Integrand size = 29, antiderivative size = 856 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=-\frac {d \left (a^3 d^2 f^2 h+a^2 b d f (5 d f g+6 d e h-14 c f h)+b^3 \left (32 d^2 e^2 g-4 c d e (7 f g+6 e h)+c^2 f (f g+18 e h)\right )-a b^2 \left (19 c^2 f^2 h+4 d^2 e (9 f g+2 e h)-2 c d f (13 f g+20 e h)\right )\right ) \sqrt {e+f x}}{8 b^2 (b c-a d)^4 (b e-a f) (c+d x)}-\frac {\left (a^2 d f h-b^2 (8 d e g-3 c f g-6 c e h)+a b (5 d f g+2 d e h-9 c f h)\right ) \sqrt {e+f x}}{12 b^2 (b c-a d)^2 (a+b x)^2 (c+d x)}-\frac {\left (a^3 d^2 f^2 h+a^2 b d f (5 d f g+8 d e h-16 c f h)+b^3 \left (48 d^2 e^2 g+3 c^2 f (f g+10 e h)-2 c d e (23 f g+18 e h)\right )-a b^2 \left (33 c^2 f^2 h+2 d^2 e (25 f g+6 e h)-2 c d f (20 f g+29 e h)\right )\right ) \sqrt {e+f x}}{24 b^2 (b c-a d)^3 (b e-a f) (a+b x) (c+d x)}-\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)}-\frac {\left (a^4 d^3 f^3 h+a^3 b d^2 f^2 (5 d f g+6 d e h-15 c f h)-3 a^2 b^2 d f \left (15 c^2 f^2 h-15 c d f (f g+2 e h)+4 d^2 e (5 f g+2 e h)\right )-b^4 \left (64 d^3 e^3 g+c^3 f^2 (f g-6 e h)-24 c d^2 e^2 (3 f g+2 e h)+12 c^2 d e f (f g+4 e h)\right )-a b^3 \left (5 c^3 f^3 h+120 c d^2 e f (f g+e h)-8 d^3 e^2 (15 f g+2 e h)-15 c^2 d f^2 (f g+6 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 b^{3/2} (b c-a d)^5 (b e-a f)^{3/2}}+\frac {\sqrt {d} \sqrt {d e-c f} \left (a d (3 d f g+2 d e h-5 c f h)-b \left (8 d^2 e g+3 c^2 f h-c d (5 f g+6 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d)^5} \] Output:
-1/8*d*(a^3*d^2*f^2*h+a^2*b*d*f*(-14*c*f*h+6*d*e*h+5*d*f*g)+b^3*(32*d^2*e^ 2*g-4*c*d*e*(6*e*h+7*f*g)+c^2*f*(18*e*h+f*g))-a*b^2*(19*c^2*f^2*h+4*d^2*e* (2*e*h+9*f*g)-2*c*d*f*(20*e*h+13*f*g)))*(f*x+e)^(1/2)/b^2/(-a*d+b*c)^4/(-a *f+b*e)/(d*x+c)-1/12*(a^2*d*f*h-b^2*(-6*c*e*h-3*c*f*g+8*d*e*g)+a*b*(-9*c*f *h+2*d*e*h+5*d*f*g))*(f*x+e)^(1/2)/b^2/(-a*d+b*c)^2/(b*x+a)^2/(d*x+c)-1/24 *(a^3*d^2*f^2*h+a^2*b*d*f*(-16*c*f*h+8*d*e*h+5*d*f*g)+b^3*(48*d^2*e^2*g+3* c^2*f*(10*e*h+f*g)-2*c*d*e*(18*e*h+23*f*g))-a*b^2*(33*c^2*f^2*h+2*d^2*e*(6 *e*h+25*f*g)-2*c*d*f*(29*e*h+20*f*g)))*(f*x+e)^(1/2)/b^2/(-a*d+b*c)^3/(-a* f+b*e)/(b*x+a)/(d*x+c)-1/3*(-a*h+b*g)*(f*x+e)^(3/2)/b/(-a*d+b*c)/(b*x+a)^3 /(d*x+c)-1/8*(a^4*d^3*f^3*h+a^3*b*d^2*f^2*(-15*c*f*h+6*d*e*h+5*d*f*g)-3*a^ 2*b^2*d*f*(15*c^2*f^2*h-15*c*d*f*(2*e*h+f*g)+4*d^2*e*(2*e*h+5*f*g))-b^4*(6 4*d^3*e^3*g+c^3*f^2*(-6*e*h+f*g)-24*c*d^2*e^2*(2*e*h+3*f*g)+12*c^2*d*e*f*( 4*e*h+f*g))-a*b^3*(5*c^3*f^3*h+120*c*d^2*e*f*(e*h+f*g)-8*d^3*e^2*(2*e*h+15 *f*g)-15*c^2*d*f^2*(6*e*h+f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^ (1/2))/b^(3/2)/(-a*d+b*c)^5/(-a*f+b*e)^(3/2)+d^(1/2)*(-c*f+d*e)^(1/2)*(a*d *(-5*c*f*h+2*d*e*h+3*d*f*g)-b*(8*d^2*e*g+3*c^2*f*h-c*d*(6*e*h+5*f*g)))*arc tanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/(-a*d+b*c)^5
Leaf count is larger than twice the leaf count of optimal. \(6288\) vs. \(2(856)=1712\).
Time = 16.36 (sec) , antiderivative size = 6288, normalized size of antiderivative = 7.35 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^(3/2)*(g + h*x))/((a + b*x)^4*(c + d*x)^2),x]
Output:
Result too large to show
Time = 2.01 (sec) , antiderivative size = 918, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {166, 27, 166, 27, 168, 27, 168, 27, 174, 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.
\(\displaystyle \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx\) |
\(\Big \downarrow \) 166 |
\(\displaystyle \frac {\int -\frac {\sqrt {e+f x} (8 b d e g-3 b c f g-6 b c e h-2 a d e h+3 a c f h+f (5 b d g-6 b c h+a d h) x)}{2 (a+b x)^3 (c+d x)^2}dx}{3 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{3 b (a+b x)^3 (c+d x) (b c-a d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {\sqrt {e+f x} (8 b d e g-a (2 d e-3 c f) h-3 b c (f g+2 e h)+f (5 b d g-6 b c h+a d h) x)}{(a+b x)^3 (c+d x)^2}dx}{6 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{3 b (a+b x)^3 (c+d x) (b c-a d)}\) |
\(\Big \downarrow \) 166 |
\(\displaystyle -\frac {\frac {\int -\frac {8 b e (d e-c f) (4 b d g-3 b c h-a d h)-(2 d e-c f) \left (d f h a^2+b (5 d f g+2 d e h-9 c f h) a-b^2 (8 d e g-3 c f g-6 c e h)\right )-f \left (-\left (\left (24 f h c^2-5 d (7 f g+6 e h) c+40 d^2 e g\right ) b^2\right )-a d (17 c f h-5 d (f g+2 e h)) b+a^2 d^2 f h\right ) x}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}dx}{2 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-9 c f h+2 d e h+5 d f g)-b^2 (-6 c e h-3 c f g+8 d e g)\right )}{2 b (a+b x)^2 (c+d x) (b c-a d)}}{6 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{3 b (a+b x)^3 (c+d x) (b c-a d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-9 c f h+2 d e h+5 d f g)-b^2 (-6 c e h-3 c f g+8 d e g)\right )}{2 b (a+b x)^2 (c+d x) (b c-a d)}-\frac {\int \frac {8 b e (d e-c f) (4 b d g-3 b c h-a d h)-(2 d e-c f) \left (d f h a^2+b (5 d f g+2 d e h-9 c f h) a-b^2 (8 d e g-3 c f g-6 c e h)\right )-f \left (-\left (\left (24 f h c^2-5 d (7 f g+6 e h) c+40 d^2 e g\right ) b^2\right )-a d (17 c f h-5 d (f g+2 e h)) b+a^2 d^2 f h\right ) x}{(a+b x)^2 (c+d x)^2 \sqrt {e+f x}}dx}{4 b (b c-a d)}}{6 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{3 b (a+b x)^3 (c+d x) (b c-a d)}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle -\frac {\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-9 c f h+2 d e h+5 d f g)-b^2 (-6 c e h-3 c f g+8 d e g)\right )}{2 b (a+b x)^2 (c+d x) (b c-a d)}-\frac {-\frac {\int \frac {3 \left (d^2 f^2 (2 d e-c f) h a^3+b d f \left (2 e (5 f g+6 e h) d^2-c f (5 f g+32 e h) d+12 c^2 f^2 h\right ) a^2+b^2 \left (-8 e^2 (9 f g+2 e h) d^3+2 c e f (37 f g+42 e h) d^2-12 c^2 f^2 (f g+5 e h) d+5 c^3 f^3 h\right ) a+b^3 \left (f^2 (f g-6 e h) c^3+12 d e f (f g+4 e h) c^2-24 d^2 e^2 (3 f g+2 e h) c+64 d^3 e^3 g\right )+d f \left (d^2 f^2 h a^3+b d f (5 d f g+8 d e h-16 c f h) a^2-b^2 \left (2 e (25 f g+6 e h) d^2-2 c f (20 f g+29 e h) d+33 c^2 f^2 h\right ) a+b^3 \left (3 f (f g+10 e h) c^2-2 d e (23 f g+18 e h) c+48 d^2 e^2 g\right )\right ) x\right )}{2 (a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (a^3 d^2 f^2 h+a^2 b d f (-16 c f h+8 d e h+5 d f g)-a b^2 \left (33 c^2 f^2 h-2 c d f (29 e h+20 f g)+2 d^2 e (6 e h+25 f g)\right )+b^3 \left (3 c^2 f (10 e h+f g)-2 c d e (18 e h+23 f g)+48 d^2 e^2 g\right )\right )}{(a+b x) (c+d x) (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{3 b (a+b x)^3 (c+d x) (b c-a d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-9 c f h+2 d e h+5 d f g)-b^2 (-6 c e h-3 c f g+8 d e g)\right )}{2 b (a+b x)^2 (c+d x) (b c-a d)}-\frac {-\frac {3 \int \frac {d^2 f^2 (2 d e-c f) h a^3+b d f \left (2 e (5 f g+6 e h) d^2-c f (5 f g+32 e h) d+12 c^2 f^2 h\right ) a^2+b^2 \left (-8 e^2 (9 f g+2 e h) d^3+2 c e f (37 f g+42 e h) d^2-12 c^2 f^2 (f g+5 e h) d+5 c^3 f^3 h\right ) a+b^3 \left (f^2 (f g-6 e h) c^3+12 d e f (f g+4 e h) c^2-24 d^2 e^2 (3 f g+2 e h) c+64 d^3 e^3 g\right )+d f \left (d^2 f^2 h a^3+b d f (5 d f g+8 d e h-16 c f h) a^2-b^2 \left (2 e (25 f g+6 e h) d^2-2 c f (20 f g+29 e h) d+33 c^2 f^2 h\right ) a+b^3 \left (3 f (f g+10 e h) c^2-2 d e (23 f g+18 e h) c+48 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (a^3 d^2 f^2 h+a^2 b d f (-16 c f h+8 d e h+5 d f g)-a b^2 \left (33 c^2 f^2 h-2 c d f (29 e h+20 f g)+2 d^2 e (6 e h+25 f g)\right )+b^3 \left (3 c^2 f (10 e h+f g)-2 c d e (18 e h+23 f g)+48 d^2 e^2 g\right )\right )}{(a+b x) (c+d x) (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{3 b (a+b x)^3 (c+d x) (b c-a d)}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)}-\frac {\frac {\left (d f h a^2+b (5 d f g+2 d e h-9 c f h) a-b^2 (8 d e g-3 c f g-6 c e h)\right ) \sqrt {e+f x}}{2 b (b c-a d) (a+b x)^2 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (5 d f g+8 d e h-16 c f h) a^2-b^2 \left (2 e (25 f g+6 e h) d^2-2 c f (20 f g+29 e h) d+33 c^2 f^2 h\right ) a+b^3 \left (3 f (f g+10 e h) c^2-2 d e (23 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}-\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (5 d f g+6 d e h-14 c f h) a^2-b^2 \left (4 e (9 f g+2 e h) d^2-2 c f (13 f g+20 e h) d+19 c^2 f^2 h\right ) a+b^3 \left (f (f g+18 e h) c^2-4 d e (7 f g+6 e h) c+32 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)}+\frac {\int \frac {b (d e-c f) \left (\left (f^2 (f g-6 e h) c^3+12 d e f (f g+4 e h) c^2-24 d^2 e^2 (3 f g+2 e h) c+64 d^3 e^3 g\right ) b^3+a \left (-8 e^2 (11 f g+2 e h) d^3+4 c e f (23 f g+24 e h) d^2-2 c^2 f^2 (7 f g+36 e h) d+5 c^3 f^3 h\right ) b^2+a^2 d f \left (8 e (3 f g+2 e h) d^2-c f (19 f g+50 e h) d+26 c^2 f^2 h\right ) b+a^3 c d^2 f^3 h+d f \left (d^2 f^2 h a^3+b d f (5 d f g+6 d e h-14 c f h) a^2-b^2 \left (4 e (9 f g+2 e h) d^2-2 c f (13 f g+20 e h) d+19 c^2 f^2 h\right ) a+b^3 \left (f (f g+18 e h) c^2-4 d e (7 f g+6 e h) c+32 d^2 e^2 g\right )\right ) x\right )}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)}-\frac {\frac {\left (d f h a^2+b (5 d f g+2 d e h-9 c f h) a-b^2 (8 d e g-3 c f g-6 c e h)\right ) \sqrt {e+f x}}{2 b (b c-a d) (a+b x)^2 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (5 d f g+8 d e h-16 c f h) a^2-b^2 \left (2 e (25 f g+6 e h) d^2-2 c f (20 f g+29 e h) d+33 c^2 f^2 h\right ) a+b^3 \left (3 f (f g+10 e h) c^2-2 d e (23 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}-\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (5 d f g+6 d e h-14 c f h) a^2-b^2 \left (4 e (9 f g+2 e h) d^2-2 c f (13 f g+20 e h) d+19 c^2 f^2 h\right ) a+b^3 \left (f (f g+18 e h) c^2-4 d e (7 f g+6 e h) c+32 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)}+\frac {b \int \frac {\left (f^2 (f g-6 e h) c^3+12 d e f (f g+4 e h) c^2-24 d^2 e^2 (3 f g+2 e h) c+64 d^3 e^3 g\right ) b^3+a \left (-8 e^2 (11 f g+2 e h) d^3+4 c e f (23 f g+24 e h) d^2-2 c^2 f^2 (7 f g+36 e h) d+5 c^3 f^3 h\right ) b^2+a^2 d f \left (8 e (3 f g+2 e h) d^2-c f (19 f g+50 e h) d+26 c^2 f^2 h\right ) b+a^3 c d^2 f^3 h+d f \left (d^2 f^2 h a^3+b d f (5 d f g+6 d e h-14 c f h) a^2-b^2 \left (4 e (9 f g+2 e h) d^2-2 c f (13 f g+20 e h) d+19 c^2 f^2 h\right ) a+b^3 \left (f (f g+18 e h) c^2-4 d e (7 f g+6 e h) c+32 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{b c-a d}\right )}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)}-\frac {\frac {\left (d f h a^2+b (5 d f g+2 d e h-9 c f h) a-b^2 (8 d e g-3 c f g-6 c e h)\right ) \sqrt {e+f x}}{2 b (b c-a d) (a+b x)^2 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (5 d f g+8 d e h-16 c f h) a^2-b^2 \left (2 e (25 f g+6 e h) d^2-2 c f (20 f g+29 e h) d+33 c^2 f^2 h\right ) a+b^3 \left (3 f (f g+10 e h) c^2-2 d e (23 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}-\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (5 d f g+6 d e h-14 c f h) a^2-b^2 \left (4 e (9 f g+2 e h) d^2-2 c f (13 f g+20 e h) d+19 c^2 f^2 h\right ) a+b^3 \left (f (f g+18 e h) c^2-4 d e (7 f g+6 e h) c+32 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)}+\frac {b \left (\frac {8 b d (b e-a f) (d e-c f) \left (a d (3 d f g+2 d e h-5 c f h)-b \left (3 f h c^2-d (5 f g+6 e h) c+8 d^2 e g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}-\frac {\left (d^3 f^3 h a^4+b d^2 f^2 (5 d f g+6 d e h-15 c f h) a^3-3 b^2 d f \left (4 e (5 f g+2 e h) d^2-15 c f (f g+2 e h) d+15 c^2 f^2 h\right ) a^2-b^3 \left (-8 e^2 (15 f g+2 e h) d^3+120 c e f (f g+e h) d^2-15 c^2 f^2 (f g+6 e h) d+5 c^3 f^3 h\right ) a-b^4 \left (f^2 (f g-6 e h) c^3+12 d e f (f g+4 e h) c^2-24 d^2 e^2 (3 f g+2 e h) c+64 d^3 e^3 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}\right )}{b c-a d}\right )}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)}-\frac {\frac {\left (d f h a^2+b (5 d f g+2 d e h-9 c f h) a-b^2 (8 d e g-3 c f g-6 c e h)\right ) \sqrt {e+f x}}{2 b (b c-a d) (a+b x)^2 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (5 d f g+8 d e h-16 c f h) a^2-b^2 \left (2 e (25 f g+6 e h) d^2-2 c f (20 f g+29 e h) d+33 c^2 f^2 h\right ) a+b^3 \left (3 f (f g+10 e h) c^2-2 d e (23 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}-\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (5 d f g+6 d e h-14 c f h) a^2-b^2 \left (4 e (9 f g+2 e h) d^2-2 c f (13 f g+20 e h) d+19 c^2 f^2 h\right ) a+b^3 \left (f (f g+18 e h) c^2-4 d e (7 f g+6 e h) c+32 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)}+\frac {b \left (\frac {16 b d (b e-a f) (d e-c f) \left (a d (3 d f g+2 d e h-5 c f h)-b \left (3 f h c^2-d (5 f g+6 e h) c+8 d^2 e g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{(b c-a d) f}-\frac {2 \left (d^3 f^3 h a^4+b d^2 f^2 (5 d f g+6 d e h-15 c f h) a^3-3 b^2 d f \left (4 e (5 f g+2 e h) d^2-15 c f (f g+2 e h) d+15 c^2 f^2 h\right ) a^2-b^3 \left (-8 e^2 (15 f g+2 e h) d^3+120 c e f (f g+e h) d^2-15 c^2 f^2 (f g+6 e h) d+5 c^3 f^3 h\right ) a-b^4 \left (f^2 (f g-6 e h) c^3+12 d e f (f g+4 e h) c^2-24 d^2 e^2 (3 f g+2 e h) c+64 d^3 e^3 g\right )\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{(b c-a d) f}\right )}{b c-a d}\right )}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)}-\frac {\frac {\left (d f h a^2+b (5 d f g+2 d e h-9 c f h) a-b^2 (8 d e g-3 c f g-6 c e h)\right ) \sqrt {e+f x}}{2 b (b c-a d) (a+b x)^2 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (5 d f g+8 d e h-16 c f h) a^2-b^2 \left (2 e (25 f g+6 e h) d^2-2 c f (20 f g+29 e h) d+33 c^2 f^2 h\right ) a+b^3 \left (3 f (f g+10 e h) c^2-2 d e (23 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}-\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (5 d f g+6 d e h-14 c f h) a^2-b^2 \left (4 e (9 f g+2 e h) d^2-2 c f (13 f g+20 e h) d+19 c^2 f^2 h\right ) a+b^3 \left (f (f g+18 e h) c^2-4 d e (7 f g+6 e h) c+32 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)}+\frac {b \left (\frac {2 \left (d^3 f^3 h a^4+b d^2 f^2 (5 d f g+6 d e h-15 c f h) a^3-3 b^2 d f \left (4 e (5 f g+2 e h) d^2-15 c f (f g+2 e h) d+15 c^2 f^2 h\right ) a^2-b^3 \left (-8 e^2 (15 f g+2 e h) d^3+120 c e f (f g+e h) d^2-15 c^2 f^2 (f g+6 e h) d+5 c^3 f^3 h\right ) a-b^4 \left (f^2 (f g-6 e h) c^3+12 d e f (f g+4 e h) c^2-24 d^2 e^2 (3 f g+2 e h) c+64 d^3 e^3 g\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b c-a d) \sqrt {b e-a f}}-\frac {16 b \sqrt {d} (b e-a f) \sqrt {d e-c f} \left (a d (3 d f g+2 d e h-5 c f h)-b \left (3 f h c^2-d (5 f g+6 e h) c+8 d^2 e g\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{b c-a d}\right )}{b c-a d}\right )}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\) |
Input:
Int[((e + f*x)^(3/2)*(g + h*x))/((a + b*x)^4*(c + d*x)^2),x]
Output:
-1/3*((b*g - a*h)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(a + b*x)^3*(c + d*x)) - (((a^2*d*f*h - b^2*(8*d*e*g - 3*c*f*g - 6*c*e*h) + a*b*(5*d*f*g + 2*d*e*h - 9*c*f*h))*Sqrt[e + f*x])/(2*b*(b*c - a*d)*(a + b*x)^2*(c + d*x)) - (-(( (a^3*d^2*f^2*h + a^2*b*d*f*(5*d*f*g + 8*d*e*h - 16*c*f*h) + b^3*(48*d^2*e^ 2*g + 3*c^2*f*(f*g + 10*e*h) - 2*c*d*e*(23*f*g + 18*e*h)) - a*b^2*(33*c^2* f^2*h + 2*d^2*e*(25*f*g + 6*e*h) - 2*c*d*f*(20*f*g + 29*e*h)))*Sqrt[e + f* x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x))) - (3*((2*d*(a^3*d^2*f^2 *h + a^2*b*d*f*(5*d*f*g + 6*d*e*h - 14*c*f*h) + b^3*(32*d^2*e^2*g - 4*c*d* e*(7*f*g + 6*e*h) + c^2*f*(f*g + 18*e*h)) - a*b^2*(19*c^2*f^2*h + 4*d^2*e* (9*f*g + 2*e*h) - 2*c*d*f*(13*f*g + 20*e*h)))*Sqrt[e + f*x])/((b*c - a*d)* (c + d*x)) + (b*((2*(a^4*d^3*f^3*h + a^3*b*d^2*f^2*(5*d*f*g + 6*d*e*h - 15 *c*f*h) - 3*a^2*b^2*d*f*(15*c^2*f^2*h - 15*c*d*f*(f*g + 2*e*h) + 4*d^2*e*( 5*f*g + 2*e*h)) - b^4*(64*d^3*e^3*g + c^3*f^2*(f*g - 6*e*h) - 24*c*d^2*e^2 *(3*f*g + 2*e*h) + 12*c^2*d*e*f*(f*g + 4*e*h)) - a*b^3*(5*c^3*f^3*h + 120* c*d^2*e*f*(f*g + e*h) - 8*d^3*e^2*(15*f*g + 2*e*h) - 15*c^2*d*f^2*(f*g + 6 *e*h)))*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b]*(b*c - a*d)*Sqrt[b*e - a*f]) - (16*b*Sqrt[d]*(b*e - a*f)*Sqrt[d*e - c*f]*(a*d*(3* d*f*g + 2*d*e*h - 5*c*f*h) - b*(8*d^2*e*g + 3*c^2*f*h - c*d*(5*f*g + 6*e*h )))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(b*c - a*d)))/(b*c - a*d)))/(2*(b*c - a*d)*(b*e - a*f)))/(4*b*(b*c - a*d)))/(6*b*(b*c - a*d...
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] :> 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_) )^(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_))^(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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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 = 14.80 (sec) , antiderivative size = 1136, normalized size of antiderivative = 1.33
method | result | size |
pseudoelliptic | \(\text {Expression too large to display}\) | \(1136\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1576\) |
default | \(\text {Expression too large to display}\) | \(1576\) |
Input:
int((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
1/8*(((c*f-d*e)*d)^(1/2)*((-64*d^3*e^3*g+48*c*(e*h+3/2*f*g)*e^2*d^2-48*c^2 *(e*h+1/4*f*g)*f*e*d+(6*e*f^2*h-f^3*g)*c^3)*b^4-5*((-16/5*e^3*h-24*g*f*e^2 )*d^3+24*c*d^2*e*f*(e*h+f*g)+(-18*e*f^2*h-3*f^3*g)*c^2*d+c^3*f^3*h)*a*b^3- 45*a^2*((8/15*e^2*h+4/3*e*f*g)*d^2+c*(-2*e*f*h-f^2*g)*d+c^2*f^2*h)*d*f*b^2 -15*a^3*((-2/5*e*h-1/3*f*g)*d+c*f*h)*d^2*f^2*b+a^4*d^3*f^3*h)*(b*x+a)^3*(d *x+c)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))-(-40*(c*f-d*e)*d*(d*x+c) *(b*x+a)^3*b*(a*f-b*e)*((8/5*d^2*e*g+c*(-6/5*e*h-f*g)*d+3/5*c^2*f*h)*b+((- 2/5*e*h-3/5*f*g)*d+c*f*h)*a*d)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)) +(a*d-b*c)*((c*f-d*e)*d)^(1/2)*(f*x+e)^(1/2)*((-32*d^3*e^2*g*x^3-16*x^2*c* ((-3/2*e*h-7/4*f*g)*x+g*e)*e*d^2+16/3*x*c^2*((-27/8*e*f*h-3/16*f^2*g)*x^2+ (9/4*e^2*h+23/8*e*f*g)*x+e^2*g)*d-8/3*((15/4*e*f*h+3/8*f^2*g)*x^2+(3/2*e^2 *h+7/4*e*f*g)*x+e^2*g)*c^3)*b^5-4/3*a*(((-6*e^2*h-27*e*f*g)*x^3+60*e^2*g*x ^2)*d^3+32*x*c*((15/16*e*f*h+39/64*f^2*g)*x^2+(-3/2*e^2*h-2*e*f*g)*x+e^2*g )*d^2-10*(57/40*f^2*h*x^3+(-19/4*e*f*h-23/20*f^2*g)*x^2+(23/10*e^2*h+51/20 *e*f*g)*x+e^2*g)*c^2*d+(-33/4*f^2*h*x^2+(11/2*e*f*h-2*f^2*g)*x+e*(e*h-1/2* f*g))*c^3)*b^4-8/3*a^2*(22*x*(9/88*(e*h+5/6*f*g)*f*x^2+(-15/44*e^2*h-137/8 8*e*f*g)*x+e^2*g)*d^3+13*c*(-21/52*f^2*h*x^3+(155/52*e*f*h+199/104*f^2*g)* x^2+(-41/26*e^2*h-141/52*e*f*g)*x+e^2*g)*d^2-4*c^2*(77/16*f^2*h*x^2+(-75/8 *e*f*h-109/32*f^2*g)*x+e*(e*h+1/4*f*g))*d+c^3*f*(-5*f*h*x-3/8*f*g+e*h))*b^ 3+5*a^3*((-1/5*f^2*h*x^3+(-46/15*e*f*h-8/3*f^2*g)*x^2+(44/15*e^2*h+206/...
Timed out. \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**(3/2)*(h*x+g)/(b*x+a)**4/(d*x+c)**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 2299 vs. \(2 (818) = 1636\).
Time = 0.28 (sec) , antiderivative size = 2299, normalized size of antiderivative = 2.69 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^2,x, algorithm="giac")
Output:
-1/8*(64*b^4*d^3*e^3*g - 72*b^4*c*d^2*e^2*f*g - 120*a*b^3*d^3*e^2*f*g + 12 *b^4*c^2*d*e*f^2*g + 120*a*b^3*c*d^2*e*f^2*g + 60*a^2*b^2*d^3*e*f^2*g + b^ 4*c^3*f^3*g - 15*a*b^3*c^2*d*f^3*g - 45*a^2*b^2*c*d^2*f^3*g - 5*a^3*b*d^3* f^3*g - 48*b^4*c*d^2*e^3*h - 16*a*b^3*d^3*e^3*h + 48*b^4*c^2*d*e^2*f*h + 1 20*a*b^3*c*d^2*e^2*f*h + 24*a^2*b^2*d^3*e^2*f*h - 6*b^4*c^3*e*f^2*h - 90*a *b^3*c^2*d*e*f^2*h - 90*a^2*b^2*c*d^2*e*f^2*h - 6*a^3*b*d^3*e*f^2*h + 5*a* b^3*c^3*f^3*h + 45*a^2*b^2*c^2*d*f^3*h + 15*a^3*b*c*d^2*f^3*h - a^4*d^3*f^ 3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^7*c^5*e - 5*a*b^6*c^ 4*d*e + 10*a^2*b^5*c^3*d^2*e - 10*a^3*b^4*c^2*d^3*e + 5*a^4*b^3*c*d^4*e - a^5*b^2*d^5*e - a*b^6*c^5*f + 5*a^2*b^5*c^4*d*f - 10*a^3*b^4*c^3*d^2*f + 1 0*a^4*b^3*c^2*d^3*f - 5*a^5*b^2*c*d^4*f + a^6*b*d^5*f)*sqrt(-b^2*e + a*b*f )) + (8*b*d^4*e^2*g - 13*b*c*d^3*e*f*g - 3*a*d^4*e*f*g + 5*b*c^2*d^2*f^2*g + 3*a*c*d^3*f^2*g - 6*b*c*d^3*e^2*h - 2*a*d^4*e^2*h + 9*b*c^2*d^2*e*f*h + 7*a*c*d^3*e*f*h - 3*b*c^3*d*f^2*h - 5*a*c^2*d^2*f^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*sqrt(-d^2*e + c*d*f)) - (s qrt(f*x + e)*d^3*e*f*g - sqrt(f*x + e)*c*d^2*f^2*g - sqrt(f*x + e)*c*d^2*e *f*h + sqrt(f*x + e)*c^2*d*f^2*h)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^ 2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*((f*x + e)*d - d*e + c*f)) - 1/24*(72*(f* x + e)^(5/2)*b^5*d^2*e^2*f*g - 144*(f*x + e)^(3/2)*b^5*d^2*e^3*f*g + 72...
Time = 31.47 (sec) , antiderivative size = 671141, normalized size of antiderivative = 784.04 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=\text {Too large to display} \] Input:
int(((e + f*x)^(3/2)*(g + h*x))/((a + b*x)^4*(c + d*x)^2),x)
Output:
atan(((((32*a*b^15*c^13*d^2*f^7*g - 608*a^13*b^3*c*d^14*f^7*g + 32*a^14*b^ 2*c*d^14*f^7*h + 608*a^13*b^3*d^15*e*f^6*g - 32*a^14*b^2*d^15*e*f^6*h - 32 *b^16*c^13*d^2*e*f^6*g - 768*a^2*b^14*c^12*d^3*f^7*g + 5312*a^3*b^13*c^11* d^4*f^7*g - 17920*a^4*b^12*c^10*d^5*f^7*g + 33120*a^5*b^11*c^9*d^6*f^7*g - 29184*a^6*b^10*c^8*d^7*f^7*g - 8064*a^7*b^9*c^7*d^8*f^7*g + 55296*a^8*b^8 *c^6*d^9*f^7*g - 72480*a^9*b^7*c^5*d^10*f^7*g + 52480*a^10*b^6*c^4*d^11*f^ 7*g - 22848*a^11*b^5*c^3*d^12*f^7*g + 5632*a^12*b^4*c^2*d^13*f^7*g + 160*a ^2*b^14*c^13*d^2*f^7*h - 768*a^3*b^13*c^12*d^3*f^7*h - 1088*a^4*b^12*c^11* d^4*f^7*h + 17920*a^5*b^11*c^10*d^5*f^7*h - 64800*a^6*b^10*c^9*d^6*f^7*h + 130560*a^7*b^9*c^8*d^7*f^7*h - 169344*a^8*b^8*c^7*d^8*f^7*h + 147456*a^9* b^7*c^6*d^9*f^7*h - 85920*a^10*b^6*c^5*d^10*f^7*h + 32000*a^11*b^5*c^4*d^1 1*f^7*h - 6720*a^12*b^4*c^3*d^12*f^7*h + 512*a^13*b^3*c^2*d^13*f^7*h - 102 4*a^10*b^6*d^15*e^4*f^3*g + 2688*a^11*b^5*d^15*e^3*f^4*g - 2272*a^12*b^4*d ^15*e^2*f^5*g + 256*a^11*b^5*d^15*e^4*f^3*h - 576*a^12*b^4*d^15*e^3*f^4*h + 352*a^13*b^3*d^15*e^2*f^5*h - 1024*b^16*c^10*d^5*e^4*f^3*g + 1408*b^16*c ^11*d^4*e^3*f^4*g - 352*b^16*c^12*d^3*e^2*f^5*g + 768*b^16*c^11*d^4*e^4*f^ 3*h - 960*b^16*c^12*d^3*e^3*f^4*h + 192*b^16*c^13*d^2*e^2*f^5*h - 46080*a^ 2*b^14*c^8*d^7*e^4*f^3*g + 36480*a^2*b^14*c^9*d^6*e^3*f^4*g + 17088*a^2*b^ 14*c^10*d^5*e^2*f^5*g + 122880*a^3*b^13*c^7*d^8*e^4*f^3*g - 48000*a^3*b^13 *c^8*d^7*e^3*f^4*g - 93440*a^3*b^13*c^9*d^6*e^2*f^5*g - 215040*a^4*b^12...
Time = 0.55 (sec) , antiderivative size = 18011, normalized size of antiderivative = 21.04 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^2,x)
Output:
(3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e) ))*a**7*c*d**3*f**3*h + 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/( sqrt(b)*sqrt(a*f - b*e)))*a**7*d**4*f**3*h*x - 45*sqrt(b)*sqrt(a*f - b*e)* atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**2*d**2*f**3*h + 18*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b* e)))*a**6*b*c*d**3*e*f**2*h + 15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f* x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c*d**3*f**3*g - 36*sqrt(b)*sqrt(a* f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c*d**3*f **3*h*x + 18*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt( a*f - b*e)))*a**6*b*d**4*e*f**2*h*x + 15*sqrt(b)*sqrt(a*f - b*e)*atan((sqr t(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*d**4*f**3*g*x + 9*sqrt(b)* sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*d **4*f**3*h*x**2 - 135*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt (b)*sqrt(a*f - b*e)))*a**5*b**2*c**3*d*f**3*h + 270*sqrt(b)*sqrt(a*f - b*e )*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*b**2*c**2*d**2*e* f**2*h + 135*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt( a*f - b*e)))*a**5*b**2*c**2*d**2*f**3*g - 270*sqrt(b)*sqrt(a*f - b*e)*atan ((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*b**2*c**2*d**2*f**3*h*x - 72*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b *e)))*a**5*b**2*c*d**3*e**2*f*h - 180*sqrt(b)*sqrt(a*f - b*e)*atan((sqr...