Integrand size = 32, antiderivative size = 824 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\frac {\left (32 c^3 d g^3-8 c^2 g \left (2 b g (e g+3 d h)+a \left (f g^2-6 e g h+3 d h^2\right )\right )-b h \left (16 a^2 f h^2-2 a b h (6 f g+5 e h)+b^2 \left (3 f g^2+3 e g h+7 d h^2\right )\right )+2 c \left (4 a^2 h^2 (6 f g-e h)-6 a b h \left (3 f g^2+3 e g h-d h^2\right )+b^2 g \left (5 f g^2+6 e g h+15 d h^2\right )\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{128 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^2}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {\left (2 c g \left (3 f g^2+h (2 e g-7 d h)\right )+h \left (10 a h (2 f g-e h)-b \left (13 f g^2-3 e g h-7 d h^2\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{40 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}+\frac {\left (4 c^2 g^2 \left (3 f g^2+h (2 e g-27 d h)\right )-5 h^2 \left (16 a^2 f h^2-2 a b h (6 f g+5 e h)+b^2 \left (3 f g^2+3 e g h+7 d h^2\right )\right )-2 c h \left (b g \left (16 f g^2-21 e g h-54 d h^2\right )-2 a h \left (18 f g^2-33 e g h+8 d h^2\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{240 h \left (c g^2-b g h+a h^2\right )^3 (g+h x)^3}-\frac {\left (b^2-4 a c\right ) \left (32 c^3 d g^3-8 c^2 g \left (2 b g (e g+3 d h)+a \left (f g^2-6 e g h+3 d h^2\right )\right )-b h \left (16 a^2 f h^2-2 a b h (6 f g+5 e h)+b^2 \left (3 f g^2+3 e g h+7 d h^2\right )\right )+2 c \left (4 a^2 h^2 (6 f g-e h)-6 a b h \left (3 f g^2+3 e g h-d h^2\right )+b^2 g \left (5 f g^2+6 e g h+15 d h^2\right )\right )\right ) \text {arctanh}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c g^2-b g h+a h^2\right )^{9/2}} \]
-1/5*(f*g^2-h*(-d*h+e*g))*(c*x^2+b*x+a)^(3/2)/h/(a*h^2-b*g*h+c*g^2)/(h*x+g )^5+1/40*(2*c*g*(3*f*g^2+h*(-7*d*h+2*e*g))+h*(10*a*h*(-e*h+2*f*g)-b*(-7*d* h^2-3*e*g*h+13*f*g^2)))*(c*x^2+b*x+a)^(3/2)/h/(a*h^2-b*g*h+c*g^2)^2/(h*x+g )^4+1/240*(4*c^2*g^2*(3*f*g^2+h*(-27*d*h+2*e*g))-5*h^2*(16*a^2*f*h^2-2*a*b *h*(5*e*h+6*f*g)+b^2*(7*d*h^2+3*e*g*h+3*f*g^2))-2*c*h*(b*g*(-54*d*h^2-21*e *g*h+16*f*g^2)-2*a*h*(8*d*h^2-33*e*g*h+18*f*g^2)))*(c*x^2+b*x+a)^(3/2)/h/( a*h^2-b*g*h+c*g^2)^3/(h*x+g)^3-1/256*(-4*a*c+b^2)*(32*c^3*d*g^3-8*c^2*g*(2 *b*g*(3*d*h+e*g)+a*(3*d*h^2-6*e*g*h+f*g^2))-b*h*(16*a^2*f*h^2-2*a*b*h*(5*e *h+6*f*g)+b^2*(7*d*h^2+3*e*g*h+3*f*g^2))+2*c*(4*a^2*h^2*(-e*h+6*f*g)-6*a*b *h*(-d*h^2+3*e*g*h+3*f*g^2)+b^2*g*(15*d*h^2+6*e*g*h+5*f*g^2)))*arctanh(1/2 *(b*g-2*a*h+(-b*h+2*c*g)*x)/(a*h^2-b*g*h+c*g^2)^(1/2)/(c*x^2+b*x+a)^(1/2)) /(a*h^2-b*g*h+c*g^2)^(9/2)+1/128*(32*c^3*d*g^3-8*c^2*g*(2*b*g*(3*d*h+e*g)+ a*(3*d*h^2-6*e*g*h+f*g^2))-b*h*(16*a^2*f*h^2-2*a*b*h*(5*e*h+6*f*g)+b^2*(7* d*h^2+3*e*g*h+3*f*g^2))+2*c*(4*a^2*h^2*(-e*h+6*f*g)-6*a*b*h*(-d*h^2+3*e*g* h+3*f*g^2)+b^2*g*(15*d*h^2+6*e*g*h+5*f*g^2)))*(b*g-2*a*h+(-b*h+2*c*g)*x)*( c*x^2+b*x+a)^(1/2)/(a*h^2-b*g*h+c*g^2)^4/(h*x+g)^2
Time = 16.24 (sec) , antiderivative size = 1334, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right ) \sqrt {a+x (b+c x)}}{5 h \left (c g^2-h (b g-a h)\right ) (g+h x)^5}+\frac {(2 f g-e h) \left (a+b x+c x^2\right ) \sqrt {a+x (b+c x)}}{4 h \left (c g^2-h (b g-a h)\right ) (g+h x)^4}-\frac {f \left (a+b x+c x^2\right ) \sqrt {a+x (b+c x)}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}-\frac {(-2 f g+e h) \sqrt {a+x (b+c x)} \left (\frac {\left (c g h-\frac {1}{2} h (-8 c g+5 b h)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 \left (c g^2-b g h+a h^2\right ) (g+h x)^3}-\frac {\left (-2 \left (a c h^2+\frac {1}{2} c g (-8 c g+5 b h)\right )+b \left (c g h+\frac {1}{2} h (-8 c g+5 b h)\right )\right ) \left (\frac {(b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{4 \left (c g^2-b g h+a h^2\right ) (g+h x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b g+2 a h-(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c g^2-b g h+a h^2} \left (4 c g^2-4 b g h+4 a h^2\right )}\right )}{2 \left (c g^2-b g h+a h^2\right )}\right )}{4 h^2 \left (c g^2-b g h+a h^2\right ) \sqrt {a+b x+c x^2}}-\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+x (b+c x)} \left (-\frac {\left (-2 c g h+\frac {1}{2} h (-10 c g+7 b h)\right ) \left (a+b x+c x^2\right )^{3/2}}{4 \left (c g^2-b g h+a h^2\right ) (g+h x)^4}-\frac {\frac {\left (-\frac {7}{2} c g h (2 c g-b h)-\frac {1}{4} h \left (80 c^2 g^2+35 b^2 h^2-2 c h (47 b g+16 a h)\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{3 \left (c g^2-b g h+a h^2\right ) (g+h x)^3}-\frac {\left (-2 \left (-\frac {7}{2} a c h^2 (2 c g-b h)+\frac {1}{4} c g \left (80 c^2 g^2+35 b^2 h^2-2 c h (47 b g+16 a h)\right )\right )+b \left (-\frac {7}{2} c g h (2 c g-b h)+\frac {1}{4} h \left (80 c^2 g^2+35 b^2 h^2-2 c h (47 b g+16 a h)\right )\right )\right ) \left (\frac {(b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{4 \left (c g^2-b g h+a h^2\right ) (g+h x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b g+2 a h-(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c g^2-b g h+a h^2} \left (4 c g^2-4 b g h+4 a h^2\right )}\right )}{2 \left (c g^2-b g h+a h^2\right )}}{4 \left (c g^2-b g h+a h^2\right )}\right )}{5 h^2 \left (c g^2-b g h+a h^2\right ) \sqrt {a+b x+c x^2}}+\frac {f (2 c g-b h) \sqrt {a+x (b+c x)} \left (\frac {2 (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{\left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-h (b g-a h)} \sqrt {a+b x+c x^2}}\right )}{\left (c g^2-h (b g-a h)\right )^{3/2}}\right )}{16 h^2 \left (c g^2-h (b g-a h)\right ) \sqrt {a+b x+c x^2}} \]
-1/5*((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)*Sqrt[a + x*(b + c*x)])/(h* (c*g^2 - h*(b*g - a*h))*(g + h*x)^5) + ((2*f*g - e*h)*(a + b*x + c*x^2)*Sq rt[a + x*(b + c*x)])/(4*h*(c*g^2 - h*(b*g - a*h))*(g + h*x)^4) - (f*(a + b *x + c*x^2)*Sqrt[a + x*(b + c*x)])/(3*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^ 3) - ((-2*f*g + e*h)*Sqrt[a + x*(b + c*x)]*(((c*g*h - (h*(-8*c*g + 5*b*h)) /2)*(a + b*x + c*x^2)^(3/2))/(3*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^3) - ((- 2*(a*c*h^2 + (c*g*(-8*c*g + 5*b*h))/2) + b*(c*g*h + (h*(-8*c*g + 5*b*h))/2 ))*(((b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*g^2 - b* g*h + a*h^2)*(g + h*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*g) + 2*a*h - (2*c* g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(2*Sqr t[c*g^2 - b*g*h + a*h^2]*(4*c*g^2 - 4*b*g*h + 4*a*h^2))))/(2*(c*g^2 - b*g* h + a*h^2))))/(4*h^2*(c*g^2 - b*g*h + a*h^2)*Sqrt[a + b*x + c*x^2]) - ((f* g^2 - e*g*h + d*h^2)*Sqrt[a + x*(b + c*x)]*(-1/4*((-2*c*g*h + (h*(-10*c*g + 7*b*h))/2)*(a + b*x + c*x^2)^(3/2))/((c*g^2 - b*g*h + a*h^2)*(g + h*x)^4 ) - ((((-7*c*g*h*(2*c*g - b*h))/2 - (h*(80*c^2*g^2 + 35*b^2*h^2 - 2*c*h*(4 7*b*g + 16*a*h)))/4)*(a + b*x + c*x^2)^(3/2))/(3*(c*g^2 - b*g*h + a*h^2)*( g + h*x)^3) - ((-2*((-7*a*c*h^2*(2*c*g - b*h))/2 + (c*g*(80*c^2*g^2 + 35*b ^2*h^2 - 2*c*h*(47*b*g + 16*a*h)))/4) + b*((-7*c*g*h*(2*c*g - b*h))/2 + (h *(80*c^2*g^2 + 35*b^2*h^2 - 2*c*h*(47*b*g + 16*a*h)))/4))*(((b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*g^2 - b*g*h + a*h^2)*(g ...
Time = 1.52 (sec) , antiderivative size = 730, normalized size of antiderivative = 0.89, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2181, 27, 1237, 27, 1228, 1152, 1154, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {\int -\frac {\left (\frac {3 b f g^2}{h}+10 c d g-3 b e g-10 a f g-7 b d h+10 a e h+2 \left (\frac {3 c f g^2}{h}+2 c e g-5 b f g-2 c d h+5 a f h\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (g+h x)^5}dx}{5 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (10 c d g-b \left (-\frac {3 f g^2}{h}+3 e g+7 d h\right )-10 a (f g-e h)-2 \left (5 b f g-5 a f h-c \left (\frac {3 f g^2}{h}+2 e g-2 d h\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{(g+h x)^5}dx}{10 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (10 a h^2 (2 f g-e h)-b h \left (13 f g^2-h (7 d h+3 e g)\right )+2 c g h (2 e g-7 d h)+6 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}-\frac {\int -\frac {\left (h \left (5 \left (3 f g^2+h (3 e g+7 d h)\right ) b^2-2 c g \left (-\frac {3 f g^2}{h}+18 e g+47 d h\right ) b-10 a h (6 f g+5 e h) b+80 c^2 d g^2+80 a^2 f h^2-16 a c \left (2 f g^2-h (7 e g-2 d h)\right )\right )+2 c \left (6 c f g^3+2 c h (2 e g-7 d h) g+10 a h^2 (2 f g-e h)-b h \left (13 f g^2-h (3 e g+7 d h)\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{2 h (g+h x)^4}dx}{4 \left (a h^2-b g h+c g^2\right )}}{10 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (h \left (5 \left (3 f g^2+h (3 e g+7 d h)\right ) b^2-2 c g \left (-\frac {3 f g^2}{h}+18 e g+47 d h\right ) b-10 a h (6 f g+5 e h) b+80 c^2 d g^2+80 a^2 f h^2-16 a c \left (2 f g^2-h (7 e g-2 d h)\right )\right )+2 c \left (6 c f g^3+2 c h (2 e g-7 d h) g+10 a h^2 (2 f g-e h)-b h \left (13 f g^2-h (3 e g+7 d h)\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{(g+h x)^4}dx}{8 h \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (10 a h^2 (2 f g-e h)-b h \left (13 f g^2-h (7 d h+3 e g)\right )+2 c g h (2 e g-7 d h)+6 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}}{10 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\frac {\frac {5 h \left (2 c \left (4 a^2 h^2 (6 f g-e h)-6 a b h \left (h (3 e g-d h)+3 f g^2\right )+b^2 \left (3 g h (5 d h+2 e g)+5 f g^3\right )\right )-b h \left (16 a^2 f h^2-2 a b h (5 e h+6 f g)+b^2 \left (h (7 d h+3 e g)+3 f g^2\right )\right )-8 c^2 g \left (-3 a h (2 e g-d h)+a f g^2+2 b g (3 d h+e g)\right )+32 c^3 d g^3\right ) \int \frac {\sqrt {c x^2+b x+a}}{(g+h x)^3}dx}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-5 h^2 \left (16 a^2 f h^2-2 a b h (5 e h+6 f g)+b^2 \left (7 d h^2+3 e g h+3 f g^2\right )\right )-2 c h \left (b g \left (-54 d h^2-21 e g h+16 f g^2\right )-2 a h \left (8 d h^2-33 e g h+18 f g^2\right )\right )+4 c^2 \left (g^2 h (2 e g-27 d h)+3 f g^4\right )\right )}{3 (g+h x)^3 \left (a h^2-b g h+c g^2\right )}}{8 h \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (10 a h^2 (2 f g-e h)-b h \left (13 f g^2-h (7 d h+3 e g)\right )+2 c g h (2 e g-7 d h)+6 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}}{10 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {\frac {\frac {5 h \left (2 c \left (4 a^2 h^2 (6 f g-e h)-6 a b h \left (h (3 e g-d h)+3 f g^2\right )+b^2 \left (3 g h (5 d h+2 e g)+5 f g^3\right )\right )-b h \left (16 a^2 f h^2-2 a b h (5 e h+6 f g)+b^2 \left (h (7 d h+3 e g)+3 f g^2\right )\right )-8 c^2 g \left (-3 a h (2 e g-d h)+a f g^2+2 b g (3 d h+e g)\right )+32 c^3 d g^3\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{8 \left (a h^2-b g h+c g^2\right )}\right )}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-5 h^2 \left (16 a^2 f h^2-2 a b h (5 e h+6 f g)+b^2 \left (7 d h^2+3 e g h+3 f g^2\right )\right )-2 c h \left (b g \left (-54 d h^2-21 e g h+16 f g^2\right )-2 a h \left (8 d h^2-33 e g h+18 f g^2\right )\right )+4 c^2 \left (g^2 h (2 e g-27 d h)+3 f g^4\right )\right )}{3 (g+h x)^3 \left (a h^2-b g h+c g^2\right )}}{8 h \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (10 a h^2 (2 f g-e h)-b h \left (13 f g^2-h (7 d h+3 e g)\right )+2 c g h (2 e g-7 d h)+6 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}}{10 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {\frac {5 h \left (2 c \left (4 a^2 h^2 (6 f g-e h)-6 a b h \left (h (3 e g-d h)+3 f g^2\right )+b^2 \left (3 g h (5 d h+2 e g)+5 f g^3\right )\right )-b h \left (16 a^2 f h^2-2 a b h (5 e h+6 f g)+b^2 \left (h (7 d h+3 e g)+3 f g^2\right )\right )-8 c^2 g \left (-3 a h (2 e g-d h)+a f g^2+2 b g (3 d h+e g)\right )+32 c^3 d g^3\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c g^2-b h g+a h^2\right )-\frac {(b g-2 a h+(2 c g-b h) x)^2}{c x^2+b x+a}}d\left (-\frac {b g-2 a h+(2 c g-b h) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a h^2-b g h+c g^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\right )}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-5 h^2 \left (16 a^2 f h^2-2 a b h (5 e h+6 f g)+b^2 \left (7 d h^2+3 e g h+3 f g^2\right )\right )-2 c h \left (b g \left (-54 d h^2-21 e g h+16 f g^2\right )-2 a h \left (8 d h^2-33 e g h+18 f g^2\right )\right )+4 c^2 \left (g^2 h (2 e g-27 d h)+3 f g^4\right )\right )}{3 (g+h x)^3 \left (a h^2-b g h+c g^2\right )}}{8 h \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (10 a h^2 (2 f g-e h)-b h \left (13 f g^2-h (7 d h+3 e g)\right )+2 c g h (2 e g-7 d h)+6 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}}{10 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {5 h \left (\frac {\sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right )}{8 \left (a h^2-b g h+c g^2\right )^{3/2}}\right ) \left (2 c \left (4 a^2 h^2 (6 f g-e h)-6 a b h \left (h (3 e g-d h)+3 f g^2\right )+b^2 \left (3 g h (5 d h+2 e g)+5 f g^3\right )\right )-b h \left (16 a^2 f h^2-2 a b h (5 e h+6 f g)+b^2 \left (h (7 d h+3 e g)+3 f g^2\right )\right )-8 c^2 g \left (-3 a h (2 e g-d h)+a f g^2+2 b g (3 d h+e g)\right )+32 c^3 d g^3\right )}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-5 h^2 \left (16 a^2 f h^2-2 a b h (5 e h+6 f g)+b^2 \left (7 d h^2+3 e g h+3 f g^2\right )\right )-2 c h \left (b g \left (-54 d h^2-21 e g h+16 f g^2\right )-2 a h \left (8 d h^2-33 e g h+18 f g^2\right )\right )+4 c^2 \left (g^2 h (2 e g-27 d h)+3 f g^4\right )\right )}{3 (g+h x)^3 \left (a h^2-b g h+c g^2\right )}}{8 h \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (10 a h^2 (2 f g-e h)-b h \left (13 f g^2-h (7 d h+3 e g)\right )+2 c g h (2 e g-7 d h)+6 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}}{10 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}\) |
-1/5*((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(3/2))/(h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^5) + (((6*c*f*g^3 + 2*c*g*h*(2*e*g - 7*d*h) + 10*a*h^2*( 2*f*g - e*h) - b*h*(13*f*g^2 - h*(3*e*g + 7*d*h)))*(a + b*x + c*x^2)^(3/2) )/(4*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^4) + (((4*c^2*(3*f*g^4 + g^2*h*(2 *e*g - 27*d*h)) - 5*h^2*(16*a^2*f*h^2 - 2*a*b*h*(6*f*g + 5*e*h) + b^2*(3*f *g^2 + 3*e*g*h + 7*d*h^2)) - 2*c*h*(b*g*(16*f*g^2 - 21*e*g*h - 54*d*h^2) - 2*a*h*(18*f*g^2 - 33*e*g*h + 8*d*h^2)))*(a + b*x + c*x^2)^(3/2))/(3*(c*g^ 2 - b*g*h + a*h^2)*(g + h*x)^3) + (5*h*(32*c^3*d*g^3 - 8*c^2*g*(a*f*g^2 - 3*a*h*(2*e*g - d*h) + 2*b*g*(e*g + 3*d*h)) + 2*c*(4*a^2*h^2*(6*f*g - e*h) - 6*a*b*h*(3*f*g^2 + h*(3*e*g - d*h)) + b^2*(5*f*g^3 + 3*g*h*(2*e*g + 5*d* h))) - b*h*(16*a^2*f*h^2 - 2*a*b*h*(6*f*g + 5*e*h) + b^2*(3*f*g^2 + h*(3*e *g + 7*d*h))))*(((b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/(4 *(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) - ((b^2 - 4*a*c)*ArcTanh[(b*g - 2*a* h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2]) ])/(8*(c*g^2 - b*g*h + a*h^2)^(3/2))))/(2*(c*g^2 - b*g*h + a*h^2)))/(8*h*( c*g^2 - b*g*h + a*h^2)))/(10*(c*g^2 - b*g*h + a*h^2))
3.2.95.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(7713\) vs. \(2(798)=1596\).
Time = 1.55 (sec) , antiderivative size = 7714, normalized size of antiderivative = 9.36
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\int \frac {\sqrt {a + b x + c x^{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{6}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*h^2-b*g*h>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 28577 vs. \(2 (798) = 1596\).
Time = 1.87 (sec) , antiderivative size = 28577, normalized size of antiderivative = 34.68 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\text {Too large to display} \]
-1/128*(32*b^2*c^3*d*g^3 - 128*a*c^4*d*g^3 - 16*b^3*c^2*e*g^3 + 64*a*b*c^3 *e*g^3 + 10*b^4*c*f*g^3 - 48*a*b^2*c^2*f*g^3 + 32*a^2*c^3*f*g^3 - 48*b^3*c ^2*d*g^2*h + 192*a*b*c^3*d*g^2*h + 12*b^4*c*e*g^2*h - 192*a^2*c^3*e*g^2*h - 3*b^5*f*g^2*h - 24*a*b^3*c*f*g^2*h + 144*a^2*b*c^2*f*g^2*h + 30*b^4*c*d* g*h^2 - 144*a*b^2*c^2*d*g*h^2 + 96*a^2*c^3*d*g*h^2 - 3*b^5*e*g*h^2 - 24*a* b^3*c*e*g*h^2 + 144*a^2*b*c^2*e*g*h^2 + 12*a*b^4*f*g*h^2 - 192*a^3*c^2*f*g *h^2 - 7*b^5*d*h^3 + 40*a*b^3*c*d*h^3 - 48*a^2*b*c^2*d*h^3 + 10*a*b^4*e*h^ 3 - 48*a^2*b^2*c*e*h^3 + 32*a^3*c^2*e*h^3 - 16*a^2*b^3*f*h^3 + 64*a^3*b*c* f*h^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*h + sqrt(c)*g)/sqrt(-c *g^2 + b*g*h - a*h^2))/((c^4*g^8 - 4*b*c^3*g^7*h + 6*b^2*c^2*g^6*h^2 + 4*a *c^3*g^6*h^2 - 4*b^3*c*g^5*h^3 - 12*a*b*c^2*g^5*h^3 + b^4*g^4*h^4 + 12*a*b ^2*c*g^4*h^4 + 6*a^2*c^2*g^4*h^4 - 4*a*b^3*g^3*h^5 - 12*a^2*b*c*g^3*h^5 + 6*a^2*b^2*g^2*h^6 + 4*a^3*c*g^2*h^6 - 4*a^3*b*g*h^7 + a^4*h^8)*sqrt(-c*g^2 + b*g*h - a*h^2)) + 1/1920*(480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^2 *c^3*d*g^3*h^8 - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*c^4*d*g^3*h^ 8 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c^2*e*g^3*h^8 + 960*(sqr t(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b*c^3*e*g^3*h^8 + 150*(sqrt(c)*x - sqr t(c*x^2 + b*x + a))^9*b^4*c*f*g^3*h^8 - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^2*c^2*f*g^3*h^8 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^ 2*c^3*f*g^3*h^8 - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c^2*d*g...
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^6} \,d x \]