x\right ) \sqrt {c x^2+b x+a}dx}{4 c}}{14 c h}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{3/2} (11 b f h-14 c e h+6 c f g)}{6 c}-\frac {\frac {\int -\frac {1}{2} (g+h x) \left (99 f g h^2 b^3+2 \left (66 a f h^3-c g h (79 f g+63 e h)\right ) b^2+4 c \left (6 c f g^3+14 c h (4 e g+3 d h) g-a h^2 (95 f g+42 e h)\right ) b-8 c h \left (70 c^2 d g^2+16 a^2 f h^2-a c \left (19 f g^2+7 h (7 e g+4 d h)\right )\right )+\left (16 \left (3 f g^3-7 g h (e g+7 d h)\right ) c^3+8 h \left (5 b f g^2+7 b h (9 e g+7 d h)+a h (41 f g+35 e h)\right ) c^2-6 b h^2 (59 b f g+49 b e h+74 a f h) c+231 b^3 f h^3\right ) x\right ) \sqrt {c x^2+b x+a}dx}{5 c}+\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (-2 c h (16 a f h+21 b e h+8 b f g)+33 b^2 f h^2-4 c^2 \left (3 f g^2-7 h (2 d h+e g)\right )\right )}{5 c}}{4 c}}{14 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{3/2} (11 b f h-14 c e h+6 c f g)}{6 c}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (-2 c h (16 a f h+21 b e h+8 b f g)+33 b^2 f h^2-4 c^2 \left (3 f g^2-7 h (2 d h+e g)\right )\right )}{5 c}-\frac {\int (g+h x) \left (99 f g h^2 b^3+2 \left (66 a f h^3-c g h (79 f g+63 e h)\right ) b^2+4 c \left (6 c f g^3+14 c h (4 e g+3 d h) g-a h^2 (95 f g+42 e h)\right ) b-8 c h \left (70 c^2 d g^2+16 a^2 f h^2-a c \left (19 f g^2+7 h (7 e g+4 d h)\right )\right )+\left (16 \left (3 f g^3-7 g h (e g+7 d h)\right ) c^3+8 h \left (5 b f g^2+7 b h (9 e g+7 d h)+a h (41 f g+35 e h)\right ) c^2-6 b h^2 (59 b f g+49 b e h+74 a f h) c+231 b^3 f h^3\right ) x\right ) \sqrt {c x^2+b x+a}dx}{10 c}}{4 c}}{14 c h}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{3/2} (11 b f h-14 c e h+6 c f g)}{6 c}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (-2 c h (16 a f h+21 b e h+8 b f g)+33 b^2 f h^2-4 c^2 \left (3 f g^2-7 h (2 d h+e g)\right )\right )}{5 c}-\frac {-\frac {35 h \left (16 c^3 \left (2 a^2 h^2 (e h+3 f g)+6 a b h \left (h (d h+3 e g)+3 f g^2\right )+5 b^2 g \left (3 h (d h+e g)+f g^2\right )\right )-8 b c^2 h \left (10 a^2 f h^2+14 a b h (e h+3 f g)+7 b^2 \left (d h^2+3 e g h+3 f g^2\right )\right )+6 b^3 c h^2 (20 a f h+7 b (e h+3 f g))-64 c^4 g \left (3 a h (d h+e g)+a f g^2+2 b g (3 d h+e g)\right )-33 b^5 f h^3+256 c^5 d g^3\right ) \int \sqrt {c x^2+b x+a}dx}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (8 c^2 h^2 \left (128 a^2 f h^2+343 a b h (e h+3 f g)+b^2 \left (245 h (d h+3 e g)+537 f g^2\right )\right )-6 c h x \left (8 c^2 h \left (a h (35 e h+41 f g)+7 b h (7 d h+9 e g)+5 b f g^2\right )-6 b c h^2 (74 a f h+49 b e h+59 b f g)+231 b^3 f h^3+16 c^3 \left (3 f g^3-7 g h (7 d h+e g)\right )\right )-42 b^2 c h^3 (78 a f h+35 b (e h+3 f g))-16 c^3 h \left (16 a h \left (7 h (d h+3 e g)+15 f g^2\right )+b g \left (21 h (25 d h+19 e g)+17 f g^2\right )\right )+1155 b^4 f h^4-128 c^4 \left (3 f g^4-7 g^2 h (12 d h+e g)\right )\right )}{24 c^2}}{10 c}}{4 c}}{14 c h}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{3/2} (11 b f h-14 c e h+6 c f g)}{6 c}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (-2 c h (16 a f h+21 b e h+8 b f g)+33 b^2 f h^2-4 c^2 \left (3 f g^2-7 h (2 d h+e g)\right )\right )}{5 c}-\frac {-\frac {35 h \left (16 c^3 \left (2 a^2 h^2 (e h+3 f g)+6 a b h \left (h (d h+3 e g)+3 f g^2\right )+5 b^2 g \left (3 h (d h+e g)+f g^2\right )\right )-8 b c^2 h \left (10 a^2 f h^2+14 a b h (e h+3 f g)+7 b^2 \left (d h^2+3 e g h+3 f g^2\right )\right )+6 b^3 c h^2 (20 a f h+7 b (e h+3 f g))-64 c^4 g \left (3 a h (d h+e g)+a f g^2+2 b g (3 d h+e g)\right )-33 b^5 f h^3+256 c^5 d g^3\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (8 c^2 h^2 \left (128 a^2 f h^2+343 a b h (e h+3 f g)+b^2 \left (245 h (d h+3 e g)+537 f g^2\right )\right )-6 c h x \left (8 c^2 h \left (a h (35 e h+41 f g)+7 b h (7 d h+9 e g)+5 b f g^2\right )-6 b c h^2 (74 a f h+49 b e h+59 b f g)+231 b^3 f h^3+16 c^3 \left (3 f g^3-7 g h (7 d h+e g)\right )\right )-42 b^2 c h^3 (78 a f h+35 b (e h+3 f g))-16 c^3 h \left (16 a h \left (7 h (d h+3 e g)+15 f g^2\right )+b g \left (21 h (25 d h+19 e g)+17 f g^2\right )\right )+1155 b^4 f h^4-128 c^4 \left (3 f g^4-7 g^2 h (12 d h+e g)\right )\right )}{24 c^2}}{10 c}}{4 c}}{14 c h}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{3/2} (11 b f h-14 c e h+6 c f g)}{6 c}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (-2 c h (16 a f h+21 b e h+8 b f g)+33 b^2 f h^2-4 c^2 \left (3 f g^2-7 h (2 d h+e g)\right )\right )}{5 c}-\frac {-\frac {35 h \left (16 c^3 \left (2 a^2 h^2 (e h+3 f g)+6 a b h \left (h (d h+3 e g)+3 f g^2\right )+5 b^2 g \left (3 h (d h+e g)+f g^2\right )\right )-8 b c^2 h \left (10 a^2 f h^2+14 a b h (e h+3 f g)+7 b^2 \left (d h^2+3 e g h+3 f g^2\right )\right )+6 b^3 c h^2 (20 a f h+7 b (e h+3 f g))-64 c^4 g \left (3 a h (d h+e g)+a f g^2+2 b g (3 d h+e g)\right )-33 b^5 f h^3+256 c^5 d g^3\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (8 c^2 h^2 \left (128 a^2 f h^2+343 a b h (e h+3 f g)+b^2 \left (245 h (d h+3 e g)+537 f g^2\right )\right )-6 c h x \left (8 c^2 h \left (a h (35 e h+41 f g)+7 b h (7 d h+9 e g)+5 b f g^2\right )-6 b c h^2 (74 a f h+49 b e h+59 b f g)+231 b^3 f h^3+16 c^3 \left (3 f g^3-7 g h (7 d h+e g)\right )\right )-42 b^2 c h^3 (78 a f h+35 b (e h+3 f g))-16 c^3 h \left (16 a h \left (7 h (d h+3 e g)+15 f g^2\right )+b g \left (21 h (25 d h+19 e g)+17 f g^2\right )\right )+1155 b^4 f h^4-128 c^4 \left (3 f g^4-7 g^2 h (12 d h+e g)\right )\right )}{24 c^2}}{10 c}}{4 c}}{14 c h}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {f (g+h x)^4 \left (a+b x+c x^2\right )^{3/2}}{7 c h}-\frac {\frac {(g+h x)^3 \left (a+b x+c x^2\right )^{3/2} (11 b f h-14 c e h+6 c f g)}{6 c}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (-2 c h (16 a f h+21 b e h+8 b f g)+33 b^2 f h^2-4 c^2 \left (3 f g^2-7 h (2 d h+e g)\right )\right )}{5 c}-\frac {-\frac {35 h \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right ) \left (16 c^3 \left (2 a^2 h^2 (e h+3 f g)+6 a b h \left (h (d h+3 e g)+3 f g^2\right )+5 b^2 g \left (3 h (d h+e g)+f g^2\right )\right )-8 b c^2 h \left (10 a^2 f h^2+14 a b h (e h+3 f g)+7 b^2 \left (d h^2+3 e g h+3 f g^2\right )\right )+6 b^3 c h^2 (20 a f h+7 b (e h+3 f g))-64 c^4 g \left (3 a h (d h+e g)+a f g^2+2 b g (3 d h+e g)\right )-33 b^5 f h^3+256 c^5 d g^3\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (8 c^2 h^2 \left (128 a^2 f h^2+343 a b h (e h+3 f g)+b^2 \left (245 h (d h+3 e g)+537 f g^2\right )\right )-6 c h x \left (8 c^2 h \left (a h (35 e h+41 f g)+7 b h (7 d h+9 e g)+5 b f g^2\right )-6 b c h^2 (74 a f h+49 b e h+59 b f g)+231 b^3 f h^3+16 c^3 \left (3 f g^3-7 g h (7 d h+e g)\right )\right )-42 b^2 c h^3 (78 a f h+35 b (e h+3 f g))-16 c^3 h \left (16 a h \left (7 h (d h+3 e g)+15 f g^2\right )+b g \left (21 h (25 d h+19 e g)+17 f g^2\right )\right )+1155 b^4 f h^4-128 c^4 \left (3 f g^4-7 g^2 h (12 d h+e g)\right )\right )}{24 c^2}}{10 c}}{4 c}}{14 c h}\) |
(f*(g + h*x)^4*(a + b*x + c*x^2)^(3/2))/(7*c*h) - (((6*c*f*g - 14*c*e*h + 11*b*f*h)*(g + h*x)^3*(a + b*x + c*x^2)^(3/2))/(6*c) - (((33*b^2*f*h^2 - 2 *c*h*(8*b*f*g + 21*b*e*h + 16*a*f*h) - 4*c^2*(3*f*g^2 - 7*h*(e*g + 2*d*h)) )*(g + h*x)^2*(a + b*x + c*x^2)^(3/2))/(5*c) - (-1/24*((1155*b^4*f*h^4 - 1 28*c^4*(3*f*g^4 - 7*g^2*h*(e*g + 12*d*h)) - 42*b^2*c*h^3*(78*a*f*h + 35*b* (3*f*g + e*h)) + 8*c^2*h^2*(128*a^2*f*h^2 + 343*a*b*h*(3*f*g + e*h) + b^2* (537*f*g^2 + 245*h*(3*e*g + d*h))) - 16*c^3*h*(16*a*h*(15*f*g^2 + 7*h*(3*e *g + d*h)) + b*g*(17*f*g^2 + 21*h*(19*e*g + 25*d*h))) - 6*c*h*(231*b^3*f*h ^3 - 6*b*c*h^2*(59*b*f*g + 49*b*e*h + 74*a*f*h) + 16*c^3*(3*f*g^3 - 7*g*h* (e*g + 7*d*h)) + 8*c^2*h*(5*b*f*g^2 + 7*b*h*(9*e*g + 7*d*h) + a*h*(41*f*g + 35*e*h)))*x)*(a + b*x + c*x^2)^(3/2))/c^2 - (35*h*(256*c^5*d*g^3 - 33*b^ 5*f*h^3 - 64*c^4*g*(a*f*g^2 + 3*a*h*(e*g + d*h) + 2*b*g*(e*g + 3*d*h)) + 6 *b^3*c*h^2*(20*a*f*h + 7*b*(3*f*g + e*h)) - 8*b*c^2*h*(10*a^2*f*h^2 + 14*a *b*h*(3*f*g + e*h) + 7*b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) + 16*c^3*(2*a^2*h^ 2*(3*f*g + e*h) + 5*b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 6*a*b*h*(3*f*g^2 + h *(3*e*g + d*h))))*(((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))/(10*c))/(4*c))/(14*c*h)
3.2.86.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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])
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]
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]
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])
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]))
Time = 1.14 (sec) , antiderivative size = 1757, normalized size of antiderivative = 1.89
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1757\) |
| default | \(\text {Expression too large to display}\) | \(2098\) |
1/107520*(15360*c^6*f*h^3*x^6+1280*b*c^5*f*h^3*x^5+17920*c^6*e*h^3*x^5+537 60*c^6*f*g*h^2*x^5+3072*a*c^5*f*h^3*x^4-1408*b^2*c^4*f*h^3*x^4+1792*b*c^5* e*h^3*x^4+5376*b*c^5*f*g*h^2*x^4+21504*c^6*d*h^3*x^4+64512*c^6*e*g*h^2*x^4 +64512*c^6*f*g^2*h*x^4-5056*a*b*c^4*f*h^3*x^3+4480*a*c^5*e*h^3*x^3+13440*a *c^5*f*g*h^2*x^3+1584*b^3*c^3*f*h^3*x^3-2016*b^2*c^4*e*h^3*x^3-6048*b^2*c^ 4*f*g*h^2*x^3+2688*b*c^5*d*h^3*x^3+8064*b*c^5*e*g*h^2*x^3+8064*b*c^5*f*g^2 *h*x^3+80640*c^6*d*g*h^2*x^3+80640*c^6*e*g^2*h*x^3+26880*c^6*f*g^3*x^3-409 6*a^2*c^4*f*h^3*x^2+7776*a*b^2*c^3*f*h^3*x^2-7616*a*b*c^4*e*h^3*x^2-22848* a*b*c^4*f*g*h^2*x^2+7168*a*c^5*d*h^3*x^2+21504*a*c^5*e*g*h^2*x^2+21504*a*c ^5*f*g^2*h*x^2-1848*b^4*c^2*f*h^3*x^2+2352*b^3*c^3*e*h^3*x^2+7056*b^3*c^3* f*g*h^2*x^2-3136*b^2*c^4*d*h^3*x^2-9408*b^2*c^4*e*g*h^2*x^2-9408*b^2*c^4*f *g^2*h*x^2+13440*b*c^5*d*g*h^2*x^2+13440*b*c^5*e*g^2*h*x^2+4480*b*c^5*f*g^ 3*x^2+107520*c^6*d*g^2*h*x^2+35840*c^6*e*g^3*x^2+12704*a^2*b*c^3*f*h^3*x-6 720*a^2*c^4*e*h^3*x-20160*a^2*c^4*f*g*h^2*x-12096*a*b^3*c^2*f*h^3*x+12544* a*b^2*c^3*e*h^3*x+37632*a*b^2*c^3*f*g*h^2*x-12992*a*b*c^4*d*h^3*x-38976*a* b*c^4*e*g*h^2*x-38976*a*b*c^4*f*g^2*h*x+40320*a*c^5*d*g*h^2*x+40320*a*c^5* e*g^2*h*x+13440*a*c^5*f*g^3*x+2310*b^5*c*f*h^3*x-2940*b^4*c^2*e*h^3*x-8820 *b^4*c^2*f*g*h^2*x+3920*b^3*c^3*d*h^3*x+11760*b^3*c^3*e*g*h^2*x+11760*b^3* c^3*f*g^2*h*x-16800*b^2*c^4*d*g*h^2*x-16800*b^2*c^4*e*g^2*h*x-5600*b^2*c^4 *f*g^3*x+26880*b*c^5*d*g^2*h*x+8960*b*c^5*e*g^3*x+53760*c^6*d*g^3*x+819...
Time = 1.12 (sec) , antiderivative size = 2817, normalized size of antiderivative = 3.03 \[ \int (g+h x)^3 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
[1/430080*(105*(16*(16*(b^2*c^5 - 4*a*c^6)*d - 8*(b^3*c^4 - 4*a*b*c^5)*e + (5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*f)*g^3 - 24*(16*(b^3*c^4 - 4*a*b* c^5)*d - 2*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*e + (7*b^5*c^2 - 40*a*b ^3*c^3 + 48*a^2*b*c^4)*f)*g^2*h + 6*(8*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2* c^5)*d - 4*(7*b^5*c^2 - 40*a*b^3*c^3 + 48*a^2*b*c^4)*e + (21*b^6*c - 140*a *b^4*c^2 + 240*a^2*b^2*c^3 - 64*a^3*c^4)*f)*g*h^2 - (8*(7*b^5*c^2 - 40*a*b ^3*c^3 + 48*a^2*b*c^4)*d - 2*(21*b^6*c - 140*a*b^4*c^2 + 240*a^2*b^2*c^3 - 64*a^3*c^4)*e + (33*b^7 - 252*a*b^5*c + 560*a^2*b^3*c^2 - 320*a^3*b*c^3)* f)*h^3)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*( 2*c*x + b)*sqrt(c) - 4*a*c) + 4*(15360*c^7*f*h^3*x^6 + 1280*(42*c^7*f*g*h^ 2 + (14*c^7*e + b*c^6*f)*h^3)*x^5 + 128*(504*c^7*f*g^2*h + 42*(12*c^7*e + b*c^6*f)*g*h^2 + (168*c^7*d + 14*b*c^6*e - (11*b^2*c^5 - 24*a*c^6)*f)*h^3) *x^4 + 560*(48*b*c^6*d - 8*(3*b^2*c^5 - 8*a*c^6)*e + (15*b^3*c^4 - 52*a*b* c^5)*f)*g^3 - 168*(80*(3*b^2*c^5 - 8*a*c^6)*d - 10*(15*b^3*c^4 - 52*a*b*c^ 5)*e + (105*b^4*c^3 - 460*a*b^2*c^4 + 256*a^2*c^5)*f)*g^2*h + 42*(40*(15*b ^3*c^4 - 52*a*b*c^5)*d - 4*(105*b^4*c^3 - 460*a*b^2*c^4 + 256*a^2*c^5)*e + (315*b^5*c^2 - 1680*a*b^3*c^3 + 1808*a^2*b*c^4)*f)*g*h^2 - (56*(105*b^4*c ^3 - 460*a*b^2*c^4 + 256*a^2*c^5)*d - 14*(315*b^5*c^2 - 1680*a*b^3*c^3 + 1 808*a^2*b*c^4)*e + (3465*b^6*c - 21840*a*b^4*c^2 + 34608*a^2*b^2*c^3 - 819 2*a^3*c^4)*f)*h^3 + 16*(1680*c^7*f*g^3 + 504*(10*c^7*e + b*c^6*f)*g^2*h...
Leaf count of result is larger than twice the leaf count of optimal. 4820 vs. \(2 (966) = 1932\).
Time = 1.38 (sec) , antiderivative size = 4820, normalized size of antiderivative = 5.18 \[ \int (g+h x)^3 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
Piecewise((sqrt(a + b*x + c*x**2)*(f*h**3*x**6/7 + x**5*(b*f*h**3/14 + c*e *h**3 + 3*c*f*g*h**2)/(6*c) + x**4*(a*f*h**3/7 + b*e*h**3 + 3*b*f*g*h**2 - 11*b*(b*f*h**3/14 + c*e*h**3 + 3*c*f*g*h**2)/(12*c) + c*d*h**3 + 3*c*e*g* h**2 + 3*c*f*g**2*h)/(5*c) + x**3*(a*e*h**3 + 3*a*f*g*h**2 - 5*a*(b*f*h**3 /14 + c*e*h**3 + 3*c*f*g*h**2)/(6*c) + b*d*h**3 + 3*b*e*g*h**2 + 3*b*f*g** 2*h - 9*b*(a*f*h**3/7 + b*e*h**3 + 3*b*f*g*h**2 - 11*b*(b*f*h**3/14 + c*e* h**3 + 3*c*f*g*h**2)/(12*c) + c*d*h**3 + 3*c*e*g*h**2 + 3*c*f*g**2*h)/(10* c) + 3*c*d*g*h**2 + 3*c*e*g**2*h + c*f*g**3)/(4*c) + x**2*(a*d*h**3 + 3*a* e*g*h**2 + 3*a*f*g**2*h - 4*a*(a*f*h**3/7 + b*e*h**3 + 3*b*f*g*h**2 - 11*b *(b*f*h**3/14 + c*e*h**3 + 3*c*f*g*h**2)/(12*c) + c*d*h**3 + 3*c*e*g*h**2 + 3*c*f*g**2*h)/(5*c) + 3*b*d*g*h**2 + 3*b*e*g**2*h + b*f*g**3 - 7*b*(a*e* h**3 + 3*a*f*g*h**2 - 5*a*(b*f*h**3/14 + c*e*h**3 + 3*c*f*g*h**2)/(6*c) + b*d*h**3 + 3*b*e*g*h**2 + 3*b*f*g**2*h - 9*b*(a*f*h**3/7 + b*e*h**3 + 3*b* f*g*h**2 - 11*b*(b*f*h**3/14 + c*e*h**3 + 3*c*f*g*h**2)/(12*c) + c*d*h**3 + 3*c*e*g*h**2 + 3*c*f*g**2*h)/(10*c) + 3*c*d*g*h**2 + 3*c*e*g**2*h + c*f* g**3)/(8*c) + 3*c*d*g**2*h + c*e*g**3)/(3*c) + x*(3*a*d*g*h**2 + 3*a*e*g** 2*h + a*f*g**3 - 3*a*(a*e*h**3 + 3*a*f*g*h**2 - 5*a*(b*f*h**3/14 + c*e*h** 3 + 3*c*f*g*h**2)/(6*c) + b*d*h**3 + 3*b*e*g*h**2 + 3*b*f*g**2*h - 9*b*(a* f*h**3/7 + b*e*h**3 + 3*b*f*g*h**2 - 11*b*(b*f*h**3/14 + c*e*h**3 + 3*c*f* g*h**2)/(12*c) + c*d*h**3 + 3*c*e*g*h**2 + 3*c*f*g**2*h)/(10*c) + 3*c*d...
Exception generated. \[ \int (g+h x)^3 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.29 (sec) , antiderivative size = 1657, normalized size of antiderivative = 1.78 \[ \int (g+h x)^3 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*f*h^3*x + (42*c^6*f*g*h ^2 + 14*c^6*e*h^3 + b*c^5*f*h^3)/c^6)*x + (504*c^6*f*g^2*h + 504*c^6*e*g*h ^2 + 42*b*c^5*f*g*h^2 + 168*c^6*d*h^3 + 14*b*c^5*e*h^3 - 11*b^2*c^4*f*h^3 + 24*a*c^5*f*h^3)/c^6)*x + (1680*c^6*f*g^3 + 5040*c^6*e*g^2*h + 504*b*c^5* f*g^2*h + 5040*c^6*d*g*h^2 + 504*b*c^5*e*g*h^2 - 378*b^2*c^4*f*g*h^2 + 840 *a*c^5*f*g*h^2 + 168*b*c^5*d*h^3 - 126*b^2*c^4*e*h^3 + 280*a*c^5*e*h^3 + 9 9*b^3*c^3*f*h^3 - 316*a*b*c^4*f*h^3)/c^6)*x + (4480*c^6*e*g^3 + 560*b*c^5* f*g^3 + 13440*c^6*d*g^2*h + 1680*b*c^5*e*g^2*h - 1176*b^2*c^4*f*g^2*h + 26 88*a*c^5*f*g^2*h + 1680*b*c^5*d*g*h^2 - 1176*b^2*c^4*e*g*h^2 + 2688*a*c^5* e*g*h^2 + 882*b^3*c^3*f*g*h^2 - 2856*a*b*c^4*f*g*h^2 - 392*b^2*c^4*d*h^3 + 896*a*c^5*d*h^3 + 294*b^3*c^3*e*h^3 - 952*a*b*c^4*e*h^3 - 231*b^4*c^2*f*h ^3 + 972*a*b^2*c^3*f*h^3 - 512*a^2*c^4*f*h^3)/c^6)*x + (26880*c^6*d*g^3 + 4480*b*c^5*e*g^3 - 2800*b^2*c^4*f*g^3 + 6720*a*c^5*f*g^3 + 13440*b*c^5*d*g ^2*h - 8400*b^2*c^4*e*g^2*h + 20160*a*c^5*e*g^2*h + 5880*b^3*c^3*f*g^2*h - 19488*a*b*c^4*f*g^2*h - 8400*b^2*c^4*d*g*h^2 + 20160*a*c^5*d*g*h^2 + 5880 *b^3*c^3*e*g*h^2 - 19488*a*b*c^4*e*g*h^2 - 4410*b^4*c^2*f*g*h^2 + 18816*a* b^2*c^3*f*g*h^2 - 10080*a^2*c^4*f*g*h^2 + 1960*b^3*c^3*d*h^3 - 6496*a*b*c^ 4*d*h^3 - 1470*b^4*c^2*e*h^3 + 6272*a*b^2*c^3*e*h^3 - 3360*a^2*c^4*e*h^3 + 1155*b^5*c*f*h^3 - 6048*a*b^3*c^2*f*h^3 + 6352*a^2*b*c^3*f*h^3)/c^6)*x + (26880*b*c^5*d*g^3 - 13440*b^2*c^4*e*g^3 + 35840*a*c^5*e*g^3 + 8400*b^3...
Time = 25.11 (sec) , antiderivative size = 3262, normalized size of antiderivative = 3.51 \[ \int (g+h x)^3 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
d*g^3*(x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (8*a^3*f*h^3*(a + b*x + c* x^2)^(1/2))/(105*c^3) - (33*b^6*f*h^3*(a + b*x + c*x^2)^(1/2))/(1024*c^6) + (d*h^3*x^2*(a + b*x + c*x^2)^(3/2))/(5*c) + (e*h^3*x^3*(a + b*x + c*x^2) ^(3/2))/(6*c) + (f*h^3*x^4*(a + b*x + c*x^2)^(3/2))/(7*c) - (a*f*g^3*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c) + (d*g^3*log((b/2 + c*x )/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2)) + (e*g^3*l og((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c ^(5/2)) - (2*a*d*h^3*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2) )*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(5*c) - (5*b*f*g^3*((log((b + 2*c*x)/c^(1 /2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) + (e *g^3*(8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2) + (33*b^7*f*h^3*log(b + 2*c^(1/2)*(a + b*x + c*x^2)^(1/2) + 2*c*x))/(2048 *c^(13/2)) + (f*g^3*x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*e*h^3*((5*b*((lo g((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^ (5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24 *c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1...