Integrand size = 29, antiderivative size = 321 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx=\frac {(b c-a d) \left (9 a^2 d f h+b^2 (4 d e g+3 c f g+2 c e h)-a b (7 d f g+6 d e h+5 c f h)\right ) \sqrt {e+f x}}{b^5}+\frac {2 (b c-a d) (2 b d g+b c h-3 a d h) (e+f x)^{3/2}}{3 b^4}-\frac {(b c-a d)^2 (b g-a h) (e+f x)^{3/2}}{b^4 (a+b x)}-\frac {2 d (2 a d f h-b (d f g-d e h+2 c f h)) (e+f x)^{5/2}}{5 b^3 f^2}+\frac {2 d^2 h (e+f x)^{7/2}}{7 b^2 f^2}-\frac {(b c-a d) \sqrt {b e-a f} \left (9 a^2 d f h+b^2 (4 d e g+3 c f g+2 c e h)-a b (7 d f g+6 d e h+5 c f h)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{11/2}} \] Output:
(-a*d+b*c)*(9*a^2*d*f*h+b^2*(2*c*e*h+3*c*f*g+4*d*e*g)-a*b*(5*c*f*h+6*d*e*h +7*d*f*g))*(f*x+e)^(1/2)/b^5+2/3*(-a*d+b*c)*(-3*a*d*h+b*c*h+2*b*d*g)*(f*x+ e)^(3/2)/b^4-(-a*d+b*c)^2*(-a*h+b*g)*(f*x+e)^(3/2)/b^4/(b*x+a)-2/5*d*(2*a* d*f*h-b*(2*c*f*h-d*e*h+d*f*g))*(f*x+e)^(5/2)/b^3/f^2+2/7*d^2*h*(f*x+e)^(7/ 2)/b^2/f^2-(-a*d+b*c)*(-a*f+b*e)^(1/2)*(9*a^2*d*f*h+b^2*(2*c*e*h+3*c*f*g+4 *d*e*g)-a*b*(5*c*f*h+6*d*e*h+7*d*f*g))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f +b*e)^(1/2))/b^(11/2)
Time = 0.99 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.46 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx=\frac {\sqrt {e+f x} \left (-945 a^4 d^2 f^3 h+105 a^3 b d f^2 (14 c f h+d (7 f g+9 e h-6 f h x))+b^4 \left (6 d^2 x (e+f x)^2 (7 f g-2 e h+5 f h x)+28 c d f x \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )+35 c^2 f^2 (2 f x (3 g+h x)+e (-3 g+8 h x))\right )-7 a^2 b^2 f \left (75 c^2 f^2 h+10 c d f (15 f g+19 e h-14 f h x)+d^2 \left (12 e^2 h+e f (95 g-96 h x)-2 f^2 x (35 g+9 h x)\right )\right )-a b^3 \left (-35 c^2 f^2 (9 f g+11 e h-10 f h x)+2 d^2 \left (6 e^3 h+f^3 x^2 (49 g+27 h x)+2 e f^2 x (119 g+30 h x)+e^2 f (-21 g+39 h x)\right )+14 c d f \left (-6 e^2 h+2 f^2 x (25 g+7 h x)+e f (-55 g+68 h x)\right )\right )\right )}{105 b^5 f^2 (a+b x)}-\frac {(b c-a d) \sqrt {-b e+a f} \left (9 a^2 d f h+b^2 (4 d e g+3 c f g+2 c e h)-a b (7 d f g+6 d e h+5 c f h)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{b^{11/2}} \] Input:
Integrate[((c + d*x)^2*(e + f*x)^(3/2)*(g + h*x))/(a + b*x)^2,x]
Output:
(Sqrt[e + f*x]*(-945*a^4*d^2*f^3*h + 105*a^3*b*d*f^2*(14*c*f*h + d*(7*f*g + 9*e*h - 6*f*h*x)) + b^4*(6*d^2*x*(e + f*x)^2*(7*f*g - 2*e*h + 5*f*h*x) + 28*c*d*f*x*(3*e^2*h + f^2*x*(5*g + 3*h*x) + e*f*(20*g + 6*h*x)) + 35*c^2* f^2*(2*f*x*(3*g + h*x) + e*(-3*g + 8*h*x))) - 7*a^2*b^2*f*(75*c^2*f^2*h + 10*c*d*f*(15*f*g + 19*e*h - 14*f*h*x) + d^2*(12*e^2*h + e*f*(95*g - 96*h*x ) - 2*f^2*x*(35*g + 9*h*x))) - a*b^3*(-35*c^2*f^2*(9*f*g + 11*e*h - 10*f*h *x) + 2*d^2*(6*e^3*h + f^3*x^2*(49*g + 27*h*x) + 2*e*f^2*x*(119*g + 30*h*x ) + e^2*f*(-21*g + 39*h*x)) + 14*c*d*f*(-6*e^2*h + 2*f^2*x*(25*g + 7*h*x) + e*f*(-55*g + 68*h*x)))))/(105*b^5*f^2*(a + b*x)) - ((b*c - a*d)*Sqrt[-(b *e) + a*f]*(9*a^2*d*f*h + b^2*(4*d*e*g + 3*c*f*g + 2*c*e*h) - a*b*(7*d*f*g + 6*d*e*h + 5*c*f*h))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]]) /b^(11/2)
Time = 0.50 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {166, 27, 25, 164, 60, 60, 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.
| \(\displaystyle \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {\int -\frac {(c+d x) (e+f x)^{3/2} \left (a (4 d e+5 c f) h-2 b \left (2 d e g+\frac {3 c f g}{2}+c e h\right )-d (7 b f g+2 b e h-9 a f h) x\right )}{2 (a+b x)}dx}{b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -\frac {(c+d x) (e+f x)^{3/2} (4 b d e g+3 b c f g+2 b c e h-4 a d e h-5 a c f h+d (7 b f g+2 b e h-9 a f h) x)}{a+b x}dx}{2 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(c+d x) (e+f x)^{3/2} (4 b d e g+3 b c f g+2 b c e h-4 a d e h-5 a c f h+d (7 b f g+2 b e h-9 a f h) x)}{a+b x}dx}{2 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (9 a^2 d f h-a b (5 c f h+6 d e h+7 d f g)+b^2 (2 c e h+3 c f g+4 d e g)\right ) \int \frac {(e+f x)^{3/2}}{a+b x}dx}{b^2}+\frac {2 d (e+f x)^{5/2} \left (63 a^2 d f^2 h-a b f (98 c f h+24 d e h+49 d f g)+5 b d f x (-9 a f h+2 b e h+7 b f g)+2 b^2 (7 c f (2 e h+5 f g)+d e (7 f g-2 e h))\right )}{35 b^2 f^2}}{2 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (9 a^2 d f h-a b (5 c f h+6 d e h+7 d f g)+b^2 (2 c e h+3 c f g+4 d e g)\right ) \left (\frac {(b e-a f) \int \frac {\sqrt {e+f x}}{a+b x}dx}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{b^2}+\frac {2 d (e+f x)^{5/2} \left (63 a^2 d f^2 h-a b f (98 c f h+24 d e h+49 d f g)+5 b d f x (-9 a f h+2 b e h+7 b f g)+2 b^2 (7 c f (2 e h+5 f g)+d e (7 f g-2 e h))\right )}{35 b^2 f^2}}{2 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (9 a^2 d f h-a b (5 c f h+6 d e h+7 d f g)+b^2 (2 c e h+3 c f g+4 d e g)\right ) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b}+\frac {2 \sqrt {e+f x}}{b}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{b^2}+\frac {2 d (e+f x)^{5/2} \left (63 a^2 d f^2 h-a b f (98 c f h+24 d e h+49 d f g)+5 b d f x (-9 a f h+2 b e h+7 b f g)+2 b^2 (7 c f (2 e h+5 f g)+d e (7 f g-2 e h))\right )}{35 b^2 f^2}}{2 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (9 a^2 d f h-a b (5 c f h+6 d e h+7 d f g)+b^2 (2 c e h+3 c f g+4 d e g)\right ) \left (\frac {(b e-a f) \left (\frac {2 (b e-a f) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b f}+\frac {2 \sqrt {e+f x}}{b}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{b^2}+\frac {2 d (e+f x)^{5/2} \left (63 a^2 d f^2 h-a b f (98 c f h+24 d e h+49 d f g)+5 b d f x (-9 a f h+2 b e h+7 b f g)+2 b^2 (7 c f (2 e h+5 f g)+d e (7 f g-2 e h))\right )}{35 b^2 f^2}}{2 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \left (\frac {2 \sqrt {e+f x}}{b}-\frac {2 \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right ) \left (9 a^2 d f h-a b (5 c f h+6 d e h+7 d f g)+b^2 (2 c e h+3 c f g+4 d e g)\right )}{b^2}+\frac {2 d (e+f x)^{5/2} \left (63 a^2 d f^2 h-a b f (98 c f h+24 d e h+49 d f g)+5 b d f x (-9 a f h+2 b e h+7 b f g)+2 b^2 (7 c f (2 e h+5 f g)+d e (7 f g-2 e h))\right )}{35 b^2 f^2}}{2 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\) |
Input:
Int[((c + d*x)^2*(e + f*x)^(3/2)*(g + h*x))/(a + b*x)^2,x]
Output:
-(((b*g - a*h)*(c + d*x)^2*(e + f*x)^(5/2))/(b*(b*e - a*f)*(a + b*x))) + ( (2*d*(e + f*x)^(5/2)*(63*a^2*d*f^2*h - a*b*f*(49*d*f*g + 24*d*e*h + 98*c*f *h) + 2*b^2*(d*e*(7*f*g - 2*e*h) + 7*c*f*(5*f*g + 2*e*h)) + 5*b*d*f*(7*b*f *g + 2*b*e*h - 9*a*f*h)*x))/(35*b^2*f^2) + ((b*c - a*d)*(9*a^2*d*f*h + b^2 *(4*d*e*g + 3*c*f*g + 2*c*e*h) - a*b*(7*d*f*g + 6*d*e*h + 5*c*f*h))*((2*(e + f*x)^(3/2))/(3*b) + ((b*e - a*f)*((2*Sqrt[e + f*x])/b - (2*Sqrt[b*e - a *f]*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/b^(3/2)))/b))/b^2)/( 2*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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
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 = 0.60 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.54
| method | result | size |
| pseudoelliptic | \(\frac {9 \left (a d -b c \right ) \left (b x +a \right ) \left (\frac {\left (c f g +\frac {2 e \left (c h +2 d g \right )}{3}\right ) b^{2}}{3}-\frac {5 a \left (\left (c h +\frac {7 d g}{5}\right ) f +\frac {6 d e h}{5}\right ) b}{9}+a^{2} d f h \right ) f^{2} \left (a f -b e \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )-9 \sqrt {\left (a f -b e \right ) b}\, \sqrt {f x +e}\, \left (\frac {\left (-2 x \left (\frac {x^{2} \left (\frac {5 h x}{7}+g \right ) d^{2}}{5}+\frac {2 x c \left (\frac {3 h x}{5}+g \right ) d}{3}+c^{2} \left (\frac {h x}{3}+g \right )\right ) f^{3}+e \left (-\frac {4 x^{2} \left (\frac {4 h x}{7}+g \right ) d^{2}}{5}-\frac {16 \left (\frac {3 h x}{10}+g \right ) x c d}{3}+c^{2} \left (-\frac {8 h x}{3}+g \right )\right ) f^{2}-\frac {4 x d \left (\frac {\left (\frac {h x}{7}+g \right ) d}{2}+c h \right ) e^{2} f}{5}+\frac {4 d^{2} e^{3} h x}{35}\right ) b^{4}}{9}-\frac {11 a \left (\frac {\left (-\frac {14 x^{2} \left (\frac {27 h x}{49}+g \right ) d^{2}}{5}-20 x c \left (\frac {7 h x}{25}+g \right ) d +9 c^{2} \left (-\frac {10 h x}{9}+g \right )\right ) f^{3}}{11}+\left (\frac {4 \left (-\frac {6}{7} h \,x^{2}-\frac {17}{5} g x \right ) d^{2}}{11}+2 c \left (-\frac {68 h x}{55}+g \right ) d +h \,c^{2}\right ) e \,f^{2}+\frac {12 \left (\frac {\left (-\frac {13 h x}{7}+g \right ) d}{2}+c h \right ) d \,e^{2} f}{55}-\frac {12 d^{2} e^{3} h}{385}\right ) b^{3}}{27}+\frac {5 a^{2} \left (\left (-\frac {14 x \left (\frac {9 h x}{35}+g \right ) d^{2}}{15}+2 \left (-\frac {14 h x}{15}+g \right ) c d +h \,c^{2}\right ) f^{2}+\frac {38 d e \left (\left (-\frac {48 h x}{95}+\frac {g}{2}\right ) d +c h \right ) f}{15}+\frac {4 d^{2} e^{2} h}{25}\right ) f \,b^{2}}{9}-\frac {14 a^{3} d \left (\left (\left (-\frac {3 h x}{7}+\frac {g}{2}\right ) d +c h \right ) f +\frac {9 d e h}{14}\right ) f^{2} b}{9}+a^{4} d^{2} f^{3} h \right )}{f^{2} \left (b x +a \right ) b^{5} \sqrt {\left (a f -b e \right ) b}}\) | \(495\) |
| risch | \(-\frac {2 \left (-15 d^{2} h \,x^{3} b^{3} f^{3}+42 a \,b^{2} d^{2} f^{3} h \,x^{2}-42 b^{3} c d \,f^{3} h \,x^{2}-24 b^{3} d^{2} e \,f^{2} h \,x^{2}-21 b^{3} d^{2} f^{3} g \,x^{2}-105 a^{2} b \,d^{2} f^{3} h x +140 a \,b^{2} c d \,f^{3} h x +84 a \,b^{2} d^{2} e \,f^{2} h x +70 a \,b^{2} d^{2} f^{3} g x -35 b^{3} c^{2} f^{3} h x -84 b^{3} c d e \,f^{2} h x -70 b^{3} c d \,f^{3} g x -3 b^{3} d^{2} e^{2} f h x -42 b^{3} d^{2} e \,f^{2} g x +420 a^{3} d^{2} f^{3} h -630 a^{2} b c d \,f^{3} h -420 a^{2} b \,d^{2} e \,f^{2} h -315 a^{2} b \,d^{2} f^{3} g +210 a \,b^{2} c^{2} f^{3} h +560 a \,b^{2} c d e \,f^{2} h +420 a \,b^{2} c d \,f^{3} g +42 a \,b^{2} d^{2} e^{2} f h +280 a \,b^{2} d^{2} e \,f^{2} g -140 b^{3} c^{2} e \,f^{2} h -105 b^{3} c^{2} f^{3} g -42 b^{3} c d \,e^{2} f h -280 b^{3} c d e \,f^{2} g +6 b^{3} d^{2} e^{3} h -21 b^{3} d^{2} e^{2} f g \right ) \sqrt {f x +e}}{105 f^{2} b^{5}}+\frac {\left (2 a^{2} d f -2 a b c f -2 a b d e +2 c e \,b^{2}\right ) \left (\frac {\left (-\frac {1}{2} a^{2} d f h +\frac {1}{2} a b c f h +\frac {1}{2} a b d f g -\frac {1}{2} b^{2} c f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (9 a^{2} d f h -5 a b c f h -6 a b d e h -7 a b d f g +2 b^{2} c e h +3 b^{2} c f g +4 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{b^{5}}\) | \(585\) |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {d^{2} h \left (f x +e \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 a \,b^{2} d^{2} f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} c d f h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{3} d^{2} e h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} d^{2} f g \left (f x +e \right )^{\frac {5}{2}}}{5}-a^{2} b \,d^{2} f^{2} h \left (f x +e \right )^{\frac {3}{2}}+\frac {4 a \,b^{2} c d \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {2 a \,b^{2} d^{2} f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{3} c^{2} f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {2 b^{3} c d \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}+4 a^{3} d^{2} f^{3} h \sqrt {f x +e}-6 a^{2} b c d \,f^{3} h \sqrt {f x +e}-3 a^{2} b \,d^{2} e \,f^{2} h \sqrt {f x +e}-3 a^{2} b \,d^{2} f^{3} g \sqrt {f x +e}+2 a \,b^{2} c^{2} f^{3} h \sqrt {f x +e}+4 a \,b^{2} c d e \,f^{2} h \sqrt {f x +e}+4 a \,b^{2} c d \,f^{3} g \sqrt {f x +e}+2 a \,b^{2} d^{2} e \,f^{2} g \sqrt {f x +e}-b^{3} c^{2} e \,f^{2} h \sqrt {f x +e}-b^{3} c^{2} f^{3} g \sqrt {f x +e}-2 b^{3} c d e \,f^{2} g \sqrt {f x +e}\right )}{b^{5}}+\frac {2 f^{2} \left (\frac {\left (-\frac {1}{2} a^{4} d^{2} f^{2} h +a^{3} b c d \,f^{2} h +\frac {1}{2} a^{3} b \,d^{2} e f h +\frac {1}{2} a^{3} b \,d^{2} f^{2} g -\frac {1}{2} a^{2} b^{2} c^{2} f^{2} h -a^{2} b^{2} c d e f h -a^{2} b^{2} c d \,f^{2} g -\frac {1}{2} a^{2} b^{2} d^{2} e f g +\frac {1}{2} a \,b^{3} c^{2} e f h +\frac {1}{2} a \,b^{3} c^{2} f^{2} g +a \,b^{3} c d e f g -\frac {1}{2} b^{4} c^{2} e f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (9 a^{4} d^{2} f^{2} h -14 a^{3} b c d \,f^{2} h -15 a^{3} b \,d^{2} e f h -7 a^{3} b \,d^{2} f^{2} g +5 a^{2} b^{2} c^{2} f^{2} h +22 a^{2} b^{2} c d e f h +10 a^{2} b^{2} c d \,f^{2} g +6 a^{2} b^{2} d^{2} e^{2} h +11 a^{2} b^{2} d^{2} e f g -7 a \,b^{3} c^{2} e f h -3 a \,b^{3} c^{2} f^{2} g -8 a \,b^{3} c d \,e^{2} h -14 a \,b^{3} c d e f g -4 a \,b^{3} d^{2} e^{2} g +2 b^{4} c^{2} e^{2} h +3 b^{4} c^{2} e f g +4 b^{4} c d \,e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{b^{5}}}{f^{2}}\) | \(846\) |
| default | \(\frac {-\frac {2 \left (-\frac {d^{2} h \left (f x +e \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 a \,b^{2} d^{2} f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} c d f h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{3} d^{2} e h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} d^{2} f g \left (f x +e \right )^{\frac {5}{2}}}{5}-a^{2} b \,d^{2} f^{2} h \left (f x +e \right )^{\frac {3}{2}}+\frac {4 a \,b^{2} c d \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {2 a \,b^{2} d^{2} f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{3} c^{2} f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {2 b^{3} c d \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}+4 a^{3} d^{2} f^{3} h \sqrt {f x +e}-6 a^{2} b c d \,f^{3} h \sqrt {f x +e}-3 a^{2} b \,d^{2} e \,f^{2} h \sqrt {f x +e}-3 a^{2} b \,d^{2} f^{3} g \sqrt {f x +e}+2 a \,b^{2} c^{2} f^{3} h \sqrt {f x +e}+4 a \,b^{2} c d e \,f^{2} h \sqrt {f x +e}+4 a \,b^{2} c d \,f^{3} g \sqrt {f x +e}+2 a \,b^{2} d^{2} e \,f^{2} g \sqrt {f x +e}-b^{3} c^{2} e \,f^{2} h \sqrt {f x +e}-b^{3} c^{2} f^{3} g \sqrt {f x +e}-2 b^{3} c d e \,f^{2} g \sqrt {f x +e}\right )}{b^{5}}+\frac {2 f^{2} \left (\frac {\left (-\frac {1}{2} a^{4} d^{2} f^{2} h +a^{3} b c d \,f^{2} h +\frac {1}{2} a^{3} b \,d^{2} e f h +\frac {1}{2} a^{3} b \,d^{2} f^{2} g -\frac {1}{2} a^{2} b^{2} c^{2} f^{2} h -a^{2} b^{2} c d e f h -a^{2} b^{2} c d \,f^{2} g -\frac {1}{2} a^{2} b^{2} d^{2} e f g +\frac {1}{2} a \,b^{3} c^{2} e f h +\frac {1}{2} a \,b^{3} c^{2} f^{2} g +a \,b^{3} c d e f g -\frac {1}{2} b^{4} c^{2} e f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (9 a^{4} d^{2} f^{2} h -14 a^{3} b c d \,f^{2} h -15 a^{3} b \,d^{2} e f h -7 a^{3} b \,d^{2} f^{2} g +5 a^{2} b^{2} c^{2} f^{2} h +22 a^{2} b^{2} c d e f h +10 a^{2} b^{2} c d \,f^{2} g +6 a^{2} b^{2} d^{2} e^{2} h +11 a^{2} b^{2} d^{2} e f g -7 a \,b^{3} c^{2} e f h -3 a \,b^{3} c^{2} f^{2} g -8 a \,b^{3} c d \,e^{2} h -14 a \,b^{3} c d e f g -4 a \,b^{3} d^{2} e^{2} g +2 b^{4} c^{2} e^{2} h +3 b^{4} c^{2} e f g +4 b^{4} c d \,e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{b^{5}}}{f^{2}}\) | \(846\) |
Input:
int((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
9/((a*f-b*e)*b)^(1/2)*((a*d-b*c)*(b*x+a)*(1/3*(c*f*g+2/3*e*(c*h+2*d*g))*b^ 2-5/9*a*((c*h+7/5*d*g)*f+6/5*d*e*h)*b+a^2*d*f*h)*f^2*(a*f-b*e)*arctan(b*(f *x+e)^(1/2)/((a*f-b*e)*b)^(1/2))-((a*f-b*e)*b)^(1/2)*(f*x+e)^(1/2)*(1/9*(- 2*x*(1/5*x^2*(5/7*h*x+g)*d^2+2/3*x*c*(3/5*h*x+g)*d+c^2*(1/3*h*x+g))*f^3+e* (-4/5*x^2*(4/7*h*x+g)*d^2-16/3*(3/10*h*x+g)*x*c*d+c^2*(-8/3*h*x+g))*f^2-4/ 5*x*d*(1/2*(1/7*h*x+g)*d+c*h)*e^2*f+4/35*d^2*e^3*h*x)*b^4-11/27*a*(1/11*(- 14/5*x^2*(27/49*h*x+g)*d^2-20*x*c*(7/25*h*x+g)*d+9*c^2*(-10/9*h*x+g))*f^3+ (4/11*(-6/7*h*x^2-17/5*g*x)*d^2+2*c*(-68/55*h*x+g)*d+h*c^2)*e*f^2+12/55*(1 /2*(-13/7*h*x+g)*d+c*h)*d*e^2*f-12/385*d^2*e^3*h)*b^3+5/9*a^2*((-14/15*x*( 9/35*h*x+g)*d^2+2*(-14/15*h*x+g)*c*d+h*c^2)*f^2+38/15*d*e*((-48/95*h*x+1/2 *g)*d+c*h)*f+4/25*d^2*e^2*h)*f*b^2-14/9*a^3*d*(((-3/7*h*x+1/2*g)*d+c*h)*f+ 9/14*d*e*h)*f^2*b+a^4*d^2*f^3*h))/f^2/b^5/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (295) = 590\).
Time = 0.15 (sec) , antiderivative size = 1822, normalized size of antiderivative = 5.68 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^2,x, algorithm="fricas")
Output:
[-1/210*(105*((4*(a*b^3*c*d - a^2*b^2*d^2)*e*f^2 + (3*a*b^3*c^2 - 10*a^2*b ^2*c*d + 7*a^3*b*d^2)*f^3)*g + (2*(a*b^3*c^2 - 4*a^2*b^2*c*d + 3*a^3*b*d^2 )*e*f^2 - (5*a^2*b^2*c^2 - 14*a^3*b*c*d + 9*a^4*d^2)*f^3)*h + ((4*(b^4*c*d - a*b^3*d^2)*e*f^2 + (3*b^4*c^2 - 10*a*b^3*c*d + 7*a^2*b^2*d^2)*f^3)*g + (2*(b^4*c^2 - 4*a*b^3*c*d + 3*a^2*b^2*d^2)*e*f^2 - (5*a*b^3*c^2 - 14*a^2*b ^2*c*d + 9*a^3*b*d^2)*f^3)*h)*x)*sqrt((b*e - a*f)/b)*log((b*f*x + 2*b*e - a*f + 2*sqrt(f*x + e)*b*sqrt((b*e - a*f)/b))/(b*x + a)) - 2*(30*b^4*d^2*f^ 3*h*x^4 + 6*(7*b^4*d^2*f^3*g + (8*b^4*d^2*e*f^2 + (14*b^4*c*d - 9*a*b^3*d^ 2)*f^3)*h)*x^3 + 2*(7*(6*b^4*d^2*e*f^2 + (10*b^4*c*d - 7*a*b^3*d^2)*f^3)*g + (3*b^4*d^2*e^2*f + 12*(7*b^4*c*d - 5*a*b^3*d^2)*e*f^2 + 7*(5*b^4*c^2 - 14*a*b^3*c*d + 9*a^2*b^2*d^2)*f^3)*h)*x^2 + 7*(6*a*b^3*d^2*e^2*f - 5*(3*b^ 4*c^2 - 22*a*b^3*c*d + 19*a^2*b^2*d^2)*e*f^2 + 15*(3*a*b^3*c^2 - 10*a^2*b^ 2*c*d + 7*a^3*b*d^2)*f^3)*g - (12*a*b^3*d^2*e^3 - 84*(a*b^3*c*d - a^2*b^2* d^2)*e^2*f - 35*(11*a*b^3*c^2 - 38*a^2*b^2*c*d + 27*a^3*b*d^2)*e*f^2 + 105 *(5*a^2*b^2*c^2 - 14*a^3*b*c*d + 9*a^4*d^2)*f^3)*h + 2*(7*(3*b^4*d^2*e^2*f + 2*(20*b^4*c*d - 17*a*b^3*d^2)*e*f^2 + 5*(3*b^4*c^2 - 10*a*b^3*c*d + 7*a ^2*b^2*d^2)*f^3)*g - (6*b^4*d^2*e^3 - 3*(14*b^4*c*d - 13*a*b^3*d^2)*e^2*f - 28*(5*b^4*c^2 - 17*a*b^3*c*d + 12*a^2*b^2*d^2)*e*f^2 + 35*(5*a*b^3*c^2 - 14*a^2*b^2*c*d + 9*a^3*b*d^2)*f^3)*h)*x)*sqrt(f*x + e))/(b^6*f^2*x + a*b^ 5*f^2), -1/105*(105*((4*(a*b^3*c*d - a^2*b^2*d^2)*e*f^2 + (3*a*b^3*c^2 ...
Timed out. \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**2*(f*x+e)**(3/2)*(h*x+g)/(b*x+a)**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^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. 941 vs. \(2 (295) = 590\).
Time = 0.15 (sec) , antiderivative size = 941, normalized size of antiderivative = 2.93 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^2,x, algorithm="giac")
Output:
(4*b^4*c*d*e^2*g - 4*a*b^3*d^2*e^2*g + 3*b^4*c^2*e*f*g - 14*a*b^3*c*d*e*f* g + 11*a^2*b^2*d^2*e*f*g - 3*a*b^3*c^2*f^2*g + 10*a^2*b^2*c*d*f^2*g - 7*a^ 3*b*d^2*f^2*g + 2*b^4*c^2*e^2*h - 8*a*b^3*c*d*e^2*h + 6*a^2*b^2*d^2*e^2*h - 7*a*b^3*c^2*e*f*h + 22*a^2*b^2*c*d*e*f*h - 15*a^3*b*d^2*e*f*h + 5*a^2*b^ 2*c^2*f^2*h - 14*a^3*b*c*d*f^2*h + 9*a^4*d^2*f^2*h)*arctan(sqrt(f*x + e)*b /sqrt(-b^2*e + a*b*f))/(sqrt(-b^2*e + a*b*f)*b^5) - (sqrt(f*x + e)*b^4*c^2 *e*f*g - 2*sqrt(f*x + e)*a*b^3*c*d*e*f*g + sqrt(f*x + e)*a^2*b^2*d^2*e*f*g - sqrt(f*x + e)*a*b^3*c^2*f^2*g + 2*sqrt(f*x + e)*a^2*b^2*c*d*f^2*g - sqr t(f*x + e)*a^3*b*d^2*f^2*g - sqrt(f*x + e)*a*b^3*c^2*e*f*h + 2*sqrt(f*x + e)*a^2*b^2*c*d*e*f*h - sqrt(f*x + e)*a^3*b*d^2*e*f*h + sqrt(f*x + e)*a^2*b ^2*c^2*f^2*h - 2*sqrt(f*x + e)*a^3*b*c*d*f^2*h + sqrt(f*x + e)*a^4*d^2*f^2 *h)/(((f*x + e)*b - b*e + a*f)*b^5) + 2/105*(21*(f*x + e)^(5/2)*b^12*d^2*f ^13*g + 70*(f*x + e)^(3/2)*b^12*c*d*f^14*g - 70*(f*x + e)^(3/2)*a*b^11*d^2 *f^14*g + 210*sqrt(f*x + e)*b^12*c*d*e*f^14*g - 210*sqrt(f*x + e)*a*b^11*d ^2*e*f^14*g + 105*sqrt(f*x + e)*b^12*c^2*f^15*g - 420*sqrt(f*x + e)*a*b^11 *c*d*f^15*g + 315*sqrt(f*x + e)*a^2*b^10*d^2*f^15*g + 15*(f*x + e)^(7/2)*b ^12*d^2*f^12*h - 21*(f*x + e)^(5/2)*b^12*d^2*e*f^12*h + 42*(f*x + e)^(5/2) *b^12*c*d*f^13*h - 42*(f*x + e)^(5/2)*a*b^11*d^2*f^13*h + 35*(f*x + e)^(3/ 2)*b^12*c^2*f^14*h - 140*(f*x + e)^(3/2)*a*b^11*c*d*f^14*h + 105*(f*x + e) ^(3/2)*a^2*b^10*d^2*f^14*h + 105*sqrt(f*x + e)*b^12*c^2*e*f^14*h - 420*...
Time = 2.64 (sec) , antiderivative size = 715, normalized size of antiderivative = 2.23 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx={\left (e+f\,x\right )}^{5/2}\,\left (\frac {2\,d^2\,f\,g-6\,d^2\,e\,h+4\,c\,d\,f\,h}{5\,b^2\,f^2}-\frac {4\,d^2\,h\,\left (a\,f-b\,e\right )}{5\,b^3\,f^2}\right )-{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,\left (\frac {2\,d^2\,f\,g-6\,d^2\,e\,h+4\,c\,d\,f\,h}{b^2\,f^2}-\frac {4\,d^2\,h\,\left (a\,f-b\,e\right )}{b^3\,f^2}\right )\,\left (a\,f-b\,e\right )}{3\,b}-\frac {2\,\left (c\,f-d\,e\right )\,\left (c\,f\,h-3\,d\,e\,h+2\,d\,f\,g\right )}{3\,b^2\,f^2}+\frac {2\,d^2\,h\,{\left (a\,f-b\,e\right )}^2}{3\,b^4\,f^2}\right )-\sqrt {e+f\,x}\,\left (\frac {\left (\frac {2\,d^2\,f\,g-6\,d^2\,e\,h+4\,c\,d\,f\,h}{b^2\,f^2}-\frac {4\,d^2\,h\,\left (a\,f-b\,e\right )}{b^3\,f^2}\right )\,{\left (a\,f-b\,e\right )}^2}{b^2}-\frac {2\,\left (a\,f-b\,e\right )\,\left (\frac {2\,\left (\frac {2\,d^2\,f\,g-6\,d^2\,e\,h+4\,c\,d\,f\,h}{b^2\,f^2}-\frac {4\,d^2\,h\,\left (a\,f-b\,e\right )}{b^3\,f^2}\right )\,\left (a\,f-b\,e\right )}{b}-\frac {2\,\left (c\,f-d\,e\right )\,\left (c\,f\,h-3\,d\,e\,h+2\,d\,f\,g\right )}{b^2\,f^2}+\frac {2\,d^2\,h\,{\left (a\,f-b\,e\right )}^2}{b^4\,f^2}\right )}{b}+\frac {2\,{\left (c\,f-d\,e\right )}^2\,\left (e\,h-f\,g\right )}{b^2\,f^2}\right )-\frac {\sqrt {e+f\,x}\,\left (h\,a^4\,d^2\,f^2-2\,h\,a^3\,b\,c\,d\,f^2-g\,a^3\,b\,d^2\,f^2-e\,h\,a^3\,b\,d^2\,f+h\,a^2\,b^2\,c^2\,f^2+2\,g\,a^2\,b^2\,c\,d\,f^2+2\,e\,h\,a^2\,b^2\,c\,d\,f+e\,g\,a^2\,b^2\,d^2\,f-g\,a\,b^3\,c^2\,f^2-e\,h\,a\,b^3\,c^2\,f-2\,e\,g\,a\,b^3\,c\,d\,f+e\,g\,b^4\,c^2\,f\right )}{b^6\,\left (e+f\,x\right )-b^6\,e+a\,b^5\,f}+\frac {2\,d^2\,h\,{\left (e+f\,x\right )}^{7/2}}{7\,b^2\,f^2}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e+f\,x}\,1{}\mathrm {i}}{\sqrt {b\,e-a\,f}}\right )\,\left (a\,d-b\,c\right )\,\sqrt {b\,e-a\,f}\,\left (2\,b^2\,c\,e\,h+3\,b^2\,c\,f\,g+4\,b^2\,d\,e\,g+9\,a^2\,d\,f\,h-5\,a\,b\,c\,f\,h-6\,a\,b\,d\,e\,h-7\,a\,b\,d\,f\,g\right )\,1{}\mathrm {i}}{b^{11/2}} \] Input:
int(((e + f*x)^(3/2)*(g + h*x)*(c + d*x)^2)/(a + b*x)^2,x)
Output:
(e + f*x)^(5/2)*((2*d^2*f*g - 6*d^2*e*h + 4*c*d*f*h)/(5*b^2*f^2) - (4*d^2* h*(a*f - b*e))/(5*b^3*f^2)) - (e + f*x)^(3/2)*((2*((2*d^2*f*g - 6*d^2*e*h + 4*c*d*f*h)/(b^2*f^2) - (4*d^2*h*(a*f - b*e))/(b^3*f^2))*(a*f - b*e))/(3* b) - (2*(c*f - d*e)*(c*f*h - 3*d*e*h + 2*d*f*g))/(3*b^2*f^2) + (2*d^2*h*(a *f - b*e)^2)/(3*b^4*f^2)) - (e + f*x)^(1/2)*((((2*d^2*f*g - 6*d^2*e*h + 4* c*d*f*h)/(b^2*f^2) - (4*d^2*h*(a*f - b*e))/(b^3*f^2))*(a*f - b*e)^2)/b^2 - (2*(a*f - b*e)*((2*((2*d^2*f*g - 6*d^2*e*h + 4*c*d*f*h)/(b^2*f^2) - (4*d^ 2*h*(a*f - b*e))/(b^3*f^2))*(a*f - b*e))/b - (2*(c*f - d*e)*(c*f*h - 3*d*e *h + 2*d*f*g))/(b^2*f^2) + (2*d^2*h*(a*f - b*e)^2)/(b^4*f^2)))/b + (2*(c*f - d*e)^2*(e*h - f*g))/(b^2*f^2)) - ((e + f*x)^(1/2)*(a^4*d^2*f^2*h + b^4* c^2*e*f*g - a*b^3*c^2*f^2*g - a^3*b*d^2*f^2*g + a^2*b^2*c^2*f^2*h - 2*a^3* b*c*d*f^2*h - a*b^3*c^2*e*f*h - a^3*b*d^2*e*f*h + 2*a^2*b^2*c*d*f^2*g + a^ 2*b^2*d^2*e*f*g - 2*a*b^3*c*d*e*f*g + 2*a^2*b^2*c*d*e*f*h))/(b^6*(e + f*x) - b^6*e + a*b^5*f) - (atan((b^(1/2)*(e + f*x)^(1/2)*1i)/(b*e - a*f)^(1/2) )*(a*d - b*c)*(b*e - a*f)^(1/2)*(2*b^2*c*e*h + 3*b^2*c*f*g + 4*b^2*d*e*g + 9*a^2*d*f*h - 5*a*b*c*f*h - 6*a*b*d*e*h - 7*a*b*d*f*g)*1i)/b^(11/2) + (2* d^2*h*(e + f*x)^(7/2))/(7*b^2*f^2)
Time = 0.17 (sec) , antiderivative size = 2025, normalized size of antiderivative = 6.31 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^2,x)
Output:
(945*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b* e)))*a**4*d**2*f**3*h - 1470*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b )/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*c*d*f**3*h - 630*sqrt(b)*sqrt(a*f - b* e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*d**2*e*f**2*h - 735*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b *e)))*a**3*b*d**2*f**3*g + 945*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x) *b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*d**2*f**3*h*x + 525*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c**2*f **3*h + 840*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a *f - b*e)))*a**2*b**2*c*d*e*f**2*h + 1050*sqrt(b)*sqrt(a*f - b*e)*atan((sq rt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c*d*f**3*g - 1470*sqrt (b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2 *b**2*c*d*f**3*h*x + 420*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(s qrt(b)*sqrt(a*f - b*e)))*a**2*b**2*d**2*e*f**2*g - 630*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*d**2*e*f* *2*h*x - 735*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt( a*f - b*e)))*a**2*b**2*d**2*f**3*g*x - 210*sqrt(b)*sqrt(a*f - b*e)*atan((s qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**3*c**2*e*f**2*h - 315*sqrt (b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b* *3*c**2*f**3*g + 525*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sq...