Integrand size = 32, antiderivative size = 753 \[ \int (g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=-\frac {\left (b^2-4 a c\right ) \left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (3 b g (e g+2 d h)+a \left (f g^2+2 e g h+d h^2\right )\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+2 e g h+d h^2\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^6}+\frac {\left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (3 b g (e g+2 d h)+a \left (f g^2+2 e g h+d h^2\right )\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+2 e g h+d h^2\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}-\frac {(10 c f g-16 c e h+11 b f h) (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2 h}+\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac {\left (693 b^3 f h^3+96 c^3 g \left (5 f g^2-8 h (e g+7 d h)\right )-36 b c h^2 (31 a f h+28 b (2 f g+e h))+8 c^2 h \left (96 a h (2 f g+e h)+b \left (31 f g^2+196 h (2 e g+d h)\right )\right )-10 c h \left (99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (2 e g+7 d h)\right )-12 c h (7 a f h+2 b (f g+6 e h))\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4 h}+\frac {\left (b^2-4 a c\right )^2 \left (768 c^4 d g^2+99 b^4 f h^2-72 b^2 c h (4 b f g+2 b e h+3 a f h)-128 c^3 \left (3 b g (e g+2 d h)+a \left (f g^2+2 e g h+d h^2\right )\right )+16 c^2 \left (3 a^2 f h^2+12 a b h (2 f g+e h)+14 b^2 \left (f g^2+2 e g h+d h^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2}} \]
1/6144*(768*c^4*d*g^2+99*b^4*f*h^2-72*b^2*c*h*(3*a*f*h+2*b*e*h+4*b*f*g)-12 8*c^3*(3*b*g*(2*d*h+e*g)+a*(d*h^2+2*e*g*h+f*g^2))+16*c^2*(3*a^2*f*h^2+12*a *b*h*(e*h+2*f*g)+14*b^2*(d*h^2+2*e*g*h+f*g^2)))*(2*c*x+b)*(c*x^2+b*x+a)^(3 /2)/c^5-1/112*(11*b*f*h-16*c*e*h+10*c*f*g)*(h*x+g)^2*(c*x^2+b*x+a)^(5/2)/c ^2/h+1/8*f*(h*x+g)^3*(c*x^2+b*x+a)^(5/2)/c/h-1/13440*(693*b^3*f*h^3+96*c^3 *g*(5*f*g^2-8*h*(7*d*h+e*g))-36*b*c*h^2*(31*a*f*h+28*b*(e*h+2*f*g))+8*c^2* h*(96*a*h*(e*h+2*f*g)+b*(31*f*g^2+196*h*(d*h+2*e*g)))-10*c*h*(99*b^2*f*h^2 -8*c^2*(5*f*g^2-4*h*(7*d*h+2*e*g))-12*c*h*(7*a*f*h+2*b*(6*e*h+f*g)))*x)*(c *x^2+b*x+a)^(5/2)/c^4/h+1/32768*(-4*a*c+b^2)^2*(768*c^4*d*g^2+99*b^4*f*h^2 -72*b^2*c*h*(3*a*f*h+2*b*e*h+4*b*f*g)-128*c^3*(3*b*g*(2*d*h+e*g)+a*(d*h^2+ 2*e*g*h+f*g^2))+16*c^2*(3*a^2*f*h^2+12*a*b*h*(e*h+2*f*g)+14*b^2*(d*h^2+2*e *g*h+f*g^2)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(13/2)- 1/16384*(-4*a*c+b^2)*(768*c^4*d*g^2+99*b^4*f*h^2-72*b^2*c*h*(3*a*f*h+2*b*e *h+4*b*f*g)-128*c^3*(3*b*g*(2*d*h+e*g)+a*(d*h^2+2*e*g*h+f*g^2))+16*c^2*(3* a^2*f*h^2+12*a*b*h*(e*h+2*f*g)+14*b^2*(d*h^2+2*e*g*h+f*g^2)))*(2*c*x+b)*(c *x^2+b*x+a)^(1/2)/c^6
Time = 11.44 (sec) , antiderivative size = 877, normalized size of antiderivative = 1.16 \[ \int (g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\frac {430080 d g^2 (b+2 c x) (a+x (b+c x))^{3/2}+688128 g (e g+2 d h) (a+x (b+c x))^{5/2}+573440 \left (f g^2+h (2 e g+d h)\right ) x (a+x (b+c x))^{5/2}+491520 h (2 f g+e h) x^2 (a+x (b+c x))^{5/2}+430080 f h^2 x^3 (a+x (b+c x))^{5/2}+\frac {80640 \left (b^2-4 a c\right ) d g^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 {13440 b g (e g+2 d h) \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 {48 h (2 f g+e h) \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 (f g^2+h (2 e g+d h)\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 h^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} \]
(430080*d*g^2*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) + 688128*g*(e*g + 2*d*h) *(a + x*(b + c*x))^(5/2) + 573440*(f*g^2 + h*(2*e*g + d*h))*x*(a + x*(b + c*x))^(5/2) + 491520*h*(2*f*g + e*h)*x^2*(a + x*(b + c*x))^(5/2) + 430080* f*h^2*x^3*(a + x*(b + c*x))^(5/2) + (80640*(b^2 - 4*a*c)*d*g^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) - (13440*b*g*(e*g + 2*d*h)*(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*Sqr t[c]*Sqrt[a + x*(b + c*x)])])))/c^(5/2) + (48*h*(2*f*g + e*h)*(-256*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*(f*g^2 + h*(2*e* g + d*h))*(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*Sq rt[c]*Sqrt[a + x*(b + c*x)])]))))/c^(7/2) - (3*f*h^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)*...
Time = 1.20 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.70, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2184, 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.
| \(\displaystyle \int (g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {\int -\frac {1}{2} h (g+h x)^2 (5 b f g-16 c d h+6 a f h+(10 c f g-16 c e h+11 b f h) x) \left (c x^2+b x+a\right )^{3/2}dx}{8 c h^2}+\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac {\int (g+h x)^2 (5 b f g-16 c d h+6 a f h+(10 c f g-16 c e h+11 b f h) x) \left (c x^2+b x+a\right )^{3/2}dx}{16 c h}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac {\frac {\int -\frac {1}{2} (g+h x) \left (55 f g h b^2+44 a f h^2 b-20 c g (f g+4 e h) b+4 c h (56 c d g-11 a f g-16 a e h)+\left (-8 \left (5 f g^2-4 h (2 e g+7 d h)\right ) c^2-12 h (7 a f h+2 b (f g+6 e h)) c+99 b^2 f h^2\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{7 c}+\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} (11 b f h-16 c e h+10 c f g)}{7 c}}{16 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} (11 b f h-16 c e h+10 c f g)}{7 c}-\frac {\int (g+h x) \left (55 f g h b^2+44 a f h^2 b-20 c g (f g+4 e h) b+4 c h (56 c d g-11 a f g-16 a e h)+\left (-8 \left (5 f g^2-4 h (2 e g+7 d h)\right ) c^2-12 h (7 a f h+2 b (f g+6 e h)) c+99 b^2 f h^2\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{14 c}}{16 c h}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} (11 b f h-16 c e h+10 c f g)}{7 c}-\frac {\frac {7 h \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a h (d h+2 e g)+a f g^2+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c h x \left (-12 c h (7 a f h+2 b (6 e h+f g))+99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (7 d h+2 e g)\right )\right )+8 c^2 h \left (96 a h (e h+2 f g)+196 b h (d h+2 e g)+31 b f g^2\right )-36 b c h^2 (31 a f h+28 b (e h+2 f g))+693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (7 d h+e g)\right )\right )}{60 c^2}}{14 c}}{16 c h}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} (11 b f h-16 c e h+10 c f g)}{7 c}-\frac {\frac {7 h \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a h (d h+2 e g)+a f g^2+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c h x \left (-12 c h (7 a f h+2 b (6 e h+f g))+99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (7 d h+2 e g)\right )\right )+8 c^2 h \left (96 a h (e h+2 f g)+196 b h (d h+2 e g)+31 b f g^2\right )-36 b c h^2 (31 a f h+28 b (e h+2 f g))+693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (7 d h+e g)\right )\right )}{60 c^2}}{14 c}}{16 c h}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} (11 b f h-16 c e h+10 c f g)}{7 c}-\frac {\frac {7 h \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a h (d h+2 e g)+a f g^2+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c h x \left (-12 c h (7 a f h+2 b (6 e h+f g))+99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (7 d h+2 e g)\right )\right )+8 c^2 h \left (96 a h (e h+2 f g)+196 b h (d h+2 e g)+31 b f g^2\right )-36 b c h^2 (31 a f h+28 b (e h+2 f g))+693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (7 d h+e g)\right )\right )}{60 c^2}}{14 c}}{16 c h}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} (11 b f h-16 c e h+10 c f g)}{7 c}-\frac {\frac {7 h \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a h (d h+2 e g)+a f g^2+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c h x \left (-12 c h (7 a f h+2 b (6 e h+f g))+99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (7 d h+2 e g)\right )\right )+8 c^2 h \left (96 a h (e h+2 f g)+196 b h (d h+2 e g)+31 b f g^2\right )-36 b c h^2 (31 a f h+28 b (e h+2 f g))+693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (7 d h+e g)\right )\right )}{60 c^2}}{14 c}}{16 c h}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{5/2}}{8 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{5/2} (11 b f h-16 c e h+10 c f g)}{7 c}-\frac {\frac {7 h \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right ) \left (16 c^2 \left (3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left (a h (d h+2 e g)+a f g^2+3 b g (2 d h+e g)\right )+99 b^4 f h^2+768 c^4 d g^2\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c h x \left (-12 c h (7 a f h+2 b (6 e h+f g))+99 b^2 f h^2-8 c^2 \left (5 f g^2-4 h (7 d h+2 e g)\right )\right )+8 c^2 h \left (96 a h (e h+2 f g)+196 b h (d h+2 e g)+31 b f g^2\right )-36 b c h^2 (31 a f h+28 b (e h+2 f g))+693 b^3 f h^3+96 c^3 \left (5 f g^3-8 g h (7 d h+e g)\right )\right )}{60 c^2}}{14 c}}{16 c h}\) |
(f*(g + h*x)^3*(a + b*x + c*x^2)^(5/2))/(8*c*h) - (((10*c*f*g - 16*c*e*h + 11*b*f*h)*(g + h*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c) - (-1/60*((693*b^3*f *h^3 + 96*c^3*(5*f*g^3 - 8*g*h*(e*g + 7*d*h)) - 36*b*c*h^2*(31*a*f*h + 28* b*(2*f*g + e*h)) + 8*c^2*h*(31*b*f*g^2 + 196*b*h*(2*e*g + d*h) + 96*a*h*(2 *f*g + e*h)) - 10*c*h*(99*b^2*f*h^2 - 8*c^2*(5*f*g^2 - 4*h*(2*e*g + 7*d*h) ) - 12*c*h*(7*a*f*h + 2*b*(f*g + 6*e*h)))*x)*(a + b*x + c*x^2)^(5/2))/c^2 + (7*h*(768*c^4*d*g^2 + 99*b^4*f*h^2 - 72*b^2*c*h*(4*b*f*g + 2*b*e*h + 3*a *f*h) - 128*c^3*(a*f*g^2 + a*h*(2*e*g + d*h) + 3*b*g*(e*g + 2*d*h)) + 16*c ^2*(3*a^2*f*h^2 + 12*a*b*h*(2*f*g + e*h) + 14*b^2*(f*g^2 + h*(2*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)))/(24*c^2))/(14*c ))/(16*c*h)
3.2.97.3.1 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]))
Leaf count of result is larger than twice the leaf count of optimal. \(1679\) vs. \(2(723)=1446\).
Time = 0.85 (sec) , antiderivative size = 1680, normalized size of antiderivative = 2.23
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1680\) |
| risch | \(\text {Expression too large to display}\) | \(1933\) |
d*g^2*(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))))+f*h^2*(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+c*x)/ c^(1/2)+(c*x^2+b*x+a)^(1/2))))))+(e*h^2+2*f*g*h)*(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*(...
Leaf count of result is larger than twice the leaf count of optimal. 1571 vs. \(2 (723) = 1446\).
Time = 1.50 (sec) , antiderivative size = 3145, normalized size of antiderivative = 4.18 \[ \int (g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
[1/6881280*(105*(32*(24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d - 12*(b^5*c ^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*e + (7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b ^2*c^4 - 64*a^3*c^5)*f)*g^2 - 32*(24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5 )*d - 2*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*e + 3*(3 *b^7*c - 28*a*b^5*c^2 + 80*a^2*b^3*c^3 - 64*a^3*b*c^4)*f)*g*h + (32*(7*b^6 *c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*d - 48*(3*b^7*c - 28*a *b^5*c^2 + 80*a^2*b^3*c^3 - 64*a^3*b*c^4)*e + 3*(33*b^8 - 336*a*b^6*c + 11 20*a^2*b^4*c^2 - 1280*a^3*b^2*c^3 + 256*a^4*c^4)*f)*h^2)*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*h^2*x^7 + 15360*(32*c^8*f*g*h + (16*c^8*e + 17*b*c^7* f)*h^2)*x^6 + 1280*(224*c^8*f*g^2 + 32*(14*c^8*e + 15*b*c^7*f)*g*h + (224* c^8*d + 240*b*c^7*e + 3*(b^2*c^6 + 84*a*c^7)*f)*h^2)*x^5 + 128*(224*(12*c^ 8*e + 13*b*c^7*f)*g^2 + 32*(168*c^8*d + 182*b*c^7*e + 3*(b^2*c^6 + 64*a*c^ 7)*f)*g*h + (2912*b*c^7*d + 48*(b^2*c^6 + 64*a*c^7)*e - 3*(11*b^3*c^5 - 52 *a*b*c^6)*f)*h^2)*x^4 + 16*(224*(120*c^8*d + 132*b*c^7*e + (3*b^2*c^6 + 14 0*a*c^7)*f)*g^2 + 32*(1848*b*c^7*d + 14*(3*b^2*c^6 + 140*a*c^7)*e - 3*(9*b ^3*c^5 - 44*a*b*c^6)*f)*g*h + (224*(3*b^2*c^6 + 140*a*c^7)*d - 48*(9*b^3*c ^5 - 44*a*b*c^6)*e + 3*(99*b^4*c^4 - 568*a*b^2*c^5 + 560*a^2*c^6)*f)*h^2)* x^3 - 224*(120*(3*b^3*c^5 - 20*a*b*c^6)*d - 12*(15*b^4*c^4 - 100*a*b^2*c^5 + 128*a^2*c^6)*e + (105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*f)*g...
Leaf count of result is larger than twice the leaf count of optimal. 9687 vs. \(2 (790) = 1580\).
Time = 1.33 (sec) , antiderivative size = 9687, normalized size of antiderivative = 12.86 \[ \int (g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
Piecewise((sqrt(a + b*x + c*x**2)*(c*f*h**2*x**7/8 + x**6*(17*b*c*f*h**2/1 6 + c**2*e*h**2 + 2*c**2*f*g*h)/(7*c) + x**5*(9*a*c*f*h**2/8 + b**2*f*h**2 + 2*b*c*e*h**2 + 4*b*c*f*g*h - 13*b*(17*b*c*f*h**2/16 + c**2*e*h**2 + 2*c **2*f*g*h)/(14*c) + c**2*d*h**2 + 2*c**2*e*g*h + c**2*f*g**2)/(6*c) + x**4 *(2*a*b*f*h**2 + 2*a*c*e*h**2 + 4*a*c*f*g*h - 6*a*(17*b*c*f*h**2/16 + c**2 *e*h**2 + 2*c**2*f*g*h)/(7*c) + b**2*e*h**2 + 2*b**2*f*g*h + 2*b*c*d*h**2 + 4*b*c*e*g*h + 2*b*c*f*g**2 - 11*b*(9*a*c*f*h**2/8 + b**2*f*h**2 + 2*b*c* e*h**2 + 4*b*c*f*g*h - 13*b*(17*b*c*f*h**2/16 + c**2*e*h**2 + 2*c**2*f*g*h )/(14*c) + c**2*d*h**2 + 2*c**2*e*g*h + c**2*f*g**2)/(12*c) + 2*c**2*d*g*h + c**2*e*g**2)/(5*c) + x**3*(a**2*f*h**2 + 2*a*b*e*h**2 + 4*a*b*f*g*h + 2 *a*c*d*h**2 + 4*a*c*e*g*h + 2*a*c*f*g**2 - 5*a*(9*a*c*f*h**2/8 + b**2*f*h* *2 + 2*b*c*e*h**2 + 4*b*c*f*g*h - 13*b*(17*b*c*f*h**2/16 + c**2*e*h**2 + 2 *c**2*f*g*h)/(14*c) + c**2*d*h**2 + 2*c**2*e*g*h + c**2*f*g**2)/(6*c) + b* *2*d*h**2 + 2*b**2*e*g*h + b**2*f*g**2 + 4*b*c*d*g*h + 2*b*c*e*g**2 - 9*b* (2*a*b*f*h**2 + 2*a*c*e*h**2 + 4*a*c*f*g*h - 6*a*(17*b*c*f*h**2/16 + c**2* e*h**2 + 2*c**2*f*g*h)/(7*c) + b**2*e*h**2 + 2*b**2*f*g*h + 2*b*c*d*h**2 + 4*b*c*e*g*h + 2*b*c*f*g**2 - 11*b*(9*a*c*f*h**2/8 + b**2*f*h**2 + 2*b*c*e *h**2 + 4*b*c*f*g*h - 13*b*(17*b*c*f*h**2/16 + c**2*e*h**2 + 2*c**2*f*g*h) /(14*c) + c**2*d*h**2 + 2*c**2*e*g*h + c**2*f*g**2)/(12*c) + 2*c**2*d*g*h + c**2*e*g**2)/(10*c) + c**2*d*g**2)/(4*c) + x**2*(a**2*e*h**2 + 2*a**2...
Exception generated. \[ \int (g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 1802 vs. \(2 (723) = 1446\).
Time = 0.32 (sec) , antiderivative size = 1802, normalized size of antiderivative = 2.39 \[ \int (g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
1/1720320*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*(14*c*f*h^2*x + (32*c^ 8*f*g*h + 16*c^8*e*h^2 + 17*b*c^7*f*h^2)/c^7)*x + (224*c^8*f*g^2 + 448*c^8 *e*g*h + 480*b*c^7*f*g*h + 224*c^8*d*h^2 + 240*b*c^7*e*h^2 + 3*b^2*c^6*f*h ^2 + 252*a*c^7*f*h^2)/c^7)*x + (2688*c^8*e*g^2 + 2912*b*c^7*f*g^2 + 5376*c ^8*d*g*h + 5824*b*c^7*e*g*h + 96*b^2*c^6*f*g*h + 6144*a*c^7*f*g*h + 2912*b *c^7*d*h^2 + 48*b^2*c^6*e*h^2 + 3072*a*c^7*e*h^2 - 33*b^3*c^5*f*h^2 + 156* a*b*c^6*f*h^2)/c^7)*x + (26880*c^8*d*g^2 + 29568*b*c^7*e*g^2 + 672*b^2*c^6 *f*g^2 + 31360*a*c^7*f*g^2 + 59136*b*c^7*d*g*h + 1344*b^2*c^6*e*g*h + 6272 0*a*c^7*e*g*h - 864*b^3*c^5*f*g*h + 4224*a*b*c^6*f*g*h + 672*b^2*c^6*d*h^2 + 31360*a*c^7*d*h^2 - 432*b^3*c^5*e*h^2 + 2112*a*b*c^6*e*h^2 + 297*b^4*c^ 4*f*h^2 - 1704*a*b^2*c^5*f*h^2 + 1680*a^2*c^6*f*h^2)/c^7)*x + (80640*b*c^7 *d*g^2 + 2688*b^2*c^6*e*g^2 + 86016*a*c^7*e*g^2 - 1568*b^3*c^5*f*g^2 + 806 4*a*b*c^6*f*g^2 + 5376*b^2*c^6*d*g*h + 172032*a*c^7*d*g*h - 3136*b^3*c^5*e *g*h + 16128*a*b*c^6*e*g*h + 2016*b^4*c^4*f*g*h - 11904*a*b^2*c^5*f*g*h + 12288*a^2*c^6*f*g*h - 1568*b^3*c^5*d*h^2 + 8064*a*b*c^6*d*h^2 + 1008*b^4*c ^4*e*h^2 - 5952*a*b^2*c^5*e*h^2 + 6144*a^2*c^6*e*h^2 - 693*b^5*c^3*f*h^2 + 4680*a*b^3*c^4*f*h^2 - 7248*a^2*b*c^5*f*h^2)/c^7)*x + (26880*b^2*c^6*d*g^ 2 + 537600*a*c^7*d*g^2 - 13440*b^3*c^5*e*g^2 + 75264*a*b*c^6*e*g^2 + 7840* b^4*c^4*f*g^2 - 48384*a*b^2*c^5*f*g^2 + 53760*a^2*c^6*f*g^2 - 26880*b^3*c^ 5*d*g*h + 150528*a*b*c^6*d*g*h + 15680*b^4*c^4*e*g*h - 96768*a*b^2*c^5*...
Timed out. \[ \int (g+h x)^2 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\int {\left (g+h\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right ) \,d x \]