Integrand size = 29, antiderivative size = 517 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=-\frac {2 d (4 a d f h-b (d f g+d e h+2 c f h)) \sqrt {e+f x}}{b^5}-\frac {(b c-a d) \left (7 a^2 d f h+b^2 (4 d e g+c f g+2 c e h)-a b (5 d f g+6 d e h+3 c f h)\right ) \sqrt {e+f x}}{4 b^5 (a+b x)^2}+\frac {\left (55 a^3 d^2 f^2 h-a^2 b d f (29 d f g+78 d e h+58 c f h)-b^3 \left (8 d^2 e^2 g+4 c d e (5 f g+4 e h)+c^2 f (f g+10 e h)\right )+a b^2 \left (11 c^2 f^2 h+12 d^2 e (3 f g+2 e h)+2 c d f (11 f g+36 e h)\right )\right ) \sqrt {e+f x}}{8 b^5 (b e-a f) (a+b x)}+\frac {2 d^2 h (e+f x)^{3/2}}{3 b^4}-\frac {(b c-a d)^2 (b g-a h) (e+f x)^{3/2}}{3 b^4 (a+b x)^3}+\frac {\left (105 a^3 d^2 f^3 h-35 a^2 b d f^2 (d f g+6 d e h+2 c f h)+b^3 \left (c^2 f^2 (f g-6 e h)-8 d^2 e^2 (3 f g+2 e h)-12 c d e f (f g+4 e h)\right )+5 a b^2 f \left (c^2 f^2 h+12 d^2 e (f g+2 e h)+2 c d f (f g+12 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 b^{11/2} (b e-a f)^{3/2}} \] Output:
-2*d*(4*a*d*f*h-b*(2*c*f*h+d*e*h+d*f*g))*(f*x+e)^(1/2)/b^5-1/4*(-a*d+b*c)* (7*a^2*d*f*h+b^2*(2*c*e*h+c*f*g+4*d*e*g)-a*b*(3*c*f*h+6*d*e*h+5*d*f*g))*(f *x+e)^(1/2)/b^5/(b*x+a)^2+1/8*(55*a^3*d^2*f^2*h-a^2*b*d*f*(58*c*f*h+78*d*e *h+29*d*f*g)-b^3*(8*d^2*e^2*g+4*c*d*e*(4*e*h+5*f*g)+c^2*f*(10*e*h+f*g))+a* b^2*(11*c^2*f^2*h+12*d^2*e*(2*e*h+3*f*g)+2*c*d*f*(36*e*h+11*f*g)))*(f*x+e) ^(1/2)/b^5/(-a*f+b*e)/(b*x+a)+2/3*d^2*h*(f*x+e)^(3/2)/b^4-1/3*(-a*d+b*c)^2 *(-a*h+b*g)*(f*x+e)^(3/2)/b^4/(b*x+a)^3+1/8*(105*a^3*d^2*f^3*h-35*a^2*b*d* f^2*(2*c*f*h+6*d*e*h+d*f*g)+b^3*(c^2*f^2*(-6*e*h+f*g)-8*d^2*e^2*(2*e*h+3*f *g)-12*c*d*e*f*(4*e*h+f*g))+5*a*b^2*f*(c^2*f^2*h+12*d^2*e*(2*e*h+f*g)+2*c* d*f*(12*e*h+f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(11/2 )/(-a*f+b*e)^(3/2)
Time = 2.36 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.30 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=\frac {\sqrt {e+f x} \left (315 a^5 d^2 f^2 h-105 a^4 b d f (4 d e h+2 c f h+d f (g-8 h x))-b^5 \left (12 c d e x (f x (5 g-8 h x)+2 e (g+2 h x))-8 d^2 e x^2 (2 f x (3 g+h x)+e (-3 g+8 h x))+c^2 \left (3 f^2 g x^2+4 e^2 (2 g+3 h x)+2 e f x (7 g+15 h x)\right )\right )-a b^4 \left (2 c d \left (2 e f x (11 g-126 h x)+4 e^2 (g+6 h x)+3 f^2 x^2 (-11 g+16 h x)\right )+c^2 \left (4 e^2 h-2 e f (g-11 h x)-f^2 x (8 g+33 h x)\right )+4 d^2 x \left (6 e^2 (g-11 h x)+4 f^2 x^2 (3 g+h x)+e f x (-63 g+52 h x)\right )\right )+a^3 b^2 \left (15 c^2 f^2 h+10 c d f (3 f g+22 e h-56 f h x)+d^2 \left (108 e^2 h+2 e f (55 g-567 h x)+7 f^2 x (-40 g+99 h x)\right )\right )+a^2 b^3 \left (c^2 f (3 f g-8 e h+40 f h x)-2 c d \left (8 e^2 h+e f (8 g-298 h x)+f^2 x (-40 g+231 h x)\right )+d^2 \left (2 e f x (149 g-477 h x)+3 f^2 x^2 (-77 g+48 h x)+e^2 (-8 g+300 h x)\right )\right )\right )}{24 b^5 (b e-a f) (a+b x)^3}-\frac {\left (-105 a^3 d^2 f^3 h+35 a^2 b d f^2 (d f g+6 d e h+2 c f h)+b^3 \left (8 d^2 e^2 (3 f g+2 e h)+12 c d e f (f g+4 e h)+c^2 f^2 (-f g+6 e h)\right )-5 a b^2 f \left (c^2 f^2 h+12 d^2 e (f g+2 e h)+2 c d f (f g+12 e h)\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{8 b^{11/2} (-b e+a f)^{3/2}} \] Input:
Integrate[((c + d*x)^2*(e + f*x)^(3/2)*(g + h*x))/(a + b*x)^4,x]
Output:
(Sqrt[e + f*x]*(315*a^5*d^2*f^2*h - 105*a^4*b*d*f*(4*d*e*h + 2*c*f*h + d*f *(g - 8*h*x)) - b^5*(12*c*d*e*x*(f*x*(5*g - 8*h*x) + 2*e*(g + 2*h*x)) - 8* d^2*e*x^2*(2*f*x*(3*g + h*x) + e*(-3*g + 8*h*x)) + c^2*(3*f^2*g*x^2 + 4*e^ 2*(2*g + 3*h*x) + 2*e*f*x*(7*g + 15*h*x))) - a*b^4*(2*c*d*(2*e*f*x*(11*g - 126*h*x) + 4*e^2*(g + 6*h*x) + 3*f^2*x^2*(-11*g + 16*h*x)) + c^2*(4*e^2*h - 2*e*f*(g - 11*h*x) - f^2*x*(8*g + 33*h*x)) + 4*d^2*x*(6*e^2*(g - 11*h*x ) + 4*f^2*x^2*(3*g + h*x) + e*f*x*(-63*g + 52*h*x))) + a^3*b^2*(15*c^2*f^2 *h + 10*c*d*f*(3*f*g + 22*e*h - 56*f*h*x) + d^2*(108*e^2*h + 2*e*f*(55*g - 567*h*x) + 7*f^2*x*(-40*g + 99*h*x))) + a^2*b^3*(c^2*f*(3*f*g - 8*e*h + 4 0*f*h*x) - 2*c*d*(8*e^2*h + e*f*(8*g - 298*h*x) + f^2*x*(-40*g + 231*h*x)) + d^2*(2*e*f*x*(149*g - 477*h*x) + 3*f^2*x^2*(-77*g + 48*h*x) + e^2*(-8*g + 300*h*x)))))/(24*b^5*(b*e - a*f)*(a + b*x)^3) - ((-105*a^3*d^2*f^3*h + 35*a^2*b*d*f^2*(d*f*g + 6*d*e*h + 2*c*f*h) + b^3*(8*d^2*e^2*(3*f*g + 2*e*h ) + 12*c*d*e*f*(f*g + 4*e*h) + c^2*f^2*(-(f*g) + 6*e*h)) - 5*a*b^2*f*(c^2* f^2*h + 12*d^2*e*(f*g + 2*e*h) + 2*c*d*f*(f*g + 12*e*h)))*ArcTan[(Sqrt[b]* Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(8*b^(11/2)*(-(b*e) + a*f)^(3/2))
Time = 0.81 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {166, 27, 162, 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.
| \(\displaystyle \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {\int \frac {(c+d x) (e+f x)^{3/2} (4 b d e g-b c f g+6 b c e h-4 a d e h-5 a c f h+3 d (b f g+2 b e h-3 a f h) x)}{2 (a+b x)^3}dx}{3 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x) (e+f x)^{3/2} (4 b d e g-b c f g+6 b c e h-4 a d e h-5 a c f h+3 d (b f g+2 b e h-3 a f h) x)}{(a+b x)^3}dx}{6 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {\frac {(e+f x)^{5/2} \left (63 a^4 d^2 f^2 h-3 a^3 b d f (14 c f h+36 d e h+7 d f g)-a^2 b^2 \left (5 c^2 f^2 h-6 c d f (10 e h+f g)-2 d^2 e (22 e h+15 f g)\right )+b x \left (81 a^3 d^2 f^2 h-a^2 b d f (70 c f h+146 d e h+27 d f g)+a b^2 \left (5 c^2 f^2 h+10 c d f (12 e h+f g)+4 d^2 e (16 e h+11 f g)\right )-b^3 \left (c^2 (-f) (f g-6 e h)+12 c d e (4 e h+f g)+16 d^2 e^2 g\right )\right )-a b^3 \left (c^2 f (f g-16 e h)+16 c d e^2 h+8 d^2 e^2 g\right )-2 b^4 c e (6 c e h-c f g+4 d e g)\right )}{4 b^2 (a+b x)^2 (b e-a f)^2}-\frac {3 \left (105 a^3 d^2 f^3 h-35 a^2 b d f^2 (2 c f h+6 d e h+d f g)+5 a b^2 f \left (c^2 f^2 h+2 c d f (12 e h+f g)+12 d^2 e (2 e h+f g)\right )+b^3 \left (c^2 f^2 (f g-6 e h)-12 c d e f (4 e h+f g)-8 d^2 e^2 (2 e h+3 f g)\right )\right ) \int \frac {(e+f x)^{3/2}}{a+b x}dx}{8 b^2 (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {(e+f x)^{5/2} \left (63 a^4 d^2 f^2 h-3 a^3 b d f (14 c f h+36 d e h+7 d f g)-a^2 b^2 \left (5 c^2 f^2 h-6 c d f (10 e h+f g)-2 d^2 e (22 e h+15 f g)\right )+b x \left (81 a^3 d^2 f^2 h-a^2 b d f (70 c f h+146 d e h+27 d f g)+a b^2 \left (5 c^2 f^2 h+10 c d f (12 e h+f g)+4 d^2 e (16 e h+11 f g)\right )-b^3 \left (c^2 (-f) (f g-6 e h)+12 c d e (4 e h+f g)+16 d^2 e^2 g\right )\right )-a b^3 \left (c^2 f (f g-16 e h)+16 c d e^2 h+8 d^2 e^2 g\right )-2 b^4 c e (6 c e h-c f g+4 d e g)\right )}{4 b^2 (a+b x)^2 (b e-a f)^2}-\frac {3 \left (105 a^3 d^2 f^3 h-35 a^2 b d f^2 (2 c f h+6 d e h+d f g)+5 a b^2 f \left (c^2 f^2 h+2 c d f (12 e h+f g)+12 d^2 e (2 e h+f g)\right )+b^3 \left (c^2 f^2 (f g-6 e h)-12 c d e f (4 e h+f g)-8 d^2 e^2 (2 e h+3 f g)\right )\right ) \left (\frac {(b e-a f) \int \frac {\sqrt {e+f x}}{a+b x}dx}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{8 b^2 (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {(e+f x)^{5/2} \left (63 a^4 d^2 f^2 h-3 a^3 b d f (14 c f h+36 d e h+7 d f g)-a^2 b^2 \left (5 c^2 f^2 h-6 c d f (10 e h+f g)-2 d^2 e (22 e h+15 f g)\right )+b x \left (81 a^3 d^2 f^2 h-a^2 b d f (70 c f h+146 d e h+27 d f g)+a b^2 \left (5 c^2 f^2 h+10 c d f (12 e h+f g)+4 d^2 e (16 e h+11 f g)\right )-b^3 \left (c^2 (-f) (f g-6 e h)+12 c d e (4 e h+f g)+16 d^2 e^2 g\right )\right )-a b^3 \left (c^2 f (f g-16 e h)+16 c d e^2 h+8 d^2 e^2 g\right )-2 b^4 c e (6 c e h-c f g+4 d e g)\right )}{4 b^2 (a+b x)^2 (b e-a f)^2}-\frac {3 \left (105 a^3 d^2 f^3 h-35 a^2 b d f^2 (2 c f h+6 d e h+d f g)+5 a b^2 f \left (c^2 f^2 h+2 c d f (12 e h+f g)+12 d^2 e (2 e h+f g)\right )+b^3 \left (c^2 f^2 (f g-6 e h)-12 c d e f (4 e h+f g)-8 d^2 e^2 (2 e h+3 f g)\right )\right ) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b}+\frac {2 \sqrt {e+f x}}{b}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{8 b^2 (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {(e+f x)^{5/2} \left (63 a^4 d^2 f^2 h-3 a^3 b d f (14 c f h+36 d e h+7 d f g)-a^2 b^2 \left (5 c^2 f^2 h-6 c d f (10 e h+f g)-2 d^2 e (22 e h+15 f g)\right )+b x \left (81 a^3 d^2 f^2 h-a^2 b d f (70 c f h+146 d e h+27 d f g)+a b^2 \left (5 c^2 f^2 h+10 c d f (12 e h+f g)+4 d^2 e (16 e h+11 f g)\right )-b^3 \left (c^2 (-f) (f g-6 e h)+12 c d e (4 e h+f g)+16 d^2 e^2 g\right )\right )-a b^3 \left (c^2 f (f g-16 e h)+16 c d e^2 h+8 d^2 e^2 g\right )-2 b^4 c e (6 c e h-c f g+4 d e g)\right )}{4 b^2 (a+b x)^2 (b e-a f)^2}-\frac {3 \left (105 a^3 d^2 f^3 h-35 a^2 b d f^2 (2 c f h+6 d e h+d f g)+5 a b^2 f \left (c^2 f^2 h+2 c d f (12 e h+f g)+12 d^2 e (2 e h+f g)\right )+b^3 \left (c^2 f^2 (f g-6 e h)-12 c d e f (4 e h+f g)-8 d^2 e^2 (2 e h+3 f g)\right )\right ) \left (\frac {(b e-a f) \left (\frac {2 (b e-a f) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b f}+\frac {2 \sqrt {e+f x}}{b}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{8 b^2 (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {(e+f x)^{5/2} \left (63 a^4 d^2 f^2 h-3 a^3 b d f (14 c f h+36 d e h+7 d f g)-a^2 b^2 \left (5 c^2 f^2 h-6 c d f (10 e h+f g)-2 d^2 e (22 e h+15 f g)\right )+b x \left (81 a^3 d^2 f^2 h-a^2 b d f (70 c f h+146 d e h+27 d f g)+a b^2 \left (5 c^2 f^2 h+10 c d f (12 e h+f g)+4 d^2 e (16 e h+11 f g)\right )-b^3 \left (c^2 (-f) (f g-6 e h)+12 c d e (4 e h+f g)+16 d^2 e^2 g\right )\right )-a b^3 \left (c^2 f (f g-16 e h)+16 c d e^2 h+8 d^2 e^2 g\right )-2 b^4 c e (6 c e h-c f g+4 d e g)\right )}{4 b^2 (a+b x)^2 (b e-a f)^2}-\frac {3 \left (\frac {(b e-a f) \left (\frac {2 \sqrt {e+f x}}{b}-\frac {2 \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right ) \left (105 a^3 d^2 f^3 h-35 a^2 b d f^2 (2 c f h+6 d e h+d f g)+5 a b^2 f \left (c^2 f^2 h+2 c d f (12 e h+f g)+12 d^2 e (2 e h+f g)\right )+b^3 \left (c^2 f^2 (f g-6 e h)-12 c d e f (4 e h+f g)-8 d^2 e^2 (2 e h+3 f g)\right )\right )}{8 b^2 (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{3 b (a+b x)^3 (b e-a f)}\) |
Input:
Int[((c + d*x)^2*(e + f*x)^(3/2)*(g + h*x))/(a + b*x)^4,x]
Output:
-1/3*((b*g - a*h)*(c + d*x)^2*(e + f*x)^(5/2))/(b*(b*e - a*f)*(a + b*x)^3) + (((e + f*x)^(5/2)*(63*a^4*d^2*f^2*h - 2*b^4*c*e*(4*d*e*g - c*f*g + 6*c* e*h) - 3*a^3*b*d*f*(7*d*f*g + 36*d*e*h + 14*c*f*h) - a*b^3*(8*d^2*e^2*g + 16*c*d*e^2*h + c^2*f*(f*g - 16*e*h)) - a^2*b^2*(5*c^2*f^2*h - 6*c*d*f*(f*g + 10*e*h) - 2*d^2*e*(15*f*g + 22*e*h)) + b*(81*a^3*d^2*f^2*h - a^2*b*d*f* (27*d*f*g + 146*d*e*h + 70*c*f*h) - b^3*(16*d^2*e^2*g - c^2*f*(f*g - 6*e*h ) + 12*c*d*e*(f*g + 4*e*h)) + a*b^2*(5*c^2*f^2*h + 10*c*d*f*(f*g + 12*e*h) + 4*d^2*e*(11*f*g + 16*e*h)))*x))/(4*b^2*(b*e - a*f)^2*(a + b*x)^2) - (3* (105*a^3*d^2*f^3*h - 35*a^2*b*d*f^2*(d*f*g + 6*d*e*h + 2*c*f*h) + b^3*(c^2 *f^2*(f*g - 6*e*h) - 8*d^2*e^2*(3*f*g + 2*e*h) - 12*c*d*e*f*(f*g + 4*e*h)) + 5*a*b^2*f*(c^2*f^2*h + 12*d^2*e*(f*g + 2*e*h) + 2*c*d*f*(f*g + 12*e*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) )/(8*b^2*(b*e - a*f)^2))/(6*b*(b*e - a*f))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> 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[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.72 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.30
| method | result | size |
| pseudoelliptic | \(\frac {\frac {105 \left (b x +a \right )^{3} \left (\left (\frac {c^{2} g \,f^{3}}{105}-\frac {2 c e \left (c h +2 d g \right ) f^{2}}{35}-\frac {16 \left (c h +\frac {d g}{2}\right ) d \,e^{2} f}{35}-\frac {16 d^{2} e^{3} h}{105}\right ) b^{3}+\frac {a \left (\left (h \,c^{2}+2 c d g \right ) f^{2}+24 \left (c h +\frac {d g}{2}\right ) d e f +24 d^{2} e^{2} h \right ) f \,b^{2}}{21}-\frac {2 a^{2} d \left (\left (c h +\frac {d g}{2}\right ) f +3 d e h \right ) f^{2} b}{3}+a^{3} d^{2} f^{3} h \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8}-\frac {105 \sqrt {\left (a f -b e \right ) b}\, \sqrt {f x +e}\, \left (\left (-\frac {c^{2} f^{2} g \,x^{2}}{105}-\frac {2 x \left (-\frac {24 x^{2} \left (\frac {h x}{3}+g \right ) d^{2}}{7}+\frac {30 x \left (-\frac {8 h x}{5}+g \right ) c d}{7}+c^{2} \left (\frac {15 h x}{7}+g \right )\right ) e f}{45}-\frac {8 \left (\left (-8 h \,x^{3}+3 g \,x^{2}\right ) d^{2}+3 c x \left (2 h x +g \right ) d +c^{2} \left (\frac {3 h x}{2}+g \right )\right ) e^{2}}{315}\right ) b^{5}-\frac {4 a \left (-2 x \left (\left (-2 h \,x^{3}-6 g \,x^{2}\right ) d^{2}+\frac {33 x c \left (-\frac {16 h x}{11}+g \right ) d}{4}+c^{2} \left (\frac {33 h x}{8}+g \right )\right ) f^{2}-\frac {\left (\left (-104 h \,x^{3}+126 g \,x^{2}\right ) d^{2}-22 x c \left (-\frac {126 h x}{11}+g \right ) d +c^{2} \left (-11 h x +g \right )\right ) e f}{2}+\left (\left (-66 h \,x^{2}+6 g x \right ) d^{2}+2 c \left (6 h x +g \right ) d +h \,c^{2}\right ) e^{2}\right ) b^{4}}{315}-\frac {8 a^{2} \left (\left (\left (-18 h \,x^{3}+\frac {231}{8} g \,x^{2}\right ) d^{2}-10 x c \left (-\frac {231 h x}{40}+g \right ) d -\frac {3 c^{2} \left (\frac {40 h x}{3}+g \right )}{8}\right ) f^{2}+\left (\left (\frac {477}{4} h \,x^{2}-\frac {149}{4} g x \right ) d^{2}+2 c \left (-\frac {149 h x}{4}+g \right ) d +h \,c^{2}\right ) e f +2 d \left (\left (-\frac {75 h x}{4}+\frac {g}{2}\right ) d +c h \right ) e^{2}\right ) b^{3}}{315}+\frac {a^{3} \left (\left (-\frac {56 \left (-\frac {99 h x}{40}+g \right ) x \,d^{2}}{3}+2 c \left (-\frac {56 h x}{3}+g \right ) d +h \,c^{2}\right ) f^{2}+\frac {44 d \left (\left (-\frac {567 h x}{110}+\frac {g}{2}\right ) d +c h \right ) e f}{3}+\frac {36 d^{2} e^{2} h}{5}\right ) b^{2}}{21}-\frac {2 a^{4} d \left (\left (\left (-4 h x +\frac {g}{2}\right ) d +c h \right ) f +2 d e h \right ) f b}{3}+a^{5} d^{2} f^{2} h \right )}{8}}{\left (b x +a \right )^{3} b^{5} \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}\) | \(673\) |
| risch | \(-\frac {2 d \left (-h x d b f +12 a d f h -6 b c f h -4 h e d b -3 b d f g \right ) \sqrt {f x +e}}{3 b^{5}}+\frac {\frac {-\frac {b^{2} f \left (55 a^{3} d^{2} f^{2} h -58 a^{2} b c d \,f^{2} h -78 a^{2} b \,d^{2} e f h -29 a^{2} b \,d^{2} f^{2} g +11 a \,b^{2} c^{2} f^{2} h +72 a \,b^{2} c d e f h +22 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +36 a \,b^{2} d^{2} e f g -10 b^{3} c^{2} e f h -b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -20 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \left (f x +e \right )^{\frac {5}{2}}}{8 \left (a f -b e \right )}-\frac {\left (35 a^{3} d^{2} f^{2} h -34 a^{2} b c d \,f^{2} h -54 a^{2} b \,d^{2} e f h -17 a^{2} b \,d^{2} f^{2} g +5 a \,b^{2} c^{2} f^{2} h +48 a \,b^{2} c d e f h +10 a \,b^{2} c d \,f^{2} g +18 a \,b^{2} d^{2} e^{2} h +24 a \,b^{2} d^{2} e f g -6 b^{3} c^{2} e f h +b^{3} c^{2} f^{2} g -12 b^{3} c d \,e^{2} h -12 b^{3} c d e f g -6 b^{3} d^{2} e^{2} g \right ) f b \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {\left (41 a^{4} d^{2} f^{3} h -38 a^{3} b c d \,f^{3} h -107 a^{3} b \,d^{2} e \,f^{2} h -19 a^{3} b \,d^{2} f^{3} g +5 a^{2} b^{2} c^{2} f^{3} h +94 a^{2} b^{2} c d e \,f^{2} h +10 a^{2} b^{2} c d \,f^{3} g +90 a^{2} b^{2} d^{2} e^{2} f h +47 a^{2} b^{2} d^{2} e \,f^{2} g -11 a \,b^{3} c^{2} e \,f^{2} h +a \,b^{3} c^{2} f^{3} g -72 a \,b^{3} c d \,e^{2} f h -22 a \,b^{3} c d e \,f^{2} g -24 a \,b^{3} d^{2} e^{3} h -36 a \,b^{3} d^{2} e^{2} f g +6 b^{4} c^{2} e^{2} f h -b^{4} c^{2} e \,f^{2} g +16 b^{4} c d \,e^{3} h +12 b^{4} c d \,e^{2} f g +8 b^{4} d^{2} e^{3} g \right ) f \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (105 a^{3} d^{2} f^{3} h -70 a^{2} b c d \,f^{3} h -210 a^{2} b \,d^{2} e \,f^{2} h -35 a^{2} b \,d^{2} f^{3} g +5 a \,b^{2} c^{2} f^{3} h +120 a \,b^{2} c d e \,f^{2} h +10 a \,b^{2} c d \,f^{3} g +120 a \,b^{2} d^{2} e^{2} f h +60 a \,b^{2} d^{2} e \,f^{2} g -6 b^{3} c^{2} e \,f^{2} h +b^{3} c^{2} f^{3} g -48 b^{3} c d \,e^{2} f h -12 b^{3} c d e \,f^{2} g -16 b^{3} d^{2} e^{3} h -24 b^{3} d^{2} e^{2} f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}}{b^{5}}\) | \(954\) |
| derivativedivides | \(-\frac {2 d \left (-\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+4 a d f h \sqrt {f x +e}-2 b c f h \sqrt {f x +e}-b d e h \sqrt {f x +e}-b d f g \sqrt {f x +e}\right )}{b^{5}}+\frac {\frac {2 \left (-\frac {b^{2} f \left (55 a^{3} d^{2} f^{2} h -58 a^{2} b c d \,f^{2} h -78 a^{2} b \,d^{2} e f h -29 a^{2} b \,d^{2} f^{2} g +11 a \,b^{2} c^{2} f^{2} h +72 a \,b^{2} c d e f h +22 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +36 a \,b^{2} d^{2} e f g -10 b^{3} c^{2} e f h -b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -20 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 \left (a f -b e \right )}-\frac {\left (35 a^{3} d^{2} f^{2} h -34 a^{2} b c d \,f^{2} h -54 a^{2} b \,d^{2} e f h -17 a^{2} b \,d^{2} f^{2} g +5 a \,b^{2} c^{2} f^{2} h +48 a \,b^{2} c d e f h +10 a \,b^{2} c d \,f^{2} g +18 a \,b^{2} d^{2} e^{2} h +24 a \,b^{2} d^{2} e f g -6 b^{3} c^{2} e f h +b^{3} c^{2} f^{2} g -12 b^{3} c d \,e^{2} h -12 b^{3} c d e f g -6 b^{3} d^{2} e^{2} g \right ) f b \left (f x +e \right )^{\frac {3}{2}}}{6}-\frac {\left (41 a^{4} d^{2} f^{3} h -38 a^{3} b c d \,f^{3} h -107 a^{3} b \,d^{2} e \,f^{2} h -19 a^{3} b \,d^{2} f^{3} g +5 a^{2} b^{2} c^{2} f^{3} h +94 a^{2} b^{2} c d e \,f^{2} h +10 a^{2} b^{2} c d \,f^{3} g +90 a^{2} b^{2} d^{2} e^{2} f h +47 a^{2} b^{2} d^{2} e \,f^{2} g -11 a \,b^{3} c^{2} e \,f^{2} h +a \,b^{3} c^{2} f^{3} g -72 a \,b^{3} c d \,e^{2} f h -22 a \,b^{3} c d e \,f^{2} g -24 a \,b^{3} d^{2} e^{3} h -36 a \,b^{3} d^{2} e^{2} f g +6 b^{4} c^{2} e^{2} f h -b^{4} c^{2} e \,f^{2} g +16 b^{4} c d \,e^{3} h +12 b^{4} c d \,e^{2} f g +8 b^{4} d^{2} e^{3} g \right ) f \sqrt {f x +e}}{16}\right )}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (105 a^{3} d^{2} f^{3} h -70 a^{2} b c d \,f^{3} h -210 a^{2} b \,d^{2} e \,f^{2} h -35 a^{2} b \,d^{2} f^{3} g +5 a \,b^{2} c^{2} f^{3} h +120 a \,b^{2} c d e \,f^{2} h +10 a \,b^{2} c d \,f^{3} g +120 a \,b^{2} d^{2} e^{2} f h +60 a \,b^{2} d^{2} e \,f^{2} g -6 b^{3} c^{2} e \,f^{2} h +b^{3} c^{2} f^{3} g -48 b^{3} c d \,e^{2} f h -12 b^{3} c d e \,f^{2} g -16 b^{3} d^{2} e^{3} h -24 b^{3} d^{2} e^{2} f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}}{b^{5}}\) | \(980\) |
| default | \(-\frac {2 d \left (-\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+4 a d f h \sqrt {f x +e}-2 b c f h \sqrt {f x +e}-b d e h \sqrt {f x +e}-b d f g \sqrt {f x +e}\right )}{b^{5}}+\frac {\frac {2 \left (-\frac {b^{2} f \left (55 a^{3} d^{2} f^{2} h -58 a^{2} b c d \,f^{2} h -78 a^{2} b \,d^{2} e f h -29 a^{2} b \,d^{2} f^{2} g +11 a \,b^{2} c^{2} f^{2} h +72 a \,b^{2} c d e f h +22 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +36 a \,b^{2} d^{2} e f g -10 b^{3} c^{2} e f h -b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -20 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 \left (a f -b e \right )}-\frac {\left (35 a^{3} d^{2} f^{2} h -34 a^{2} b c d \,f^{2} h -54 a^{2} b \,d^{2} e f h -17 a^{2} b \,d^{2} f^{2} g +5 a \,b^{2} c^{2} f^{2} h +48 a \,b^{2} c d e f h +10 a \,b^{2} c d \,f^{2} g +18 a \,b^{2} d^{2} e^{2} h +24 a \,b^{2} d^{2} e f g -6 b^{3} c^{2} e f h +b^{3} c^{2} f^{2} g -12 b^{3} c d \,e^{2} h -12 b^{3} c d e f g -6 b^{3} d^{2} e^{2} g \right ) f b \left (f x +e \right )^{\frac {3}{2}}}{6}-\frac {\left (41 a^{4} d^{2} f^{3} h -38 a^{3} b c d \,f^{3} h -107 a^{3} b \,d^{2} e \,f^{2} h -19 a^{3} b \,d^{2} f^{3} g +5 a^{2} b^{2} c^{2} f^{3} h +94 a^{2} b^{2} c d e \,f^{2} h +10 a^{2} b^{2} c d \,f^{3} g +90 a^{2} b^{2} d^{2} e^{2} f h +47 a^{2} b^{2} d^{2} e \,f^{2} g -11 a \,b^{3} c^{2} e \,f^{2} h +a \,b^{3} c^{2} f^{3} g -72 a \,b^{3} c d \,e^{2} f h -22 a \,b^{3} c d e \,f^{2} g -24 a \,b^{3} d^{2} e^{3} h -36 a \,b^{3} d^{2} e^{2} f g +6 b^{4} c^{2} e^{2} f h -b^{4} c^{2} e \,f^{2} g +16 b^{4} c d \,e^{3} h +12 b^{4} c d \,e^{2} f g +8 b^{4} d^{2} e^{3} g \right ) f \sqrt {f x +e}}{16}\right )}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (105 a^{3} d^{2} f^{3} h -70 a^{2} b c d \,f^{3} h -210 a^{2} b \,d^{2} e \,f^{2} h -35 a^{2} b \,d^{2} f^{3} g +5 a \,b^{2} c^{2} f^{3} h +120 a \,b^{2} c d e \,f^{2} h +10 a \,b^{2} c d \,f^{3} g +120 a \,b^{2} d^{2} e^{2} f h +60 a \,b^{2} d^{2} e \,f^{2} g -6 b^{3} c^{2} e \,f^{2} h +b^{3} c^{2} f^{3} g -48 b^{3} c d \,e^{2} f h -12 b^{3} c d e \,f^{2} g -16 b^{3} d^{2} e^{3} h -24 b^{3} d^{2} e^{2} f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}}{b^{5}}\) | \(980\) |
Input:
int((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
105/8/((a*f-b*e)*b)^(1/2)*((b*x+a)^3*((1/105*c^2*g*f^3-2/35*c*e*(c*h+2*d*g )*f^2-16/35*(c*h+1/2*d*g)*d*e^2*f-16/105*d^2*e^3*h)*b^3+1/21*a*((c^2*h+2*c *d*g)*f^2+24*(c*h+1/2*d*g)*d*e*f+24*d^2*e^2*h)*f*b^2-2/3*a^2*d*((c*h+1/2*d *g)*f+3*d*e*h)*f^2*b+a^3*d^2*f^3*h)*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)*((-1/105*c^2*f^2*g*x^2-2/45*x*(-24 /7*x^2*(1/3*h*x+g)*d^2+30/7*x*(-8/5*h*x+g)*c*d+c^2*(15/7*h*x+g))*e*f-8/315 *((-8*h*x^3+3*g*x^2)*d^2+3*c*x*(2*h*x+g)*d+c^2*(3/2*h*x+g))*e^2)*b^5-4/315 *a*(-2*x*((-2*h*x^3-6*g*x^2)*d^2+33/4*x*c*(-16/11*h*x+g)*d+c^2*(33/8*h*x+g ))*f^2-1/2*((-104*h*x^3+126*g*x^2)*d^2-22*x*c*(-126/11*h*x+g)*d+c^2*(-11*h *x+g))*e*f+((-66*h*x^2+6*g*x)*d^2+2*c*(6*h*x+g)*d+h*c^2)*e^2)*b^4-8/315*a^ 2*(((-18*h*x^3+231/8*g*x^2)*d^2-10*x*c*(-231/40*h*x+g)*d-3/8*c^2*(40/3*h*x +g))*f^2+((477/4*h*x^2-149/4*g*x)*d^2+2*c*(-149/4*h*x+g)*d+h*c^2)*e*f+2*d* ((-75/4*h*x+1/2*g)*d+c*h)*e^2)*b^3+1/21*a^3*((-56/3*(-99/40*h*x+g)*x*d^2+2 *c*(-56/3*h*x+g)*d+h*c^2)*f^2+44/3*d*((-567/110*h*x+1/2*g)*d+c*h)*e*f+36/5 *d^2*e^2*h)*b^2-2/3*a^4*d*(((-4*h*x+1/2*g)*d+c*h)*f+2*d*e*h)*f*b+a^5*d^2*f ^2*h))/(b*x+a)^3/b^5/(a*f-b*e)
Leaf count of result is larger than twice the leaf count of optimal. 1964 vs. \(2 (489) = 978\).
Time = 0.38 (sec) , antiderivative size = 3942, normalized size of antiderivative = 7.62 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**2*(f*x+e)**(3/2)*(h*x+g)/(b*x+a)**4,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4,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. 1653 vs. \(2 (489) = 978\).
Time = 0.17 (sec) , antiderivative size = 1653, normalized size of antiderivative = 3.20 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4,x, algorithm="giac")
Output:
1/8*(24*b^3*d^2*e^2*f*g + 12*b^3*c*d*e*f^2*g - 60*a*b^2*d^2*e*f^2*g - b^3* c^2*f^3*g - 10*a*b^2*c*d*f^3*g + 35*a^2*b*d^2*f^3*g + 16*b^3*d^2*e^3*h + 4 8*b^3*c*d*e^2*f*h - 120*a*b^2*d^2*e^2*f*h + 6*b^3*c^2*e*f^2*h - 120*a*b^2* c*d*e*f^2*h + 210*a^2*b*d^2*e*f^2*h - 5*a*b^2*c^2*f^3*h + 70*a^2*b*c*d*f^3 *h - 105*a^3*d^2*f^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^6 *e - a*b^5*f)*sqrt(-b^2*e + a*b*f)) - 1/24*(24*(f*x + e)^(5/2)*b^5*d^2*e^2 *f*g - 48*(f*x + e)^(3/2)*b^5*d^2*e^3*f*g + 24*sqrt(f*x + e)*b^5*d^2*e^4*f *g + 60*(f*x + e)^(5/2)*b^5*c*d*e*f^2*g - 108*(f*x + e)^(5/2)*a*b^4*d^2*e* f^2*g - 96*(f*x + e)^(3/2)*b^5*c*d*e^2*f^2*g + 240*(f*x + e)^(3/2)*a*b^4*d ^2*e^2*f^2*g + 36*sqrt(f*x + e)*b^5*c*d*e^3*f^2*g - 132*sqrt(f*x + e)*a*b^ 4*d^2*e^3*f^2*g + 3*(f*x + e)^(5/2)*b^5*c^2*f^3*g - 66*(f*x + e)^(5/2)*a*b ^4*c*d*f^3*g + 87*(f*x + e)^(5/2)*a^2*b^3*d^2*f^3*g + 8*(f*x + e)^(3/2)*b^ 5*c^2*e*f^3*g + 176*(f*x + e)^(3/2)*a*b^4*c*d*e*f^3*g - 328*(f*x + e)^(3/2 )*a^2*b^3*d^2*e*f^3*g - 3*sqrt(f*x + e)*b^5*c^2*e^2*f^3*g - 102*sqrt(f*x + e)*a*b^4*c*d*e^2*f^3*g + 249*sqrt(f*x + e)*a^2*b^3*d^2*e^2*f^3*g - 8*(f*x + e)^(3/2)*a*b^4*c^2*f^4*g - 80*(f*x + e)^(3/2)*a^2*b^3*c*d*f^4*g + 136*( f*x + e)^(3/2)*a^3*b^2*d^2*f^4*g + 6*sqrt(f*x + e)*a*b^4*c^2*e*f^4*g + 96* sqrt(f*x + e)*a^2*b^3*c*d*e*f^4*g - 198*sqrt(f*x + e)*a^3*b^2*d^2*e*f^4*g - 3*sqrt(f*x + e)*a^2*b^3*c^2*f^5*g - 30*sqrt(f*x + e)*a^3*b^2*c*d*f^5*g + 57*sqrt(f*x + e)*a^4*b*d^2*f^5*g + 48*(f*x + e)^(5/2)*b^5*c*d*e^2*f*h ...
Time = 0.41 (sec) , antiderivative size = 1109, normalized size of antiderivative = 2.15 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:
int(((e + f*x)^(3/2)*(g + h*x)*(c + d*x)^2)/(a + b*x)^4,x)
Output:
(e + f*x)^(1/2)*((2*d^2*f*g - 6*d^2*e*h + 4*c*d*f*h)/b^4 + (2*d^2*h*(4*b^4 *e - 4*a*b^3*f))/b^8) - ((e + f*x)^(1/2)*((41*a^4*d^2*f^4*h)/8 + (a*b^3*c^ 2*f^4*g)/8 - (19*a^3*b*d^2*f^4*g)/8 - (b^4*c^2*e*f^3*g)/8 + b^4*d^2*e^3*f* g + (5*a^2*b^2*c^2*f^4*h)/8 + (3*b^4*c^2*e^2*f^2*h)/4 - (9*a*b^3*d^2*e^2*f ^2*g)/2 + (47*a^2*b^2*d^2*e*f^3*g)/8 - (19*a^3*b*c*d*f^4*h)/4 + 2*b^4*c*d* e^3*f*h + (45*a^2*b^2*d^2*e^2*f^2*h)/4 + (5*a^2*b^2*c*d*f^4*g)/4 - (11*a*b ^3*c^2*e*f^3*h)/8 - 3*a*b^3*d^2*e^3*f*h - (107*a^3*b*d^2*e*f^3*h)/8 + (3*b ^4*c*d*e^2*f^2*g)/2 - 9*a*b^3*c*d*e^2*f^2*h + (47*a^2*b^2*c*d*e*f^3*h)/4 - (11*a*b^3*c*d*e*f^3*g)/4) + (e + f*x)^(3/2)*((b^4*c^2*f^3*g)/3 + (5*a*b^3 *c^2*f^3*h)/3 + (35*a^3*b*d^2*f^3*h)/3 - 2*b^4*c^2*e*f^2*h - 2*b^4*d^2*e^2 *f*g - (17*a^2*b^2*d^2*f^3*g)/3 - 18*a^2*b^2*d^2*e*f^2*h + (10*a*b^3*c*d*f ^3*g)/3 - 4*b^4*c*d*e*f^2*g - 4*b^4*c*d*e^2*f*h - (34*a^2*b^2*c*d*f^3*h)/3 + 8*a*b^3*d^2*e*f^2*g + 6*a*b^3*d^2*e^2*f*h + 16*a*b^3*c*d*e*f^2*h) - ((e + f*x)^(5/2)*(b^5*c^2*f^3*g - 11*a*b^4*c^2*f^3*h + 10*b^5*c^2*e*f^2*h + 8 *b^5*d^2*e^2*f*g + 29*a^2*b^3*d^2*f^3*g - 55*a^3*b^2*d^2*f^3*h + 78*a^2*b^ 3*d^2*e*f^2*h - 22*a*b^4*c*d*f^3*g + 20*b^5*c*d*e*f^2*g + 16*b^5*c*d*e^2*f *h + 58*a^2*b^3*c*d*f^3*h - 36*a*b^4*d^2*e*f^2*g - 24*a*b^4*d^2*e^2*f*h - 72*a*b^4*c*d*e*f^2*h))/(8*(a*f - b*e)))/(b^8*(e + f*x)^3 - (e + f*x)^2*(3* b^8*e - 3*a*b^7*f) + (e + f*x)*(3*b^8*e^2 + 3*a^2*b^6*f^2 - 6*a*b^7*e*f) - b^8*e^3 + a^3*b^5*f^3 - 3*a^2*b^6*e*f^2 + 3*a*b^7*e^2*f) + (2*d^2*h*(e...
Time = 0.20 (sec) , antiderivative size = 4947, normalized size of antiderivative = 9.57 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4,x)
Output:
(315*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b* e)))*a**6*d**2*f**3*h - 210*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b) /(sqrt(b)*sqrt(a*f - b*e)))*a**5*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**5*b*d**2*e*f**2*h - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b* e)))*a**5*b*d**2*f**3*g + 945*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)* b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*b*d**2*f**3*h*x + 15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*c**2*f** 3*h + 360*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*c*d*e*f**2*h + 30*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*c*d*f**3*g - 630*sqrt(b)*s qrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2 *c*d*f**3*h*x + 360*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b )*sqrt(a*f - b*e)))*a**4*b**2*d**2*e**2*f*h + 180*sqrt(b)*sqrt(a*f - b*e)* atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d**2*e*f**2*g - 1890*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d**2*e*f**2*h*x - 315*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt( e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d**2*f**3*g*x + 945*sqrt( b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4* b**2*d**2*f**3*h*x**2 - 18*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*...