Integrand size = 32, antiderivative size = 459 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=-\frac {\left (b f h^2 (b g-a h)+4 c^2 g \left (3 f g^2-h (2 e g-d h)\right )+c h \left (4 a h (2 f g-e h)-b \left (13 f g^2-8 e g h+4 d h^2\right )\right )+2 c h^2 \left (2 c e g+b f g-\frac {3 c f g^2}{h}-2 c d h-a f h\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c h^3 \left (c g^2-b g h+a h^2\right )}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{3/2}}{h \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac {\left (b^2 f h^2+4 c h (2 b f g-b e h-a f h)-8 c^2 \left (3 f g^2-h (2 e g-d h)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2} h^4}-\frac {\left (2 c g \left (3 f g^2-h (2 e g-d h)\right )+h \left (2 a h (2 f g-e h)-b \left (5 f g^2-3 e g h+d h^2\right )\right )\right ) \text {arctanh}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{2 h^4 \sqrt {c g^2-b g h+a h^2}} \]
-(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)-1/ 8*(b^2*f*h^2+4*c*h*(-a*f*h-b*e*h+2*b*f*g)-8*c^2*(3*f*g^2-h*(-d*h+2*e*g)))* arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/h^4-1/2*(2*c*g* (3*f*g^2-h*(-d*h+2*e*g))+h*(2*a*h*(-e*h+2*f*g)-b*(d*h^2-3*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))/h^4/(a*h^2-b*g*h+c*g^2)^(1/2)-1/4*(b*f*h^2*(-a*h+b*g)+4*c^2*g *(3*f*g^2-h*(-d*h+2*e*g))+c*h*(4*a*h*(-e*h+2*f*g)-b*(4*d*h^2-8*e*g*h+13*f* g^2))+2*c*h^2*(2*c*e*g+b*f*g-3*c*f*g^2/h-2*c*d*h-a*f*h)*x)*(c*x^2+b*x+a)^( 1/2)/c/h^3/(a*h^2-b*g*h+c*g^2)
Time = 10.51 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\frac {-8 (2 f g-e h) \sqrt {a+x (b+c x)}+\frac {2 f h (b+2 c x) \sqrt {a+x (b+c x)}}{c}-\frac {8 \left (f g^2+h (-e g+d h)\right ) \sqrt {a+x (b+c x)}}{g+h x}+\frac {\left (-b^2+4 a c\right ) f h \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{3/2}}+\frac {4 \left (f g^2+h (-e g+d h)\right ) \left (2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-\frac {(2 c g-b h) \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c g^2+h (-b g+a h)}}\right )}{h}+\frac {4 (2 f g-e h) \left ((2 c g-b h) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} \sqrt {c g^2+h (-b g+a h)} \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )\right )}{\sqrt {c} h}}{8 h^3} \]
(-8*(2*f*g - e*h)*Sqrt[a + x*(b + c*x)] + (2*f*h*(b + 2*c*x)*Sqrt[a + x*(b + c*x)])/c - (8*(f*g^2 + h*(-(e*g) + d*h))*Sqrt[a + x*(b + c*x)])/(g + h* x) + ((-b^2 + 4*a*c)*f*h*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c* x)])])/c^(3/2) + (4*(f*g^2 + h*(-(e*g) + d*h))*(2*Sqrt[c]*ArcTanh[(b + 2*c *x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - ((2*c*g - b*h)*ArcTanh[(-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)])])/Sqrt[c*g^2 + h*(-(b*g) + a*h)]))/h + (4*(2*f*g - e*h)*((2*c*g - b* h)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - 2*Sqrt[c]*Sqrt [c*g^2 + h*(-(b*g) + a*h)]*ArcTanh[(-2*a*h + 2*c*g*x + b*(g - h*x))/(2*Sqr t[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])]))/(Sqrt[c]*h))/(8*h^3)
Time = 1.13 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2181, 27, 1231, 27, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {\int -\frac {\left (\frac {3 b f g^2}{h}+2 c d g-3 b e g-2 a f g+b d h+2 a e h-2 \left (-\frac {3 c f g^2}{h}+2 c e g+b f g-2 c d h-a f h\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (g+h x)}dx}{a h^2-b g h+c g^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{h (g+h x) \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (2 c d g-2 a f g+2 a e h-b \left (-\frac {3 f g^2}{h}+3 e g-d h\right )-2 \left (-\frac {3 c f g^2}{h}+2 c e g+b f g-2 c d h-a f h\right ) x\right ) \sqrt {c x^2+b x+a}}{g+h x}dx}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{h (g+h x) \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (c g^2-b h g+a h^2\right ) \left (f g h b^2-4 c \left (3 f g^2-h (2 e g-d h)\right ) b+4 a c h (3 f g-2 e h)+\left (-8 \left (3 f g^2-h (2 e g-d h)\right ) c^2+4 h (2 b f g-b e h-a f h) c+b^2 f h^2\right ) x\right )}{h (g+h x) \sqrt {c x^2+b x+a}}dx}{4 c h^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c h x \left (-a f h+b f g-2 c d h+2 c e g-\frac {3 c f g^2}{h}\right )+b f h (b g-a h)+4 a c h (2 f g-e h)-b c \left (13 f g^2-4 h (2 e g-d h)\right )-4 c^2 g \left (-d h+2 e g-\frac {3 f g^2}{h}\right )\right )}{2 c h^2}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{h (g+h x) \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\left (a h^2-b g h+c g^2\right ) \int \frac {f g h b^2-4 c \left (3 f g^2-h (2 e g-d h)\right ) b+4 a c h (3 f g-2 e h)+\left (-8 \left (3 f g^2-h (2 e g-d h)\right ) c^2+4 h (2 b f g-b e h-a f h) c+b^2 f h^2\right ) x}{(g+h x) \sqrt {c x^2+b x+a}}dx}{4 c h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 c h x \left (-a f h+b f g-2 c d h+2 c e g-\frac {3 c f g^2}{h}\right )+b f h (b g-a h)+4 a c h (2 f g-e h)-b c \left (13 f g^2-4 h (2 e g-d h)\right )-4 c^2 g \left (-d h+2 e g-\frac {3 f g^2}{h}\right )\right )}{2 c h^2}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{h (g+h x) \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {-\frac {\left (a h^2-b g h+c g^2\right ) \left (\frac {\left (4 c h (-a f h-b e h+2 b f g)+b^2 f h^2-8 c^2 \left (3 f g^2-h (2 e g-d h)\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{h}+\frac {4 c \left (2 c \left (3 f g^3-g h (2 e g-d h)\right )-h \left (-2 a h (2 f g-e h)-b h (3 e g-d h)+5 b f g^2\right )\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}\right )}{4 c h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 c h x \left (-a f h+b f g-2 c d h+2 c e g-\frac {3 c f g^2}{h}\right )+b f h (b g-a h)+4 a c h (2 f g-e h)-b c \left (13 f g^2-4 h (2 e g-d h)\right )-4 c^2 g \left (-d h+2 e g-\frac {3 f g^2}{h}\right )\right )}{2 c h^2}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{h (g+h x) \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {-\frac {\left (a h^2-b g h+c g^2\right ) \left (\frac {2 \left (4 c h (-a f h-b e h+2 b f g)+b^2 f h^2-8 c^2 \left (3 f g^2-h (2 e g-d h)\right )\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{h}+\frac {4 c \left (2 c \left (3 f g^3-g h (2 e g-d h)\right )-h \left (-2 a h (2 f g-e h)-b h (3 e g-d h)+5 b f g^2\right )\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}\right )}{4 c h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 c h x \left (-a f h+b f g-2 c d h+2 c e g-\frac {3 c f g^2}{h}\right )+b f h (b g-a h)+4 a c h (2 f g-e h)-b c \left (13 f g^2-4 h (2 e g-d h)\right )-4 c^2 g \left (-d h+2 e g-\frac {3 f g^2}{h}\right )\right )}{2 c h^2}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{h (g+h x) \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\left (a h^2-b g h+c g^2\right ) \left (\frac {4 c \left (2 c \left (3 f g^3-g h (2 e g-d h)\right )-h \left (-2 a h (2 f g-e h)-b h (3 e g-d h)+5 b f g^2\right )\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c h (-a f h-b e h+2 b f g)+b^2 f h^2-8 c^2 \left (3 f g^2-h (2 e g-d h)\right )\right )}{\sqrt {c} h}\right )}{4 c h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 c h x \left (-a f h+b f g-2 c d h+2 c e g-\frac {3 c f g^2}{h}\right )+b f h (b g-a h)+4 a c h (2 f g-e h)-b c \left (13 f g^2-4 h (2 e g-d h)\right )-4 c^2 g \left (-d h+2 e g-\frac {3 f g^2}{h}\right )\right )}{2 c h^2}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{h (g+h x) \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {-\frac {\left (a h^2-b g h+c g^2\right ) \left (\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c h (-a f h-b e h+2 b f g)+b^2 f h^2-8 c^2 \left (3 f g^2-h (2 e g-d h)\right )\right )}{\sqrt {c} h}-\frac {8 c \left (2 c \left (3 f g^3-g h (2 e g-d h)\right )-h \left (-2 a h (2 f g-e h)-b h (3 e g-d h)+5 b f g^2\right )\right ) \int \frac {1}{4 \left (c g^2-b h g+a h^2\right )-\frac {(b g-2 a h+(2 c g-b h) x)^2}{c x^2+b x+a}}d\left (-\frac {b g-2 a h+(2 c g-b h) x}{\sqrt {c x^2+b x+a}}\right )}{h}\right )}{4 c h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 c h x \left (-a f h+b f g-2 c d h+2 c e g-\frac {3 c f g^2}{h}\right )+b f h (b g-a h)+4 a c h (2 f g-e h)-b c \left (13 f g^2-4 h (2 e g-d h)\right )-4 c^2 g \left (-d h+2 e g-\frac {3 f g^2}{h}\right )\right )}{2 c h^2}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{h (g+h x) \left (a h^2-b g h+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\left (a h^2-b g h+c g^2\right ) \left (\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c h (-a f h-b e h+2 b f g)+b^2 f h^2-8 c^2 \left (3 f g^2-h (2 e g-d h)\right )\right )}{\sqrt {c} h}+\frac {4 c \text {arctanh}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right ) \left (2 c \left (3 f g^3-g h (2 e g-d h)\right )-h \left (-2 a h (2 f g-e h)-b h (3 e g-d h)+5 b f g^2\right )\right )}{h \sqrt {a h^2-b g h+c g^2}}\right )}{4 c h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 c h x \left (-a f h+b f g-2 c d h+2 c e g-\frac {3 c f g^2}{h}\right )+b f h (b g-a h)+4 a c h (2 f g-e h)-b c \left (13 f g^2-4 h (2 e g-d h)\right )-4 c^2 g \left (-d h+2 e g-\frac {3 f g^2}{h}\right )\right )}{2 c h^2}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{h (g+h x) \left (a h^2-b g h+c g^2\right )}\) |
-(((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))) + (-1/2*((b*f*h*(b*g - a*h) - 4*c^2*g*(2*e*g - (3*f*g^2)/ h - d*h) + 4*a*c*h*(2*f*g - e*h) - b*c*(13*f*g^2 - 4*h*(2*e*g - d*h)) + 2* c*h*(2*c*e*g + b*f*g - (3*c*f*g^2)/h - 2*c*d*h - a*f*h)*x)*Sqrt[a + b*x + c*x^2])/(c*h^2) - ((c*g^2 - b*g*h + a*h^2)*(((b^2*f*h^2 + 4*c*h*(2*b*f*g - b*e*h - a*f*h) - 8*c^2*(3*f*g^2 - h*(2*e*g - d*h)))*ArcTanh[(b + 2*c*x)/( 2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*h) + (4*c*(2*c*(3*f*g^3 - g*h* (2*e*g - d*h)) - h*(5*b*f*g^2 - b*h*(3*e*g - d*h) - 2*a*h*(2*f*g - e*h)))* ArcTanh[(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqr t[a + b*x + c*x^2])])/(h*Sqrt[c*g^2 - b*g*h + a*h^2])))/(4*c*h^3))/(2*(c*g ^2 - b*g*h + a*h^2))
3.2.91.3.1 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)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[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]
Time = 0.85 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.53
| method | result | size |
| risch | \(\frac {\left (2 c f h x +b f h +4 e h c -8 c f g \right ) \sqrt {c \,x^{2}+b x +a}}{4 c \,h^{3}}+\frac {\frac {\left (4 a c f \,h^{2}-b^{2} f \,h^{2}+4 b c e \,h^{2}-8 b c f g h +8 c^{2} d \,h^{2}-16 c^{2} e g h +24 c^{2} f \,g^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{h \sqrt {c}}-\frac {8 c \left (a e \,h^{3}-2 a f g \,h^{2}+b d \,h^{3}-2 b e g \,h^{2}+3 b f \,g^{2} h -2 c d g \,h^{2}+3 c e \,g^{2} h -4 c f \,g^{3}\right ) \ln \left (\frac {\frac {2 a \,h^{2}-2 b g h +2 c \,g^{2}}{h^{2}}+\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c +\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{h^{2} \sqrt {\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}+\frac {8 c \left (a d \,h^{4}-a e g \,h^{3}+a f \,g^{2} h^{2}-b d g \,h^{3}+b e \,g^{2} h^{2}-b f \,g^{3} h +c d \,g^{2} h^{2}-g^{3} c e h +g^{4} c f \right ) \left (-\frac {h^{2} \sqrt {\left (x +\frac {g}{h}\right )^{2} c +\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}{\left (a \,h^{2}-b g h +c \,g^{2}\right ) \left (x +\frac {g}{h}\right )}+\frac {\left (b h -2 c g \right ) h \ln \left (\frac {\frac {2 a \,h^{2}-2 b g h +2 c \,g^{2}}{h^{2}}+\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c +\frac {\left (b h -2 c g \right ) \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{2 \left (a \,h^{2}-b g h +c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}-b g h +c \,g^{2}}{h^{2}}}}\right )}{h^{3}}}{8 c \,h^{3}}\) | \(704\) |
| default | \(\text {Expression too large to display}\) | \(1097\) |
1/4*(2*c*f*h*x+b*f*h+4*c*e*h-8*c*f*g)/c*(c*x^2+b*x+a)^(1/2)/h^3+1/8/c/h^3* ((4*a*c*f*h^2-b^2*f*h^2+4*b*c*e*h^2-8*b*c*f*g*h+8*c^2*d*h^2-16*c^2*e*g*h+2 4*c^2*f*g^2)/h*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-8*c/h^2 *(a*e*h^3-2*a*f*g*h^2+b*d*h^3-2*b*e*g*h^2+3*b*f*g^2*h-2*c*d*g*h^2+3*c*e*g^ 2*h-4*c*f*g^3)/((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))+8*c*( a*d*h^4-a*e*g*h^3+a*f*g^2*h^2-b*d*g*h^3+b*e*g^2*h^2-b*f*g^3*h+c*d*g^2*h^2- c*e*g^3*h+c*f*g^4)/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)) ))
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\int \frac {\sqrt {a + b x + c x^{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*h-2*c*g>0)', see `assume?` for more deta
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^2} \,d x \]