Integrand size = 29, antiderivative size = 475 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=-\frac {(b c-a d)^2 (b g-a h) \sqrt {e+f x}}{3 b^3 (b e-a f) (a+b x)^3}-\frac {(b c-a d) \left (13 a^2 d f h+b^2 (12 d e g-5 c f g+6 c e h)-a b (7 d f g+18 d e h+c f h)\right ) \sqrt {e+f x}}{12 b^3 (b e-a f)^2 (a+b x)^2}+\frac {\left (11 a^3 d^2 f^2 h-a^2 b d f (d f g+30 d e h+2 c f h)-b^3 \left (8 d^2 e^2 g+c^2 f (5 f g-6 e h)-4 c d e (3 f g-4 e h)\right )-a b^2 \left (c^2 f^2 h+2 c d f (f g-4 e h)-4 d^2 e (f g+6 e h)\right )\right ) \sqrt {e+f x}}{8 b^3 (b e-a f)^3 (a+b x)}+\frac {\left (5 a^3 d^2 f^3 h+a^2 b d f^2 (d f g-18 d e h+2 c f h)+a b^2 f \left (c^2 f^2 h-4 d^2 e (f g-6 e h)+2 c d f (f g-4 e h)\right )+b^3 \left (c^2 f^2 (5 f g-6 e h)-4 c d e f (3 f g-4 e h)+8 d^2 e^2 (f g-2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 b^{7/2} (b e-a f)^{7/2}} \] Output:
-1/3*(-a*d+b*c)^2*(-a*h+b*g)*(f*x+e)^(1/2)/b^3/(-a*f+b*e)/(b*x+a)^3-1/12*( -a*d+b*c)*(13*a^2*d*f*h+b^2*(6*c*e*h-5*c*f*g+12*d*e*g)-a*b*(c*f*h+18*d*e*h +7*d*f*g))*(f*x+e)^(1/2)/b^3/(-a*f+b*e)^2/(b*x+a)^2+1/8*(11*a^3*d^2*f^2*h- a^2*b*d*f*(2*c*f*h+30*d*e*h+d*f*g)-b^3*(8*d^2*e^2*g+c^2*f*(-6*e*h+5*f*g)-4 *c*d*e*(-4*e*h+3*f*g))-a*b^2*(c^2*f^2*h+2*c*d*f*(-4*e*h+f*g)-4*d^2*e*(6*e* h+f*g)))*(f*x+e)^(1/2)/b^3/(-a*f+b*e)^3/(b*x+a)+1/8*(5*a^3*d^2*f^3*h+a^2*b *d*f^2*(2*c*f*h-18*d*e*h+d*f*g)+a*b^2*f*(c^2*f^2*h-4*d^2*e*(-6*e*h+f*g)+2* c*d*f*(-4*e*h+f*g))+b^3*(c^2*f^2*(-6*e*h+5*f*g)-4*c*d*e*f*(-4*e*h+3*f*g)+8 *d^2*e^2*(-2*e*h+f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^ (7/2)/(-a*f+b*e)^(7/2)
Time = 1.87 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.30 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\frac {\sqrt {e+f x} \left (15 a^5 d^2 f^2 h+a^4 b d f (6 c f h+d (3 f g-44 e h+40 f h x))-b^5 \left (24 d^2 e^2 g x^2+12 c d e x (-3 f g x+2 e (g+2 h x))+c^2 \left (15 f^2 g x^2+4 e^2 (2 g+3 h x)-2 e f x (5 g+9 h x)\right )\right )-a^2 b^3 \left (c^2 f (33 f g-16 e h+8 f h x)+2 c d \left (8 e^2 h-2 e f (8 g-7 h x)+f^2 x (8 g+3 h x)\right )+d^2 \left (3 f^2 g x^2+4 e^2 (2 g-27 h x)+2 e f x (7 g+45 h x)\right )\right )+a b^4 \left (12 d^2 e x (-2 e g+f g x+6 e h x)-2 c d \left (3 f^2 g x^2+4 e^2 (g+6 h x)-2 e f x (25 g+6 h x)\right )+c^2 \left (-4 e^2 h-f^2 x (40 g+3 h x)+e f (26 g+50 h x)\right )\right )+a^3 b^2 \left (3 c^2 f^2 h+2 c d f (3 f g-10 e h+8 f h x)+d^2 \left (44 e^2 h+f^2 x (8 g+33 h x)-2 e f (5 g+59 h x)\right )\right )\right )}{24 b^3 (b e-a f)^3 (a+b x)^3}-\frac {\left (-5 a^3 d^2 f^3 h-a^2 b d f^2 (d f g-18 d e h+2 c f h)+b^3 \left (8 d^2 e^2 (-f g+2 e h)-4 c d e f (-3 f g+4 e h)+c^2 f^2 (-5 f g+6 e h)\right )-a b^2 f \left (c^2 f^2 h+2 c d f (f g-4 e h)+4 d^2 e (-f g+6 e h)\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{8 b^{7/2} (-b e+a f)^{7/2}} \] Input:
Integrate[((c + d*x)^2*(g + h*x))/((a + b*x)^4*Sqrt[e + f*x]),x]
Output:
(Sqrt[e + f*x]*(15*a^5*d^2*f^2*h + a^4*b*d*f*(6*c*f*h + d*(3*f*g - 44*e*h + 40*f*h*x)) - b^5*(24*d^2*e^2*g*x^2 + 12*c*d*e*x*(-3*f*g*x + 2*e*(g + 2*h *x)) + c^2*(15*f^2*g*x^2 + 4*e^2*(2*g + 3*h*x) - 2*e*f*x*(5*g + 9*h*x))) - a^2*b^3*(c^2*f*(33*f*g - 16*e*h + 8*f*h*x) + 2*c*d*(8*e^2*h - 2*e*f*(8*g - 7*h*x) + f^2*x*(8*g + 3*h*x)) + d^2*(3*f^2*g*x^2 + 4*e^2*(2*g - 27*h*x) + 2*e*f*x*(7*g + 45*h*x))) + a*b^4*(12*d^2*e*x*(-2*e*g + f*g*x + 6*e*h*x) - 2*c*d*(3*f^2*g*x^2 + 4*e^2*(g + 6*h*x) - 2*e*f*x*(25*g + 6*h*x)) + c^2*( -4*e^2*h - f^2*x*(40*g + 3*h*x) + e*f*(26*g + 50*h*x))) + a^3*b^2*(3*c^2*f ^2*h + 2*c*d*f*(3*f*g - 10*e*h + 8*f*h*x) + d^2*(44*e^2*h + f^2*x*(8*g + 3 3*h*x) - 2*e*f*(5*g + 59*h*x)))))/(24*b^3*(b*e - a*f)^3*(a + b*x)^3) - ((- 5*a^3*d^2*f^3*h - a^2*b*d*f^2*(d*f*g - 18*d*e*h + 2*c*f*h) + b^3*(8*d^2*e^ 2*(-(f*g) + 2*e*h) - 4*c*d*e*f*(-3*f*g + 4*e*h) + c^2*f^2*(-5*f*g + 6*e*h) ) - a*b^2*f*(c^2*f^2*h + 2*c*d*f*(f*g - 4*e*h) + 4*d^2*e*(-(f*g) + 6*e*h)) )*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(8*b^(7/2)*(-(b*e) + a*f)^(7/2))
Time = 0.76 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {166, 27, 25, 162, 73, 221}
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.
| \(\displaystyle \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {\int \frac {(c+d x) ((4 d e+c f) (b g-a h)-6 b c (f g-e h)-d (b f g-6 b e h+5 a f h) x)}{2 (a+b x)^3 \sqrt {e+f x}}dx}{3 b (b e-a f)}-\frac {(c+d x)^2 \sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {(c+d x) (a (4 d e+c f) h-b (4 d e g-5 c f g+6 c e h)+d (b f g-6 b e h+5 a f h) x)}{(a+b x)^3 \sqrt {e+f x}}dx}{6 b (b e-a f)}-\frac {(c+d x)^2 \sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {(c+d x) (a (4 d e+c f) h-b (4 d e g-5 c f g+6 c e h)+d (b f g-6 b e h+5 a f h) x)}{(a+b x)^3 \sqrt {e+f x}}dx}{6 b (b e-a f)}-\frac {(c+d x)^2 \sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle -\frac {\frac {3 \left (5 a^3 d^2 f^3 h+a^2 b d f^2 (2 c f h-18 d e h+d f g)+a b^2 f \left (c^2 f^2 h+2 c d f (f g-4 e h)-4 d^2 e (f g-6 e h)\right )+b^3 \left (c^2 f^2 (5 f g-6 e h)-4 c d e f (3 f g-4 e h)+8 d^2 e^2 (f g-2 e h)\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{8 b^2 (b e-a f)^2}-\frac {\sqrt {e+f x} \left (15 a^4 d^2 f^2 h+a^3 b d f (6 c f h-44 d e h+3 d f g)-a^2 b^2 \left (5 c^2 f^2 h-2 c d f (3 f g-10 e h)+2 d^2 e (5 f g-22 e h)\right )+b x \left (25 a^3 d^2 f^2 h+a^2 b d f (-6 c f h-74 d e h+5 d f g)-a b^2 \left (3 c^2 f^2 h+6 c d f (f g-4 e h)+4 d^2 e (f g-16 e h)\right )-b^3 \left (3 c^2 f (5 f g-6 e h)-12 c d e (3 f g-4 e h)+16 d^2 e^2 g\right )\right )-a b^3 \left (c^2 f (25 f g-32 e h)-16 c d e (2 f g-e h)+8 d^2 e^2 g\right )-2 b^4 c e (6 c e h-5 c f g+4 d e g)\right )}{4 b^2 (a+b x)^2 (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 \sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {3 \left (5 a^3 d^2 f^3 h+a^2 b d f^2 (2 c f h-18 d e h+d f g)+a b^2 f \left (c^2 f^2 h+2 c d f (f g-4 e h)-4 d^2 e (f g-6 e h)\right )+b^3 \left (c^2 f^2 (5 f g-6 e h)-4 c d e f (3 f g-4 e h)+8 d^2 e^2 (f g-2 e h)\right )\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{4 b^2 f (b e-a f)^2}-\frac {\sqrt {e+f x} \left (15 a^4 d^2 f^2 h+a^3 b d f (6 c f h-44 d e h+3 d f g)-a^2 b^2 \left (5 c^2 f^2 h-2 c d f (3 f g-10 e h)+2 d^2 e (5 f g-22 e h)\right )+b x \left (25 a^3 d^2 f^2 h+a^2 b d f (-6 c f h-74 d e h+5 d f g)-a b^2 \left (3 c^2 f^2 h+6 c d f (f g-4 e h)+4 d^2 e (f g-16 e h)\right )-b^3 \left (3 c^2 f (5 f g-6 e h)-12 c d e (3 f g-4 e h)+16 d^2 e^2 g\right )\right )-a b^3 \left (c^2 f (25 f g-32 e h)-16 c d e (2 f g-e h)+8 d^2 e^2 g\right )-2 b^4 c e (6 c e h-5 c f g+4 d e g)\right )}{4 b^2 (a+b x)^2 (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 \sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right ) \left (5 a^3 d^2 f^3 h+a^2 b d f^2 (2 c f h-18 d e h+d f g)+a b^2 f \left (c^2 f^2 h+2 c d f (f g-4 e h)-4 d^2 e (f g-6 e h)\right )+b^3 \left (c^2 f^2 (5 f g-6 e h)-4 c d e f (3 f g-4 e h)+8 d^2 e^2 (f g-2 e h)\right )\right )}{4 b^{5/2} (b e-a f)^{5/2}}-\frac {\sqrt {e+f x} \left (15 a^4 d^2 f^2 h+a^3 b d f (6 c f h-44 d e h+3 d f g)-a^2 b^2 \left (5 c^2 f^2 h-2 c d f (3 f g-10 e h)+2 d^2 e (5 f g-22 e h)\right )+b x \left (25 a^3 d^2 f^2 h+a^2 b d f (-6 c f h-74 d e h+5 d f g)-a b^2 \left (3 c^2 f^2 h+6 c d f (f g-4 e h)+4 d^2 e (f g-16 e h)\right )-b^3 \left (3 c^2 f (5 f g-6 e h)-12 c d e (3 f g-4 e h)+16 d^2 e^2 g\right )\right )-a b^3 \left (c^2 f (25 f g-32 e h)-16 c d e (2 f g-e h)+8 d^2 e^2 g\right )-2 b^4 c e (6 c e h-5 c f g+4 d e g)\right )}{4 b^2 (a+b x)^2 (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 \sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
Input:
Int[((c + d*x)^2*(g + h*x))/((a + b*x)^4*Sqrt[e + f*x]),x]
Output:
-1/3*((b*g - a*h)*(c + d*x)^2*Sqrt[e + f*x])/(b*(b*e - a*f)*(a + b*x)^3) - (-1/4*(Sqrt[e + f*x]*(15*a^4*d^2*f^2*h - 2*b^4*c*e*(4*d*e*g - 5*c*f*g + 6 *c*e*h) + a^3*b*d*f*(3*d*f*g - 44*d*e*h + 6*c*f*h) - a^2*b^2*(5*c^2*f^2*h + 2*d^2*e*(5*f*g - 22*e*h) - 2*c*d*f*(3*f*g - 10*e*h)) - a*b^3*(8*d^2*e^2* g + c^2*f*(25*f*g - 32*e*h) - 16*c*d*e*(2*f*g - e*h)) + b*(25*a^3*d^2*f^2* h + a^2*b*d*f*(5*d*f*g - 74*d*e*h - 6*c*f*h) - a*b^2*(3*c^2*f^2*h + 4*d^2* e*(f*g - 16*e*h) + 6*c*d*f*(f*g - 4*e*h)) - b^3*(16*d^2*e^2*g + 3*c^2*f*(5 *f*g - 6*e*h) - 12*c*d*e*(3*f*g - 4*e*h)))*x))/(b^2*(b*e - a*f)^2*(a + b*x )^2) - (3*(5*a^3*d^2*f^3*h + a^2*b*d*f^2*(d*f*g - 18*d*e*h + 2*c*f*h) + a* b^2*f*(c^2*f^2*h - 4*d^2*e*(f*g - 6*e*h) + 2*c*d*f*(f*g - 4*e*h)) + b^3*(c ^2*f^2*(5*f*g - 6*e*h) - 4*c*d*e*f*(3*f*g - 4*e*h) + 8*d^2*e^2*(f*g - 2*e* h)))*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(4*b^(5/2)*(b*e - a *f)^(5/2)))/(6*b*(b*e - a*f))
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.63 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.27
| method | result | size |
| pseudoelliptic | \(\frac {\frac {5 \left (\left (c^{2} g \,f^{3}-\frac {6 c e \left (c h +2 d g \right ) f^{2}}{5}+\frac {16 \left (c h +\frac {d g}{2}\right ) d \,e^{2} f}{5}-\frac {16 d^{2} e^{3} h}{5}\right ) b^{3}+\frac {a \left (\left (h \,c^{2}+2 c d g \right ) f^{2}+4 \left (-2 c d h -d^{2} g \right ) e f +24 d^{2} e^{2} h \right ) f \,b^{2}}{5}+\frac {2 a^{2} d \left (\left (c h +\frac {d g}{2}\right ) f -9 d e h \right ) f^{2} b}{5}+a^{3} d^{2} f^{3} h \right ) \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8}-\frac {5 \left (\left (-c^{2} f^{2} g \,x^{2}+\frac {2 x c \left (\frac {18 d g x}{5}+c \left (\frac {9 h x}{5}+g \right )\right ) e f}{3}-\frac {8 \left (3 d^{2} g \,x^{2}+3 c x \left (2 h x +g \right ) d +c^{2} \left (\frac {3 h x}{2}+g \right )\right ) e^{2}}{15}\right ) b^{5}-\frac {4 a \left (10 x c \left (\frac {3 d g x}{20}+c \left (\frac {3 h x}{40}+g \right )\right ) f^{2}-\frac {13 \left (\frac {6 d^{2} g \,x^{2}}{13}+\frac {50 x c \left (\frac {6 h x}{25}+g \right ) d}{13}+c^{2} \left (\frac {25 h x}{13}+g \right )\right ) e f}{2}+e^{2} \left (6 x \left (-3 h x +g \right ) d^{2}+2 c \left (6 h x +g \right ) d +h \,c^{2}\right )\right ) b^{4}}{15}+\frac {16 a^{2} \left (\left (-\frac {3 d^{2} g \,x^{2}}{16}-x c \left (\frac {3 h x}{8}+g \right ) d -\frac {33 c^{2} \left (\frac {8 h x}{33}+g \right )}{16}\right ) f^{2}+\left (-\frac {7 x \left (\frac {45 h x}{7}+g \right ) d^{2}}{8}+2 c \left (-\frac {7 h x}{8}+g \right ) d +h \,c^{2}\right ) e f -d \,e^{2} \left (\frac {\left (-\frac {27 h x}{2}+g \right ) d}{2}+c h \right )\right ) b^{3}}{15}+\frac {a^{3} \left (\left (\left (11 h \,x^{2}+\frac {8}{3} g x \right ) d^{2}+2 c \left (\frac {8 h x}{3}+g \right ) d +h \,c^{2}\right ) f^{2}-\frac {20 d e \left (\frac {\left (\frac {59 h x}{5}+g \right ) d}{2}+c h \right ) f}{3}+\frac {44 d^{2} e^{2} h}{3}\right ) b^{2}}{5}+\frac {2 a^{4} \left (\left (\left (\frac {20 h x}{3}+\frac {g}{2}\right ) d +c h \right ) f -\frac {22 d e h}{3}\right ) d f b}{5}+a^{5} d^{2} f^{2} h \right ) \sqrt {f x +e}\, \sqrt {\left (a f -b e \right ) b}}{8}}{\sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{3} \left (a f -b e \right )^{3} b^{3}}\) | \(601\) |
| derivativedivides | \(-\frac {2 \left (\frac {f \left (11 a^{3} d^{2} f^{2} h -2 a^{2} b c d \,f^{2} h -30 a^{2} b \,d^{2} e f h -a^{2} b \,d^{2} f^{2} g -a \,b^{2} c^{2} f^{2} h +8 a \,b^{2} c d e f h -2 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +4 a \,b^{2} d^{2} e f g +6 b^{3} c^{2} e f h -5 b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h +12 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 b \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}+\frac {\left (5 a^{3} d^{2} f^{2} h +2 a^{2} b c d \,f^{2} h -18 a^{2} b \,d^{2} e f h +a^{2} b \,d^{2} f^{2} g -a \,b^{2} c^{2} f^{2} h -2 a \,b^{2} c d \,f^{2} g +18 a \,b^{2} d^{2} e^{2} h +6 b^{3} c^{2} e f h -5 b^{3} c^{2} f^{2} g -12 b^{3} c d \,e^{2} h +12 b^{3} c d e f g -6 b^{3} d^{2} e^{2} g \right ) f \left (f x +e \right )^{\frac {3}{2}}}{6 b^{2} \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (5 a^{3} d^{2} f^{2} h +2 a^{2} b c d \,f^{2} h -18 a^{2} b \,d^{2} e f h +a^{2} b \,d^{2} f^{2} g +a \,b^{2} c^{2} f^{2} h -8 a \,b^{2} c d e f h +2 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h -4 a \,b^{2} d^{2} e f g +10 b^{3} c^{2} e f h -11 b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h +20 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) f \sqrt {f x +e}}{16 b^{3} \left (a f -b e \right )}\right )}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (5 a^{3} d^{2} f^{3} h +2 a^{2} b c d \,f^{3} h -18 a^{2} b \,d^{2} e \,f^{2} h +a^{2} b \,d^{2} f^{3} g +a \,b^{2} c^{2} f^{3} h -8 a \,b^{2} c d e \,f^{2} h +2 a \,b^{2} c d \,f^{3} g +24 a \,b^{2} d^{2} e^{2} f h -4 a \,b^{2} d^{2} e \,f^{2} g -6 b^{3} c^{2} e \,f^{2} h +5 b^{3} c^{2} f^{3} g +16 b^{3} c d \,e^{2} f h -12 b^{3} c d e \,f^{2} g -16 b^{3} d^{2} h \,e^{3}+8 b^{3} d^{2} e^{2} f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 b^{3} \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \sqrt {\left (a f -b e \right ) b}}\) | \(874\) |
| default | \(-\frac {2 \left (\frac {f \left (11 a^{3} d^{2} f^{2} h -2 a^{2} b c d \,f^{2} h -30 a^{2} b \,d^{2} e f h -a^{2} b \,d^{2} f^{2} g -a \,b^{2} c^{2} f^{2} h +8 a \,b^{2} c d e f h -2 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +4 a \,b^{2} d^{2} e f g +6 b^{3} c^{2} e f h -5 b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h +12 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 b \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}+\frac {\left (5 a^{3} d^{2} f^{2} h +2 a^{2} b c d \,f^{2} h -18 a^{2} b \,d^{2} e f h +a^{2} b \,d^{2} f^{2} g -a \,b^{2} c^{2} f^{2} h -2 a \,b^{2} c d \,f^{2} g +18 a \,b^{2} d^{2} e^{2} h +6 b^{3} c^{2} e f h -5 b^{3} c^{2} f^{2} g -12 b^{3} c d \,e^{2} h +12 b^{3} c d e f g -6 b^{3} d^{2} e^{2} g \right ) f \left (f x +e \right )^{\frac {3}{2}}}{6 b^{2} \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (5 a^{3} d^{2} f^{2} h +2 a^{2} b c d \,f^{2} h -18 a^{2} b \,d^{2} e f h +a^{2} b \,d^{2} f^{2} g +a \,b^{2} c^{2} f^{2} h -8 a \,b^{2} c d e f h +2 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h -4 a \,b^{2} d^{2} e f g +10 b^{3} c^{2} e f h -11 b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h +20 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) f \sqrt {f x +e}}{16 b^{3} \left (a f -b e \right )}\right )}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (5 a^{3} d^{2} f^{3} h +2 a^{2} b c d \,f^{3} h -18 a^{2} b \,d^{2} e \,f^{2} h +a^{2} b \,d^{2} f^{3} g +a \,b^{2} c^{2} f^{3} h -8 a \,b^{2} c d e \,f^{2} h +2 a \,b^{2} c d \,f^{3} g +24 a \,b^{2} d^{2} e^{2} f h -4 a \,b^{2} d^{2} e \,f^{2} g -6 b^{3} c^{2} e \,f^{2} h +5 b^{3} c^{2} f^{3} g +16 b^{3} c d \,e^{2} f h -12 b^{3} c d e \,f^{2} g -16 b^{3} d^{2} h \,e^{3}+8 b^{3} d^{2} e^{2} f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 b^{3} \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \sqrt {\left (a f -b e \right ) b}}\) | \(874\) |
Input:
int((d*x+c)^2*(h*x+g)/(b*x+a)^4/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
Output:
5/8*(((c^2*g*f^3-6/5*c*e*(c*h+2*d*g)*f^2+16/5*(c*h+1/2*d*g)*d*e^2*f-16/5*d ^2*e^3*h)*b^3+1/5*a*((c^2*h+2*c*d*g)*f^2+4*(-2*c*d*h-d^2*g)*e*f+24*d^2*e^2 *h)*f*b^2+2/5*a^2*d*((c*h+1/2*d*g)*f-9*d*e*h)*f^2*b+a^3*d^2*f^3*h)*(b*x+a) ^3*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))-((-c^2*f^2*g*x^2+2/3*x*c*(1 8/5*d*g*x+c*(9/5*h*x+g))*e*f-8/15*(3*d^2*g*x^2+3*c*x*(2*h*x+g)*d+c^2*(3/2* h*x+g))*e^2)*b^5-4/15*a*(10*x*c*(3/20*d*g*x+c*(3/40*h*x+g))*f^2-13/2*(6/13 *d^2*g*x^2+50/13*x*c*(6/25*h*x+g)*d+c^2*(25/13*h*x+g))*e*f+e^2*(6*x*(-3*h* x+g)*d^2+2*c*(6*h*x+g)*d+h*c^2))*b^4+16/15*a^2*((-3/16*d^2*g*x^2-x*c*(3/8* h*x+g)*d-33/16*c^2*(8/33*h*x+g))*f^2+(-7/8*x*(45/7*h*x+g)*d^2+2*c*(-7/8*h* x+g)*d+h*c^2)*e*f-d*e^2*(1/2*(-27/2*h*x+g)*d+c*h))*b^3+1/5*a^3*(((11*h*x^2 +8/3*g*x)*d^2+2*c*(8/3*h*x+g)*d+h*c^2)*f^2-20/3*d*e*(1/2*(59/5*h*x+g)*d+c* h)*f+44/3*d^2*e^2*h)*b^2+2/5*a^4*(((20/3*h*x+1/2*g)*d+c*h)*f-22/3*d*e*h)*d *f*b+a^5*d^2*f^2*h)*(f*x+e)^(1/2)*((a*f-b*e)*b)^(1/2))/((a*f-b*e)*b)^(1/2) /(b*x+a)^3/(a*f-b*e)^3/b^3
Leaf count of result is larger than twice the leaf count of optimal. 1897 vs. \(2 (451) = 902\).
Time = 0.44 (sec) , antiderivative size = 3808, normalized size of antiderivative = 8.02 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(h*x+g)/(b*x+a)^4/(f*x+e)^(1/2),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**2*(h*x+g)/(b*x+a)**4/(f*x+e)**(1/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^2*(h*x+g)/(b*x+a)^4/(f*x+e)^(1/2),x, algorithm="maxima")
Output:
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*f-b*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1618 vs. \(2 (451) = 902\).
Time = 0.16 (sec) , antiderivative size = 1618, normalized size of antiderivative = 3.41 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(h*x+g)/(b*x+a)^4/(f*x+e)^(1/2),x, algorithm="giac")
Output:
-1/8*(8*b^3*d^2*e^2*f*g - 12*b^3*c*d*e*f^2*g - 4*a*b^2*d^2*e*f^2*g + 5*b^3 *c^2*f^3*g + 2*a*b^2*c*d*f^3*g + a^2*b*d^2*f^3*g - 16*b^3*d^2*e^3*h + 16*b ^3*c*d*e^2*f*h + 24*a*b^2*d^2*e^2*f*h - 6*b^3*c^2*e*f^2*h - 8*a*b^2*c*d*e* f^2*h - 18*a^2*b*d^2*e*f^2*h + a*b^2*c^2*f^3*h + 2*a^2*b*c*d*f^3*h + 5*a^3 *d^2*f^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^6*e^3 - 3*a*b ^5*e^2*f + 3*a^2*b^4*e*f^2 - a^3*b^3*f^3)*sqrt(-b^2*e + a*b*f)) - 1/24*(24 *(f*x + e)^(5/2)*b^5*d^2*e^2*f*g - 48*(f*x + e)^(3/2)*b^5*d^2*e^3*f*g + 24 *sqrt(f*x + e)*b^5*d^2*e^4*f*g - 36*(f*x + e)^(5/2)*b^5*c*d*e*f^2*g - 12*( f*x + e)^(5/2)*a*b^4*d^2*e*f^2*g + 96*(f*x + e)^(3/2)*b^5*c*d*e^2*f^2*g + 48*(f*x + e)^(3/2)*a*b^4*d^2*e^2*f^2*g - 60*sqrt(f*x + e)*b^5*c*d*e^3*f^2* g - 36*sqrt(f*x + e)*a*b^4*d^2*e^3*f^2*g + 15*(f*x + e)^(5/2)*b^5*c^2*f^3* g + 6*(f*x + e)^(5/2)*a*b^4*c*d*f^3*g + 3*(f*x + e)^(5/2)*a^2*b^3*d^2*f^3* g - 40*(f*x + e)^(3/2)*b^5*c^2*e*f^3*g - 112*(f*x + e)^(3/2)*a*b^4*c*d*e*f ^3*g + 8*(f*x + e)^(3/2)*a^2*b^3*d^2*e*f^3*g + 33*sqrt(f*x + e)*b^5*c^2*e^ 2*f^3*g + 114*sqrt(f*x + e)*a*b^4*c*d*e^2*f^3*g - 3*sqrt(f*x + e)*a^2*b^3* d^2*e^2*f^3*g + 40*(f*x + e)^(3/2)*a*b^4*c^2*f^4*g + 16*(f*x + e)^(3/2)*a^ 2*b^3*c*d*f^4*g - 8*(f*x + e)^(3/2)*a^3*b^2*d^2*f^4*g - 66*sqrt(f*x + e)*a *b^4*c^2*e*f^4*g - 48*sqrt(f*x + e)*a^2*b^3*c*d*e*f^4*g + 18*sqrt(f*x + e) *a^3*b^2*d^2*e*f^4*g + 33*sqrt(f*x + e)*a^2*b^3*c^2*f^5*g - 6*sqrt(f*x + e )*a^3*b^2*c*d*f^5*g - 3*sqrt(f*x + e)*a^4*b*d^2*f^5*g + 48*(f*x + e)^(5...
Time = 2.93 (sec) , antiderivative size = 908, normalized size of antiderivative = 1.91 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\frac {\frac {{\left (e+f\,x\right )}^{5/2}\,\left (-11\,h\,a^3\,d^2\,f^3+2\,h\,a^2\,b\,c\,d\,f^3+30\,h\,a^2\,b\,d^2\,e\,f^2+g\,a^2\,b\,d^2\,f^3+h\,a\,b^2\,c^2\,f^3-8\,h\,a\,b^2\,c\,d\,e\,f^2+2\,g\,a\,b^2\,c\,d\,f^3-24\,h\,a\,b^2\,d^2\,e^2\,f-4\,g\,a\,b^2\,d^2\,e\,f^2-6\,h\,b^3\,c^2\,e\,f^2+5\,g\,b^3\,c^2\,f^3+16\,h\,b^3\,c\,d\,e^2\,f-12\,g\,b^3\,c\,d\,e\,f^2+8\,g\,b^3\,d^2\,e^2\,f\right )}{8\,b\,{\left (a\,f-b\,e\right )}^3}-\frac {\sqrt {e+f\,x}\,\left (5\,h\,a^3\,d^2\,f^3+2\,h\,a^2\,b\,c\,d\,f^3-18\,h\,a^2\,b\,d^2\,e\,f^2+g\,a^2\,b\,d^2\,f^3+h\,a\,b^2\,c^2\,f^3-8\,h\,a\,b^2\,c\,d\,e\,f^2+2\,g\,a\,b^2\,c\,d\,f^3+24\,h\,a\,b^2\,d^2\,e^2\,f-4\,g\,a\,b^2\,d^2\,e\,f^2+10\,h\,b^3\,c^2\,e\,f^2-11\,g\,b^3\,c^2\,f^3-16\,h\,b^3\,c\,d\,e^2\,f+20\,g\,b^3\,c\,d\,e\,f^2-8\,g\,b^3\,d^2\,e^2\,f\right )}{8\,b^3\,\left (a\,f-b\,e\right )}+\frac {{\left (e+f\,x\right )}^{3/2}\,\left (-5\,h\,a^3\,d^2\,f^3-2\,h\,a^2\,b\,c\,d\,f^3+18\,h\,a^2\,b\,d^2\,e\,f^2-g\,a^2\,b\,d^2\,f^3+h\,a\,b^2\,c^2\,f^3+2\,g\,a\,b^2\,c\,d\,f^3-18\,h\,a\,b^2\,d^2\,e^2\,f-6\,h\,b^3\,c^2\,e\,f^2+5\,g\,b^3\,c^2\,f^3+12\,h\,b^3\,c\,d\,e^2\,f-12\,g\,b^3\,c\,d\,e\,f^2+6\,g\,b^3\,d^2\,e^2\,f\right )}{3\,b^2\,{\left (a\,f-b\,e\right )}^2}}{\left (e+f\,x\right )\,\left (3\,a^2\,b\,f^2-6\,a\,b^2\,e\,f+3\,b^3\,e^2\right )+b^3\,{\left (e+f\,x\right )}^3-{\left (e+f\,x\right )}^2\,\left (3\,b^3\,e-3\,a\,b^2\,f\right )+a^3\,f^3-b^3\,e^3+3\,a\,b^2\,e^2\,f-3\,a^2\,b\,e\,f^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e+f\,x}}{\sqrt {a\,f-b\,e}}\right )\,\left (5\,h\,a^3\,d^2\,f^3+2\,h\,a^2\,b\,c\,d\,f^3-18\,h\,a^2\,b\,d^2\,e\,f^2+g\,a^2\,b\,d^2\,f^3+h\,a\,b^2\,c^2\,f^3-8\,h\,a\,b^2\,c\,d\,e\,f^2+2\,g\,a\,b^2\,c\,d\,f^3+24\,h\,a\,b^2\,d^2\,e^2\,f-4\,g\,a\,b^2\,d^2\,e\,f^2-6\,h\,b^3\,c^2\,e\,f^2+5\,g\,b^3\,c^2\,f^3+16\,h\,b^3\,c\,d\,e^2\,f-12\,g\,b^3\,c\,d\,e\,f^2-16\,h\,b^3\,d^2\,e^3+8\,g\,b^3\,d^2\,e^2\,f\right )}{8\,b^{7/2}\,{\left (a\,f-b\,e\right )}^{7/2}} \] Input:
int(((g + h*x)*(c + d*x)^2)/((e + f*x)^(1/2)*(a + b*x)^4),x)
Output:
(((e + f*x)^(5/2)*(5*b^3*c^2*f^3*g - 11*a^3*d^2*f^3*h + a*b^2*c^2*f^3*h + a^2*b*d^2*f^3*g - 6*b^3*c^2*e*f^2*h + 8*b^3*d^2*e^2*f*g + 2*a*b^2*c*d*f^3* g + 2*a^2*b*c*d*f^3*h - 12*b^3*c*d*e*f^2*g + 16*b^3*c*d*e^2*f*h - 4*a*b^2* d^2*e*f^2*g - 24*a*b^2*d^2*e^2*f*h + 30*a^2*b*d^2*e*f^2*h - 8*a*b^2*c*d*e* f^2*h))/(8*b*(a*f - b*e)^3) - ((e + f*x)^(1/2)*(5*a^3*d^2*f^3*h - 11*b^3*c ^2*f^3*g + a*b^2*c^2*f^3*h + a^2*b*d^2*f^3*g + 10*b^3*c^2*e*f^2*h - 8*b^3* d^2*e^2*f*g + 2*a*b^2*c*d*f^3*g + 2*a^2*b*c*d*f^3*h + 20*b^3*c*d*e*f^2*g - 16*b^3*c*d*e^2*f*h - 4*a*b^2*d^2*e*f^2*g + 24*a*b^2*d^2*e^2*f*h - 18*a^2* b*d^2*e*f^2*h - 8*a*b^2*c*d*e*f^2*h))/(8*b^3*(a*f - b*e)) + ((e + f*x)^(3/ 2)*(5*b^3*c^2*f^3*g - 5*a^3*d^2*f^3*h + a*b^2*c^2*f^3*h - a^2*b*d^2*f^3*g - 6*b^3*c^2*e*f^2*h + 6*b^3*d^2*e^2*f*g + 2*a*b^2*c*d*f^3*g - 2*a^2*b*c*d* f^3*h - 12*b^3*c*d*e*f^2*g + 12*b^3*c*d*e^2*f*h - 18*a*b^2*d^2*e^2*f*h + 1 8*a^2*b*d^2*e*f^2*h))/(3*b^2*(a*f - b*e)^2))/((e + f*x)*(3*b^3*e^2 + 3*a^2 *b*f^2 - 6*a*b^2*e*f) + b^3*(e + f*x)^3 - (e + f*x)^2*(3*b^3*e - 3*a*b^2*f ) + a^3*f^3 - b^3*e^3 + 3*a*b^2*e^2*f - 3*a^2*b*e*f^2) + (atan((b^(1/2)*(e + f*x)^(1/2))/(a*f - b*e)^(1/2))*(5*b^3*c^2*f^3*g + 5*a^3*d^2*f^3*h - 16* b^3*d^2*e^3*h + a*b^2*c^2*f^3*h + a^2*b*d^2*f^3*g - 6*b^3*c^2*e*f^2*h + 8* b^3*d^2*e^2*f*g + 2*a*b^2*c*d*f^3*g + 2*a^2*b*c*d*f^3*h - 12*b^3*c*d*e*f^2 *g + 16*b^3*c*d*e^2*f*h - 4*a*b^2*d^2*e*f^2*g + 24*a*b^2*d^2*e^2*f*h - 18* a^2*b*d^2*e*f^2*h - 8*a*b^2*c*d*e*f^2*h))/(8*b^(7/2)*(a*f - b*e)^(7/2))
Time = 0.21 (sec) , antiderivative size = 4777, normalized size of antiderivative = 10.06 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^2*(h*x+g)/(b*x+a)^4/(f*x+e)^(1/2),x)
Output:
(15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e )))*a**6*d**2*f**3*h + 6*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(s qrt(b)*sqrt(a*f - b*e)))*a**5*b*c*d*f**3*h - 54*sqrt(b)*sqrt(a*f - b*e)*at an((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*b*d**2*e*f**2*h + 3*s qrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a **5*b*d**2*f**3*g + 45*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqr t(b)*sqrt(a*f - b*e)))*a**5*b*d**2*f**3*h*x + 3*sqrt(b)*sqrt(a*f - b*e)*at an((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*c**2*f**3*h - 24 *sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e))) *a**4*b**2*c*d*e*f**2*h + 6*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b) /(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*c*d*f**3*g + 18*sqrt(b)*sqrt(a*f - b *e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*c*d*f**3*h *x + 72*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d**2*e**2*f*h - 12*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d**2*e*f**2*g - 162*sqrt(b) *sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b* *2*d**2*e*f**2*h*x + 9*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqr t(b)*sqrt(a*f - b*e)))*a**4*b**2*d**2*f**3*g*x + 45*sqrt(b)*sqrt(a*f - b*e )*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d**2*f**3*h* x**2 - 18*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(...