Integrand size = 32, antiderivative size = 1097 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\frac {\left (64 c^3 g^4 (5 f g-e h)-16 c^2 g^2 h (b g (41 f g-7 e h)-8 a h (5 f g-e h))+4 c h^2 \left (2 b^2 g^2 (46 f g-5 e h)+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e g h-3 d h^2\right )\right )-b h^3 \left (48 a^2 f h^2-8 a b h (10 f g+e h)+b^2 \left (35 f g^2+5 e g h+3 d h^2\right )\right )+2 c h \left (16 c^2 g^3 (5 f g-e h)-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (48 a^2 f h^2-8 a b h (14 f g-e h)+b^2 \left (61 f g^2-h (5 e g+3 d h)\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 h^5 \left (c g^2-b g h+a h^2\right )^2 (g+h x)}-\frac {\left (16 c^2 g^4 (5 f g-e h)-h^2 \left (16 a^2 h^2 (f g-2 e h)-b^2 g \left (35 f g^2+5 e g h+3 d h^2\right )+4 a b h \left (7 f g^2+7 e g h+3 d h^2\right )\right )-4 c g h \left (b g \left (31 f g^2-5 e g h+3 d h^2\right )-a h \left (25 f g^2-5 e g h+9 d h^2\right )\right )+3 h \left (8 c^2 g^2 \left (5 f g^2-h (e g+d h)\right )+h^2 \left (16 a^2 f h^2-8 a b h (6 f g-e h)+b^2 \left (29 f g^2-5 e g h-3 d h^2\right )\right )-4 c h \left (2 b g \left (9 f g^2-2 e g h-d h^2\right )-a h \left (17 f g^2-5 e g h+d h^2\right )\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^3}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{4 h \left (c g^2-b g h+a h^2\right ) (g+h x)^4}-\frac {\sqrt {c} (10 c f g-2 c e h-3 b f h) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 h^6}+\frac {\left (128 c^4 g^5 (5 f g-e h)-64 c^3 g^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h))+8 c h^3 \left (24 a^3 f h^3-12 a^2 b h^2 (10 f g-e h)-5 b^3 g^2 (14 f g-e h)+3 a b^2 h \left (55 f g^2-5 e g h-d h^2\right )\right )-48 c^2 h^2 \left (10 a b g^2 h (6 f g-e h)-5 b^2 g^3 (7 f g-e h)-a^2 h^2 \left (25 f g^2-5 e g h+d h^2\right )\right )+b^2 h^4 \left (48 a^2 f h^2-8 a b h (10 f g+e h)+b^2 \left (35 f g^2+5 e g h+3 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 )}{128 h^6 \left (c g^2-b g h+a h^2\right )^{5/2}} \]
-1/96*(16*c^2*g^4*(-e*h+5*f*g)-h^2*(16*a^2*h^2*(-2*e*h+f*g)-b^2*g*(3*d*h^2 +5*e*g*h+35*f*g^2)+4*a*b*h*(3*d*h^2+7*e*g*h+7*f*g^2))-4*c*g*h*(b*g*(3*d*h^ 2-5*e*g*h+31*f*g^2)-a*h*(9*d*h^2-5*e*g*h+25*f*g^2))+3*h*(8*c^2*g^2*(5*f*g^ 2-h*(d*h+e*g))+h^2*(16*a^2*f*h^2-8*a*b*h*(-e*h+6*f*g)+b^2*(-3*d*h^2-5*e*g* h+29*f*g^2))-4*c*h*(2*b*g*(-d*h^2-2*e*g*h+9*f*g^2)-a*h*(d*h^2-5*e*g*h+17*f *g^2)))*x)*(c*x^2+b*x+a)^(3/2)/h^3/(a*h^2-b*g*h+c*g^2)^2/(h*x+g)^3-1/4*(f* g^2-h*(-d*h+e*g))*(c*x^2+b*x+a)^(5/2)/h/(a*h^2-b*g*h+c*g^2)/(h*x+g)^4+1/12 8*(128*c^4*g^5*(-e*h+5*f*g)-64*c^3*g^3*h*(b*g*(-5*e*h+28*f*g)-5*a*h*(-e*h+ 5*f*g))+8*c*h^3*(24*a^3*f*h^3-12*a^2*b*h^2*(-e*h+10*f*g)-5*b^3*g^2*(-e*h+1 4*f*g)+3*a*b^2*h*(-d*h^2-5*e*g*h+55*f*g^2))-48*c^2*h^2*(10*a*b*g^2*h*(-e*h +6*f*g)-5*b^2*g^3*(-e*h+7*f*g)-a^2*h^2*(d*h^2-5*e*g*h+25*f*g^2))+b^2*h^4*( 48*a^2*f*h^2-8*a*b*h*(e*h+10*f*g)+b^2*(3*d*h^2+5*e*g*h+35*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))/h^6/(a*h^2-b*g*h+c*g^2)^(5/2)-1/2*(-3*b*f*h-2*c*e*h+10*c*f*g)*arctanh (1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*c^(1/2)/h^6+1/64*(64*c^3*g^4*( -e*h+5*f*g)-16*c^2*g^2*h*(b*g*(-7*e*h+41*f*g)-8*a*h*(-e*h+5*f*g))+4*c*h^2* (2*b^2*g^2*(-5*e*h+46*f*g)+16*a^2*h^2*(-e*h+5*f*g)-a*b*h*(-3*d*h^2-25*e*g* h+173*f*g^2))-b*h^3*(48*a^2*f*h^2-8*a*b*h*(e*h+10*f*g)+b^2*(3*d*h^2+5*e*g* h+35*f*g^2))+2*c*h*(16*c^2*g^3*(-e*h+5*f*g)-4*c*h*(6*b*g^2*(-e*h+6*f*g)-a* h*(35*f*g^2-h*(-3*d*h+7*e*g)))+h^2*(48*a^2*f*h^2-8*a*b*h*(-e*h+14*f*g)+...
Time = 15.59 (sec) , antiderivative size = 1005, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\frac {\frac {128 (2 f g-e h) (a+x (b+c x))^{3/2}}{(g+h x)^3}-\frac {192 f (a+x (b+c x))^{3/2}}{(g+h x)^2}+\frac {48 h \left (f g^2+h (-e g+d h)\right ) (a+x (b+c x))^{3/2} (-2 a h+2 c g x+b (g-h x))}{\left (c g^2+h (-b g+a h)\right ) (g+h x)^4}+\frac {288 f \left (\frac {(-2 c g+b h) (a+x (b+c x))^{3/2}}{g+h x}-\frac {\sqrt {a+x (b+c x)} \left (b^2 h^2+2 c^2 g (2 g-h x)+c h (-5 b g+2 a h+b h x)\right )}{h^2}+\frac {4 \sqrt {c} (2 c g-b h) \left (c g^2+h (-b g+a h)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+\left (8 c^2 g^2+b^2 h^2+4 c h (-2 b g+a h)\right ) \sqrt {c g^2+h (-b g+a h)} \text {arctanh}\left (\frac {-b g+2 a h-2 c g x+b h x}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )}{2 h^3}\right )}{-c g^2+h (b g-a h)}+\frac {24 (-2 f g+e h) \left (\frac {4 (2 c g-b h) \left (c g^2+h (-b g+a h)\right ) (a+x (b+c x))^{3/2}}{(g+h x)^2}+\frac {2 \left (-4 c^2 g^2+b^2 h^2+4 c h (b g-2 a h)\right ) (a+x (b+c x))^{3/2}}{g+h x}+\frac {-2 c h \sqrt {a+x (b+c x)} \left (b^3 h^3+4 c^3 g^2 (2 g-h x)+b c h^2 (5 b g-10 a h+b h x)-2 c^2 h (b g (7 g-2 h x)+2 a h (-3 g+2 h x))\right )+16 c^{5/2} \left (c g^2+h (-b g+a h)\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+c (2 c g-b h) \sqrt {c g^2+h (-b g+a h)} \left (8 c^2 g^2-b^2 h^2+4 c h (-2 b g+3 a h)\right ) \text {arctanh}\left (\frac {-b g+2 a h-2 c g x+b h x}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )}{c h^3}\right )}{\left (c g^2+h (-b g+a h)\right )^2}-\frac {9 \left (b^2-4 a c\right ) h \left (f g^2+h (-e g+d h)\right ) \left (\frac {2 \sqrt {a+x (b+c x)} (-2 a h+2 c g x+b (g-h x))}{\left (c g^2+h (-b g+a h)\right ) (g+h x)^2}+\frac {\left (-b^2+4 a c\right ) \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )}{\left (c g^2+h (-b g+a h)\right )^{3/2}}\right )}{c g^2+h (-b g+a h)}}{384 h^3} \]
((128*(2*f*g - e*h)*(a + x*(b + c*x))^(3/2))/(g + h*x)^3 - (192*f*(a + x*( b + c*x))^(3/2))/(g + h*x)^2 + (48*h*(f*g^2 + h*(-(e*g) + d*h))*(a + x*(b + c*x))^(3/2)*(-2*a*h + 2*c*g*x + b*(g - h*x)))/((c*g^2 + h*(-(b*g) + a*h) )*(g + h*x)^4) + (288*f*(((-2*c*g + b*h)*(a + x*(b + c*x))^(3/2))/(g + h*x ) - (Sqrt[a + x*(b + c*x)]*(b^2*h^2 + 2*c^2*g*(2*g - h*x) + c*h*(-5*b*g + 2*a*h + b*h*x)))/h^2 + (4*Sqrt[c]*(2*c*g - b*h)*(c*g^2 + h*(-(b*g) + a*h)) *ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + (8*c^2*g^2 + b^2 *h^2 + 4*c*h*(-2*b*g + a*h))*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*ArcTanh[(-(b*g ) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x* (b + c*x)])])/(2*h^3)))/(-(c*g^2) + h*(b*g - a*h)) + (24*(-2*f*g + e*h)*(( 4*(2*c*g - b*h)*(c*g^2 + h*(-(b*g) + a*h))*(a + x*(b + c*x))^(3/2))/(g + h *x)^2 + (2*(-4*c^2*g^2 + b^2*h^2 + 4*c*h*(b*g - 2*a*h))*(a + x*(b + c*x))^ (3/2))/(g + h*x) + (-2*c*h*Sqrt[a + x*(b + c*x)]*(b^3*h^3 + 4*c^3*g^2*(2*g - h*x) + b*c*h^2*(5*b*g - 10*a*h + b*h*x) - 2*c^2*h*(b*g*(7*g - 2*h*x) + 2*a*h*(-3*g + 2*h*x))) + 16*c^(5/2)*(c*g^2 + h*(-(b*g) + a*h))^2*ArcTanh[( b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + c*(2*c*g - b*h)*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*(8*c^2*g^2 - b^2*h^2 + 4*c*h*(-2*b*g + 3*a*h))*ArcTan h[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqr t[a + x*(b + c*x)])])/(c*h^3)))/(c*g^2 + h*(-(b*g) + a*h))^2 - (9*(b^2 - 4 *a*c)*h*(f*g^2 + h*(-(e*g) + d*h))*((2*Sqrt[a + x*(b + c*x)]*(-2*a*h + ...
Time = 2.66 (sec) , antiderivative size = 1146, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2181, 27, 1229, 27, 1230, 25, 1269, 1092, 219, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {\int -\frac {\left (\frac {5 b f g^2}{h}+8 c d g-5 b e g-8 a f g-3 b d h+8 a e h-2 \left (-\frac {5 c f g^2}{h}+c e g+4 b f g-c d h-4 a f h\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 (g+h x)^4}dx}{4 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (8 c d g-8 a f g+8 a e h-b \left (-\frac {5 f g^2}{h}+5 e g+3 d h\right )-2 \left (-\frac {5 c f g^2}{h}+c e g+4 b f g-c d h-4 a f h\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{(g+h x)^4}dx}{8 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (h \left (h \left (35 f g^2+h (5 e g+3 d h)\right ) b^3-4 \left (31 c f g^3-c h (5 e g-3 d h) g+2 a h^2 (10 f g+e h)\right ) b^2+48 a^2 f h^3 b+\frac {16 c^2 g^3 (5 f g-e h) b}{h}+4 a c h \left (61 f g^2-h (17 e g+3 d h)\right ) b-16 a c \left (5 c f g^3-c h (e g+3 d h) g+4 a h^2 (2 f g-e h)\right )\right )+2 c \left (16 c^2 (5 f g-e h) g^3+h^2 \left (\left (61 f g^2-5 e h g-3 d h^2\right ) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right )-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-7 e h g+3 d h^2\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{2 h (g+h x)^2}dx}{4 h^2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 h x \left (h \left (16 a^2 f h^2-8 a b h (6 f g-e h)+b^2 \left (-3 d h^2-5 e g h+29 f g^2\right )\right )+4 a c h \left (17 f g^2-h (5 e g-d h)\right )-8 b c g \left (9 f g^2-h (d h+2 e g)\right )+\frac {8 c^2 \left (5 f g^4-g^2 h (d h+e g)\right )}{h}\right )-h \left (16 a^2 h^2 (f g-2 e h)+4 a b h \left (3 d h^2+7 e g h+7 f g^2\right )+b^2 (-g) \left (3 d h^2+5 e g h+35 f g^2\right )\right )-4 c g \left (b g \left (3 d h^2-5 e g h+31 f g^2\right )-a h \left (9 d h^2-5 e g h+25 f g^2\right )\right )+\frac {16 c^2 g^4 (5 f g-e h)}{h}\right )}{12 h^2 (g+h x)^3 \left (a h^2-b g h+c g^2\right )}}{8 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (h^2 \left (35 f g^2+h (5 e g+3 d h)\right ) b^3-4 h \left (31 c f g^3-c h (5 e g-3 d h) g+2 a h^2 (10 f g+e h)\right ) b^2+4 \left (12 a^2 f h^4+a c \left (61 f g^2-h (17 e g+3 d h)\right ) h^2+4 c^2 g^3 (5 f g-e h)\right ) b-16 a c h \left (5 c f g^3-c h (e g+3 d h) g+4 a h^2 (2 f g-e h)\right )+2 c \left (16 c^2 (5 f g-e h) g^3-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (\left (61 f g^2-h (5 e g+3 d h)\right ) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{(g+h x)^2}dx}{8 h^3 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 h x \left (h \left (16 a^2 f h^2-8 a b h (6 f g-e h)+b^2 \left (-3 d h^2-5 e g h+29 f g^2\right )\right )+4 a c h \left (17 f g^2-h (5 e g-d h)\right )-8 b c g \left (9 f g^2-h (d h+2 e g)\right )+\frac {8 c^2 \left (5 f g^4-g^2 h (d h+e g)\right )}{h}\right )-h \left (16 a^2 h^2 (f g-2 e h)+4 a b h \left (3 d h^2+7 e g h+7 f g^2\right )+b^2 (-g) \left (3 d h^2+5 e g h+35 f g^2\right )\right )-4 c g \left (b g \left (3 d h^2-5 e g h+31 f g^2\right )-a h \left (9 d h^2-5 e g h+25 f g^2\right )\right )+\frac {16 c^2 g^4 (5 f g-e h)}{h}\right )}{12 h^2 (g+h x)^3 \left (a h^2-b g h+c g^2\right )}}{8 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{4 h (g+h x)^4 \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\frac {\frac {\left (64 c^3 (5 f g-e h) g^4-16 c^2 h (b g (41 f g-7 e h)-8 a h (5 f g-e h)) g^2+4 c h^2 \left (2 b^2 (46 f g-5 e h) g^2+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e h g-3 d h^2\right )\right )-b h^3 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )+2 c h \left (16 c^2 (5 f g-e h) g^3-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (\left (61 f g^2-h (5 e g+3 d h)\right ) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{h^2 (g+h x)}-\frac {\int -\frac {h^3 \left (35 f g^2+h (5 e g+3 d h)\right ) b^4-8 \left (a (10 f g+e h) h^4+c g^2 (46 f g-5 e h) h^2\right ) b^3+8 \left (6 a^2 f h^5+3 a c \left (39 f g^2-5 e h g-d h^2\right ) h^3+2 c^2 g^3 (41 f g-7 e h) h\right ) b^2-32 c \left (c g^2+3 a h^2\right ) \left (2 c (5 f g-e h) g^2+a h^2 (8 f g-e h)\right ) b+16 a c h \left (12 a^2 f h^4+a c \left (35 f g^2-7 e h g+3 d h^2\right ) h^2+4 c^2 g^3 (5 f g-e h)\right )-64 c (10 c f g-2 c e h-3 b f h) \left (c g^2-b h g+a h^2\right )^2 x}{(g+h x) \sqrt {c x^2+b x+a}}dx}{2 h^2}}{8 h^3 \left (c g^2-b h g+a h^2\right )}-\frac {\left (\frac {16 c^2 (5 f g-e h) g^4}{h}-4 c \left (b g \left (31 f g^2-5 e h g+3 d h^2\right )-a h \left (25 f g^2-5 e h g+9 d h^2\right )\right ) g-h \left (-g \left (35 f g^2+5 e h g+3 d h^2\right ) b^2+4 a h \left (7 f g^2+7 e h g+3 d h^2\right ) b+16 a^2 h^2 (f g-2 e h)\right )+3 h \left (\frac {8 \left (5 f g^4-g^2 h (e g+d h)\right ) c^2}{h}+4 a h \left (17 f g^2-h (5 e g-d h)\right ) c-8 b g \left (9 f g^2-h (2 e g+d h)\right ) c+h \left (\left (29 f g^2-5 e h g-3 d h^2\right ) b^2-8 a h (6 f g-e h) b+16 a^2 f h^2\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)^3}}{8 \left (c g^2-b h g+a h^2\right )}-\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{4 h \left (c g^2-b h g+a h^2\right ) (g+h x)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (5 f g-e h) g^4-16 c^2 h (b g (41 f g-7 e h)-8 a h (5 f g-e h)) g^2+4 c h^2 \left (2 b^2 (46 f g-5 e h) g^2+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e h g-3 d h^2\right )\right )-b h^3 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )+2 c h \left (16 c^2 (5 f g-e h) g^3-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (\left (61 f g^2-h (5 e g+3 d h)\right ) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right )\right ) x\right )}{h^2 (g+h x)}+\frac {\int \frac {h^3 \left (35 f g^2+h (5 e g+3 d h)\right ) b^4-8 \left (a (10 f g+e h) h^4+c g^2 (46 f g-5 e h) h^2\right ) b^3+8 \left (6 a^2 f h^5+3 a c \left (39 f g^2-5 e h g-d h^2\right ) h^3+2 c^2 g^3 (41 f g-7 e h) h\right ) b^2-32 c \left (c g^2+3 a h^2\right ) \left (2 c (5 f g-e h) g^2+a h^2 (8 f g-e h)\right ) b+16 a c h \left (12 a^2 f h^4+a c \left (35 f g^2-7 e h g+3 d h^2\right ) h^2+4 c^2 g^3 (5 f g-e h)\right )-64 c (10 c f g-2 c e h-3 b f h) \left (c g^2-b h g+a h^2\right )^2 x}{(g+h x) \sqrt {c x^2+b x+a}}dx}{2 h^2}}{8 h^3 \left (c g^2-b h g+a h^2\right )}-\frac {\left (\frac {16 c^2 (5 f g-e h) g^4}{h}-4 c \left (b g \left (31 f g^2-5 e h g+3 d h^2\right )-a h \left (25 f g^2-5 e h g+9 d h^2\right )\right ) g-h \left (-g \left (35 f g^2+5 e h g+3 d h^2\right ) b^2+4 a h \left (7 f g^2+7 e h g+3 d h^2\right ) b+16 a^2 h^2 (f g-2 e h)\right )+3 h \left (\frac {8 \left (5 f g^4-g^2 h (e g+d h)\right ) c^2}{h}+4 a h \left (17 f g^2-h (5 e g-d h)\right ) c-8 b g \left (9 f g^2-h (2 e g+d h)\right ) c+h \left (\left (29 f g^2-5 e h g-3 d h^2\right ) b^2-8 a h (6 f g-e h) b+16 a^2 f h^2\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)^3}}{8 \left (c g^2-b h g+a h^2\right )}-\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{4 h \left (c g^2-b h g+a h^2\right ) (g+h x)^4}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (5 f g-e h) g^4-16 c^2 h (b g (41 f g-7 e h)-8 a h (5 f g-e h)) g^2+4 c h^2 \left (2 b^2 (46 f g-5 e h) g^2+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e h g-3 d h^2\right )\right )-b h^3 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )+2 c h \left (16 c^2 (5 f g-e h) g^3-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (\left (61 f g^2-h (5 e g+3 d h)\right ) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right )\right ) x\right )}{h^2 (g+h x)}+\frac {\frac {\left (128 c^4 (5 f g-e h) g^5-64 c^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h)) g^3+8 c h^3 \left (-5 g^2 (14 f g-e h) b^3+3 a h \left (55 f g^2-5 e h g-d h^2\right ) b^2-12 a^2 h^2 (10 f g-e h) b+24 a^3 f h^3\right )-48 c^2 h^2 \left (-5 b^2 (7 f g-e h) g^3+10 a b h (6 f g-e h) g^2-a^2 h^2 \left (25 f g^2-5 e h g+d h^2\right )\right )+b^2 h^4 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}-\frac {64 c (10 c f g-2 c e h-3 b f h) \left (c g^2-b h g+a h^2\right )^2 \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{h}}{2 h^2}}{8 h^3 \left (c g^2-b h g+a h^2\right )}-\frac {\left (\frac {16 c^2 (5 f g-e h) g^4}{h}-4 c \left (b g \left (31 f g^2-5 e h g+3 d h^2\right )-a h \left (25 f g^2-5 e h g+9 d h^2\right )\right ) g-h \left (-g \left (35 f g^2+5 e h g+3 d h^2\right ) b^2+4 a h \left (7 f g^2+7 e h g+3 d h^2\right ) b+16 a^2 h^2 (f g-2 e h)\right )+3 h \left (\frac {8 \left (5 f g^4-g^2 h (e g+d h)\right ) c^2}{h}+4 a h \left (17 f g^2-h (5 e g-d h)\right ) c-8 b g \left (9 f g^2-h (2 e g+d h)\right ) c+h \left (\left (29 f g^2-5 e h g-3 d h^2\right ) b^2-8 a h (6 f g-e h) b+16 a^2 f h^2\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)^3}}{8 \left (c g^2-b h g+a h^2\right )}-\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{4 h \left (c g^2-b h g+a h^2\right ) (g+h x)^4}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (5 f g-e h) g^4-16 c^2 h (b g (41 f g-7 e h)-8 a h (5 f g-e h)) g^2+4 c h^2 \left (2 b^2 (46 f g-5 e h) g^2+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e h g-3 d h^2\right )\right )-b h^3 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )+2 c h \left (16 c^2 (5 f g-e h) g^3-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (\left (61 f g^2-h (5 e g+3 d h)\right ) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right )\right ) x\right )}{h^2 (g+h x)}+\frac {\frac {\left (128 c^4 (5 f g-e h) g^5-64 c^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h)) g^3+8 c h^3 \left (-5 g^2 (14 f g-e h) b^3+3 a h \left (55 f g^2-5 e h g-d h^2\right ) b^2-12 a^2 h^2 (10 f g-e h) b+24 a^3 f h^3\right )-48 c^2 h^2 \left (-5 b^2 (7 f g-e h) g^3+10 a b h (6 f g-e h) g^2-a^2 h^2 \left (25 f g^2-5 e h g+d h^2\right )\right )+b^2 h^4 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}-\frac {128 c (10 c f g-2 c e h-3 b f h) \left (c g^2-b h g+a h^2\right )^2 \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}}}{h}}{2 h^2}}{8 h^3 \left (c g^2-b h g+a h^2\right )}-\frac {\left (\frac {16 c^2 (5 f g-e h) g^4}{h}-4 c \left (b g \left (31 f g^2-5 e h g+3 d h^2\right )-a h \left (25 f g^2-5 e h g+9 d h^2\right )\right ) g-h \left (-g \left (35 f g^2+5 e h g+3 d h^2\right ) b^2+4 a h \left (7 f g^2+7 e h g+3 d h^2\right ) b+16 a^2 h^2 (f g-2 e h)\right )+3 h \left (\frac {8 \left (5 f g^4-g^2 h (e g+d h)\right ) c^2}{h}+4 a h \left (17 f g^2-h (5 e g-d h)\right ) c-8 b g \left (9 f g^2-h (2 e g+d h)\right ) c+h \left (\left (29 f g^2-5 e h g-3 d h^2\right ) b^2-8 a h (6 f g-e h) b+16 a^2 f h^2\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)^3}}{8 \left (c g^2-b h g+a h^2\right )}-\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{4 h \left (c g^2-b h g+a h^2\right ) (g+h x)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (5 f g-e h) g^4-16 c^2 h (b g (41 f g-7 e h)-8 a h (5 f g-e h)) g^2+4 c h^2 \left (2 b^2 (46 f g-5 e h) g^2+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e h g-3 d h^2\right )\right )-b h^3 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )+2 c h \left (16 c^2 (5 f g-e h) g^3-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (\left (61 f g^2-h (5 e g+3 d h)\right ) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right )\right ) x\right )}{h^2 (g+h x)}+\frac {\frac {\left (128 c^4 (5 f g-e h) g^5-64 c^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h)) g^3+8 c h^3 \left (-5 g^2 (14 f g-e h) b^3+3 a h \left (55 f g^2-5 e h g-d h^2\right ) b^2-12 a^2 h^2 (10 f g-e h) b+24 a^3 f h^3\right )-48 c^2 h^2 \left (-5 b^2 (7 f g-e h) g^3+10 a b h (6 f g-e h) g^2-a^2 h^2 \left (25 f g^2-5 e h g+d h^2\right )\right )+b^2 h^4 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}-\frac {64 \sqrt {c} (10 c f g-2 c e h-3 b f h) \left (c g^2-b h g+a h^2\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{h}}{2 h^2}}{8 h^3 \left (c g^2-b h g+a h^2\right )}-\frac {\left (\frac {16 c^2 (5 f g-e h) g^4}{h}-4 c \left (b g \left (31 f g^2-5 e h g+3 d h^2\right )-a h \left (25 f g^2-5 e h g+9 d h^2\right )\right ) g-h \left (-g \left (35 f g^2+5 e h g+3 d h^2\right ) b^2+4 a h \left (7 f g^2+7 e h g+3 d h^2\right ) b+16 a^2 h^2 (f g-2 e h)\right )+3 h \left (\frac {8 \left (5 f g^4-g^2 h (e g+d h)\right ) c^2}{h}+4 a h \left (17 f g^2-h (5 e g-d h)\right ) c-8 b g \left (9 f g^2-h (2 e g+d h)\right ) c+h \left (\left (29 f g^2-5 e h g-3 d h^2\right ) b^2-8 a h (6 f g-e h) b+16 a^2 f h^2\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)^3}}{8 \left (c g^2-b h g+a h^2\right )}-\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{4 h \left (c g^2-b h g+a h^2\right ) (g+h x)^4}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (5 f g-e h) g^4-16 c^2 h (b g (41 f g-7 e h)-8 a h (5 f g-e h)) g^2+4 c h^2 \left (2 b^2 (46 f g-5 e h) g^2+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e h g-3 d h^2\right )\right )-b h^3 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )+2 c h \left (16 c^2 (5 f g-e h) g^3-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (\left (61 f g^2-h (5 e g+3 d h)\right ) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right )\right ) x\right )}{h^2 (g+h x)}+\frac {-\frac {64 \sqrt {c} (10 c f g-2 c e h-3 b f h) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) \left (c g^2-b h g+a h^2\right )^2}{h}-\frac {2 \left (128 c^4 (5 f g-e h) g^5-64 c^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h)) g^3+8 c h^3 \left (-5 g^2 (14 f g-e h) b^3+3 a h \left (55 f g^2-5 e h g-d h^2\right ) b^2-12 a^2 h^2 (10 f g-e h) b+24 a^3 f h^3\right )-48 c^2 h^2 \left (-5 b^2 (7 f g-e h) g^3+10 a b h (6 f g-e h) g^2-a^2 h^2 \left (25 f g^2-5 e h g+d h^2\right )\right )+b^2 h^4 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )\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 )}{h}}{2 h^2}}{8 h^3 \left (c g^2-b h g+a h^2\right )}-\frac {\left (\frac {16 c^2 (5 f g-e h) g^4}{h}-4 c \left (b g \left (31 f g^2-5 e h g+3 d h^2\right )-a h \left (25 f g^2-5 e h g+9 d h^2\right )\right ) g-h \left (-g \left (35 f g^2+5 e h g+3 d h^2\right ) b^2+4 a h \left (7 f g^2+7 e h g+3 d h^2\right ) b+16 a^2 h^2 (f g-2 e h)\right )+3 h \left (\frac {8 \left (5 f g^4-g^2 h (e g+d h)\right ) c^2}{h}+4 a h \left (17 f g^2-h (5 e g-d h)\right ) c-8 b g \left (9 f g^2-h (2 e g+d h)\right ) c+h \left (\left (29 f g^2-5 e h g-3 d h^2\right ) b^2-8 a h (6 f g-e h) b+16 a^2 f h^2\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)^3}}{8 \left (c g^2-b h g+a h^2\right )}-\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{4 h \left (c g^2-b h g+a h^2\right ) (g+h x)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (5 f g-e h) g^4-16 c^2 h (b g (41 f g-7 e h)-8 a h (5 f g-e h)) g^2+4 c h^2 \left (2 b^2 (46 f g-5 e h) g^2+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e h g-3 d h^2\right )\right )-b h^3 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )+2 c h \left (16 c^2 (5 f g-e h) g^3-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (\left (61 f g^2-h (5 e g+3 d h)\right ) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right )\right ) x\right )}{h^2 (g+h x)}+\frac {\frac {\left (128 c^4 (5 f g-e h) g^5-64 c^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h)) g^3+8 c h^3 \left (-5 g^2 (14 f g-e h) b^3+3 a h \left (55 f g^2-5 e h g-d h^2\right ) b^2-12 a^2 h^2 (10 f g-e h) b+24 a^3 f h^3\right )-48 c^2 h^2 \left (-5 b^2 (7 f g-e h) g^3+10 a b h (6 f g-e h) g^2-a^2 h^2 \left (25 f g^2-5 e h g+d h^2\right )\right )+b^2 h^4 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )\right ) \text {arctanh}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b h g+a h^2} \sqrt {c x^2+b x+a}}\right )}{h \sqrt {c g^2-b h g+a h^2}}-\frac {64 \sqrt {c} (10 c f g-2 c e h-3 b f h) \left (c g^2-b h g+a h^2\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{h}}{2 h^2}}{8 h^3 \left (c g^2-b h g+a h^2\right )}-\frac {\left (\frac {16 c^2 (5 f g-e h) g^4}{h}-4 c \left (b g \left (31 f g^2-5 e h g+3 d h^2\right )-a h \left (25 f g^2-5 e h g+9 d h^2\right )\right ) g-h \left (-g \left (35 f g^2+5 e h g+3 d h^2\right ) b^2+4 a h \left (7 f g^2+7 e h g+3 d h^2\right ) b+16 a^2 h^2 (f g-2 e h)\right )+3 h \left (\frac {8 \left (5 f g^4-g^2 h (e g+d h)\right ) c^2}{h}+4 a h \left (17 f g^2-h (5 e g-d h)\right ) c-8 b g \left (9 f g^2-h (2 e g+d h)\right ) c+h \left (\left (29 f g^2-5 e h g-3 d h^2\right ) b^2-8 a h (6 f g-e h) b+16 a^2 f h^2\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)^3}}{8 \left (c g^2-b h g+a h^2\right )}-\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{4 h \left (c g^2-b h g+a h^2\right ) (g+h x)^4}\) |
-1/4*((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(5/2))/(h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^4) + (-1/12*(((16*c^2*g^4*(5*f*g - e*h))/h - h*(16*a^2*h ^2*(f*g - 2*e*h) - b^2*g*(35*f*g^2 + 5*e*g*h + 3*d*h^2) + 4*a*b*h*(7*f*g^2 + 7*e*g*h + 3*d*h^2)) - 4*c*g*(b*g*(31*f*g^2 - 5*e*g*h + 3*d*h^2) - a*h*( 25*f*g^2 - 5*e*g*h + 9*d*h^2)) + 3*h*(4*a*c*h*(17*f*g^2 - h*(5*e*g - d*h)) + (8*c^2*(5*f*g^4 - g^2*h*(e*g + d*h)))/h - 8*b*c*g*(9*f*g^2 - h*(2*e*g + d*h)) + h*(16*a^2*f*h^2 - 8*a*b*h*(6*f*g - e*h) + b^2*(29*f*g^2 - 5*e*g*h - 3*d*h^2)))*x)*(a + b*x + c*x^2)^(3/2))/(h^2*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^3) + (((64*c^3*g^4*(5*f*g - e*h) - 16*c^2*g^2*h*(b*g*(41*f*g - 7*e* h) - 8*a*h*(5*f*g - e*h)) + 4*c*h^2*(2*b^2*g^2*(46*f*g - 5*e*h) + 16*a^2*h ^2*(5*f*g - e*h) - a*b*h*(173*f*g^2 - 25*e*g*h - 3*d*h^2)) - b*h^3*(48*a^2 *f*h^2 - 8*a*b*h*(10*f*g + e*h) + b^2*(35*f*g^2 + 5*e*g*h + 3*d*h^2)) + 2* c*h*(16*c^2*g^3*(5*f*g - e*h) - 4*c*h*(6*b*g^2*(6*f*g - e*h) - a*h*(35*f*g ^2 - h*(7*e*g - 3*d*h))) + h^2*(48*a^2*f*h^2 - 8*a*b*h*(14*f*g - e*h) + b^ 2*(61*f*g^2 - h*(5*e*g + 3*d*h))))*x)*Sqrt[a + b*x + c*x^2])/(h^2*(g + h*x )) + ((-64*Sqrt[c]*(10*c*f*g - 2*c*e*h - 3*b*f*h)*(c*g^2 - b*g*h + a*h^2)^ 2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/h + ((128*c^4*g^ 5*(5*f*g - e*h) - 64*c^3*g^3*h*(b*g*(28*f*g - 5*e*h) - 5*a*h*(5*f*g - e*h) ) + 8*c*h^3*(24*a^3*f*h^3 - 12*a^2*b*h^2*(10*f*g - e*h) - 5*b^3*g^2*(14*f* g - e*h) + 3*a*b^2*h*(55*f*g^2 - 5*e*g*h - d*h^2)) - 48*c^2*h^2*(10*a*b...
3.3.4.3.1 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[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[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[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
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. \(4358\) vs. \(2(1065)=2130\).
Time = 1.21 (sec) , antiderivative size = 4359, normalized size of antiderivative = 3.97
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(4359\) |
| default | \(\text {Expression too large to display}\) | \(10038\) |
f/h^5*(c*x^2+b*x+a)^(1/2)*c+1/2/h^5*(c^(1/2)*(3*b*f*h+2*c*e*h-10*c*f*g)/h* ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-(4*a*c*f*h^2+2*b^2*f*h^2+4*b*c *e*h^2-20*b*c*f*g*h+2*c^2*d*h^2-10*c^2*e*g*h+30*c^2*f*g^2)/h^2/((a*h^2-b*g *h+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+1/h*g) +2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+1/h*g)^2*c+(b*h-2*c*g)/h*(x+1/h*g)+ (a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+1/h*g))+(4*a*b*f*h^3+4*a*c*e*h^3-16*a*c *f*g*h^2+2*b^2*e*h^3-8*b^2*f*g*h^2+4*b*c*d*h^3-16*b*c*e*g*h^2+40*b*c*f*g^2 *h-8*c^2*d*g*h^2+20*c^2*e*g^2*h-40*c^2*f*g^3)/h^3*(-1/(a*h^2-b*g*h+c*g^2)* h^2/(x+1/h*g)*((x+1/h*g)^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a*h^2-b*g*h+c*g^2)/h ^2)^(1/2)+1/2*(b*h-2*c*g)*h/(a*h^2-b*g*h+c*g^2)/((a*h^2-b*g*h+c*g^2)/h^2)^ (1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+1/h*g)+2*((a*h^2-b*g* h+c*g^2)/h^2)^(1/2)*((x+1/h*g)^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a*h^2-b*g*h+c* g^2)/h^2)^(1/2))/(x+1/h*g)))+(2*a^2*f*h^4+4*a*b*e*h^4-12*a*b*f*g*h^3+4*a*c *d*h^4-12*a*c*e*g*h^3+24*a*c*f*g^2*h^2+2*b^2*d*h^4-6*b^2*e*g*h^3+12*b^2*f* g^2*h^2-12*b*c*d*g*h^3+24*b*c*e*g^2*h^2-40*b*c*f*g^3*h+12*c^2*d*g^2*h^2-20 *c^2*e*g^3*h+30*c^2*f*g^4)/h^4*(-1/2/(a*h^2-b*g*h+c*g^2)*h^2/(x+1/h*g)^2*( (x+1/h*g)^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)-3/4*( b*h-2*c*g)*h/(a*h^2-b*g*h+c*g^2)*(-1/(a*h^2-b*g*h+c*g^2)*h^2/(x+1/h*g)*((x +1/h*g)^2*c+(b*h-2*c*g)/h*(x+1/h*g)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)+1/2*(b* h-2*c*g)*h/(a*h^2-b*g*h+c*g^2)/((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*ln((2*(a...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{5}}\, dx \]
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\text {Exception raised: ValueError} \]
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
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^5} \,d x \]