Integrand size = 27, antiderivative size = 245 \[ \int \frac {(a+b x)^2 (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=-\frac {2 (b e-a f)^2 (d e-c f) (f g-e h) \sqrt {e+f x}}{f^5}+\frac {2 (b e-a f) (b d e (3 f g-4 e h)-b c f (2 f g-3 e h)-a f (d f g-2 d e h+c f h)) (e+f x)^{3/2}}{3 f^5}+\frac {2 \left (a^2 d f^2 h+2 a b f (d f g-3 d e h+c f h)+b^2 (c f (f g-3 e h)-3 d e (f g-2 e h))\right ) (e+f x)^{5/2}}{5 f^5}+\frac {2 b (2 a d f h+b (d f g-4 d e h+c f h)) (e+f x)^{7/2}}{7 f^5}+\frac {2 b^2 d h (e+f x)^{9/2}}{9 f^5} \] Output:
-2*(-a*f+b*e)^2*(-c*f+d*e)*(-e*h+f*g)*(f*x+e)^(1/2)/f^5+2/3*(-a*f+b*e)*(b* d*e*(-4*e*h+3*f*g)-b*c*f*(-3*e*h+2*f*g)-a*f*(c*f*h-2*d*e*h+d*f*g))*(f*x+e) ^(3/2)/f^5+2/5*(a^2*d*f^2*h+2*a*b*f*(c*f*h-3*d*e*h+d*f*g)+b^2*(c*f*(-3*e*h +f*g)-3*d*e*(-2*e*h+f*g)))*(f*x+e)^(5/2)/f^5+2/7*b*(2*a*d*f*h+b*(c*f*h-4*d *e*h+d*f*g))*(f*x+e)^(7/2)/f^5+2/9*b^2*d*h*(f*x+e)^(9/2)/f^5
Time = 0.23 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b x)^2 (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \sqrt {e+f x} \left (21 a^2 f^2 \left (5 c f (3 f g-2 e h+f h x)+d \left (8 e^2 h-2 e f (5 g+2 h x)+f^2 x (5 g+3 h x)\right )\right )+6 a b f \left (7 c f \left (8 e^2 h-2 e f (5 g+2 h x)+f^2 x (5 g+3 h x)\right )+d \left (-48 e^3 h+8 e^2 f (7 g+3 h x)+3 f^3 x^2 (7 g+5 h x)-2 e f^2 x (14 g+9 h x)\right )\right )+b^2 \left (3 c f \left (-48 e^3 h+8 e^2 f (7 g+3 h x)+3 f^3 x^2 (7 g+5 h x)-2 e f^2 x (14 g+9 h x)\right )+d \left (128 e^4 h+24 e^2 f^2 x (3 g+2 h x)-16 e^3 f (9 g+4 h x)+5 f^4 x^3 (9 g+7 h x)-2 e f^3 x^2 (27 g+20 h x)\right )\right )\right )}{315 f^5} \] Input:
Integrate[((a + b*x)^2*(c + d*x)*(g + h*x))/Sqrt[e + f*x],x]
Output:
(2*Sqrt[e + f*x]*(21*a^2*f^2*(5*c*f*(3*f*g - 2*e*h + f*h*x) + d*(8*e^2*h - 2*e*f*(5*g + 2*h*x) + f^2*x*(5*g + 3*h*x))) + 6*a*b*f*(7*c*f*(8*e^2*h - 2 *e*f*(5*g + 2*h*x) + f^2*x*(5*g + 3*h*x)) + d*(-48*e^3*h + 8*e^2*f*(7*g + 3*h*x) + 3*f^3*x^2*(7*g + 5*h*x) - 2*e*f^2*x*(14*g + 9*h*x))) + b^2*(3*c*f *(-48*e^3*h + 8*e^2*f*(7*g + 3*h*x) + 3*f^3*x^2*(7*g + 5*h*x) - 2*e*f^2*x* (14*g + 9*h*x)) + d*(128*e^4*h + 24*e^2*f^2*x*(3*g + 2*h*x) - 16*e^3*f*(9* g + 4*h*x) + 5*f^4*x^3*(9*g + 7*h*x) - 2*e*f^3*x^2*(27*g + 20*h*x)))))/(31 5*f^5)
Time = 0.52 (sec) , antiderivative size = 245, 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.
| \(\displaystyle \int \frac {(a+b x)^2 (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 159 |
| \(\displaystyle \int \left (\frac {(e+f x)^{3/2} \left (a^2 d f^2 h+2 a b f (c f h-3 d e h+d f g)+b^2 (c f (f g-3 e h)-3 d e (f g-2 e h))\right )}{f^4}+\frac {b (e+f x)^{5/2} (2 a d f h+b (c f h-4 d e h+d f g))}{f^4}+\frac {\sqrt {e+f x} (b e-a f) (-a f (c f h-2 d e h+d f g)-b c f (2 f g-3 e h)+b d e (3 f g-4 e h))}{f^4}+\frac {(a f-b e)^2 (c f-d e) (f g-e h)}{f^4 \sqrt {e+f x}}+\frac {b^2 d h (e+f x)^{7/2}}{f^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (e+f x)^{5/2} \left (a^2 d f^2 h+2 a b f (c f h-3 d e h+d f g)+b^2 (c f (f g-3 e h)-3 d e (f g-2 e h))\right )}{5 f^5}+\frac {2 b (e+f x)^{7/2} (2 a d f h+b (c f h-4 d e h+d f g))}{7 f^5}+\frac {2 (e+f x)^{3/2} (b e-a f) (-a f (c f h-2 d e h+d f g)-b c f (2 f g-3 e h)+b d e (3 f g-4 e h))}{3 f^5}-\frac {2 \sqrt {e+f x} (b e-a f)^2 (d e-c f) (f g-e h)}{f^5}+\frac {2 b^2 d h (e+f x)^{9/2}}{9 f^5}\) |
Input:
Int[((a + b*x)^2*(c + d*x)*(g + h*x))/Sqrt[e + f*x],x]
Output:
(-2*(b*e - a*f)^2*(d*e - c*f)*(f*g - e*h)*Sqrt[e + f*x])/f^5 + (2*(b*e - a *f)*(b*d*e*(3*f*g - 4*e*h) - b*c*f*(2*f*g - 3*e*h) - a*f*(d*f*g - 2*d*e*h + c*f*h))*(e + f*x)^(3/2))/(3*f^5) + (2*(a^2*d*f^2*h + 2*a*b*f*(d*f*g - 3* d*e*h + c*f*h) + b^2*(c*f*(f*g - 3*e*h) - 3*d*e*(f*g - 2*e*h)))*(e + f*x)^ (5/2))/(5*f^5) + (2*b*(2*a*d*f*h + b*(d*f*g - 4*d*e*h + c*f*h))*(e + f*x)^ (7/2))/(7*f^5) + (2*b^2*d*h*(e + f*x)^(9/2))/(9*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 = 0.92 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(\frac {\frac {2 d h \,b^{2} \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (2 b \left (a f -b e \right ) d +b^{2} \left (c f -d e \right )\right ) h +d \,b^{2} \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (\left (a f -b e \right )^{2} d +2 b \left (a f -b e \right ) \left (c f -d e \right )\right ) h +\left (2 b \left (a f -b e \right ) d +b^{2} \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a f -b e \right )^{2} \left (c f -d e \right ) h +\left (\left (a f -b e \right )^{2} d +2 b \left (a f -b e \right ) \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+2 \left (a f -b e \right )^{2} \left (c f -d e \right ) \left (-e h +f g \right ) \sqrt {f x +e}}{f^{5}}\) | \(254\) |
| default | \(\frac {\frac {2 d h \,b^{2} \left (f x +e \right )^{\frac {9}{2}}}{9}-\frac {2 \left (-\left (2 b \left (a f -b e \right ) d +b^{2} \left (c f -d e \right )\right ) h +d \,b^{2} \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {2 \left (-\left (\left (a f -b e \right )^{2} d +2 b \left (a f -b e \right ) \left (c f -d e \right )\right ) h +\left (2 b \left (a f -b e \right ) d +b^{2} \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 \left (-\left (a f -b e \right )^{2} \left (c f -d e \right ) h +\left (\left (a f -b e \right )^{2} d +2 b \left (a f -b e \right ) \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}-2 \left (a f -b e \right )^{2} \left (c f -d e \right ) \left (e h -f g \right ) \sqrt {f x +e}}{f^{5}}\) | \(258\) |
| pseudoelliptic | \(-\frac {4 \sqrt {f x +e}\, \left (\left (-\frac {3 x^{2} \left (\frac {5 d h \,x^{2}}{9}+\frac {5 \left (c h +d g \right ) x}{7}+c g \right ) b^{2}}{10}-a x \left (\frac {3 d h \,x^{2}}{7}+\frac {3 \left (c h +d g \right ) x}{5}+c g \right ) b -\frac {3 \left (\frac {d h \,x^{2}}{5}+\frac {\left (c h +d g \right ) x}{3}+c g \right ) a^{2}}{2}\right ) f^{4}+\left (\left (\frac {4 d h \,x^{3}}{21}+\frac {9 \left (c h +d g \right ) x^{2}}{35}+\frac {2 c g x}{5}\right ) b^{2}+2 a \left (\frac {9 d h \,x^{2}}{35}+\frac {2 \left (c h +d g \right ) x}{5}+c g \right ) b +a^{2} \left (c h +d g +\frac {2}{5} d h x \right )\right ) e \,f^{3}-\frac {4 \left (\left (\frac {2 d h \,x^{2}}{7}+\frac {3 \left (c h +d g \right ) x}{7}+c g \right ) b^{2}+2 a \left (\frac {3}{7} d h x +c h +d g \right ) b +a^{2} d h \right ) e^{2} f^{2}}{5}+\frac {48 \left (\left (\frac {2}{9} d h x +\frac {1}{2} c h +\frac {1}{2} d g \right ) b +a d h \right ) b \,e^{3} f}{35}-\frac {64 b^{2} d \,e^{4} h}{105}\right )}{3 f^{5}}\) | \(270\) |
| gosper | \(-\frac {2 \sqrt {f x +e}\, \left (-35 d h \,b^{2} x^{4} f^{4}-90 a b d \,f^{4} h \,x^{3}-45 b^{2} c \,f^{4} h \,x^{3}+40 b^{2} d e \,f^{3} h \,x^{3}-45 b^{2} d \,f^{4} g \,x^{3}-63 a^{2} d \,f^{4} h \,x^{2}-126 a b c \,f^{4} h \,x^{2}+108 a b d e \,f^{3} h \,x^{2}-126 a b d \,f^{4} g \,x^{2}+54 b^{2} c e \,f^{3} h \,x^{2}-63 b^{2} c \,f^{4} g \,x^{2}-48 b^{2} d \,e^{2} f^{2} h \,x^{2}+54 b^{2} d e \,f^{3} g \,x^{2}-105 a^{2} c \,f^{4} h x +84 a^{2} d e \,f^{3} h x -105 a^{2} d \,f^{4} g x +168 a b c e \,f^{3} h x -210 a b c \,f^{4} g x -144 a b d \,e^{2} f^{2} h x +168 a b d e \,f^{3} g x -72 b^{2} c \,e^{2} f^{2} h x +84 b^{2} c e \,f^{3} g x +64 b^{2} d \,e^{3} f h x -72 b^{2} d \,e^{2} f^{2} g x +210 a^{2} c e \,f^{3} h -315 c g \,a^{2} f^{4}-168 a^{2} d \,e^{2} f^{2} h +210 a^{2} d e \,f^{3} g -336 a b c \,e^{2} f^{2} h +420 a b c e \,f^{3} g +288 a b d \,e^{3} f h -336 a b d \,e^{2} f^{2} g +144 b^{2} c \,e^{3} f h -168 b^{2} c \,e^{2} f^{2} g -128 b^{2} d \,e^{4} h +144 b^{2} d \,e^{3} f g \right )}{315 f^{5}}\) | \(451\) |
| trager | \(-\frac {2 \sqrt {f x +e}\, \left (-35 d h \,b^{2} x^{4} f^{4}-90 a b d \,f^{4} h \,x^{3}-45 b^{2} c \,f^{4} h \,x^{3}+40 b^{2} d e \,f^{3} h \,x^{3}-45 b^{2} d \,f^{4} g \,x^{3}-63 a^{2} d \,f^{4} h \,x^{2}-126 a b c \,f^{4} h \,x^{2}+108 a b d e \,f^{3} h \,x^{2}-126 a b d \,f^{4} g \,x^{2}+54 b^{2} c e \,f^{3} h \,x^{2}-63 b^{2} c \,f^{4} g \,x^{2}-48 b^{2} d \,e^{2} f^{2} h \,x^{2}+54 b^{2} d e \,f^{3} g \,x^{2}-105 a^{2} c \,f^{4} h x +84 a^{2} d e \,f^{3} h x -105 a^{2} d \,f^{4} g x +168 a b c e \,f^{3} h x -210 a b c \,f^{4} g x -144 a b d \,e^{2} f^{2} h x +168 a b d e \,f^{3} g x -72 b^{2} c \,e^{2} f^{2} h x +84 b^{2} c e \,f^{3} g x +64 b^{2} d \,e^{3} f h x -72 b^{2} d \,e^{2} f^{2} g x +210 a^{2} c e \,f^{3} h -315 c g \,a^{2} f^{4}-168 a^{2} d \,e^{2} f^{2} h +210 a^{2} d e \,f^{3} g -336 a b c \,e^{2} f^{2} h +420 a b c e \,f^{3} g +288 a b d \,e^{3} f h -336 a b d \,e^{2} f^{2} g +144 b^{2} c \,e^{3} f h -168 b^{2} c \,e^{2} f^{2} g -128 b^{2} d \,e^{4} h +144 b^{2} d \,e^{3} f g \right )}{315 f^{5}}\) | \(451\) |
| risch | \(-\frac {2 \sqrt {f x +e}\, \left (-35 d h \,b^{2} x^{4} f^{4}-90 a b d \,f^{4} h \,x^{3}-45 b^{2} c \,f^{4} h \,x^{3}+40 b^{2} d e \,f^{3} h \,x^{3}-45 b^{2} d \,f^{4} g \,x^{3}-63 a^{2} d \,f^{4} h \,x^{2}-126 a b c \,f^{4} h \,x^{2}+108 a b d e \,f^{3} h \,x^{2}-126 a b d \,f^{4} g \,x^{2}+54 b^{2} c e \,f^{3} h \,x^{2}-63 b^{2} c \,f^{4} g \,x^{2}-48 b^{2} d \,e^{2} f^{2} h \,x^{2}+54 b^{2} d e \,f^{3} g \,x^{2}-105 a^{2} c \,f^{4} h x +84 a^{2} d e \,f^{3} h x -105 a^{2} d \,f^{4} g x +168 a b c e \,f^{3} h x -210 a b c \,f^{4} g x -144 a b d \,e^{2} f^{2} h x +168 a b d e \,f^{3} g x -72 b^{2} c \,e^{2} f^{2} h x +84 b^{2} c e \,f^{3} g x +64 b^{2} d \,e^{3} f h x -72 b^{2} d \,e^{2} f^{2} g x +210 a^{2} c e \,f^{3} h -315 c g \,a^{2} f^{4}-168 a^{2} d \,e^{2} f^{2} h +210 a^{2} d e \,f^{3} g -336 a b c \,e^{2} f^{2} h +420 a b c e \,f^{3} g +288 a b d \,e^{3} f h -336 a b d \,e^{2} f^{2} g +144 b^{2} c \,e^{3} f h -168 b^{2} c \,e^{2} f^{2} g -128 b^{2} d \,e^{4} h +144 b^{2} d \,e^{3} f g \right )}{315 f^{5}}\) | \(451\) |
| orering | \(-\frac {2 \sqrt {f x +e}\, \left (-35 d h \,b^{2} x^{4} f^{4}-90 a b d \,f^{4} h \,x^{3}-45 b^{2} c \,f^{4} h \,x^{3}+40 b^{2} d e \,f^{3} h \,x^{3}-45 b^{2} d \,f^{4} g \,x^{3}-63 a^{2} d \,f^{4} h \,x^{2}-126 a b c \,f^{4} h \,x^{2}+108 a b d e \,f^{3} h \,x^{2}-126 a b d \,f^{4} g \,x^{2}+54 b^{2} c e \,f^{3} h \,x^{2}-63 b^{2} c \,f^{4} g \,x^{2}-48 b^{2} d \,e^{2} f^{2} h \,x^{2}+54 b^{2} d e \,f^{3} g \,x^{2}-105 a^{2} c \,f^{4} h x +84 a^{2} d e \,f^{3} h x -105 a^{2} d \,f^{4} g x +168 a b c e \,f^{3} h x -210 a b c \,f^{4} g x -144 a b d \,e^{2} f^{2} h x +168 a b d e \,f^{3} g x -72 b^{2} c \,e^{2} f^{2} h x +84 b^{2} c e \,f^{3} g x +64 b^{2} d \,e^{3} f h x -72 b^{2} d \,e^{2} f^{2} g x +210 a^{2} c e \,f^{3} h -315 c g \,a^{2} f^{4}-168 a^{2} d \,e^{2} f^{2} h +210 a^{2} d e \,f^{3} g -336 a b c \,e^{2} f^{2} h +420 a b c e \,f^{3} g +288 a b d \,e^{3} f h -336 a b d \,e^{2} f^{2} g +144 b^{2} c \,e^{3} f h -168 b^{2} c \,e^{2} f^{2} g -128 b^{2} d \,e^{4} h +144 b^{2} d \,e^{3} f g \right )}{315 f^{5}}\) | \(451\) |
Input:
int((b*x+a)^2*(d*x+c)*(h*x+g)/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/f^5*(1/9*d*h*b^2*(f*x+e)^(9/2)+1/7*((2*b*(a*f-b*e)*d+b^2*(c*f-d*e))*h+d* b^2*(-e*h+f*g))*(f*x+e)^(7/2)+1/5*(((a*f-b*e)^2*d+2*b*(a*f-b*e)*(c*f-d*e)) *h+(2*b*(a*f-b*e)*d+b^2*(c*f-d*e))*(-e*h+f*g))*(f*x+e)^(5/2)+1/3*((a*f-b*e )^2*(c*f-d*e)*h+((a*f-b*e)^2*d+2*b*(a*f-b*e)*(c*f-d*e))*(-e*h+f*g))*(f*x+e )^(3/2)+(a*f-b*e)^2*(c*f-d*e)*(-e*h+f*g)*(f*x+e)^(1/2))
Time = 0.09 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^2 (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \, {\left (35 \, b^{2} d f^{4} h x^{4} + 5 \, {\left (9 \, b^{2} d f^{4} g - {\left (8 \, b^{2} d e f^{3} - 9 \, {\left (b^{2} c + 2 \, a b d\right )} f^{4}\right )} h\right )} x^{3} - 3 \, {\left (3 \, {\left (6 \, b^{2} d e f^{3} - 7 \, {\left (b^{2} c + 2 \, a b d\right )} f^{4}\right )} g - {\left (16 \, b^{2} d e^{2} f^{2} - 18 \, {\left (b^{2} c + 2 \, a b d\right )} e f^{3} + 21 \, {\left (2 \, a b c + a^{2} d\right )} f^{4}\right )} h\right )} x^{2} - 3 \, {\left (48 \, b^{2} d e^{3} f - 105 \, a^{2} c f^{4} - 56 \, {\left (b^{2} c + 2 \, a b d\right )} e^{2} f^{2} + 70 \, {\left (2 \, a b c + a^{2} d\right )} e f^{3}\right )} g + 2 \, {\left (64 \, b^{2} d e^{4} - 105 \, a^{2} c e f^{3} - 72 \, {\left (b^{2} c + 2 \, a b d\right )} e^{3} f + 84 \, {\left (2 \, a b c + a^{2} d\right )} e^{2} f^{2}\right )} h + {\left (3 \, {\left (24 \, b^{2} d e^{2} f^{2} - 28 \, {\left (b^{2} c + 2 \, a b d\right )} e f^{3} + 35 \, {\left (2 \, a b c + a^{2} d\right )} f^{4}\right )} g - {\left (64 \, b^{2} d e^{3} f - 105 \, a^{2} c f^{4} - 72 \, {\left (b^{2} c + 2 \, a b d\right )} e^{2} f^{2} + 84 \, {\left (2 \, a b c + a^{2} d\right )} e f^{3}\right )} h\right )} x\right )} \sqrt {f x + e}}{315 \, f^{5}} \] Input:
integrate((b*x+a)^2*(d*x+c)*(h*x+g)/(f*x+e)^(1/2),x, algorithm="fricas")
Output:
2/315*(35*b^2*d*f^4*h*x^4 + 5*(9*b^2*d*f^4*g - (8*b^2*d*e*f^3 - 9*(b^2*c + 2*a*b*d)*f^4)*h)*x^3 - 3*(3*(6*b^2*d*e*f^3 - 7*(b^2*c + 2*a*b*d)*f^4)*g - (16*b^2*d*e^2*f^2 - 18*(b^2*c + 2*a*b*d)*e*f^3 + 21*(2*a*b*c + a^2*d)*f^4 )*h)*x^2 - 3*(48*b^2*d*e^3*f - 105*a^2*c*f^4 - 56*(b^2*c + 2*a*b*d)*e^2*f^ 2 + 70*(2*a*b*c + a^2*d)*e*f^3)*g + 2*(64*b^2*d*e^4 - 105*a^2*c*e*f^3 - 72 *(b^2*c + 2*a*b*d)*e^3*f + 84*(2*a*b*c + a^2*d)*e^2*f^2)*h + (3*(24*b^2*d* e^2*f^2 - 28*(b^2*c + 2*a*b*d)*e*f^3 + 35*(2*a*b*c + a^2*d)*f^4)*g - (64*b ^2*d*e^3*f - 105*a^2*c*f^4 - 72*(b^2*c + 2*a*b*d)*e^2*f^2 + 84*(2*a*b*c + a^2*d)*e*f^3)*h)*x)*sqrt(f*x + e)/f^5
Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (260) = 520\).
Time = 1.50 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.53 \[ \int \frac {(a+b x)^2 (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{2} d h \left (e + f x\right )^{\frac {9}{2}}}{9 f^{4}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (2 a b d f h + b^{2} c f h - 4 b^{2} d e h + b^{2} d f g\right )}{7 f^{4}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (a^{2} d f^{2} h + 2 a b c f^{2} h - 6 a b d e f h + 2 a b d f^{2} g - 3 b^{2} c e f h + b^{2} c f^{2} g + 6 b^{2} d e^{2} h - 3 b^{2} d e f g\right )}{5 f^{4}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (a^{2} c f^{3} h - 2 a^{2} d e f^{2} h + a^{2} d f^{3} g - 4 a b c e f^{2} h + 2 a b c f^{3} g + 6 a b d e^{2} f h - 4 a b d e f^{2} g + 3 b^{2} c e^{2} f h - 2 b^{2} c e f^{2} g - 4 b^{2} d e^{3} h + 3 b^{2} d e^{2} f g\right )}{3 f^{4}} + \frac {\sqrt {e + f x} \left (- a^{2} c e f^{3} h + a^{2} c f^{4} g + a^{2} d e^{2} f^{2} h - a^{2} d e f^{3} g + 2 a b c e^{2} f^{2} h - 2 a b c e f^{3} g - 2 a b d e^{3} f h + 2 a b d e^{2} f^{2} g - b^{2} c e^{3} f h + b^{2} c e^{2} f^{2} g + b^{2} d e^{4} h - b^{2} d e^{3} f g\right )}{f^{4}}\right )}{f} & \text {for}\: f \neq 0 \\\frac {a^{2} c g x + \frac {b^{2} d h x^{5}}{5} + \frac {x^{4} \cdot \left (2 a b d h + b^{2} c h + b^{2} d g\right )}{4} + \frac {x^{3} \left (a^{2} d h + 2 a b c h + 2 a b d g + b^{2} c g\right )}{3} + \frac {x^{2} \left (a^{2} c h + a^{2} d g + 2 a b c g\right )}{2}}{\sqrt {e}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**2*(d*x+c)*(h*x+g)/(f*x+e)**(1/2),x)
Output:
Piecewise((2*(b**2*d*h*(e + f*x)**(9/2)/(9*f**4) + (e + f*x)**(7/2)*(2*a*b *d*f*h + b**2*c*f*h - 4*b**2*d*e*h + b**2*d*f*g)/(7*f**4) + (e + f*x)**(5/ 2)*(a**2*d*f**2*h + 2*a*b*c*f**2*h - 6*a*b*d*e*f*h + 2*a*b*d*f**2*g - 3*b* *2*c*e*f*h + b**2*c*f**2*g + 6*b**2*d*e**2*h - 3*b**2*d*e*f*g)/(5*f**4) + (e + f*x)**(3/2)*(a**2*c*f**3*h - 2*a**2*d*e*f**2*h + a**2*d*f**3*g - 4*a* b*c*e*f**2*h + 2*a*b*c*f**3*g + 6*a*b*d*e**2*f*h - 4*a*b*d*e*f**2*g + 3*b* *2*c*e**2*f*h - 2*b**2*c*e*f**2*g - 4*b**2*d*e**3*h + 3*b**2*d*e**2*f*g)/( 3*f**4) + sqrt(e + f*x)*(-a**2*c*e*f**3*h + a**2*c*f**4*g + a**2*d*e**2*f* *2*h - a**2*d*e*f**3*g + 2*a*b*c*e**2*f**2*h - 2*a*b*c*e*f**3*g - 2*a*b*d* e**3*f*h + 2*a*b*d*e**2*f**2*g - b**2*c*e**3*f*h + b**2*c*e**2*f**2*g + b* *2*d*e**4*h - b**2*d*e**3*f*g)/f**4)/f, Ne(f, 0)), ((a**2*c*g*x + b**2*d*h *x**5/5 + x**4*(2*a*b*d*h + b**2*c*h + b**2*d*g)/4 + x**3*(a**2*d*h + 2*a* b*c*h + 2*a*b*d*g + b**2*c*g)/3 + x**2*(a**2*c*h + a**2*d*g + 2*a*b*c*g)/2 )/sqrt(e), True))
Time = 0.04 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b x)^2 (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \, {\left (35 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{2} d h + 45 \, {\left (b^{2} d f g - {\left (4 \, b^{2} d e - {\left (b^{2} c + 2 \, a b d\right )} f\right )} h\right )} {\left (f x + e\right )}^{\frac {7}{2}} - 63 \, {\left ({\left (3 \, b^{2} d e f - {\left (b^{2} c + 2 \, a b d\right )} f^{2}\right )} g - {\left (6 \, b^{2} d e^{2} - 3 \, {\left (b^{2} c + 2 \, a b d\right )} e f + {\left (2 \, a b c + a^{2} d\right )} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {5}{2}} + 105 \, {\left ({\left (3 \, b^{2} d e^{2} f - 2 \, {\left (b^{2} c + 2 \, a b d\right )} e f^{2} + {\left (2 \, a b c + a^{2} d\right )} f^{3}\right )} g - {\left (4 \, b^{2} d e^{3} - a^{2} c f^{3} - 3 \, {\left (b^{2} c + 2 \, a b d\right )} e^{2} f + 2 \, {\left (2 \, a b c + a^{2} d\right )} e f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {3}{2}} - 315 \, {\left ({\left (b^{2} d e^{3} f - a^{2} c f^{4} - {\left (b^{2} c + 2 \, a b d\right )} e^{2} f^{2} + {\left (2 \, a b c + a^{2} d\right )} e f^{3}\right )} g - {\left (b^{2} d e^{4} - a^{2} c e f^{3} - {\left (b^{2} c + 2 \, a b d\right )} e^{3} f + {\left (2 \, a b c + a^{2} d\right )} e^{2} f^{2}\right )} h\right )} \sqrt {f x + e}\right )}}{315 \, f^{5}} \] Input:
integrate((b*x+a)^2*(d*x+c)*(h*x+g)/(f*x+e)^(1/2),x, algorithm="maxima")
Output:
2/315*(35*(f*x + e)^(9/2)*b^2*d*h + 45*(b^2*d*f*g - (4*b^2*d*e - (b^2*c + 2*a*b*d)*f)*h)*(f*x + e)^(7/2) - 63*((3*b^2*d*e*f - (b^2*c + 2*a*b*d)*f^2) *g - (6*b^2*d*e^2 - 3*(b^2*c + 2*a*b*d)*e*f + (2*a*b*c + a^2*d)*f^2)*h)*(f *x + e)^(5/2) + 105*((3*b^2*d*e^2*f - 2*(b^2*c + 2*a*b*d)*e*f^2 + (2*a*b*c + a^2*d)*f^3)*g - (4*b^2*d*e^3 - a^2*c*f^3 - 3*(b^2*c + 2*a*b*d)*e^2*f + 2*(2*a*b*c + a^2*d)*e*f^2)*h)*(f*x + e)^(3/2) - 315*((b^2*d*e^3*f - a^2*c* f^4 - (b^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*g - (b^2*d*e^4 - a^2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2)*h)*sq rt(f*x + e))/f^5
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (229) = 458\).
Time = 0.12 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b x)^2 (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {f x + e} a^{2} c g + \frac {210 \, {\left ({\left (f x + e\right )}^{\frac {3}{2}} - 3 \, \sqrt {f x + e} e\right )} a b c g}{f} + \frac {105 \, {\left ({\left (f x + e\right )}^{\frac {3}{2}} - 3 \, \sqrt {f x + e} e\right )} a^{2} d g}{f} + \frac {105 \, {\left ({\left (f x + e\right )}^{\frac {3}{2}} - 3 \, \sqrt {f x + e} e\right )} a^{2} c h}{f} + \frac {21 \, {\left (3 \, {\left (f x + e\right )}^{\frac {5}{2}} - 10 \, {\left (f x + e\right )}^{\frac {3}{2}} e + 15 \, \sqrt {f x + e} e^{2}\right )} b^{2} c g}{f^{2}} + \frac {42 \, {\left (3 \, {\left (f x + e\right )}^{\frac {5}{2}} - 10 \, {\left (f x + e\right )}^{\frac {3}{2}} e + 15 \, \sqrt {f x + e} e^{2}\right )} a b d g}{f^{2}} + \frac {42 \, {\left (3 \, {\left (f x + e\right )}^{\frac {5}{2}} - 10 \, {\left (f x + e\right )}^{\frac {3}{2}} e + 15 \, \sqrt {f x + e} e^{2}\right )} a b c h}{f^{2}} + \frac {21 \, {\left (3 \, {\left (f x + e\right )}^{\frac {5}{2}} - 10 \, {\left (f x + e\right )}^{\frac {3}{2}} e + 15 \, \sqrt {f x + e} e^{2}\right )} a^{2} d h}{f^{2}} + \frac {9 \, {\left (5 \, {\left (f x + e\right )}^{\frac {7}{2}} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} e + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} e^{2} - 35 \, \sqrt {f x + e} e^{3}\right )} b^{2} d g}{f^{3}} + \frac {9 \, {\left (5 \, {\left (f x + e\right )}^{\frac {7}{2}} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} e + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} e^{2} - 35 \, \sqrt {f x + e} e^{3}\right )} b^{2} c h}{f^{3}} + \frac {18 \, {\left (5 \, {\left (f x + e\right )}^{\frac {7}{2}} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} e + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} e^{2} - 35 \, \sqrt {f x + e} e^{3}\right )} a b d h}{f^{3}} + \frac {{\left (35 \, {\left (f x + e\right )}^{\frac {9}{2}} - 180 \, {\left (f x + e\right )}^{\frac {7}{2}} e + 378 \, {\left (f x + e\right )}^{\frac {5}{2}} e^{2} - 420 \, {\left (f x + e\right )}^{\frac {3}{2}} e^{3} + 315 \, \sqrt {f x + e} e^{4}\right )} b^{2} d h}{f^{4}}\right )}}{315 \, f} \] Input:
integrate((b*x+a)^2*(d*x+c)*(h*x+g)/(f*x+e)^(1/2),x, algorithm="giac")
Output:
2/315*(315*sqrt(f*x + e)*a^2*c*g + 210*((f*x + e)^(3/2) - 3*sqrt(f*x + e)* e)*a*b*c*g/f + 105*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^2*d*g/f + 105*( (f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^2*c*h/f + 21*(3*(f*x + e)^(5/2) - 1 0*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*b^2*c*g/f^2 + 42*(3*(f*x + e)^ (5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a*b*d*g/f^2 + 42*(3*( f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a*b*c*h/f^2 + 21*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^2 *d*h/f^2 + 9*(5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2 )*e^2 - 35*sqrt(f*x + e)*e^3)*b^2*d*g/f^3 + 9*(5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2)*e^2 - 35*sqrt(f*x + e)*e^3)*b^2*c*h/f^3 + 18*(5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2)*e^2 - 35*sqrt(f*x + e)*e^3)*a*b*d*h/f^3 + (35*(f*x + e)^(9/2) - 180*(f*x + e)^( 7/2)*e + 378*(f*x + e)^(5/2)*e^2 - 420*(f*x + e)^(3/2)*e^3 + 315*sqrt(f*x + e)*e^4)*b^2*d*h/f^4)/f
Time = 0.07 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^2 (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {{\left (e+f\,x\right )}^{7/2}\,\left (2\,b^2\,c\,f\,h-8\,b^2\,d\,e\,h+2\,b^2\,d\,f\,g+4\,a\,b\,d\,f\,h\right )}{7\,f^5}+\frac {{\left (e+f\,x\right )}^{5/2}\,\left (2\,b^2\,c\,f^2\,g+2\,a^2\,d\,f^2\,h+12\,b^2\,d\,e^2\,h+4\,a\,b\,c\,f^2\,h+4\,a\,b\,d\,f^2\,g-6\,b^2\,c\,e\,f\,h-6\,b^2\,d\,e\,f\,g-12\,a\,b\,d\,e\,f\,h\right )}{5\,f^5}+\frac {2\,{\left (e+f\,x\right )}^{3/2}\,\left (a\,f-b\,e\right )\,\left (a\,c\,f^2\,h+a\,d\,f^2\,g+2\,b\,c\,f^2\,g+4\,b\,d\,e^2\,h-2\,a\,d\,e\,f\,h-3\,b\,c\,e\,f\,h-3\,b\,d\,e\,f\,g\right )}{3\,f^5}+\frac {2\,b^2\,d\,h\,{\left (e+f\,x\right )}^{9/2}}{9\,f^5}-\frac {2\,\sqrt {e+f\,x}\,{\left (a\,f-b\,e\right )}^2\,\left (c\,f-d\,e\right )\,\left (e\,h-f\,g\right )}{f^5} \] Input:
int(((g + h*x)*(a + b*x)^2*(c + d*x))/(e + f*x)^(1/2),x)
Output:
((e + f*x)^(7/2)*(2*b^2*c*f*h - 8*b^2*d*e*h + 2*b^2*d*f*g + 4*a*b*d*f*h))/ (7*f^5) + ((e + f*x)^(5/2)*(2*b^2*c*f^2*g + 2*a^2*d*f^2*h + 12*b^2*d*e^2*h + 4*a*b*c*f^2*h + 4*a*b*d*f^2*g - 6*b^2*c*e*f*h - 6*b^2*d*e*f*g - 12*a*b* d*e*f*h))/(5*f^5) + (2*(e + f*x)^(3/2)*(a*f - b*e)*(a*c*f^2*h + a*d*f^2*g + 2*b*c*f^2*g + 4*b*d*e^2*h - 2*a*d*e*f*h - 3*b*c*e*f*h - 3*b*d*e*f*g))/(3 *f^5) + (2*b^2*d*h*(e + f*x)^(9/2))/(9*f^5) - (2*(e + f*x)^(1/2)*(a*f - b* e)^2*(c*f - d*e)*(e*h - f*g))/f^5
Time = 0.16 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x)^2 (c+d x) (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \sqrt {f x +e}\, \left (35 b^{2} d \,f^{4} h \,x^{4}+90 a b d \,f^{4} h \,x^{3}+45 b^{2} c \,f^{4} h \,x^{3}-40 b^{2} d e \,f^{3} h \,x^{3}+45 b^{2} d \,f^{4} g \,x^{3}+63 a^{2} d \,f^{4} h \,x^{2}+126 a b c \,f^{4} h \,x^{2}-108 a b d e \,f^{3} h \,x^{2}+126 a b d \,f^{4} g \,x^{2}-54 b^{2} c e \,f^{3} h \,x^{2}+63 b^{2} c \,f^{4} g \,x^{2}+48 b^{2} d \,e^{2} f^{2} h \,x^{2}-54 b^{2} d e \,f^{3} g \,x^{2}+105 a^{2} c \,f^{4} h x -84 a^{2} d e \,f^{3} h x +105 a^{2} d \,f^{4} g x -168 a b c e \,f^{3} h x +210 a b c \,f^{4} g x +144 a b d \,e^{2} f^{2} h x -168 a b d e \,f^{3} g x +72 b^{2} c \,e^{2} f^{2} h x -84 b^{2} c e \,f^{3} g x -64 b^{2} d \,e^{3} f h x +72 b^{2} d \,e^{2} f^{2} g x -210 a^{2} c e \,f^{3} h +315 a^{2} c \,f^{4} g +168 a^{2} d \,e^{2} f^{2} h -210 a^{2} d e \,f^{3} g +336 a b c \,e^{2} f^{2} h -420 a b c e \,f^{3} g -288 a b d \,e^{3} f h +336 a b d \,e^{2} f^{2} g -144 b^{2} c \,e^{3} f h +168 b^{2} c \,e^{2} f^{2} g +128 b^{2} d \,e^{4} h -144 b^{2} d \,e^{3} f g \right )}{315 f^{5}} \] Input:
int((b*x+a)^2*(d*x+c)*(h*x+g)/(f*x+e)^(1/2),x)
Output:
(2*sqrt(e + f*x)*( - 210*a**2*c*e*f**3*h + 315*a**2*c*f**4*g + 105*a**2*c* f**4*h*x + 168*a**2*d*e**2*f**2*h - 210*a**2*d*e*f**3*g - 84*a**2*d*e*f**3 *h*x + 105*a**2*d*f**4*g*x + 63*a**2*d*f**4*h*x**2 + 336*a*b*c*e**2*f**2*h - 420*a*b*c*e*f**3*g - 168*a*b*c*e*f**3*h*x + 210*a*b*c*f**4*g*x + 126*a* b*c*f**4*h*x**2 - 288*a*b*d*e**3*f*h + 336*a*b*d*e**2*f**2*g + 144*a*b*d*e **2*f**2*h*x - 168*a*b*d*e*f**3*g*x - 108*a*b*d*e*f**3*h*x**2 + 126*a*b*d* f**4*g*x**2 + 90*a*b*d*f**4*h*x**3 - 144*b**2*c*e**3*f*h + 168*b**2*c*e**2 *f**2*g + 72*b**2*c*e**2*f**2*h*x - 84*b**2*c*e*f**3*g*x - 54*b**2*c*e*f** 3*h*x**2 + 63*b**2*c*f**4*g*x**2 + 45*b**2*c*f**4*h*x**3 + 128*b**2*d*e**4 *h - 144*b**2*d*e**3*f*g - 64*b**2*d*e**3*f*h*x + 72*b**2*d*e**2*f**2*g*x + 48*b**2*d*e**2*f**2*h*x**2 - 54*b**2*d*e*f**3*g*x**2 - 40*b**2*d*e*f**3* h*x**3 + 45*b**2*d*f**4*g*x**3 + 35*b**2*d*f**4*h*x**4))/(315*f**5)