Integrand size = 27, antiderivative size = 337 \[ \int \frac {(a+b x)^3 (c+d x) (g+h x)}{(e+f x)^{3/2}} \, dx=-\frac {2 (b e-a f)^3 (d e-c f) (f g-e h)}{f^6 \sqrt {e+f x}}-\frac {2 (b e-a f)^2 (b d e (4 f g-5 e h)-b c f (3 f g-4 e h)-a f (d f g-2 d e h+c f h)) \sqrt {e+f x}}{f^6}-\frac {2 (b e-a f) \left (a^2 d f^2 h+a b f (3 d f g-8 d e h+3 c f h)-b^2 (2 d e (3 f g-5 e h)-3 c f (f g-2 e h))\right ) (e+f x)^{3/2}}{3 f^6}+\frac {2 b \left (3 a^2 d f^2 h+3 a b f (d f g-4 d e h+c f h)-b^2 (2 d e (2 f g-5 e h)-c f (f g-4 e h))\right ) (e+f x)^{5/2}}{5 f^6}+\frac {2 b^2 (3 a d f h+b (d f g-5 d e h+c f h)) (e+f x)^{7/2}}{7 f^6}+\frac {2 b^3 d h (e+f x)^{9/2}}{9 f^6} \] Output:
-2*(-a*f+b*e)^3*(-c*f+d*e)*(-e*h+f*g)/f^6/(f*x+e)^(1/2)-2*(-a*f+b*e)^2*(b* d*e*(-5*e*h+4*f*g)-b*c*f*(-4*e*h+3*f*g)-a*f*(c*f*h-2*d*e*h+d*f*g))*(f*x+e) ^(1/2)/f^6-2/3*(-a*f+b*e)*(a^2*d*f^2*h+a*b*f*(3*c*f*h-8*d*e*h+3*d*f*g)-b^2 *(2*d*e*(-5*e*h+3*f*g)-3*c*f*(-2*e*h+f*g)))*(f*x+e)^(3/2)/f^6+2/5*b*(3*a^2 *d*f^2*h+3*a*b*f*(c*f*h-4*d*e*h+d*f*g)-b^2*(2*d*e*(-5*e*h+2*f*g)-c*f*(-4*e *h+f*g)))*(f*x+e)^(5/2)/f^6+2/7*b^2*(3*a*d*f*h+b*(c*f*h-5*d*e*h+d*f*g))*(f *x+e)^(7/2)/f^6+2/9*b^3*d*h*(f*x+e)^(9/2)/f^6
Time = 0.42 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x)^3 (c+d x) (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 \left (105 a^3 f^3 \left (3 c f (-f g+2 e h+f h x)+d \left (-8 e^2 h+e f (6 g-4 h x)+f^2 x (3 g+h x)\right )\right )+63 a^2 b f^2 \left (5 c f \left (-8 e^2 h+e f (6 g-4 h x)+f^2 x (3 g+h x)\right )+d \left (48 e^3 h-8 e^2 f (5 g-3 h x)+f^3 x^2 (5 g+3 h x)-2 e f^2 x (10 g+3 h x)\right )\right )+9 a b^2 f \left (7 c f \left (48 e^3 h-8 e^2 f (5 g-3 h x)+f^3 x^2 (5 g+3 h x)-2 e f^2 x (10 g+3 h x)\right )-3 d \left (128 e^4 h-16 e^3 f (7 g-4 h x)-8 e^2 f^2 x (7 g+2 h x)+2 e f^3 x^2 (7 g+4 h x)-f^4 x^3 (7 g+5 h x)\right )\right )+b^3 \left (9 c f \left (-128 e^4 h+16 e^3 f (7 g-4 h x)+8 e^2 f^2 x (7 g+2 h x)-2 e f^3 x^2 (7 g+4 h x)+f^4 x^3 (7 g+5 h x)\right )+d \left (1280 e^5 h-128 e^4 f (9 g-5 h x)+16 e^2 f^3 x^2 (9 g+5 h x)-32 e^3 f^2 x (18 g+5 h x)+5 f^5 x^4 (9 g+7 h x)-2 e f^4 x^3 (36 g+25 h x)\right )\right )\right )}{315 f^6 \sqrt {e+f x}} \] Input:
Integrate[((a + b*x)^3*(c + d*x)*(g + h*x))/(e + f*x)^(3/2),x]
Output:
(2*(105*a^3*f^3*(3*c*f*(-(f*g) + 2*e*h + f*h*x) + d*(-8*e^2*h + e*f*(6*g - 4*h*x) + f^2*x*(3*g + h*x))) + 63*a^2*b*f^2*(5*c*f*(-8*e^2*h + e*f*(6*g - 4*h*x) + f^2*x*(3*g + h*x)) + d*(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))) + 9*a*b^2*f*(7*c*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*(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))) + b^3*(9*c*f*(-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)) + d*(1280*e^5*h - 128*e^4*f*(9*g - 5*h *x) + 16*e^2*f^3*x^2*(9*g + 5*h*x) - 32*e^3*f^2*x*(18*g + 5*h*x) + 5*f^5*x ^4*(9*g + 7*h*x) - 2*e*f^4*x^3*(36*g + 25*h*x)))))/(315*f^6*Sqrt[e + f*x])
Time = 0.60 (sec) , antiderivative size = 337, 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.
| \(\displaystyle \int \frac {(a+b x)^3 (c+d x) (g+h x)}{(e+f x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 159 |
| \(\displaystyle \int \left (\frac {b (e+f x)^{3/2} \left (3 a^2 d f^2 h+3 a b f (c f h-4 d e h+d f g)-\left (b^2 (2 d e (2 f g-5 e h)-c f (f g-4 e h))\right )\right )}{f^5}+\frac {\sqrt {e+f x} (b e-a f) \left (-a^2 d f^2 h-a b f (3 c f h-8 d e h+3 d f g)+b^2 (2 d e (3 f g-5 e h)-3 c f (f g-2 e h))\right )}{f^5}+\frac {b^2 (e+f x)^{5/2} (3 a d f h+b (c f h-5 d e h+d f g))}{f^5}+\frac {(b e-a f)^2 (a f (c f h-2 d e h+d f g)+b c f (3 f g-4 e h)-b d e (4 f g-5 e h))}{f^5 \sqrt {e+f x}}+\frac {(a f-b e)^3 (c f-d e) (f g-e h)}{f^5 (e+f x)^{3/2}}+\frac {b^3 d h (e+f x)^{7/2}}{f^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b (e+f x)^{5/2} \left (3 a^2 d f^2 h+3 a b f (c f h-4 d e h+d f g)-\left (b^2 (2 d e (2 f g-5 e h)-c f (f g-4 e h))\right )\right )}{5 f^6}-\frac {2 (e+f x)^{3/2} (b e-a f) \left (a^2 d f^2 h+a b f (3 c f h-8 d e h+3 d f g)-\left (b^2 (2 d e (3 f g-5 e h)-3 c f (f g-2 e h))\right )\right )}{3 f^6}+\frac {2 b^2 (e+f x)^{7/2} (3 a d f h+b (c f h-5 d e h+d f g))}{7 f^6}-\frac {2 \sqrt {e+f x} (b e-a f)^2 (-a f (c f h-2 d e h+d f g)-b c f (3 f g-4 e h)+b d e (4 f g-5 e h))}{f^6}-\frac {2 (b e-a f)^3 (d e-c f) (f g-e h)}{f^6 \sqrt {e+f x}}+\frac {2 b^3 d h (e+f x)^{9/2}}{9 f^6}\) |
Input:
Int[((a + b*x)^3*(c + d*x)*(g + h*x))/(e + f*x)^(3/2),x]
Output:
(-2*(b*e - a*f)^3*(d*e - c*f)*(f*g - e*h))/(f^6*Sqrt[e + f*x]) - (2*(b*e - a*f)^2*(b*d*e*(4*f*g - 5*e*h) - b*c*f*(3*f*g - 4*e*h) - a*f*(d*f*g - 2*d* e*h + c*f*h))*Sqrt[e + f*x])/f^6 - (2*(b*e - a*f)*(a^2*d*f^2*h + a*b*f*(3* d*f*g - 8*d*e*h + 3*c*f*h) - b^2*(2*d*e*(3*f*g - 5*e*h) - 3*c*f*(f*g - 2*e *h)))*(e + f*x)^(3/2))/(3*f^6) + (2*b*(3*a^2*d*f^2*h + 3*a*b*f*(d*f*g - 4* d*e*h + c*f*h) - b^2*(2*d*e*(2*f*g - 5*e*h) - c*f*(f*g - 4*e*h)))*(e + f*x )^(5/2))/(5*f^6) + (2*b^2*(3*a*d*f*h + b*(d*f*g - 5*d*e*h + c*f*h))*(e + f *x)^(7/2))/(7*f^6) + (2*b^3*d*h*(e + f*x)^(9/2))/(9*f^6)
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 = 0.39 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.27
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\frac {x^{3} \left (\frac {5 d h \,x^{2}}{9}+\frac {5 \left (c h +d g \right ) x}{7}+c g \right ) b^{3}}{5}+a \,x^{2} \left (\frac {3 d h \,x^{2}}{7}+\frac {3 \left (c h +d g \right ) x}{5}+c g \right ) b^{2}+3 a^{2} x \left (\frac {d h \,x^{2}}{5}+\frac {\left (c h +d g \right ) x}{3}+c g \right ) b -a^{3} \left (-\frac {d h \,x^{2}}{3}+\left (-c h -d g \right ) x +c g \right )\right ) f^{5}+4 \left (-\frac {x^{2} \left (\frac {25 d h \,x^{2}}{63}+\frac {4 \left (c h +d g \right ) x}{7}+c g \right ) b^{3}}{5}-2 a x \left (\frac {6 d h \,x^{2}}{35}+\frac {3 \left (c h +d g \right ) x}{10}+c g \right ) b^{2}+3 a^{2} \left (-\frac {d h \,x^{2}}{5}+\frac {2 \left (-c h -d g \right ) x}{3}+c g \right ) b +a^{3} \left (c h +d g -\frac {2}{3} d h x \right )\right ) e \,f^{4}-\frac {16 \left (-\frac {3 x \left (\frac {10 d h \,x^{2}}{63}+\frac {2 \left (c h +d g \right ) x}{7}+c g \right ) b^{3}}{5}+3 a \left (-\frac {6 d h \,x^{2}}{35}+\frac {3 \left (-c h -d g \right ) x}{5}+c g \right ) b^{2}+3 a^{2} \left (-\frac {3}{5} d h x +c h +d g \right ) b +d h \,a^{3}\right ) e^{2} f^{3}}{3}+\frac {96 \left (\frac {\left (-\frac {10 d h \,x^{2}}{63}+\frac {4 \left (-c h -d g \right ) x}{7}+c g \right ) b^{2}}{3}+a \left (-\frac {4}{7} d h x +c h +d g \right ) b +a^{2} d h \right ) b \,e^{3} f^{2}}{5}-\frac {768 b^{2} e^{4} \left (\frac {\left (-\frac {5}{9} d h x +c h +d g \right ) b}{3}+a d h \right ) f}{35}+\frac {512 b^{3} d \,e^{5} h}{63}}{\sqrt {f x +e}\, f^{6}}\) | \(428\) |
| risch | \(\frac {2 \left (35 d h \,b^{3} x^{4} f^{4}+135 a \,b^{2} d \,f^{4} h \,x^{3}+45 b^{3} c \,f^{4} h \,x^{3}-85 b^{3} d e \,f^{3} h \,x^{3}+45 b^{3} d \,f^{4} g \,x^{3}+189 a^{2} b d \,f^{4} h \,x^{2}+189 a \,b^{2} c \,f^{4} h \,x^{2}-351 a \,b^{2} d e \,f^{3} h \,x^{2}+189 a \,b^{2} d \,f^{4} g \,x^{2}-117 b^{3} c e \,f^{3} h \,x^{2}+63 b^{3} c \,f^{4} g \,x^{2}+165 b^{3} d \,e^{2} f^{2} h \,x^{2}-117 b^{3} d e \,f^{3} g \,x^{2}+105 a^{3} d \,f^{4} h x +315 a^{2} b c \,f^{4} h x -567 a^{2} b d e \,f^{3} h x +315 a^{2} b d \,f^{4} g x -567 a \,b^{2} c e \,f^{3} h x +315 a \,b^{2} c \,f^{4} g x +783 a \,b^{2} d \,e^{2} f^{2} h x -567 a \,b^{2} d e \,f^{3} g x +261 b^{3} c \,e^{2} f^{2} h x -189 b^{3} c e \,f^{3} g x -325 b^{3} d \,e^{3} f h x +261 b^{3} d \,e^{2} f^{2} g x +315 a^{3} c h \,f^{4}-525 a^{3} d e \,f^{3} h +315 a^{3} d g \,f^{4}-1575 a^{2} b c e \,f^{3} h +945 c g \,a^{2} b \,f^{4}+2079 a^{2} b d \,e^{2} f^{2} h -1575 a^{2} b d e \,f^{3} g +2079 a \,b^{2} c \,e^{2} f^{2} h -1575 a \,b^{2} c e \,f^{3} g -2511 a \,b^{2} d \,e^{3} f h +2079 a \,b^{2} d \,e^{2} f^{2} g -837 b^{3} c \,e^{3} f h +693 b^{3} c \,e^{2} f^{2} g +965 b^{3} d \,e^{4} h -837 b^{3} d \,e^{3} f g \right ) \sqrt {f x +e}}{315 f^{6}}+\frac {2 a^{3} c e \,f^{4} h -2 c g \,a^{3} f^{5}-2 a^{3} d \,e^{2} f^{3} h +2 a^{3} d e \,f^{4} g -6 a^{2} b c \,e^{2} f^{3} h +6 a^{2} b c e \,f^{4} g +6 a^{2} b d \,e^{3} f^{2} h -6 a^{2} b d \,e^{2} f^{3} g +6 a \,b^{2} c \,e^{3} f^{2} h -6 a \,b^{2} c \,e^{2} f^{3} g -6 a \,b^{2} d \,e^{4} f h +6 a \,b^{2} d \,e^{3} f^{2} g -2 b^{3} c \,e^{4} f h +2 b^{3} c \,e^{3} f^{2} g +2 b^{3} d \,e^{5} h -2 b^{3} d \,e^{4} f g}{\sqrt {f x +e}\, f^{6}}\) | \(733\) |
| gosper | \(\frac {4 a^{3} d e \,f^{4} g +\frac {512}{63} b^{3} d \,e^{5} h -\frac {12}{5} a \,b^{2} d e \,f^{4} g \,x^{2}+\frac {48}{5} a^{2} b d \,e^{2} f^{3} h x +\frac {48}{5} a \,b^{2} c \,e^{2} f^{3} h x -8 a^{2} b d e \,f^{4} g x -8 a \,b^{2} c e \,f^{4} g x -\frac {384}{35} a \,b^{2} d \,e^{3} f^{2} h x +\frac {48}{5} a \,b^{2} d \,e^{2} f^{3} g x -\frac {16}{35} b^{3} c e \,f^{4} h \,x^{3}+\frac {6}{5} a \,b^{2} c \,f^{5} h \,x^{3}+\frac {6}{5} a \,b^{2} d \,f^{5} g \,x^{3}+\frac {2}{3} a^{3} d \,f^{5} h \,x^{2}+2 a^{3} c \,f^{5} h x +2 a^{3} d \,f^{5} g x -\frac {12}{5} a^{2} b d e \,f^{4} h \,x^{2}-\frac {12}{5} a \,b^{2} c e \,f^{4} h \,x^{2}-8 a^{2} b c e \,f^{4} h x +\frac {96}{35} a \,b^{2} d \,e^{2} f^{3} h \,x^{2}+\frac {2}{7} b^{3} d \,f^{5} g \,x^{4}+\frac {2}{5} b^{3} c \,f^{5} g \,x^{3}+\frac {2}{7} b^{3} c \,f^{5} h \,x^{4}+\frac {6}{7} a \,b^{2} d \,f^{5} h \,x^{4}-\frac {48}{35} a \,b^{2} d e \,f^{4} h \,x^{3}+\frac {6}{5} a^{2} b d \,f^{5} h \,x^{3}+4 a^{3} c e \,f^{4} h -\frac {16}{3} a^{3} d \,e^{2} f^{3} h -\frac {256}{35} b^{3} c \,e^{4} f h +\frac {32}{5} b^{3} c \,e^{3} f^{2} g -\frac {20}{63} b^{3} d e \,f^{4} h \,x^{4}-16 a^{2} b d \,e^{2} f^{3} g -16 a \,b^{2} c \,e^{2} f^{3} g -\frac {768}{35} a \,b^{2} d \,e^{4} f h +\frac {96}{5} a \,b^{2} d \,e^{3} f^{2} g +\frac {96}{5} a^{2} b d \,e^{3} f^{2} h +12 a^{2} b c e \,f^{4} g -16 a^{2} b c \,e^{2} f^{3} h +\frac {2}{9} d h \,b^{3} x^{5} f^{5}+\frac {96}{5} a \,b^{2} c \,e^{3} f^{2} h -\frac {256}{35} b^{3} d \,e^{4} f g -2 c g \,a^{3} f^{5}-\frac {4}{5} b^{3} c e \,f^{4} g \,x^{2}-\frac {8}{3} a^{3} d e \,f^{4} h x +2 a^{2} b d \,f^{5} g \,x^{2}+2 a \,b^{2} c \,f^{5} g \,x^{2}+\frac {32}{35} b^{3} c \,e^{2} f^{3} h \,x^{2}+6 a^{2} b c \,f^{5} g x -\frac {128}{35} b^{3} c \,e^{3} f^{2} h x +\frac {16}{5} b^{3} c \,e^{2} f^{3} g x +\frac {256}{63} b^{3} d \,e^{4} f h x -\frac {128}{35} b^{3} d \,e^{3} f^{2} g x +2 a^{2} b c \,f^{5} h \,x^{2}-\frac {16}{35} b^{3} d e \,f^{4} g \,x^{3}+\frac {32}{63} b^{3} d \,e^{2} f^{3} h \,x^{3}+\frac {32}{35} b^{3} d \,e^{2} f^{3} g \,x^{2}-\frac {64}{63} b^{3} d \,e^{3} f^{2} h \,x^{2}}{\sqrt {f x +e}\, f^{6}}\) | \(771\) |
| trager | \(\frac {4 a^{3} d e \,f^{4} g +\frac {512}{63} b^{3} d \,e^{5} h -\frac {12}{5} a \,b^{2} d e \,f^{4} g \,x^{2}+\frac {48}{5} a^{2} b d \,e^{2} f^{3} h x +\frac {48}{5} a \,b^{2} c \,e^{2} f^{3} h x -8 a^{2} b d e \,f^{4} g x -8 a \,b^{2} c e \,f^{4} g x -\frac {384}{35} a \,b^{2} d \,e^{3} f^{2} h x +\frac {48}{5} a \,b^{2} d \,e^{2} f^{3} g x -\frac {16}{35} b^{3} c e \,f^{4} h \,x^{3}+\frac {6}{5} a \,b^{2} c \,f^{5} h \,x^{3}+\frac {6}{5} a \,b^{2} d \,f^{5} g \,x^{3}+\frac {2}{3} a^{3} d \,f^{5} h \,x^{2}+2 a^{3} c \,f^{5} h x +2 a^{3} d \,f^{5} g x -\frac {12}{5} a^{2} b d e \,f^{4} h \,x^{2}-\frac {12}{5} a \,b^{2} c e \,f^{4} h \,x^{2}-8 a^{2} b c e \,f^{4} h x +\frac {96}{35} a \,b^{2} d \,e^{2} f^{3} h \,x^{2}+\frac {2}{7} b^{3} d \,f^{5} g \,x^{4}+\frac {2}{5} b^{3} c \,f^{5} g \,x^{3}+\frac {2}{7} b^{3} c \,f^{5} h \,x^{4}+\frac {6}{7} a \,b^{2} d \,f^{5} h \,x^{4}-\frac {48}{35} a \,b^{2} d e \,f^{4} h \,x^{3}+\frac {6}{5} a^{2} b d \,f^{5} h \,x^{3}+4 a^{3} c e \,f^{4} h -\frac {16}{3} a^{3} d \,e^{2} f^{3} h -\frac {256}{35} b^{3} c \,e^{4} f h +\frac {32}{5} b^{3} c \,e^{3} f^{2} g -\frac {20}{63} b^{3} d e \,f^{4} h \,x^{4}-16 a^{2} b d \,e^{2} f^{3} g -16 a \,b^{2} c \,e^{2} f^{3} g -\frac {768}{35} a \,b^{2} d \,e^{4} f h +\frac {96}{5} a \,b^{2} d \,e^{3} f^{2} g +\frac {96}{5} a^{2} b d \,e^{3} f^{2} h +12 a^{2} b c e \,f^{4} g -16 a^{2} b c \,e^{2} f^{3} h +\frac {2}{9} d h \,b^{3} x^{5} f^{5}+\frac {96}{5} a \,b^{2} c \,e^{3} f^{2} h -\frac {256}{35} b^{3} d \,e^{4} f g -2 c g \,a^{3} f^{5}-\frac {4}{5} b^{3} c e \,f^{4} g \,x^{2}-\frac {8}{3} a^{3} d e \,f^{4} h x +2 a^{2} b d \,f^{5} g \,x^{2}+2 a \,b^{2} c \,f^{5} g \,x^{2}+\frac {32}{35} b^{3} c \,e^{2} f^{3} h \,x^{2}+6 a^{2} b c \,f^{5} g x -\frac {128}{35} b^{3} c \,e^{3} f^{2} h x +\frac {16}{5} b^{3} c \,e^{2} f^{3} g x +\frac {256}{63} b^{3} d \,e^{4} f h x -\frac {128}{35} b^{3} d \,e^{3} f^{2} g x +2 a^{2} b c \,f^{5} h \,x^{2}-\frac {16}{35} b^{3} d e \,f^{4} g \,x^{3}+\frac {32}{63} b^{3} d \,e^{2} f^{3} h \,x^{3}+\frac {32}{35} b^{3} d \,e^{2} f^{3} g \,x^{2}-\frac {64}{63} b^{3} d \,e^{3} f^{2} h \,x^{2}}{\sqrt {f x +e}\, f^{6}}\) | \(771\) |
| orering | \(\frac {4 a^{3} d e \,f^{4} g +\frac {512}{63} b^{3} d \,e^{5} h -\frac {12}{5} a \,b^{2} d e \,f^{4} g \,x^{2}+\frac {48}{5} a^{2} b d \,e^{2} f^{3} h x +\frac {48}{5} a \,b^{2} c \,e^{2} f^{3} h x -8 a^{2} b d e \,f^{4} g x -8 a \,b^{2} c e \,f^{4} g x -\frac {384}{35} a \,b^{2} d \,e^{3} f^{2} h x +\frac {48}{5} a \,b^{2} d \,e^{2} f^{3} g x -\frac {16}{35} b^{3} c e \,f^{4} h \,x^{3}+\frac {6}{5} a \,b^{2} c \,f^{5} h \,x^{3}+\frac {6}{5} a \,b^{2} d \,f^{5} g \,x^{3}+\frac {2}{3} a^{3} d \,f^{5} h \,x^{2}+2 a^{3} c \,f^{5} h x +2 a^{3} d \,f^{5} g x -\frac {12}{5} a^{2} b d e \,f^{4} h \,x^{2}-\frac {12}{5} a \,b^{2} c e \,f^{4} h \,x^{2}-8 a^{2} b c e \,f^{4} h x +\frac {96}{35} a \,b^{2} d \,e^{2} f^{3} h \,x^{2}+\frac {2}{7} b^{3} d \,f^{5} g \,x^{4}+\frac {2}{5} b^{3} c \,f^{5} g \,x^{3}+\frac {2}{7} b^{3} c \,f^{5} h \,x^{4}+\frac {6}{7} a \,b^{2} d \,f^{5} h \,x^{4}-\frac {48}{35} a \,b^{2} d e \,f^{4} h \,x^{3}+\frac {6}{5} a^{2} b d \,f^{5} h \,x^{3}+4 a^{3} c e \,f^{4} h -\frac {16}{3} a^{3} d \,e^{2} f^{3} h -\frac {256}{35} b^{3} c \,e^{4} f h +\frac {32}{5} b^{3} c \,e^{3} f^{2} g -\frac {20}{63} b^{3} d e \,f^{4} h \,x^{4}-16 a^{2} b d \,e^{2} f^{3} g -16 a \,b^{2} c \,e^{2} f^{3} g -\frac {768}{35} a \,b^{2} d \,e^{4} f h +\frac {96}{5} a \,b^{2} d \,e^{3} f^{2} g +\frac {96}{5} a^{2} b d \,e^{3} f^{2} h +12 a^{2} b c e \,f^{4} g -16 a^{2} b c \,e^{2} f^{3} h +\frac {2}{9} d h \,b^{3} x^{5} f^{5}+\frac {96}{5} a \,b^{2} c \,e^{3} f^{2} h -\frac {256}{35} b^{3} d \,e^{4} f g -2 c g \,a^{3} f^{5}-\frac {4}{5} b^{3} c e \,f^{4} g \,x^{2}-\frac {8}{3} a^{3} d e \,f^{4} h x +2 a^{2} b d \,f^{5} g \,x^{2}+2 a \,b^{2} c \,f^{5} g \,x^{2}+\frac {32}{35} b^{3} c \,e^{2} f^{3} h \,x^{2}+6 a^{2} b c \,f^{5} g x -\frac {128}{35} b^{3} c \,e^{3} f^{2} h x +\frac {16}{5} b^{3} c \,e^{2} f^{3} g x +\frac {256}{63} b^{3} d \,e^{4} f h x -\frac {128}{35} b^{3} d \,e^{3} f^{2} g x +2 a^{2} b c \,f^{5} h \,x^{2}-\frac {16}{35} b^{3} d e \,f^{4} g \,x^{3}+\frac {32}{63} b^{3} d \,e^{2} f^{3} h \,x^{3}+\frac {32}{35} b^{3} d \,e^{2} f^{3} g \,x^{2}-\frac {64}{63} b^{3} d \,e^{3} f^{2} h \,x^{2}}{\sqrt {f x +e}\, f^{6}}\) | \(771\) |
| derivativedivides | \(\frac {\frac {2 d h \,b^{3} \left (f x +e \right )^{\frac {9}{2}}}{9}+12 a \,b^{2} d \,e^{2} f h \left (f x +e \right )^{\frac {3}{2}}-6 a \,b^{2} d e \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}-6 a \,b^{2} c e \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}-24 a \,b^{2} d \,e^{3} f h \sqrt {f x +e}+18 a^{2} b d \,e^{2} f^{2} h \sqrt {f x +e}+\frac {2 a^{3} d \,f^{3} h \left (f x +e \right )^{\frac {3}{2}}}{3}-12 a^{2} b d e \,f^{3} g \sqrt {f x +e}-12 a^{2} b c e \,f^{3} h \sqrt {f x +e}+18 a \,b^{2} d \,e^{2} f^{2} g \sqrt {f x +e}+18 a \,b^{2} c \,e^{2} f^{2} h \sqrt {f x +e}-12 a \,b^{2} c e \,f^{3} g \sqrt {f x +e}-\frac {24 a \,b^{2} d e f h \left (f x +e \right )^{\frac {5}{2}}}{5}-6 a^{2} b d e \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}+6 b^{3} c \,e^{2} f^{2} g \sqrt {f x +e}-2 b^{3} c e \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}+4 b^{3} d \,e^{2} f g \left (f x +e \right )^{\frac {3}{2}}+2 a^{2} b c \,f^{3} h \left (f x +e \right )^{\frac {3}{2}}+6 a^{2} b c \,f^{4} g \sqrt {f x +e}+\frac {6 a \,b^{2} d \,f^{2} g \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {20 b^{3} d \,e^{3} h \left (f x +e \right )^{\frac {3}{2}}}{3}+2 a^{3} c \,f^{4} h \sqrt {f x +e}-\frac {8 b^{3} c e f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {8 b^{3} d e f g \left (f x +e \right )^{\frac {5}{2}}}{5}+4 b^{3} c \,e^{2} f h \left (f x +e \right )^{\frac {3}{2}}+\frac {2 b^{3} c f h \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {10 b^{3} d e h \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} d f g \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} c \,f^{2} g \left (f x +e \right )^{\frac {5}{2}}}{5}+2 a^{3} d \,f^{4} g \sqrt {f x +e}-8 b^{3} c \,e^{3} f h \sqrt {f x +e}+2 a^{2} b d \,f^{3} g \left (f x +e \right )^{\frac {3}{2}}+2 a \,b^{2} c \,f^{3} g \left (f x +e \right )^{\frac {3}{2}}-8 b^{3} d \,e^{3} f g \sqrt {f x +e}+\frac {6 a \,b^{2} d f h \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} b d \,f^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} c \,f^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}-4 a^{3} d e \,f^{3} h \sqrt {f x +e}+10 b^{3} d \,e^{4} h \sqrt {f x +e}+4 b^{3} d \,e^{2} h \left (f x +e \right )^{\frac {5}{2}}-\frac {2 \left (-a^{3} c e \,f^{4} h +c g \,a^{3} f^{5}+a^{3} d \,e^{2} f^{3} h -a^{3} d e \,f^{4} g +3 a^{2} b c \,e^{2} f^{3} h -3 a^{2} b c e \,f^{4} g -3 a^{2} b d \,e^{3} f^{2} h +3 a^{2} b d \,e^{2} f^{3} g -3 a \,b^{2} c \,e^{3} f^{2} h +3 a \,b^{2} c \,e^{2} f^{3} g +3 a \,b^{2} d \,e^{4} f h -3 a \,b^{2} d \,e^{3} f^{2} g +b^{3} c \,e^{4} f h -b^{3} c \,e^{3} f^{2} g -b^{3} d \,e^{5} h +b^{3} d \,e^{4} f g \right )}{\sqrt {f x +e}}}{f^{6}}\) | \(918\) |
| default | \(\frac {\frac {2 d h \,b^{3} \left (f x +e \right )^{\frac {9}{2}}}{9}+12 a \,b^{2} d \,e^{2} f h \left (f x +e \right )^{\frac {3}{2}}-6 a \,b^{2} d e \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}-6 a \,b^{2} c e \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}-24 a \,b^{2} d \,e^{3} f h \sqrt {f x +e}+18 a^{2} b d \,e^{2} f^{2} h \sqrt {f x +e}+\frac {2 a^{3} d \,f^{3} h \left (f x +e \right )^{\frac {3}{2}}}{3}-12 a^{2} b d e \,f^{3} g \sqrt {f x +e}-12 a^{2} b c e \,f^{3} h \sqrt {f x +e}+18 a \,b^{2} d \,e^{2} f^{2} g \sqrt {f x +e}+18 a \,b^{2} c \,e^{2} f^{2} h \sqrt {f x +e}-12 a \,b^{2} c e \,f^{3} g \sqrt {f x +e}-\frac {24 a \,b^{2} d e f h \left (f x +e \right )^{\frac {5}{2}}}{5}-6 a^{2} b d e \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}+6 b^{3} c \,e^{2} f^{2} g \sqrt {f x +e}-2 b^{3} c e \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}+4 b^{3} d \,e^{2} f g \left (f x +e \right )^{\frac {3}{2}}+2 a^{2} b c \,f^{3} h \left (f x +e \right )^{\frac {3}{2}}+6 a^{2} b c \,f^{4} g \sqrt {f x +e}+\frac {6 a \,b^{2} d \,f^{2} g \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {20 b^{3} d \,e^{3} h \left (f x +e \right )^{\frac {3}{2}}}{3}+2 a^{3} c \,f^{4} h \sqrt {f x +e}-\frac {8 b^{3} c e f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {8 b^{3} d e f g \left (f x +e \right )^{\frac {5}{2}}}{5}+4 b^{3} c \,e^{2} f h \left (f x +e \right )^{\frac {3}{2}}+\frac {2 b^{3} c f h \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {10 b^{3} d e h \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} d f g \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} c \,f^{2} g \left (f x +e \right )^{\frac {5}{2}}}{5}+2 a^{3} d \,f^{4} g \sqrt {f x +e}-8 b^{3} c \,e^{3} f h \sqrt {f x +e}+2 a^{2} b d \,f^{3} g \left (f x +e \right )^{\frac {3}{2}}+2 a \,b^{2} c \,f^{3} g \left (f x +e \right )^{\frac {3}{2}}-8 b^{3} d \,e^{3} f g \sqrt {f x +e}+\frac {6 a \,b^{2} d f h \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} b d \,f^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} c \,f^{2} h \left (f x +e \right )^{\frac {5}{2}}}{5}-4 a^{3} d e \,f^{3} h \sqrt {f x +e}+10 b^{3} d \,e^{4} h \sqrt {f x +e}+4 b^{3} d \,e^{2} h \left (f x +e \right )^{\frac {5}{2}}-\frac {2 \left (-a^{3} c e \,f^{4} h +c g \,a^{3} f^{5}+a^{3} d \,e^{2} f^{3} h -a^{3} d e \,f^{4} g +3 a^{2} b c \,e^{2} f^{3} h -3 a^{2} b c e \,f^{4} g -3 a^{2} b d \,e^{3} f^{2} h +3 a^{2} b d \,e^{2} f^{3} g -3 a \,b^{2} c \,e^{3} f^{2} h +3 a \,b^{2} c \,e^{2} f^{3} g +3 a \,b^{2} d \,e^{4} f h -3 a \,b^{2} d \,e^{3} f^{2} g +b^{3} c \,e^{4} f h -b^{3} c \,e^{3} f^{2} g -b^{3} d \,e^{5} h +b^{3} d \,e^{4} f g \right )}{\sqrt {f x +e}}}{f^{6}}\) | \(918\) |
Input:
int((b*x+a)^3*(d*x+c)*(h*x+g)/(f*x+e)^(3/2),x,method=_RETURNVERBOSE)
Output:
4/(f*x+e)^(1/2)*(1/2*(1/5*x^3*(5/9*d*h*x^2+5/7*(c*h+d*g)*x+c*g)*b^3+a*x^2* (3/7*d*h*x^2+3/5*(c*h+d*g)*x+c*g)*b^2+3*a^2*x*(1/5*d*h*x^2+1/3*(c*h+d*g)*x +c*g)*b-a^3*(-1/3*d*h*x^2+(-c*h-d*g)*x+c*g))*f^5+(-1/5*x^2*(25/63*d*h*x^2+ 4/7*(c*h+d*g)*x+c*g)*b^3-2*a*x*(6/35*d*h*x^2+3/10*(c*h+d*g)*x+c*g)*b^2+3*a ^2*(-1/5*d*h*x^2+2/3*(-c*h-d*g)*x+c*g)*b+a^3*(c*h+d*g-2/3*d*h*x))*e*f^4-4/ 3*(-3/5*x*(10/63*d*h*x^2+2/7*(c*h+d*g)*x+c*g)*b^3+3*a*(-6/35*d*h*x^2+3/5*( -c*h-d*g)*x+c*g)*b^2+3*a^2*(-3/5*d*h*x+c*h+d*g)*b+d*h*a^3)*e^2*f^3+24/5*(1 /3*(-10/63*d*h*x^2+4/7*(-c*h-d*g)*x+c*g)*b^2+a*(-4/7*d*h*x+c*h+d*g)*b+a^2* d*h)*b*e^3*f^2-192/35*b^2*e^4*(1/3*(-5/9*d*h*x+c*h+d*g)*b+a*d*h)*f+128/63* b^3*d*e^5*h)/f^6
Leaf count of result is larger than twice the leaf count of optimal. 637 vs. \(2 (317) = 634\).
Time = 0.13 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.89 \[ \int \frac {(a+b x)^3 (c+d x) (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, b^{3} d f^{5} h x^{5} + 5 \, {\left (9 \, b^{3} d f^{5} g - {\left (10 \, b^{3} d e f^{4} - 9 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} f^{5}\right )} h\right )} x^{4} - {\left (9 \, {\left (8 \, b^{3} d e f^{4} - 7 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} f^{5}\right )} g - {\left (80 \, b^{3} d e^{2} f^{3} - 72 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e f^{4} + 189 \, {\left (a b^{2} c + a^{2} b d\right )} f^{5}\right )} h\right )} x^{3} + {\left (9 \, {\left (16 \, b^{3} d e^{2} f^{3} - 14 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e f^{4} + 35 \, {\left (a b^{2} c + a^{2} b d\right )} f^{5}\right )} g - {\left (160 \, b^{3} d e^{3} f^{2} - 144 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{2} f^{3} + 378 \, {\left (a b^{2} c + a^{2} b d\right )} e f^{4} - 105 \, {\left (3 \, a^{2} b c + a^{3} d\right )} f^{5}\right )} h\right )} x^{2} - 9 \, {\left (128 \, b^{3} d e^{4} f + 35 \, a^{3} c f^{5} - 112 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{3} f^{2} + 280 \, {\left (a b^{2} c + a^{2} b d\right )} e^{2} f^{3} - 70 \, {\left (3 \, a^{2} b c + a^{3} d\right )} e f^{4}\right )} g + 2 \, {\left (640 \, b^{3} d e^{5} + 315 \, a^{3} c e f^{4} - 576 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{4} f + 1512 \, {\left (a b^{2} c + a^{2} b d\right )} e^{3} f^{2} - 420 \, {\left (3 \, a^{2} b c + a^{3} d\right )} e^{2} f^{3}\right )} h - {\left (9 \, {\left (64 \, b^{3} d e^{3} f^{2} - 56 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{2} f^{3} + 140 \, {\left (a b^{2} c + a^{2} b d\right )} e f^{4} - 35 \, {\left (3 \, a^{2} b c + a^{3} d\right )} f^{5}\right )} g - {\left (640 \, b^{3} d e^{4} f + 315 \, a^{3} c f^{5} - 576 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{3} f^{2} + 1512 \, {\left (a b^{2} c + a^{2} b d\right )} e^{2} f^{3} - 420 \, {\left (3 \, a^{2} b c + a^{3} d\right )} e f^{4}\right )} h\right )} x\right )} \sqrt {f x + e}}{315 \, {\left (f^{7} x + e f^{6}\right )}} \] Input:
integrate((b*x+a)^3*(d*x+c)*(h*x+g)/(f*x+e)^(3/2),x, algorithm="fricas")
Output:
2/315*(35*b^3*d*f^5*h*x^5 + 5*(9*b^3*d*f^5*g - (10*b^3*d*e*f^4 - 9*(b^3*c + 3*a*b^2*d)*f^5)*h)*x^4 - (9*(8*b^3*d*e*f^4 - 7*(b^3*c + 3*a*b^2*d)*f^5)* g - (80*b^3*d*e^2*f^3 - 72*(b^3*c + 3*a*b^2*d)*e*f^4 + 189*(a*b^2*c + a^2* b*d)*f^5)*h)*x^3 + (9*(16*b^3*d*e^2*f^3 - 14*(b^3*c + 3*a*b^2*d)*e*f^4 + 3 5*(a*b^2*c + a^2*b*d)*f^5)*g - (160*b^3*d*e^3*f^2 - 144*(b^3*c + 3*a*b^2*d )*e^2*f^3 + 378*(a*b^2*c + a^2*b*d)*e*f^4 - 105*(3*a^2*b*c + a^3*d)*f^5)*h )*x^2 - 9*(128*b^3*d*e^4*f + 35*a^3*c*f^5 - 112*(b^3*c + 3*a*b^2*d)*e^3*f^ 2 + 280*(a*b^2*c + a^2*b*d)*e^2*f^3 - 70*(3*a^2*b*c + a^3*d)*e*f^4)*g + 2* (640*b^3*d*e^5 + 315*a^3*c*e*f^4 - 576*(b^3*c + 3*a*b^2*d)*e^4*f + 1512*(a *b^2*c + a^2*b*d)*e^3*f^2 - 420*(3*a^2*b*c + a^3*d)*e^2*f^3)*h - (9*(64*b^ 3*d*e^3*f^2 - 56*(b^3*c + 3*a*b^2*d)*e^2*f^3 + 140*(a*b^2*c + a^2*b*d)*e*f ^4 - 35*(3*a^2*b*c + a^3*d)*f^5)*g - (640*b^3*d*e^4*f + 315*a^3*c*f^5 - 57 6*(b^3*c + 3*a*b^2*d)*e^3*f^2 + 1512*(a*b^2*c + a^2*b*d)*e^2*f^3 - 420*(3* a^2*b*c + a^3*d)*e*f^4)*h)*x)*sqrt(f*x + e)/(f^7*x + e*f^6)
Leaf count of result is larger than twice the leaf count of optimal. 804 vs. \(2 (352) = 704\).
Time = 62.60 (sec) , antiderivative size = 804, normalized size of antiderivative = 2.39 \[ \int \frac {(a+b x)^3 (c+d x) (g+h x)}{(e+f x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{3} d h \left (e + f x\right )^{\frac {9}{2}}}{9 f^{5}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (3 a b^{2} d f h + b^{3} c f h - 5 b^{3} d e h + b^{3} d f g\right )}{7 f^{5}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} b d f^{2} h + 3 a b^{2} c f^{2} h - 12 a b^{2} d e f h + 3 a b^{2} d f^{2} g - 4 b^{3} c e f h + b^{3} c f^{2} g + 10 b^{3} d e^{2} h - 4 b^{3} d e f g\right )}{5 f^{5}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (a^{3} d f^{3} h + 3 a^{2} b c f^{3} h - 9 a^{2} b d e f^{2} h + 3 a^{2} b d f^{3} g - 9 a b^{2} c e f^{2} h + 3 a b^{2} c f^{3} g + 18 a b^{2} d e^{2} f h - 9 a b^{2} d e f^{2} g + 6 b^{3} c e^{2} f h - 3 b^{3} c e f^{2} g - 10 b^{3} d e^{3} h + 6 b^{3} d e^{2} f g\right )}{3 f^{5}} + \frac {\sqrt {e + f x} \left (a^{3} c f^{4} h - 2 a^{3} d e f^{3} h + a^{3} d f^{4} g - 6 a^{2} b c e f^{3} h + 3 a^{2} b c f^{4} g + 9 a^{2} b d e^{2} f^{2} h - 6 a^{2} b d e f^{3} g + 9 a b^{2} c e^{2} f^{2} h - 6 a b^{2} c e f^{3} g - 12 a b^{2} d e^{3} f h + 9 a b^{2} d e^{2} f^{2} g - 4 b^{3} c e^{3} f h + 3 b^{3} c e^{2} f^{2} g + 5 b^{3} d e^{4} h - 4 b^{3} d e^{3} f g\right )}{f^{5}} + \frac {\left (a f - b e\right )^{3} \left (c f - d e\right ) \left (e h - f g\right )}{f^{5} \sqrt {e + f x}}\right )}{f} & \text {for}\: f \neq 0 \\\frac {a^{3} c g x + \frac {b^{3} d h x^{6}}{6} + \frac {x^{5} \cdot \left (3 a b^{2} d h + b^{3} c h + b^{3} d g\right )}{5} + \frac {x^{4} \cdot \left (3 a^{2} b d h + 3 a b^{2} c h + 3 a b^{2} d g + b^{3} c g\right )}{4} + \frac {x^{3} \left (a^{3} d h + 3 a^{2} b c h + 3 a^{2} b d g + 3 a b^{2} c g\right )}{3} + \frac {x^{2} \left (a^{3} c h + a^{3} d g + 3 a^{2} b c g\right )}{2}}{e^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**3*(d*x+c)*(h*x+g)/(f*x+e)**(3/2),x)
Output:
Piecewise((2*(b**3*d*h*(e + f*x)**(9/2)/(9*f**5) + (e + f*x)**(7/2)*(3*a*b **2*d*f*h + b**3*c*f*h - 5*b**3*d*e*h + b**3*d*f*g)/(7*f**5) + (e + f*x)** (5/2)*(3*a**2*b*d*f**2*h + 3*a*b**2*c*f**2*h - 12*a*b**2*d*e*f*h + 3*a*b** 2*d*f**2*g - 4*b**3*c*e*f*h + b**3*c*f**2*g + 10*b**3*d*e**2*h - 4*b**3*d* e*f*g)/(5*f**5) + (e + f*x)**(3/2)*(a**3*d*f**3*h + 3*a**2*b*c*f**3*h - 9* a**2*b*d*e*f**2*h + 3*a**2*b*d*f**3*g - 9*a*b**2*c*e*f**2*h + 3*a*b**2*c*f **3*g + 18*a*b**2*d*e**2*f*h - 9*a*b**2*d*e*f**2*g + 6*b**3*c*e**2*f*h - 3 *b**3*c*e*f**2*g - 10*b**3*d*e**3*h + 6*b**3*d*e**2*f*g)/(3*f**5) + sqrt(e + f*x)*(a**3*c*f**4*h - 2*a**3*d*e*f**3*h + a**3*d*f**4*g - 6*a**2*b*c*e* f**3*h + 3*a**2*b*c*f**4*g + 9*a**2*b*d*e**2*f**2*h - 6*a**2*b*d*e*f**3*g + 9*a*b**2*c*e**2*f**2*h - 6*a*b**2*c*e*f**3*g - 12*a*b**2*d*e**3*f*h + 9* a*b**2*d*e**2*f**2*g - 4*b**3*c*e**3*f*h + 3*b**3*c*e**2*f**2*g + 5*b**3*d *e**4*h - 4*b**3*d*e**3*f*g)/f**5 + (a*f - b*e)**3*(c*f - d*e)*(e*h - f*g) /(f**5*sqrt(e + f*x)))/f, Ne(f, 0)), ((a**3*c*g*x + b**3*d*h*x**6/6 + x**5 *(3*a*b**2*d*h + b**3*c*h + b**3*d*g)/5 + x**4*(3*a**2*b*d*h + 3*a*b**2*c* h + 3*a*b**2*d*g + b**3*c*g)/4 + x**3*(a**3*d*h + 3*a**2*b*c*h + 3*a**2*b* d*g + 3*a*b**2*c*g)/3 + x**2*(a**3*c*h + a**3*d*g + 3*a**2*b*c*g)/2)/e**(3 /2), True))
Time = 0.04 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.85 \[ \int \frac {(a+b x)^3 (c+d x) (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{3} d h + 45 \, {\left (b^{3} d f g - {\left (5 \, b^{3} d e - {\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} h\right )} {\left (f x + e\right )}^{\frac {7}{2}} - 63 \, {\left ({\left (4 \, b^{3} d e f - {\left (b^{3} c + 3 \, a b^{2} d\right )} f^{2}\right )} g - {\left (10 \, b^{3} d e^{2} - 4 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e f + 3 \, {\left (a b^{2} c + a^{2} b d\right )} f^{2}\right )} h\right )} {\left (f x + e\right )}^{\frac {5}{2}} + 105 \, {\left (3 \, {\left (2 \, b^{3} d e^{2} f - {\left (b^{3} c + 3 \, a b^{2} d\right )} e f^{2} + {\left (a b^{2} c + a^{2} b d\right )} f^{3}\right )} g - {\left (10 \, b^{3} d e^{3} - 6 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{2} f + 9 \, {\left (a b^{2} c + a^{2} b d\right )} e f^{2} - {\left (3 \, a^{2} b c + a^{3} d\right )} f^{3}\right )} h\right )} {\left (f x + e\right )}^{\frac {3}{2}} - 315 \, {\left ({\left (4 \, b^{3} d e^{3} f - 3 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{2} f^{2} + 6 \, {\left (a b^{2} c + a^{2} b d\right )} e f^{3} - {\left (3 \, a^{2} b c + a^{3} d\right )} f^{4}\right )} g - {\left (5 \, b^{3} d e^{4} + a^{3} c f^{4} - 4 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{3} f + 9 \, {\left (a b^{2} c + a^{2} b d\right )} e^{2} f^{2} - 2 \, {\left (3 \, a^{2} b c + a^{3} d\right )} e f^{3}\right )} h\right )} \sqrt {f x + e}}{f^{5}} - \frac {315 \, {\left ({\left (b^{3} d e^{4} f + a^{3} c f^{5} - {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{3} f^{2} + 3 \, {\left (a b^{2} c + a^{2} b d\right )} e^{2} f^{3} - {\left (3 \, a^{2} b c + a^{3} d\right )} e f^{4}\right )} g - {\left (b^{3} d e^{5} + a^{3} c e f^{4} - {\left (b^{3} c + 3 \, a b^{2} d\right )} e^{4} f + 3 \, {\left (a b^{2} c + a^{2} b d\right )} e^{3} f^{2} - {\left (3 \, a^{2} b c + a^{3} d\right )} e^{2} f^{3}\right )} h\right )}}{\sqrt {f x + e} f^{5}}\right )}}{315 \, f} \] Input:
integrate((b*x+a)^3*(d*x+c)*(h*x+g)/(f*x+e)^(3/2),x, algorithm="maxima")
Output:
2/315*((35*(f*x + e)^(9/2)*b^3*d*h + 45*(b^3*d*f*g - (5*b^3*d*e - (b^3*c + 3*a*b^2*d)*f)*h)*(f*x + e)^(7/2) - 63*((4*b^3*d*e*f - (b^3*c + 3*a*b^2*d) *f^2)*g - (10*b^3*d*e^2 - 4*(b^3*c + 3*a*b^2*d)*e*f + 3*(a*b^2*c + a^2*b*d )*f^2)*h)*(f*x + e)^(5/2) + 105*(3*(2*b^3*d*e^2*f - (b^3*c + 3*a*b^2*d)*e* f^2 + (a*b^2*c + a^2*b*d)*f^3)*g - (10*b^3*d*e^3 - 6*(b^3*c + 3*a*b^2*d)*e ^2*f + 9*(a*b^2*c + a^2*b*d)*e*f^2 - (3*a^2*b*c + a^3*d)*f^3)*h)*(f*x + e) ^(3/2) - 315*((4*b^3*d*e^3*f - 3*(b^3*c + 3*a*b^2*d)*e^2*f^2 + 6*(a*b^2*c + a^2*b*d)*e*f^3 - (3*a^2*b*c + a^3*d)*f^4)*g - (5*b^3*d*e^4 + a^3*c*f^4 - 4*(b^3*c + 3*a*b^2*d)*e^3*f + 9*(a*b^2*c + a^2*b*d)*e^2*f^2 - 2*(3*a^2*b* c + a^3*d)*e*f^3)*h)*sqrt(f*x + e))/f^5 - 315*((b^3*d*e^4*f + a^3*c*f^5 - (b^3*c + 3*a*b^2*d)*e^3*f^2 + 3*(a*b^2*c + a^2*b*d)*e^2*f^3 - (3*a^2*b*c + a^3*d)*e*f^4)*g - (b^3*d*e^5 + a^3*c*e*f^4 - (b^3*c + 3*a*b^2*d)*e^4*f + 3*(a*b^2*c + a^2*b*d)*e^3*f^2 - (3*a^2*b*c + a^3*d)*e^2*f^3)*h)/(sqrt(f*x + e)*f^5))/f
Leaf count of result is larger than twice the leaf count of optimal. 965 vs. \(2 (317) = 634\).
Time = 0.15 (sec) , antiderivative size = 965, normalized size of antiderivative = 2.86 \[ \int \frac {(a+b x)^3 (c+d x) (g+h x)}{(e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^3*(d*x+c)*(h*x+g)/(f*x+e)^(3/2),x, algorithm="giac")
Output:
-2*(b^3*d*e^4*f*g - b^3*c*e^3*f^2*g - 3*a*b^2*d*e^3*f^2*g + 3*a*b^2*c*e^2* f^3*g + 3*a^2*b*d*e^2*f^3*g - 3*a^2*b*c*e*f^4*g - a^3*d*e*f^4*g + a^3*c*f^ 5*g - b^3*d*e^5*h + b^3*c*e^4*f*h + 3*a*b^2*d*e^4*f*h - 3*a*b^2*c*e^3*f^2* h - 3*a^2*b*d*e^3*f^2*h + 3*a^2*b*c*e^2*f^3*h + a^3*d*e^2*f^3*h - a^3*c*e* f^4*h)/(sqrt(f*x + e)*f^6) + 2/315*(45*(f*x + e)^(7/2)*b^3*d*f^49*g - 252* (f*x + e)^(5/2)*b^3*d*e*f^49*g + 630*(f*x + e)^(3/2)*b^3*d*e^2*f^49*g - 12 60*sqrt(f*x + e)*b^3*d*e^3*f^49*g + 63*(f*x + e)^(5/2)*b^3*c*f^50*g + 189* (f*x + e)^(5/2)*a*b^2*d*f^50*g - 315*(f*x + e)^(3/2)*b^3*c*e*f^50*g - 945* (f*x + e)^(3/2)*a*b^2*d*e*f^50*g + 945*sqrt(f*x + e)*b^3*c*e^2*f^50*g + 28 35*sqrt(f*x + e)*a*b^2*d*e^2*f^50*g + 315*(f*x + e)^(3/2)*a*b^2*c*f^51*g + 315*(f*x + e)^(3/2)*a^2*b*d*f^51*g - 1890*sqrt(f*x + e)*a*b^2*c*e*f^51*g - 1890*sqrt(f*x + e)*a^2*b*d*e*f^51*g + 945*sqrt(f*x + e)*a^2*b*c*f^52*g + 315*sqrt(f*x + e)*a^3*d*f^52*g + 35*(f*x + e)^(9/2)*b^3*d*f^48*h - 225*(f *x + e)^(7/2)*b^3*d*e*f^48*h + 630*(f*x + e)^(5/2)*b^3*d*e^2*f^48*h - 1050 *(f*x + e)^(3/2)*b^3*d*e^3*f^48*h + 1575*sqrt(f*x + e)*b^3*d*e^4*f^48*h + 45*(f*x + e)^(7/2)*b^3*c*f^49*h + 135*(f*x + e)^(7/2)*a*b^2*d*f^49*h - 252 *(f*x + e)^(5/2)*b^3*c*e*f^49*h - 756*(f*x + e)^(5/2)*a*b^2*d*e*f^49*h + 6 30*(f*x + e)^(3/2)*b^3*c*e^2*f^49*h + 1890*(f*x + e)^(3/2)*a*b^2*d*e^2*f^4 9*h - 1260*sqrt(f*x + e)*b^3*c*e^3*f^49*h - 3780*sqrt(f*x + e)*a*b^2*d*e^3 *f^49*h + 189*(f*x + e)^(5/2)*a*b^2*c*f^50*h + 189*(f*x + e)^(5/2)*a^2*...
Time = 0.15 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^3 (c+d x) (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {{\left (e+f\,x\right )}^{5/2}\,\left (2\,b^3\,c\,f^2\,g+20\,b^3\,d\,e^2\,h+6\,a\,b^2\,c\,f^2\,h+6\,a\,b^2\,d\,f^2\,g+6\,a^2\,b\,d\,f^2\,h-8\,b^3\,c\,e\,f\,h-8\,b^3\,d\,e\,f\,g-24\,a\,b^2\,d\,e\,f\,h\right )}{5\,f^6}-\frac {2\,a^3\,c\,f^5\,g-2\,b^3\,d\,e^5\,h-2\,a^3\,c\,e\,f^4\,h-2\,a^3\,d\,e\,f^4\,g+2\,b^3\,c\,e^4\,f\,h+2\,b^3\,d\,e^4\,f\,g-2\,b^3\,c\,e^3\,f^2\,g+2\,a^3\,d\,e^2\,f^3\,h-6\,a^2\,b\,c\,e\,f^4\,g+6\,a\,b^2\,d\,e^4\,f\,h+6\,a\,b^2\,c\,e^2\,f^3\,g-6\,a\,b^2\,c\,e^3\,f^2\,h-6\,a\,b^2\,d\,e^3\,f^2\,g+6\,a^2\,b\,c\,e^2\,f^3\,h+6\,a^2\,b\,d\,e^2\,f^3\,g-6\,a^2\,b\,d\,e^3\,f^2\,h}{f^6\,\sqrt {e+f\,x}}+\frac {{\left (e+f\,x\right )}^{7/2}\,\left (2\,b^3\,c\,f\,h-10\,b^3\,d\,e\,h+2\,b^3\,d\,f\,g+6\,a\,b^2\,d\,f\,h\right )}{7\,f^6}+\frac {2\,{\left (e+f\,x\right )}^{3/2}\,\left (a\,f-b\,e\right )\,\left (3\,b^2\,c\,f^2\,g+a^2\,d\,f^2\,h+10\,b^2\,d\,e^2\,h+3\,a\,b\,c\,f^2\,h+3\,a\,b\,d\,f^2\,g-6\,b^2\,c\,e\,f\,h-6\,b^2\,d\,e\,f\,g-8\,a\,b\,d\,e\,f\,h\right )}{3\,f^6}+\frac {2\,\sqrt {e+f\,x}\,{\left (a\,f-b\,e\right )}^2\,\left (a\,c\,f^2\,h+a\,d\,f^2\,g+3\,b\,c\,f^2\,g+5\,b\,d\,e^2\,h-2\,a\,d\,e\,f\,h-4\,b\,c\,e\,f\,h-4\,b\,d\,e\,f\,g\right )}{f^6}+\frac {2\,b^3\,d\,h\,{\left (e+f\,x\right )}^{9/2}}{9\,f^6} \] Input:
int(((g + h*x)*(a + b*x)^3*(c + d*x))/(e + f*x)^(3/2),x)
Output:
((e + f*x)^(5/2)*(2*b^3*c*f^2*g + 20*b^3*d*e^2*h + 6*a*b^2*c*f^2*h + 6*a*b ^2*d*f^2*g + 6*a^2*b*d*f^2*h - 8*b^3*c*e*f*h - 8*b^3*d*e*f*g - 24*a*b^2*d* e*f*h))/(5*f^6) - (2*a^3*c*f^5*g - 2*b^3*d*e^5*h - 2*a^3*c*e*f^4*h - 2*a^3 *d*e*f^4*g + 2*b^3*c*e^4*f*h + 2*b^3*d*e^4*f*g - 2*b^3*c*e^3*f^2*g + 2*a^3 *d*e^2*f^3*h - 6*a^2*b*c*e*f^4*g + 6*a*b^2*d*e^4*f*h + 6*a*b^2*c*e^2*f^3*g - 6*a*b^2*c*e^3*f^2*h - 6*a*b^2*d*e^3*f^2*g + 6*a^2*b*c*e^2*f^3*h + 6*a^2 *b*d*e^2*f^3*g - 6*a^2*b*d*e^3*f^2*h)/(f^6*(e + f*x)^(1/2)) + ((e + f*x)^( 7/2)*(2*b^3*c*f*h - 10*b^3*d*e*h + 2*b^3*d*f*g + 6*a*b^2*d*f*h))/(7*f^6) + (2*(e + f*x)^(3/2)*(a*f - b*e)*(3*b^2*c*f^2*g + a^2*d*f^2*h + 10*b^2*d*e^ 2*h + 3*a*b*c*f^2*h + 3*a*b*d*f^2*g - 6*b^2*c*e*f*h - 6*b^2*d*e*f*g - 8*a* b*d*e*f*h))/(3*f^6) + (2*(e + f*x)^(1/2)*(a*f - b*e)^2*(a*c*f^2*h + a*d*f^ 2*g + 3*b*c*f^2*g + 5*b*d*e^2*h - 2*a*d*e*f*h - 4*b*c*e*f*h - 4*b*d*e*f*g) )/f^6 + (2*b^3*d*h*(e + f*x)^(9/2))/(9*f^6)
Time = 0.27 (sec) , antiderivative size = 771, normalized size of antiderivative = 2.29 \[ \int \frac {(a+b x)^3 (c+d x) (g+h x)}{(e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^3*(d*x+c)*(h*x+g)/(f*x+e)^(3/2),x)
Output:
(2*(630*a**3*c*e*f**4*h - 315*a**3*c*f**5*g + 315*a**3*c*f**5*h*x - 840*a* *3*d*e**2*f**3*h + 630*a**3*d*e*f**4*g - 420*a**3*d*e*f**4*h*x + 315*a**3* d*f**5*g*x + 105*a**3*d*f**5*h*x**2 - 2520*a**2*b*c*e**2*f**3*h + 1890*a** 2*b*c*e*f**4*g - 1260*a**2*b*c*e*f**4*h*x + 945*a**2*b*c*f**5*g*x + 315*a* *2*b*c*f**5*h*x**2 + 3024*a**2*b*d*e**3*f**2*h - 2520*a**2*b*d*e**2*f**3*g + 1512*a**2*b*d*e**2*f**3*h*x - 1260*a**2*b*d*e*f**4*g*x - 378*a**2*b*d*e *f**4*h*x**2 + 315*a**2*b*d*f**5*g*x**2 + 189*a**2*b*d*f**5*h*x**3 + 3024* a*b**2*c*e**3*f**2*h - 2520*a*b**2*c*e**2*f**3*g + 1512*a*b**2*c*e**2*f**3 *h*x - 1260*a*b**2*c*e*f**4*g*x - 378*a*b**2*c*e*f**4*h*x**2 + 315*a*b**2* c*f**5*g*x**2 + 189*a*b**2*c*f**5*h*x**3 - 3456*a*b**2*d*e**4*f*h + 3024*a *b**2*d*e**3*f**2*g - 1728*a*b**2*d*e**3*f**2*h*x + 1512*a*b**2*d*e**2*f** 3*g*x + 432*a*b**2*d*e**2*f**3*h*x**2 - 378*a*b**2*d*e*f**4*g*x**2 - 216*a *b**2*d*e*f**4*h*x**3 + 189*a*b**2*d*f**5*g*x**3 + 135*a*b**2*d*f**5*h*x** 4 - 1152*b**3*c*e**4*f*h + 1008*b**3*c*e**3*f**2*g - 576*b**3*c*e**3*f**2* h*x + 504*b**3*c*e**2*f**3*g*x + 144*b**3*c*e**2*f**3*h*x**2 - 126*b**3*c* e*f**4*g*x**2 - 72*b**3*c*e*f**4*h*x**3 + 63*b**3*c*f**5*g*x**3 + 45*b**3* c*f**5*h*x**4 + 1280*b**3*d*e**5*h - 1152*b**3*d*e**4*f*g + 640*b**3*d*e** 4*f*h*x - 576*b**3*d*e**3*f**2*g*x - 160*b**3*d*e**3*f**2*h*x**2 + 144*b** 3*d*e**2*f**3*g*x**2 + 80*b**3*d*e**2*f**3*h*x**3 - 72*b**3*d*e*f**4*g*x** 3 - 50*b**3*d*e*f**4*h*x**4 + 45*b**3*d*f**5*g*x**4 + 35*b**3*d*f**5*h*...