Integrand size = 27, antiderivative size = 244 \[ \int \frac {(a+b x) (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 (b e-a f) (d e-c f)^2 (f g-e h)}{f^5 \sqrt {e+f x}}+\frac {2 (d e-c f) (b d e (3 f g-4 e h)-b c f (f g-2 e h)-a f (2 d f g-3 d e h+c f h)) \sqrt {e+f x}}{f^5}+\frac {2 \left (a d f (d f g-3 d e h+2 c f h)+b \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )\right ) (e+f x)^{3/2}}{3 f^5}+\frac {2 d (a d f h+b (d f g-4 d e h+2 c f h)) (e+f x)^{5/2}}{5 f^5}+\frac {2 b d^2 h (e+f x)^{7/2}}{7 f^5} \] Output:
2*(-a*f+b*e)*(-c*f+d*e)^2*(-e*h+f*g)/f^5/(f*x+e)^(1/2)+2*(-c*f+d*e)*(b*d*e *(-4*e*h+3*f*g)-b*c*f*(-2*e*h+f*g)-a*f*(c*f*h-3*d*e*h+2*d*f*g))*(f*x+e)^(1 /2)/f^5+2/3*(a*d*f*(2*c*f*h-3*d*e*h+d*f*g)+b*(c^2*f^2*h+2*c*d*f*(-3*e*h+f* g)-3*d^2*e*(-2*e*h+f*g)))*(f*x+e)^(3/2)/f^5+2/5*d*(a*d*f*h+b*(2*c*f*h-4*d* e*h+d*f*g))*(f*x+e)^(5/2)/f^5+2/7*b*d^2*h*(f*x+e)^(7/2)/f^5
Time = 0.24 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b x) (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 \left (7 a f \left (15 c^2 f^2 (-f g+2 e h+f h x)+10 c d f \left (-8 e^2 h+e f (6 g-4 h x)+f^2 x (3 g+h x)\right )+d^2 \left (48 e^3 h-8 e^2 f (5 g-3 h x)+f^3 x^2 (5 g+3 h x)-2 e f^2 x (10 g+3 h x)\right )\right )+b \left (35 c^2 f^2 \left (-8 e^2 h+e f (6 g-4 h x)+f^2 x (3 g+h x)\right )+14 c d f \left (48 e^3 h-8 e^2 f (5 g-3 h x)+f^3 x^2 (5 g+3 h x)-2 e f^2 x (10 g+3 h x)\right )-3 d^2 \left (128 e^4 h-16 e^3 f (7 g-4 h x)-8 e^2 f^2 x (7 g+2 h x)+2 e f^3 x^2 (7 g+4 h x)-f^4 x^3 (7 g+5 h x)\right )\right )\right )}{105 f^5 \sqrt {e+f x}} \] Input:
Integrate[((a + b*x)*(c + d*x)^2*(g + h*x))/(e + f*x)^(3/2),x]
Output:
(2*(7*a*f*(15*c^2*f^2*(-(f*g) + 2*e*h + f*h*x) + 10*c*d*f*(-8*e^2*h + e*f* (6*g - 4*h*x) + f^2*x*(3*g + h*x)) + d^2*(48*e^3*h - 8*e^2*f*(5*g - 3*h*x) + f^3*x^2*(5*g + 3*h*x) - 2*e*f^2*x*(10*g + 3*h*x))) + b*(35*c^2*f^2*(-8* e^2*h + e*f*(6*g - 4*h*x) + f^2*x*(3*g + h*x)) + 14*c*d*f*(48*e^3*h - 8*e^ 2*f*(5*g - 3*h*x) + f^3*x^2*(5*g + 3*h*x) - 2*e*f^2*x*(10*g + 3*h*x)) - 3* d^2*(128*e^4*h - 16*e^3*f*(7*g - 4*h*x) - 8*e^2*f^2*x*(7*g + 2*h*x) + 2*e* f^3*x^2*(7*g + 4*h*x) - f^4*x^3*(7*g + 5*h*x)))))/(105*f^5*Sqrt[e + f*x])
Time = 0.46 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {159, 2009}
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.
| \(\displaystyle \int \frac {(a+b x) (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 159 |
| \(\displaystyle \int \left (\frac {\sqrt {e+f x} \left (a d f (2 c f h-3 d e h+d f g)+b \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )\right )}{f^4}+\frac {d (e+f x)^{3/2} (a d f h+b (2 c f h-4 d e h+d f g))}{f^4}+\frac {(d e-c f) (-a f (c f h-3 d e h+2 d f g)-b c f (f g-2 e h)+b d e (3 f g-4 e h))}{f^4 \sqrt {e+f x}}+\frac {(a f-b e) (c f-d e)^2 (f g-e h)}{f^4 (e+f x)^{3/2}}+\frac {b d^2 h (e+f x)^{5/2}}{f^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (e+f x)^{3/2} \left (a d f (2 c f h-3 d e h+d f g)+b \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )\right )}{3 f^5}+\frac {2 d (e+f x)^{5/2} (a d f h+b (2 c f h-4 d e h+d f g))}{5 f^5}+\frac {2 \sqrt {e+f x} (d e-c f) (-a f (c f h-3 d e h+2 d f g)-b c f (f g-2 e h)+b d e (3 f g-4 e h))}{f^5}+\frac {2 (b e-a f) (d e-c f)^2 (f g-e h)}{f^5 \sqrt {e+f x}}+\frac {2 b d^2 h (e+f x)^{7/2}}{7 f^5}\) |
Input:
Int[((a + b*x)*(c + d*x)^2*(g + h*x))/(e + f*x)^(3/2),x]
Output:
(2*(b*e - a*f)*(d*e - c*f)^2*(f*g - e*h))/(f^5*Sqrt[e + f*x]) + (2*(d*e - c*f)*(b*d*e*(3*f*g - 4*e*h) - b*c*f*(f*g - 2*e*h) - a*f*(2*d*f*g - 3*d*e*h + c*f*h))*Sqrt[e + f*x])/f^5 + (2*(a*d*f*(d*f*g - 3*d*e*h + 2*c*f*h) + b* (c^2*f^2*h + 2*c*d*f*(f*g - 3*e*h) - 3*d^2*e*(f*g - 2*e*h)))*(e + f*x)^(3/ 2))/(3*f^5) + (2*d*(a*d*f*h + b*(d*f*g - 4*d*e*h + 2*c*f*h))*(e + f*x)^(5/ 2))/(5*f^5) + (2*b*d^2*h*(e + f*x)^(7/2))/(7*f^5)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n *(e + f*x)*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && (IGtQ [m, 0] || IntegersQ[m, n])
Time = 0.36 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.13
| method | result | size |
| pseudoelliptic | \(\frac {4 \left (\frac {x^{2} \left (\frac {3 b h \,x^{2}}{7}+\frac {3 \left (a h +b g \right ) x}{5}+g a \right ) d^{2}}{6}+x c \left (\frac {b h \,x^{2}}{5}+\frac {\left (a h +b g \right ) x}{3}+g a \right ) d -\frac {c^{2} \left (-\frac {b h \,x^{2}}{3}+\left (-a h -b g \right ) x +g a \right )}{2}\right ) f^{4}+4 \left (\left (-\frac {4 b h \,x^{3}}{35}+\frac {\left (-a h -b g \right ) x^{2}}{5}-\frac {2 a g x}{3}\right ) d^{2}+2 c \left (-\frac {b h \,x^{2}}{5}+\frac {2 \left (-a h -b g \right ) x}{3}+g a \right ) d +c^{2} \left (a h +b g -\frac {2}{3} b h x \right )\right ) e \,f^{3}-\frac {32 \left (\left (-\frac {3 b h \,x^{2}}{35}+\frac {3 \left (-a h -b g \right ) x}{10}+\frac {g a}{2}\right ) d^{2}+c \left (-\frac {3}{5} b h x +a h +b g \right ) d +\frac {b \,c^{2} h}{2}\right ) e^{2} f^{2}}{3}+\frac {32 d \left (\left (-\frac {4}{7} b h x +a h +b g \right ) d +2 b c h \right ) e^{3} f}{5}-\frac {256 b \,d^{2} e^{4} h}{35}}{\sqrt {f x +e}\, f^{5}}\) | \(276\) |
| risch | \(\frac {2 \left (15 h b \,d^{2} x^{3} f^{3}+21 a \,d^{2} f^{3} h \,x^{2}+42 b c d \,f^{3} h \,x^{2}-39 b \,d^{2} e \,f^{2} h \,x^{2}+21 b \,d^{2} f^{3} g \,x^{2}+70 a c d \,f^{3} h x -63 a \,d^{2} e \,f^{2} h x +35 a \,d^{2} f^{3} g x +35 b \,c^{2} f^{3} h x -126 b c d e \,f^{2} h x +70 b c d \,f^{3} g x +87 b \,d^{2} e^{2} f h x -63 b \,d^{2} e \,f^{2} g x +105 a \,c^{2} h \,f^{3}-350 a c d e \,f^{2} h +210 g a c d \,f^{3}+231 a \,d^{2} e^{2} f h -175 a \,d^{2} e \,f^{2} g -175 b \,c^{2} e \,f^{2} h +105 b \,c^{2} g \,f^{3}+462 b c d \,e^{2} f h -350 b c d e \,f^{2} g -279 b \,d^{2} e^{3} h +231 b \,d^{2} e^{2} f g \right ) \sqrt {f x +e}}{105 f^{5}}+\frac {2 a \,c^{2} e \,f^{3} h -2 g a \,c^{2} f^{4}-4 a c d \,e^{2} f^{2} h +4 a c d e \,f^{3} g +2 a \,d^{2} e^{3} f h -2 a \,d^{2} e^{2} f^{2} g -2 b \,c^{2} e^{2} f^{2} h +2 b \,c^{2} e \,f^{3} g +4 b c d \,e^{3} f h -4 b c d \,e^{2} f^{2} g -2 b \,d^{2} e^{4} h +2 b \,d^{2} e^{3} f g}{\sqrt {f x +e}\, f^{5}}\) | \(425\) |
| gosper | \(\frac {\frac {2}{5} a \,d^{2} f^{4} h \,x^{3}-\frac {8}{5} b c d e \,f^{3} h \,x^{2}-\frac {16}{3} a c d e \,f^{3} h x +\frac {32}{5} b c d \,e^{2} f^{2} h x -\frac {16}{3} b c d e \,f^{3} g x -\frac {32}{3} b c d \,e^{2} f^{2} g +\frac {32}{5} b \,d^{2} e^{3} f g +\frac {32}{5} a \,d^{2} e^{3} f h -\frac {16}{3} a \,d^{2} e^{2} f^{2} g -\frac {16}{3} b \,c^{2} e^{2} f^{2} h +4 b \,c^{2} e \,f^{3} g +4 a \,c^{2} e \,f^{3} h -\frac {32}{3} a c d \,e^{2} f^{2} h +8 a c d e \,f^{3} g -2 g a \,c^{2} f^{4}+2 b \,c^{2} f^{4} g x +\frac {64}{5} b c d \,e^{3} f h -\frac {4}{5} a \,d^{2} e \,f^{3} h \,x^{2}-\frac {256}{35} b \,d^{2} e^{4} h -\frac {8}{3} a \,d^{2} e \,f^{3} g x -\frac {8}{3} b \,c^{2} e \,f^{3} h x -\frac {128}{35} b \,d^{2} e^{3} f h x +\frac {16}{5} b \,d^{2} e^{2} f^{2} g x +\frac {2}{7} h b \,d^{2} x^{4} f^{4}+\frac {4}{5} b c d \,f^{4} h \,x^{3}-\frac {16}{35} b \,d^{2} e \,f^{3} h \,x^{3}+\frac {4}{3} a c d \,f^{4} h \,x^{2}+\frac {4}{3} b c d \,f^{4} g \,x^{2}+\frac {32}{35} b \,d^{2} e^{2} f^{2} h \,x^{2}-\frac {4}{5} b \,d^{2} e \,f^{3} g \,x^{2}+4 a c d \,f^{4} g x +\frac {16}{5} a \,d^{2} e^{2} f^{2} h x +\frac {2}{5} b \,d^{2} f^{4} g \,x^{3}+\frac {2}{3} a \,d^{2} f^{4} g \,x^{2}+\frac {2}{3} b \,c^{2} f^{4} h \,x^{2}+2 a \,c^{2} f^{4} h x}{\sqrt {f x +e}\, f^{5}}\) | \(451\) |
| trager | \(\frac {\frac {2}{5} a \,d^{2} f^{4} h \,x^{3}-\frac {8}{5} b c d e \,f^{3} h \,x^{2}-\frac {16}{3} a c d e \,f^{3} h x +\frac {32}{5} b c d \,e^{2} f^{2} h x -\frac {16}{3} b c d e \,f^{3} g x -\frac {32}{3} b c d \,e^{2} f^{2} g +\frac {32}{5} b \,d^{2} e^{3} f g +\frac {32}{5} a \,d^{2} e^{3} f h -\frac {16}{3} a \,d^{2} e^{2} f^{2} g -\frac {16}{3} b \,c^{2} e^{2} f^{2} h +4 b \,c^{2} e \,f^{3} g +4 a \,c^{2} e \,f^{3} h -\frac {32}{3} a c d \,e^{2} f^{2} h +8 a c d e \,f^{3} g -2 g a \,c^{2} f^{4}+2 b \,c^{2} f^{4} g x +\frac {64}{5} b c d \,e^{3} f h -\frac {4}{5} a \,d^{2} e \,f^{3} h \,x^{2}-\frac {256}{35} b \,d^{2} e^{4} h -\frac {8}{3} a \,d^{2} e \,f^{3} g x -\frac {8}{3} b \,c^{2} e \,f^{3} h x -\frac {128}{35} b \,d^{2} e^{3} f h x +\frac {16}{5} b \,d^{2} e^{2} f^{2} g x +\frac {2}{7} h b \,d^{2} x^{4} f^{4}+\frac {4}{5} b c d \,f^{4} h \,x^{3}-\frac {16}{35} b \,d^{2} e \,f^{3} h \,x^{3}+\frac {4}{3} a c d \,f^{4} h \,x^{2}+\frac {4}{3} b c d \,f^{4} g \,x^{2}+\frac {32}{35} b \,d^{2} e^{2} f^{2} h \,x^{2}-\frac {4}{5} b \,d^{2} e \,f^{3} g \,x^{2}+4 a c d \,f^{4} g x +\frac {16}{5} a \,d^{2} e^{2} f^{2} h x +\frac {2}{5} b \,d^{2} f^{4} g \,x^{3}+\frac {2}{3} a \,d^{2} f^{4} g \,x^{2}+\frac {2}{3} b \,c^{2} f^{4} h \,x^{2}+2 a \,c^{2} f^{4} h x}{\sqrt {f x +e}\, f^{5}}\) | \(451\) |
| orering | \(\frac {\frac {2}{5} a \,d^{2} f^{4} h \,x^{3}-\frac {8}{5} b c d e \,f^{3} h \,x^{2}-\frac {16}{3} a c d e \,f^{3} h x +\frac {32}{5} b c d \,e^{2} f^{2} h x -\frac {16}{3} b c d e \,f^{3} g x -\frac {32}{3} b c d \,e^{2} f^{2} g +\frac {32}{5} b \,d^{2} e^{3} f g +\frac {32}{5} a \,d^{2} e^{3} f h -\frac {16}{3} a \,d^{2} e^{2} f^{2} g -\frac {16}{3} b \,c^{2} e^{2} f^{2} h +4 b \,c^{2} e \,f^{3} g +4 a \,c^{2} e \,f^{3} h -\frac {32}{3} a c d \,e^{2} f^{2} h +8 a c d e \,f^{3} g -2 g a \,c^{2} f^{4}+2 b \,c^{2} f^{4} g x +\frac {64}{5} b c d \,e^{3} f h -\frac {4}{5} a \,d^{2} e \,f^{3} h \,x^{2}-\frac {256}{35} b \,d^{2} e^{4} h -\frac {8}{3} a \,d^{2} e \,f^{3} g x -\frac {8}{3} b \,c^{2} e \,f^{3} h x -\frac {128}{35} b \,d^{2} e^{3} f h x +\frac {16}{5} b \,d^{2} e^{2} f^{2} g x +\frac {2}{7} h b \,d^{2} x^{4} f^{4}+\frac {4}{5} b c d \,f^{4} h \,x^{3}-\frac {16}{35} b \,d^{2} e \,f^{3} h \,x^{3}+\frac {4}{3} a c d \,f^{4} h \,x^{2}+\frac {4}{3} b c d \,f^{4} g \,x^{2}+\frac {32}{35} b \,d^{2} e^{2} f^{2} h \,x^{2}-\frac {4}{5} b \,d^{2} e \,f^{3} g \,x^{2}+4 a c d \,f^{4} g x +\frac {16}{5} a \,d^{2} e^{2} f^{2} h x +\frac {2}{5} b \,d^{2} f^{4} g \,x^{3}+\frac {2}{3} a \,d^{2} f^{4} g \,x^{2}+\frac {2}{3} b \,c^{2} f^{4} h \,x^{2}+2 a \,c^{2} f^{4} h x}{\sqrt {f x +e}\, f^{5}}\) | \(451\) |
| derivativedivides | \(\frac {\frac {4 b c d f h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {4 a c d \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {2 h b \,d^{2} \left (f x +e \right )^{\frac {7}{2}}}{7}-2 b \,d^{2} e f g \left (f x +e \right )^{\frac {3}{2}}-4 a \,d^{2} e \,f^{2} g \sqrt {f x +e}-4 b c d e f h \left (f x +e \right )^{\frac {3}{2}}+2 b \,c^{2} f^{3} g \sqrt {f x +e}-8 b \,d^{2} e^{3} h \sqrt {f x +e}+2 a \,c^{2} f^{3} h \sqrt {f x +e}+\frac {2 a \,d^{2} f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {8 b \,d^{2} e h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {2 b \,d^{2} f g \left (f x +e \right )^{\frac {5}{2}}}{5}+4 a c d \,f^{3} g \sqrt {f x +e}+6 a \,d^{2} e^{2} f h \sqrt {f x +e}-2 a \,d^{2} e f h \left (f x +e \right )^{\frac {3}{2}}+\frac {2 a \,d^{2} f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {2 b \,c^{2} f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+4 b \,d^{2} e^{2} h \left (f x +e \right )^{\frac {3}{2}}-4 b \,c^{2} e \,f^{2} h \sqrt {f x +e}+6 b \,d^{2} e^{2} f g \sqrt {f x +e}+\frac {4 b c d \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}-8 a c d e \,f^{2} h \sqrt {f x +e}+12 b c d \,e^{2} f h \sqrt {f x +e}-8 b c d e \,f^{2} g \sqrt {f x +e}-\frac {2 \left (-a \,c^{2} e \,f^{3} h +g a \,c^{2} f^{4}+2 a c d \,e^{2} f^{2} h -2 a c d e \,f^{3} g -a \,d^{2} e^{3} f h +a \,d^{2} e^{2} f^{2} g +b \,c^{2} e^{2} f^{2} h -b \,c^{2} e \,f^{3} g -2 b c d \,e^{3} f h +2 b c d \,e^{2} f^{2} g +b \,d^{2} e^{4} h -b \,d^{2} e^{3} f g \right )}{\sqrt {f x +e}}}{f^{5}}\) | \(538\) |
| default | \(\frac {\frac {4 b c d f h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {4 a c d \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {2 h b \,d^{2} \left (f x +e \right )^{\frac {7}{2}}}{7}-2 b \,d^{2} e f g \left (f x +e \right )^{\frac {3}{2}}-4 a \,d^{2} e \,f^{2} g \sqrt {f x +e}-4 b c d e f h \left (f x +e \right )^{\frac {3}{2}}+2 b \,c^{2} f^{3} g \sqrt {f x +e}-8 b \,d^{2} e^{3} h \sqrt {f x +e}+2 a \,c^{2} f^{3} h \sqrt {f x +e}+\frac {2 a \,d^{2} f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {8 b \,d^{2} e h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {2 b \,d^{2} f g \left (f x +e \right )^{\frac {5}{2}}}{5}+4 a c d \,f^{3} g \sqrt {f x +e}+6 a \,d^{2} e^{2} f h \sqrt {f x +e}-2 a \,d^{2} e f h \left (f x +e \right )^{\frac {3}{2}}+\frac {2 a \,d^{2} f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {2 b \,c^{2} f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+4 b \,d^{2} e^{2} h \left (f x +e \right )^{\frac {3}{2}}-4 b \,c^{2} e \,f^{2} h \sqrt {f x +e}+6 b \,d^{2} e^{2} f g \sqrt {f x +e}+\frac {4 b c d \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}-8 a c d e \,f^{2} h \sqrt {f x +e}+12 b c d \,e^{2} f h \sqrt {f x +e}-8 b c d e \,f^{2} g \sqrt {f x +e}-\frac {2 \left (-a \,c^{2} e \,f^{3} h +g a \,c^{2} f^{4}+2 a c d \,e^{2} f^{2} h -2 a c d e \,f^{3} g -a \,d^{2} e^{3} f h +a \,d^{2} e^{2} f^{2} g +b \,c^{2} e^{2} f^{2} h -b \,c^{2} e \,f^{3} g -2 b c d \,e^{3} f h +2 b c d \,e^{2} f^{2} g +b \,d^{2} e^{4} h -b \,d^{2} e^{3} f g \right )}{\sqrt {f x +e}}}{f^{5}}\) | \(538\) |
Input:
int((b*x+a)*(d*x+c)^2*(h*x+g)/(f*x+e)^(3/2),x,method=_RETURNVERBOSE)
Output:
4/(f*x+e)^(1/2)*((1/6*x^2*(3/7*b*h*x^2+3/5*(a*h+b*g)*x+g*a)*d^2+x*c*(1/5*b *h*x^2+1/3*(a*h+b*g)*x+g*a)*d-1/2*c^2*(-1/3*b*h*x^2+(-a*h-b*g)*x+g*a))*f^4 +((-4/35*b*h*x^3+1/5*(-a*h-b*g)*x^2-2/3*a*g*x)*d^2+2*c*(-1/5*b*h*x^2+2/3*( -a*h-b*g)*x+g*a)*d+c^2*(a*h+b*g-2/3*b*h*x))*e*f^3-8/3*((-3/35*b*h*x^2+3/10 *(-a*h-b*g)*x+1/2*g*a)*d^2+c*(-3/5*b*h*x+a*h+b*g)*d+1/2*b*c^2*h)*e^2*f^2+8 /5*d*((-4/7*b*h*x+a*h+b*g)*d+2*b*c*h)*e^3*f-64/35*b*d^2*e^4*h)/f^5
Time = 0.09 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.62 \[ \int \frac {(a+b x) (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 \, {\left (15 \, b d^{2} f^{4} h x^{4} + 3 \, {\left (7 \, b d^{2} f^{4} g - {\left (8 \, b d^{2} e f^{3} - 7 \, {\left (2 \, b c d + a d^{2}\right )} f^{4}\right )} h\right )} x^{3} - {\left (7 \, {\left (6 \, b d^{2} e f^{3} - 5 \, {\left (2 \, b c d + a d^{2}\right )} f^{4}\right )} g - {\left (48 \, b d^{2} e^{2} f^{2} - 42 \, {\left (2 \, b c d + a d^{2}\right )} e f^{3} + 35 \, {\left (b c^{2} + 2 \, a c d\right )} f^{4}\right )} h\right )} x^{2} + 7 \, {\left (48 \, b d^{2} e^{3} f - 15 \, a c^{2} f^{4} - 40 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{2} + 30 \, {\left (b c^{2} + 2 \, a c d\right )} e f^{3}\right )} g - 2 \, {\left (192 \, b d^{2} e^{4} - 105 \, a c^{2} e f^{3} - 168 \, {\left (2 \, b c d + a d^{2}\right )} e^{3} f + 140 \, {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{2}\right )} h + {\left (7 \, {\left (24 \, b d^{2} e^{2} f^{2} - 20 \, {\left (2 \, b c d + a d^{2}\right )} e f^{3} + 15 \, {\left (b c^{2} + 2 \, a c d\right )} f^{4}\right )} g - {\left (192 \, b d^{2} e^{3} f - 105 \, a c^{2} f^{4} - 168 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{2} + 140 \, {\left (b c^{2} + 2 \, a c d\right )} e f^{3}\right )} h\right )} x\right )} \sqrt {f x + e}}{105 \, {\left (f^{6} x + e f^{5}\right )}} \] Input:
integrate((b*x+a)*(d*x+c)^2*(h*x+g)/(f*x+e)^(3/2),x, algorithm="fricas")
Output:
2/105*(15*b*d^2*f^4*h*x^4 + 3*(7*b*d^2*f^4*g - (8*b*d^2*e*f^3 - 7*(2*b*c*d + a*d^2)*f^4)*h)*x^3 - (7*(6*b*d^2*e*f^3 - 5*(2*b*c*d + a*d^2)*f^4)*g - ( 48*b*d^2*e^2*f^2 - 42*(2*b*c*d + a*d^2)*e*f^3 + 35*(b*c^2 + 2*a*c*d)*f^4)* h)*x^2 + 7*(48*b*d^2*e^3*f - 15*a*c^2*f^4 - 40*(2*b*c*d + a*d^2)*e^2*f^2 + 30*(b*c^2 + 2*a*c*d)*e*f^3)*g - 2*(192*b*d^2*e^4 - 105*a*c^2*e*f^3 - 168* (2*b*c*d + a*d^2)*e^3*f + 140*(b*c^2 + 2*a*c*d)*e^2*f^2)*h + (7*(24*b*d^2* e^2*f^2 - 20*(2*b*c*d + a*d^2)*e*f^3 + 15*(b*c^2 + 2*a*c*d)*f^4)*g - (192* b*d^2*e^3*f - 105*a*c^2*f^4 - 168*(2*b*c*d + a*d^2)*e^2*f^2 + 140*(b*c^2 + 2*a*c*d)*e*f^3)*h)*x)*sqrt(f*x + e)/(f^6*x + e*f^5)
Time = 23.10 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.99 \[ \int \frac {(a+b x) (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b d^{2} h \left (e + f x\right )^{\frac {7}{2}}}{7 f^{4}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (a d^{2} f h + 2 b c d f h - 4 b d^{2} e h + b d^{2} f g\right )}{5 f^{4}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \cdot \left (2 a c d f^{2} h - 3 a d^{2} e f h + a d^{2} f^{2} g + b c^{2} f^{2} h - 6 b c d e f h + 2 b c d f^{2} g + 6 b d^{2} e^{2} h - 3 b d^{2} e f g\right )}{3 f^{4}} + \frac {\sqrt {e + f x} \left (a c^{2} f^{3} h - 4 a c d e f^{2} h + 2 a c d f^{3} g + 3 a d^{2} e^{2} f h - 2 a d^{2} e f^{2} g - 2 b c^{2} e f^{2} h + b c^{2} f^{3} g + 6 b c d e^{2} f h - 4 b c d e f^{2} g - 4 b d^{2} e^{3} h + 3 b d^{2} e^{2} f g\right )}{f^{4}} + \frac {\left (a f - b e\right ) \left (c f - d e\right )^{2} \left (e h - f g\right )}{f^{4} \sqrt {e + f x}}\right )}{f} & \text {for}\: f \neq 0 \\\frac {a c^{2} g x + \frac {b d^{2} h x^{5}}{5} + \frac {x^{4} \left (a d^{2} h + 2 b c d h + b d^{2} g\right )}{4} + \frac {x^{3} \cdot \left (2 a c d h + a d^{2} g + b c^{2} h + 2 b c d g\right )}{3} + \frac {x^{2} \left (a c^{2} h + 2 a c d g + b c^{2} g\right )}{2}}{e^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)*(d*x+c)**2*(h*x+g)/(f*x+e)**(3/2),x)
Output:
Piecewise((2*(b*d**2*h*(e + f*x)**(7/2)/(7*f**4) + (e + f*x)**(5/2)*(a*d** 2*f*h + 2*b*c*d*f*h - 4*b*d**2*e*h + b*d**2*f*g)/(5*f**4) + (e + f*x)**(3/ 2)*(2*a*c*d*f**2*h - 3*a*d**2*e*f*h + a*d**2*f**2*g + b*c**2*f**2*h - 6*b* c*d*e*f*h + 2*b*c*d*f**2*g + 6*b*d**2*e**2*h - 3*b*d**2*e*f*g)/(3*f**4) + sqrt(e + f*x)*(a*c**2*f**3*h - 4*a*c*d*e*f**2*h + 2*a*c*d*f**3*g + 3*a*d** 2*e**2*f*h - 2*a*d**2*e*f**2*g - 2*b*c**2*e*f**2*h + b*c**2*f**3*g + 6*b*c *d*e**2*f*h - 4*b*c*d*e*f**2*g - 4*b*d**2*e**3*h + 3*b*d**2*e**2*f*g)/f**4 + (a*f - b*e)*(c*f - d*e)**2*(e*h - f*g)/(f**4*sqrt(e + f*x)))/f, Ne(f, 0 )), ((a*c**2*g*x + b*d**2*h*x**5/5 + x**4*(a*d**2*h + 2*b*c*d*h + b*d**2*g )/4 + x**3*(2*a*c*d*h + a*d**2*g + b*c**2*h + 2*b*c*d*g)/3 + x**2*(a*c**2* h + 2*a*c*d*g + b*c**2*g)/2)/e**(3/2), True))
Time = 0.05 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.57 \[ \int \frac {(a+b x) (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (f x + e\right )}^{\frac {7}{2}} b d^{2} h + 21 \, {\left (b d^{2} f g - {\left (4 \, b d^{2} e - {\left (2 \, b c d + a d^{2}\right )} f\right )} h\right )} {\left (f x + e\right )}^{\frac {5}{2}} - 35 \, {\left ({\left (3 \, b d^{2} e f - {\left (2 \, b c d + a d^{2}\right )} f^{2}\right )} g - {\left (6 \, b d^{2} e^{2} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e f + {\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {3}{2}} + 105 \, {\left ({\left (3 \, b d^{2} e^{2} f - 2 \, {\left (2 \, b c d + a d^{2}\right )} e f^{2} + {\left (b c^{2} + 2 \, a c d\right )} f^{3}\right )} g - {\left (4 \, b d^{2} e^{3} - a c^{2} f^{3} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f + 2 \, {\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} h\right )} \sqrt {f x + e}}{f^{4}} + \frac {105 \, {\left ({\left (b d^{2} e^{3} f - a c^{2} f^{4} - {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e f^{3}\right )} g - {\left (b d^{2} e^{4} - a c^{2} e f^{3} - {\left (2 \, b c d + a d^{2}\right )} e^{3} f + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{2}\right )} h\right )}}{\sqrt {f x + e} f^{4}}\right )}}{105 \, f} \] Input:
integrate((b*x+a)*(d*x+c)^2*(h*x+g)/(f*x+e)^(3/2),x, algorithm="maxima")
Output:
2/105*((15*(f*x + e)^(7/2)*b*d^2*h + 21*(b*d^2*f*g - (4*b*d^2*e - (2*b*c*d + a*d^2)*f)*h)*(f*x + e)^(5/2) - 35*((3*b*d^2*e*f - (2*b*c*d + a*d^2)*f^2 )*g - (6*b*d^2*e^2 - 3*(2*b*c*d + a*d^2)*e*f + (b*c^2 + 2*a*c*d)*f^2)*h)*( f*x + e)^(3/2) + 105*((3*b*d^2*e^2*f - 2*(2*b*c*d + a*d^2)*e*f^2 + (b*c^2 + 2*a*c*d)*f^3)*g - (4*b*d^2*e^3 - a*c^2*f^3 - 3*(2*b*c*d + a*d^2)*e^2*f + 2*(b*c^2 + 2*a*c*d)*e*f^2)*h)*sqrt(f*x + e))/f^4 + 105*((b*d^2*e^3*f - a* c^2*f^4 - (2*b*c*d + a*d^2)*e^2*f^2 + (b*c^2 + 2*a*c*d)*e*f^3)*g - (b*d^2* e^4 - a*c^2*e*f^3 - (2*b*c*d + a*d^2)*e^3*f + (b*c^2 + 2*a*c*d)*e^2*f^2)*h )/(sqrt(f*x + e)*f^4))/f
Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (228) = 456\).
Time = 0.14 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.35 \[ \int \frac {(a+b x) (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 \, {\left (b d^{2} e^{3} f g - 2 \, b c d e^{2} f^{2} g - a d^{2} e^{2} f^{2} g + b c^{2} e f^{3} g + 2 \, a c d e f^{3} g - a c^{2} f^{4} g - b d^{2} e^{4} h + 2 \, b c d e^{3} f h + a d^{2} e^{3} f h - b c^{2} e^{2} f^{2} h - 2 \, a c d e^{2} f^{2} h + a c^{2} e f^{3} h\right )}}{\sqrt {f x + e} f^{5}} + \frac {2 \, {\left (21 \, {\left (f x + e\right )}^{\frac {5}{2}} b d^{2} f^{31} g - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} b d^{2} e f^{31} g + 315 \, \sqrt {f x + e} b d^{2} e^{2} f^{31} g + 70 \, {\left (f x + e\right )}^{\frac {3}{2}} b c d f^{32} g + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} a d^{2} f^{32} g - 420 \, \sqrt {f x + e} b c d e f^{32} g - 210 \, \sqrt {f x + e} a d^{2} e f^{32} g + 105 \, \sqrt {f x + e} b c^{2} f^{33} g + 210 \, \sqrt {f x + e} a c d f^{33} g + 15 \, {\left (f x + e\right )}^{\frac {7}{2}} b d^{2} f^{30} h - 84 \, {\left (f x + e\right )}^{\frac {5}{2}} b d^{2} e f^{30} h + 210 \, {\left (f x + e\right )}^{\frac {3}{2}} b d^{2} e^{2} f^{30} h - 420 \, \sqrt {f x + e} b d^{2} e^{3} f^{30} h + 42 \, {\left (f x + e\right )}^{\frac {5}{2}} b c d f^{31} h + 21 \, {\left (f x + e\right )}^{\frac {5}{2}} a d^{2} f^{31} h - 210 \, {\left (f x + e\right )}^{\frac {3}{2}} b c d e f^{31} h - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a d^{2} e f^{31} h + 630 \, \sqrt {f x + e} b c d e^{2} f^{31} h + 315 \, \sqrt {f x + e} a d^{2} e^{2} f^{31} h + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b c^{2} f^{32} h + 70 \, {\left (f x + e\right )}^{\frac {3}{2}} a c d f^{32} h - 210 \, \sqrt {f x + e} b c^{2} e f^{32} h - 420 \, \sqrt {f x + e} a c d e f^{32} h + 105 \, \sqrt {f x + e} a c^{2} f^{33} h\right )}}{105 \, f^{35}} \] Input:
integrate((b*x+a)*(d*x+c)^2*(h*x+g)/(f*x+e)^(3/2),x, algorithm="giac")
Output:
2*(b*d^2*e^3*f*g - 2*b*c*d*e^2*f^2*g - a*d^2*e^2*f^2*g + b*c^2*e*f^3*g + 2 *a*c*d*e*f^3*g - a*c^2*f^4*g - b*d^2*e^4*h + 2*b*c*d*e^3*f*h + a*d^2*e^3*f *h - b*c^2*e^2*f^2*h - 2*a*c*d*e^2*f^2*h + a*c^2*e*f^3*h)/(sqrt(f*x + e)*f ^5) + 2/105*(21*(f*x + e)^(5/2)*b*d^2*f^31*g - 105*(f*x + e)^(3/2)*b*d^2*e *f^31*g + 315*sqrt(f*x + e)*b*d^2*e^2*f^31*g + 70*(f*x + e)^(3/2)*b*c*d*f^ 32*g + 35*(f*x + e)^(3/2)*a*d^2*f^32*g - 420*sqrt(f*x + e)*b*c*d*e*f^32*g - 210*sqrt(f*x + e)*a*d^2*e*f^32*g + 105*sqrt(f*x + e)*b*c^2*f^33*g + 210* sqrt(f*x + e)*a*c*d*f^33*g + 15*(f*x + e)^(7/2)*b*d^2*f^30*h - 84*(f*x + e )^(5/2)*b*d^2*e*f^30*h + 210*(f*x + e)^(3/2)*b*d^2*e^2*f^30*h - 420*sqrt(f *x + e)*b*d^2*e^3*f^30*h + 42*(f*x + e)^(5/2)*b*c*d*f^31*h + 21*(f*x + e)^ (5/2)*a*d^2*f^31*h - 210*(f*x + e)^(3/2)*b*c*d*e*f^31*h - 105*(f*x + e)^(3 /2)*a*d^2*e*f^31*h + 630*sqrt(f*x + e)*b*c*d*e^2*f^31*h + 315*sqrt(f*x + e )*a*d^2*e^2*f^31*h + 35*(f*x + e)^(3/2)*b*c^2*f^32*h + 70*(f*x + e)^(3/2)* a*c*d*f^32*h - 210*sqrt(f*x + e)*b*c^2*e*f^32*h - 420*sqrt(f*x + e)*a*c*d* e*f^32*h + 105*sqrt(f*x + e)*a*c^2*f^33*h)/f^35
Time = 0.08 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.51 \[ \int \frac {(a+b x) (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {{\left (e+f\,x\right )}^{5/2}\,\left (2\,a\,d^2\,f\,h-8\,b\,d^2\,e\,h+2\,b\,d^2\,f\,g+4\,b\,c\,d\,f\,h\right )}{5\,f^5}+\frac {{\left (e+f\,x\right )}^{3/2}\,\left (2\,a\,d^2\,f^2\,g+2\,b\,c^2\,f^2\,h+12\,b\,d^2\,e^2\,h+4\,a\,c\,d\,f^2\,h+4\,b\,c\,d\,f^2\,g-6\,a\,d^2\,e\,f\,h-6\,b\,d^2\,e\,f\,g-12\,b\,c\,d\,e\,f\,h\right )}{3\,f^5}-\frac {2\,a\,c^2\,f^4\,g+2\,b\,d^2\,e^4\,h-2\,a\,c^2\,e\,f^3\,h-2\,b\,c^2\,e\,f^3\,g-2\,a\,d^2\,e^3\,f\,h-2\,b\,d^2\,e^3\,f\,g+2\,a\,d^2\,e^2\,f^2\,g+2\,b\,c^2\,e^2\,f^2\,h-4\,a\,c\,d\,e\,f^3\,g-4\,b\,c\,d\,e^3\,f\,h+4\,a\,c\,d\,e^2\,f^2\,h+4\,b\,c\,d\,e^2\,f^2\,g}{f^5\,\sqrt {e+f\,x}}+\frac {2\,\sqrt {e+f\,x}\,\left (c\,f-d\,e\right )\,\left (a\,c\,f^2\,h+2\,a\,d\,f^2\,g+b\,c\,f^2\,g+4\,b\,d\,e^2\,h-3\,a\,d\,e\,f\,h-2\,b\,c\,e\,f\,h-3\,b\,d\,e\,f\,g\right )}{f^5}+\frac {2\,b\,d^2\,h\,{\left (e+f\,x\right )}^{7/2}}{7\,f^5} \] Input:
int(((g + h*x)*(a + b*x)*(c + d*x)^2)/(e + f*x)^(3/2),x)
Output:
((e + f*x)^(5/2)*(2*a*d^2*f*h - 8*b*d^2*e*h + 2*b*d^2*f*g + 4*b*c*d*f*h))/ (5*f^5) + ((e + f*x)^(3/2)*(2*a*d^2*f^2*g + 2*b*c^2*f^2*h + 12*b*d^2*e^2*h + 4*a*c*d*f^2*h + 4*b*c*d*f^2*g - 6*a*d^2*e*f*h - 6*b*d^2*e*f*g - 12*b*c* d*e*f*h))/(3*f^5) - (2*a*c^2*f^4*g + 2*b*d^2*e^4*h - 2*a*c^2*e*f^3*h - 2*b *c^2*e*f^3*g - 2*a*d^2*e^3*f*h - 2*b*d^2*e^3*f*g + 2*a*d^2*e^2*f^2*g + 2*b *c^2*e^2*f^2*h - 4*a*c*d*e*f^3*g - 4*b*c*d*e^3*f*h + 4*a*c*d*e^2*f^2*h + 4 *b*c*d*e^2*f^2*g)/(f^5*(e + f*x)^(1/2)) + (2*(e + f*x)^(1/2)*(c*f - d*e)*( a*c*f^2*h + 2*a*d*f^2*g + b*c*f^2*g + 4*b*d*e^2*h - 3*a*d*e*f*h - 2*b*c*e* f*h - 3*b*d*e*f*g))/f^5 + (2*b*d^2*h*(e + f*x)^(7/2))/(7*f^5)
Time = 0.17 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.85 \[ \int \frac {(a+b x) (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {-\frac {16}{3} a c d e \,f^{3} h x +\frac {32}{5} b c d \,e^{2} f^{2} h x -\frac {16}{3} b c d e \,f^{3} g x -\frac {8}{5} b c d e \,f^{3} h \,x^{2}+\frac {2}{3} a \,d^{2} f^{4} g \,x^{2}+\frac {2}{5} a \,d^{2} f^{4} h \,x^{3}-\frac {16}{3} b \,c^{2} e^{2} f^{2} h +4 b \,c^{2} e \,f^{3} g +2 b \,c^{2} f^{4} g x +\frac {2}{3} b \,c^{2} f^{4} h \,x^{2}+\frac {32}{5} b \,d^{2} e^{3} f g +\frac {2}{5} b \,d^{2} f^{4} g \,x^{3}+\frac {2}{7} b \,d^{2} f^{4} h \,x^{4}+4 a \,c^{2} e \,f^{3} h +2 a \,c^{2} f^{4} h x +\frac {32}{5} a \,d^{2} e^{3} f h -\frac {16}{3} a \,d^{2} e^{2} f^{2} g -2 a \,c^{2} f^{4} g -\frac {256}{35} b \,d^{2} e^{4} h +\frac {16}{5} b \,d^{2} e^{2} f^{2} g x +\frac {32}{35} b \,d^{2} e^{2} f^{2} h \,x^{2}-\frac {4}{5} b \,d^{2} e \,f^{3} g \,x^{2}-\frac {16}{35} b \,d^{2} e \,f^{3} h \,x^{3}-\frac {32}{3} a c d \,e^{2} f^{2} h +8 a c d e \,f^{3} g +4 a c d \,f^{4} g x +\frac {4}{3} a c d \,f^{4} h \,x^{2}+\frac {16}{5} a \,d^{2} e^{2} f^{2} h x -\frac {8}{3} a \,d^{2} e \,f^{3} g x -\frac {4}{5} a \,d^{2} e \,f^{3} h \,x^{2}-\frac {8}{3} b \,c^{2} e \,f^{3} h x +\frac {64}{5} b c d \,e^{3} f h -\frac {32}{3} b c d \,e^{2} f^{2} g +\frac {4}{3} b c d \,f^{4} g \,x^{2}+\frac {4}{5} b c d \,f^{4} h \,x^{3}-\frac {128}{35} b \,d^{2} e^{3} f h x}{\sqrt {f x +e}\, f^{5}} \] Input:
int((b*x+a)*(d*x+c)^2*(h*x+g)/(f*x+e)^(3/2),x)
Output:
(2*(210*a*c**2*e*f**3*h - 105*a*c**2*f**4*g + 105*a*c**2*f**4*h*x - 560*a* c*d*e**2*f**2*h + 420*a*c*d*e*f**3*g - 280*a*c*d*e*f**3*h*x + 210*a*c*d*f* *4*g*x + 70*a*c*d*f**4*h*x**2 + 336*a*d**2*e**3*f*h - 280*a*d**2*e**2*f**2 *g + 168*a*d**2*e**2*f**2*h*x - 140*a*d**2*e*f**3*g*x - 42*a*d**2*e*f**3*h *x**2 + 35*a*d**2*f**4*g*x**2 + 21*a*d**2*f**4*h*x**3 - 280*b*c**2*e**2*f* *2*h + 210*b*c**2*e*f**3*g - 140*b*c**2*e*f**3*h*x + 105*b*c**2*f**4*g*x + 35*b*c**2*f**4*h*x**2 + 672*b*c*d*e**3*f*h - 560*b*c*d*e**2*f**2*g + 336* b*c*d*e**2*f**2*h*x - 280*b*c*d*e*f**3*g*x - 84*b*c*d*e*f**3*h*x**2 + 70*b *c*d*f**4*g*x**2 + 42*b*c*d*f**4*h*x**3 - 384*b*d**2*e**4*h + 336*b*d**2*e **3*f*g - 192*b*d**2*e**3*f*h*x + 168*b*d**2*e**2*f**2*g*x + 48*b*d**2*e** 2*f**2*h*x**2 - 42*b*d**2*e*f**3*g*x**2 - 24*b*d**2*e*f**3*h*x**3 + 21*b*d **2*f**4*g*x**3 + 15*b*d**2*f**4*h*x**4))/(105*sqrt(e + f*x)*f**5)