\(\int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx\) [96]

Optimal result

Integrand size = 29, antiderivative size = 494 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=-\frac {2 (b c-a d) \left (a^2 d^2 f h-a b d (8 c f h-3 d (f g+e h))+b^2 \left (3 d^2 e g+10 c^2 f h-6 c d (f g+e h)\right )\right ) \sqrt {e+f x}}{d^6}-\frac {(b c-a d)^2 \left (a d (3 d f g+4 d e h-7 c f h)+b \left (12 d^2 e g+19 c^2 f h-c d (15 f g+16 e h)\right )\right ) \sqrt {e+f x}}{4 d^6 (c+d x)}-\frac {2 b (b c-a d) (b d g-2 b c h+a d h) (e+f x)^{3/2}}{d^5}+\frac {(b c-a d)^3 (d g-c h) (e+f x)^{3/2}}{2 d^5 (c+d x)^2}+\frac {2 b^2 (3 a d f h+b (d f g-d e h-3 c f h)) (e+f x)^{5/2}}{5 d^4 f^2}+\frac {2 b^3 h (e+f x)^{7/2}}{7 d^3 f^2}+\frac {3 (b c-a d) \left (a^2 d^2 f (d f g+4 d e h-5 c f h)+b^2 \left (8 d^3 e^2 g-33 c^3 f^2 h-4 c d^2 e (7 f g+4 e h)+3 c^2 d f (7 f g+16 e h)\right )+2 a b d \left (15 c^2 f^2 h+2 d^2 e (3 f g+2 e h)-c d f (7 f g+18 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 d^{13/2} \sqrt {d e-c f}} \] Output:

-2*(-a*d+b*c)*(a^2*d^2*f*h-a*b*d*(8*c*f*h-3*d*(e*h+f*g))+b^2*(3*d^2*e*g+10 
*c^2*f*h-6*c*d*(e*h+f*g)))*(f*x+e)^(1/2)/d^6-1/4*(-a*d+b*c)^2*(a*d*(-7*c*f 
*h+4*d*e*h+3*d*f*g)+b*(12*d^2*e*g+19*c^2*f*h-c*d*(16*e*h+15*f*g)))*(f*x+e) 
^(1/2)/d^6/(d*x+c)-2*b*(-a*d+b*c)*(a*d*h-2*b*c*h+b*d*g)*(f*x+e)^(3/2)/d^5+ 
1/2*(-a*d+b*c)^3*(-c*h+d*g)*(f*x+e)^(3/2)/d^5/(d*x+c)^2+2/5*b^2*(3*a*d*f*h 
+b*(-3*c*f*h-d*e*h+d*f*g))*(f*x+e)^(5/2)/d^4/f^2+2/7*b^3*h*(f*x+e)^(7/2)/d 
^3/f^2+3/4*(-a*d+b*c)*(a^2*d^2*f*(-5*c*f*h+4*d*e*h+d*f*g)+b^2*(8*d^3*e^2*g 
-33*c^3*f^2*h-4*c*d^2*e*(4*e*h+7*f*g)+3*c^2*d*f*(16*e*h+7*f*g))+2*a*b*d*(1 
5*c^2*f^2*h+2*d^2*e*(2*e*h+3*f*g)-c*d*f*(18*e*h+7*f*g)))*arctanh(d^(1/2)*( 
f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(13/2)/(-c*f+d*e)^(1/2)
 

Mathematica [A] (verified)

Time = 2.00 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.50 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\frac {\sqrt {e+f x} \left (-35 a^3 d^3 f^2 \left (-15 c^2 f h+c d (3 f g+2 e h-25 f h x)+d^2 (f x (5 g-8 h x)+2 e (g+2 h x))\right )+7 a b^2 d f \left (945 c^4 f^2 h-105 c^3 d f (6 e h+5 f (g-3 h x))+8 d^4 x^2 \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )+8 c d^3 x \left (6 e^2 h+e f (55 g-48 h x)-f^2 x (35 g+9 h x)\right )+c^2 d^2 \left (24 e^2 h+2 e f (125 g-546 h x)+7 f^2 x (-125 g+72 h x)\right )\right )+b^3 \left (-3465 c^5 f^3 h+105 c^4 d f^2 (21 f g+26 e h-55 f h x)+8 d^5 x^2 (e+f x)^2 (7 f g-2 e h+5 f h x)-8 c d^4 x \left (4 e^3 h+f^3 x^2 (21 g+11 h x)+2 e f^2 x (56 g+13 h x)+e^2 f (-14 g+19 h x)\right )-4 c^2 d^3 \left (4 e^3 h+e f^2 x (637 g-408 h x)-6 f^3 x^2 (49 g+11 h x)+e^2 f (-14 g+82 h x)\right )-21 c^3 d^2 f \left (8 e^2 h+14 e f (5 g-16 h x)+f^2 x (-175 g+88 h x)\right )\right )+35 a^2 b d^2 f^2 \left (-105 c^3 f h+5 c^2 d (9 f g+10 e h-35 f h x)+4 d^3 x (2 f x (3 g+h x)+e (-3 g+8 h x))+c d^2 (f x (75 g-56 h x)+e (-6 g+88 h x))\right )\right )}{140 d^6 f^2 (c+d x)^2}+\frac {3 (-b c+a d) \left (a^2 d^2 f (d f g+4 d e h-5 c f h)+b^2 \left (8 d^3 e^2 g-33 c^3 f^2 h-4 c d^2 e (7 f g+4 e h)+3 c^2 d f (7 f g+16 e h)\right )+2 a b d \left (15 c^2 f^2 h+2 d^2 e (3 f g+2 e h)-c d f (7 f g+18 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{4 d^{13/2} \sqrt {-d e+c f}} \] Input:

Integrate[((a + b*x)^3*(e + f*x)^(3/2)*(g + h*x))/(c + d*x)^3,x]
 

Output:

(Sqrt[e + f*x]*(-35*a^3*d^3*f^2*(-15*c^2*f*h + c*d*(3*f*g + 2*e*h - 25*f*h 
*x) + d^2*(f*x*(5*g - 8*h*x) + 2*e*(g + 2*h*x))) + 7*a*b^2*d*f*(945*c^4*f^ 
2*h - 105*c^3*d*f*(6*e*h + 5*f*(g - 3*h*x)) + 8*d^4*x^2*(3*e^2*h + f^2*x*( 
5*g + 3*h*x) + e*f*(20*g + 6*h*x)) + 8*c*d^3*x*(6*e^2*h + e*f*(55*g - 48*h 
*x) - f^2*x*(35*g + 9*h*x)) + c^2*d^2*(24*e^2*h + 2*e*f*(125*g - 546*h*x) 
+ 7*f^2*x*(-125*g + 72*h*x))) + b^3*(-3465*c^5*f^3*h + 105*c^4*d*f^2*(21*f 
*g + 26*e*h - 55*f*h*x) + 8*d^5*x^2*(e + f*x)^2*(7*f*g - 2*e*h + 5*f*h*x) 
- 8*c*d^4*x*(4*e^3*h + f^3*x^2*(21*g + 11*h*x) + 2*e*f^2*x*(56*g + 13*h*x) 
 + e^2*f*(-14*g + 19*h*x)) - 4*c^2*d^3*(4*e^3*h + e*f^2*x*(637*g - 408*h*x 
) - 6*f^3*x^2*(49*g + 11*h*x) + e^2*f*(-14*g + 82*h*x)) - 21*c^3*d^2*f*(8* 
e^2*h + 14*e*f*(5*g - 16*h*x) + f^2*x*(-175*g + 88*h*x))) + 35*a^2*b*d^2*f 
^2*(-105*c^3*f*h + 5*c^2*d*(9*f*g + 10*e*h - 35*f*h*x) + 4*d^3*x*(2*f*x*(3 
*g + h*x) + e*(-3*g + 8*h*x)) + c*d^2*(f*x*(75*g - 56*h*x) + e*(-6*g + 88* 
h*x)))))/(140*d^6*f^2*(c + d*x)^2) + (3*(-(b*c) + a*d)*(a^2*d^2*f*(d*f*g + 
 4*d*e*h - 5*c*f*h) + b^2*(8*d^3*e^2*g - 33*c^3*f^2*h - 4*c*d^2*e*(7*f*g + 
 4*e*h) + 3*c^2*d*f*(7*f*g + 16*e*h)) + 2*a*b*d*(15*c^2*f^2*h + 2*d^2*e*(3 
*f*g + 2*e*h) - c*d*f*(7*f*g + 18*e*h)))*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sq 
rt[-(d*e) + c*f]])/(4*d^(13/2)*Sqrt[-(d*e) + c*f])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.27, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {166, 27, 166, 27, 164, 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.

\(\displaystyle \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {\int \frac {(a+b x)^2 (e+f x)^{3/2} (6 b e (d g-c h)+a (d f g+4 d e h-5 c f h)+b (7 d f g+4 d e h-11 c f h) x)}{2 (c+d x)^2}dx}{2 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b x)^2 (e+f x)^{3/2} (6 b e (d g-c h)+a (d f g+4 d e h-5 c f h)+b (7 d f g+4 d e h-11 c f h) x)}{(c+d x)^2}dx}{4 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {\frac {\int \frac {(a+b x) (e+f x)^{3/2} \left (2 a d (b e-a f) (d f g+4 d e h-5 c f h)+(4 b e+5 a f) \left (a d (d f g+4 d e h-5 c f h)+b \left (11 f h c^2-d (7 f g+10 e h) c+6 d^2 e g\right )\right )+b \left (7 a d f (d f g+4 d e h-5 c f h)+b \left (8 e (7 f g+e h) d^2-c f (63 f g+100 e h) d+99 c^2 f^2 h\right )\right ) x\right )}{2 (c+d x)}dx}{d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{5/2} \left (a d (-5 c f h+4 d e h+d f g)+b \left (11 c^2 f h-c d (10 e h+7 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(a+b x) (e+f x)^{3/2} \left (2 a d (b e-a f) (d f g+4 d e h-5 c f h)+(4 b e+5 a f) \left (a d (d f g+4 d e h-5 c f h)+b \left (11 f h c^2-d (7 f g+10 e h) c+6 d^2 e g\right )\right )+b \left (7 a d f (d f g+4 d e h-5 c f h)+b \left (8 e (7 f g+e h) d^2-c f (63 f g+100 e h) d+99 c^2 f^2 h\right )\right ) x\right )}{c+d x}dx}{2 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{5/2} \left (a d (-5 c f h+4 d e h+d f g)+b \left (11 c^2 f h-c d (10 e h+7 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {\frac {\frac {2 b (e+f x)^{5/2} \left (70 a^2 d^2 f^2 (-5 c f h+4 d e h+d f g)+5 b d f x \left (7 a d f (-5 c f h+4 d e h+d f g)+b \left (99 c^2 f^2 h-c d f (100 e h+63 f g)+8 d^2 e (e h+7 f g)\right )\right )+21 a b d f \left (63 c^2 f^2 h-c d f (66 e h+35 f g)+2 d^2 e (4 e h+15 f g)\right )-\left (b^2 \left (693 c^3 f^3 h-9 c^2 d f^2 (90 e h+49 f g)+2 c d^2 e f (68 e h+231 f g)-8 d^3 e^2 (7 f g-2 e h)\right )\right )\right )}{35 d^2 f^2}-\frac {3 (b c-a d) \left (a^2 d^2 f (-5 c f h+4 d e h+d f g)+2 a b d \left (15 c^2 f^2 h-c d f (18 e h+7 f g)+2 d^2 e (2 e h+3 f g)\right )+b^2 \left (-33 c^3 f^2 h+3 c^2 d f (16 e h+7 f g)-4 c d^2 e (4 e h+7 f g)+8 d^3 e^2 g\right )\right ) \int \frac {(e+f x)^{3/2}}{c+d x}dx}{d^2}}{2 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{5/2} \left (a d (-5 c f h+4 d e h+d f g)+b \left (11 c^2 f h-c d (10 e h+7 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {2 b (e+f x)^{5/2} \left (70 a^2 d^2 f^2 (-5 c f h+4 d e h+d f g)+5 b d f x \left (7 a d f (-5 c f h+4 d e h+d f g)+b \left (99 c^2 f^2 h-c d f (100 e h+63 f g)+8 d^2 e (e h+7 f g)\right )\right )+21 a b d f \left (63 c^2 f^2 h-c d f (66 e h+35 f g)+2 d^2 e (4 e h+15 f g)\right )-\left (b^2 \left (693 c^3 f^3 h-9 c^2 d f^2 (90 e h+49 f g)+2 c d^2 e f (68 e h+231 f g)-8 d^3 e^2 (7 f g-2 e h)\right )\right )\right )}{35 d^2 f^2}-\frac {3 (b c-a d) \left (a^2 d^2 f (-5 c f h+4 d e h+d f g)+2 a b d \left (15 c^2 f^2 h-c d f (18 e h+7 f g)+2 d^2 e (2 e h+3 f g)\right )+b^2 \left (-33 c^3 f^2 h+3 c^2 d f (16 e h+7 f g)-4 c d^2 e (4 e h+7 f g)+8 d^3 e^2 g\right )\right ) \left (\frac {(d e-c f) \int \frac {\sqrt {e+f x}}{c+d x}dx}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{d^2}}{2 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{5/2} \left (a d (-5 c f h+4 d e h+d f g)+b \left (11 c^2 f h-c d (10 e h+7 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {2 b (e+f x)^{5/2} \left (70 a^2 d^2 f^2 (-5 c f h+4 d e h+d f g)+5 b d f x \left (7 a d f (-5 c f h+4 d e h+d f g)+b \left (99 c^2 f^2 h-c d f (100 e h+63 f g)+8 d^2 e (e h+7 f g)\right )\right )+21 a b d f \left (63 c^2 f^2 h-c d f (66 e h+35 f g)+2 d^2 e (4 e h+15 f g)\right )-\left (b^2 \left (693 c^3 f^3 h-9 c^2 d f^2 (90 e h+49 f g)+2 c d^2 e f (68 e h+231 f g)-8 d^3 e^2 (7 f g-2 e h)\right )\right )\right )}{35 d^2 f^2}-\frac {3 (b c-a d) \left (a^2 d^2 f (-5 c f h+4 d e h+d f g)+2 a b d \left (15 c^2 f^2 h-c d f (18 e h+7 f g)+2 d^2 e (2 e h+3 f g)\right )+b^2 \left (-33 c^3 f^2 h+3 c^2 d f (16 e h+7 f g)-4 c d^2 e (4 e h+7 f g)+8 d^3 e^2 g\right )\right ) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{d^2}}{2 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{5/2} \left (a d (-5 c f h+4 d e h+d f g)+b \left (11 c^2 f h-c d (10 e h+7 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {2 b (e+f x)^{5/2} \left (70 a^2 d^2 f^2 (-5 c f h+4 d e h+d f g)+5 b d f x \left (7 a d f (-5 c f h+4 d e h+d f g)+b \left (99 c^2 f^2 h-c d f (100 e h+63 f g)+8 d^2 e (e h+7 f g)\right )\right )+21 a b d f \left (63 c^2 f^2 h-c d f (66 e h+35 f g)+2 d^2 e (4 e h+15 f g)\right )-\left (b^2 \left (693 c^3 f^3 h-9 c^2 d f^2 (90 e h+49 f g)+2 c d^2 e f (68 e h+231 f g)-8 d^3 e^2 (7 f g-2 e h)\right )\right )\right )}{35 d^2 f^2}-\frac {3 (b c-a d) \left (a^2 d^2 f (-5 c f h+4 d e h+d f g)+2 a b d \left (15 c^2 f^2 h-c d f (18 e h+7 f g)+2 d^2 e (2 e h+3 f g)\right )+b^2 \left (-33 c^3 f^2 h+3 c^2 d f (16 e h+7 f g)-4 c d^2 e (4 e h+7 f g)+8 d^3 e^2 g\right )\right ) \left (\frac {(d e-c f) \left (\frac {2 (d e-c f) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d f}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{d^2}}{2 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{5/2} \left (a d (-5 c f h+4 d e h+d f g)+b \left (11 c^2 f h-c d (10 e h+7 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {2 b (e+f x)^{5/2} \left (70 a^2 d^2 f^2 (-5 c f h+4 d e h+d f g)+5 b d f x \left (7 a d f (-5 c f h+4 d e h+d f g)+b \left (99 c^2 f^2 h-c d f (100 e h+63 f g)+8 d^2 e (e h+7 f g)\right )\right )+21 a b d f \left (63 c^2 f^2 h-c d f (66 e h+35 f g)+2 d^2 e (4 e h+15 f g)\right )-\left (b^2 \left (693 c^3 f^3 h-9 c^2 d f^2 (90 e h+49 f g)+2 c d^2 e f (68 e h+231 f g)-8 d^3 e^2 (7 f g-2 e h)\right )\right )\right )}{35 d^2 f^2}-\frac {3 (b c-a d) \left (\frac {(d e-c f) \left (\frac {2 \sqrt {e+f x}}{d}-\frac {2 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2}}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right ) \left (a^2 d^2 f (-5 c f h+4 d e h+d f g)+2 a b d \left (15 c^2 f^2 h-c d f (18 e h+7 f g)+2 d^2 e (2 e h+3 f g)\right )+b^2 \left (-33 c^3 f^2 h+3 c^2 d f (16 e h+7 f g)-4 c d^2 e (4 e h+7 f g)+8 d^3 e^2 g\right )\right )}{d^2}}{2 d (d e-c f)}-\frac {(a+b x)^2 (e+f x)^{5/2} \left (a d (-5 c f h+4 d e h+d f g)+b \left (11 c^2 f h-c d (10 e h+7 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

Input:

Int[((a + b*x)^3*(e + f*x)^(3/2)*(g + h*x))/(c + d*x)^3,x]
 

Output:

-1/2*((d*g - c*h)*(a + b*x)^3*(e + f*x)^(5/2))/(d*(d*e - c*f)*(c + d*x)^2) 
 + (-(((a*d*(d*f*g + 4*d*e*h - 5*c*f*h) + b*(6*d^2*e*g + 11*c^2*f*h - c*d* 
(7*f*g + 10*e*h)))*(a + b*x)^2*(e + f*x)^(5/2))/(d*(d*e - c*f)*(c + d*x))) 
 + ((2*b*(e + f*x)^(5/2)*(70*a^2*d^2*f^2*(d*f*g + 4*d*e*h - 5*c*f*h) + 21* 
a*b*d*f*(63*c^2*f^2*h + 2*d^2*e*(15*f*g + 4*e*h) - c*d*f*(35*f*g + 66*e*h) 
) - b^2*(693*c^3*f^3*h - 8*d^3*e^2*(7*f*g - 2*e*h) + 2*c*d^2*e*f*(231*f*g 
+ 68*e*h) - 9*c^2*d*f^2*(49*f*g + 90*e*h)) + 5*b*d*f*(7*a*d*f*(d*f*g + 4*d 
*e*h - 5*c*f*h) + b*(99*c^2*f^2*h + 8*d^2*e*(7*f*g + e*h) - c*d*f*(63*f*g 
+ 100*e*h)))*x))/(35*d^2*f^2) - (3*(b*c - a*d)*(a^2*d^2*f*(d*f*g + 4*d*e*h 
 - 5*c*f*h) + b^2*(8*d^3*e^2*g - 33*c^3*f^2*h - 4*c*d^2*e*(7*f*g + 4*e*h) 
+ 3*c^2*d*f*(7*f*g + 16*e*h)) + 2*a*b*d*(15*c^2*f^2*h + 2*d^2*e*(3*f*g + 2 
*e*h) - c*d*f*(7*f*g + 18*e*h)))*((2*(e + f*x)^(3/2))/(3*d) + ((d*e - c*f) 
*((2*Sqrt[e + f*x])/d - (2*Sqrt[d*e - c*f]*ArcTanh[(Sqrt[d]*Sqrt[e + f*x]) 
/Sqrt[d*e - c*f]])/d^(3/2)))/d))/d^2)/(2*d*(d*e - c*f)))/(4*d*(d*e - c*f))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

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]
 

rule 73

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]
 

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 166

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]
 

rule 221

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]
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 758, normalized size of antiderivative = 1.53

method	result	size
pseudoelliptic	\(\frac {-\frac {15 \left (a d -b c \right ) \left (x d +c \right )^{2} f^{2} \left (\frac {\left (-a^{2} f^{2} g -4 a e \left (a h +3 b g \right ) f -8 b \,e^{2} \left (a h +b g \right )\right ) d^{3}}{5}+c \left (a \left (a h +\frac {14 b g}{5}\right ) f^{2}+\frac {36 \left (a h +\frac {7 b g}{9}\right ) b e f}{5}+\frac {16 b^{2} e^{2} h}{5}\right ) d^{2}-6 c^{2} \left (\left (a h +\frac {7 b g}{10}\right ) f +\frac {8 e h b}{5}\right ) b f d +\frac {33 b^{2} c^{3} f^{2} h}{5}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{4}+\frac {15 \sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}\, \left (\left (-\frac {\left (-\frac {8 x^{3} \left (\frac {5 h x}{7}+g \right ) b^{3}}{25}-\frac {8 a \,x^{2} \left (\frac {3 h x}{5}+g \right ) b^{2}}{5}-\frac {24 a^{2} x \left (\frac {h x}{3}+g \right ) b}{5}+a^{3} \left (-\frac {8 h x}{5}+g \right )\right ) x \,f^{3}}{3}-\frac {2 \left (-\frac {8 x^{3} \left (\frac {4 h x}{7}+g \right ) b^{3}}{5}-16 a \,x^{2} \left (\frac {3 h x}{10}+g \right ) b^{2}+6 a^{2} x \left (-\frac {8 h x}{3}+g \right ) b +a^{3} \left (2 h x +g \right )\right ) e \,f^{2}}{15}+\frac {8 x^{2} \left (\frac {\left (\frac {h x}{7}+g \right ) b}{3}+a h \right ) b^{2} e^{2} f}{25}-\frac {16 b^{3} e^{3} h \,x^{2}}{525}\right ) d^{5}-\frac {2 c \left (\left (\frac {12 x^{3} \left (\frac {11 h x}{21}+g \right ) b^{3}}{5}+28 a \,x^{2} \left (\frac {9 h x}{35}+g \right ) b^{2}-\frac {75 \left (-\frac {56 h x}{75}+g \right ) a^{2} x b}{2}+\frac {3 a^{3} \left (-\frac {25 h x}{3}+g \right )}{2}\right ) f^{3}+\left (\frac {64 x^{2} \left (\frac {13 h x}{56}+g \right ) b^{3}}{5}-44 a x \left (-\frac {48 h x}{55}+g \right ) b^{2}+3 a^{2} \left (-\frac {44 h x}{3}+g \right ) b +h \,a^{3}\right ) e \,f^{2}-\frac {24 x \left (\frac {\left (-\frac {19 h x}{14}+g \right ) b}{3}+a h \right ) b^{2} e^{2} f}{5}+\frac {16 b^{3} e^{3} h x}{35}\right ) d^{4}}{15}+c^{2} \left (\left (\frac {56 x^{2} \left (\frac {11 h x}{49}+g \right ) b^{3}}{25}-\frac {35 a x \left (-\frac {72 h x}{125}+g \right ) b^{2}}{3}+3 a^{2} \left (-\frac {35 h x}{9}+g \right ) b +h \,a^{3}\right ) f^{3}+\frac {10 b \left (-\frac {182 x \left (-\frac {408 h x}{637}+g \right ) b^{2}}{125}+a \left (-\frac {546 h x}{125}+g \right ) b +a^{2} h \right ) e \,f^{2}}{3}+\frac {8 \left (\frac {\left (-\frac {41 h x}{7}+g \right ) b}{3}+a h \right ) b^{2} e^{2} f}{25}-\frac {16 b^{3} e^{3} h}{525}\right ) d^{3}-7 c^{3} \left (\left (\left (\frac {88}{175} h \,x^{2}-g x \right ) b^{2}+a \left (-3 h x +g \right ) b +a^{2} h \right ) f^{2}+\frac {6 b \left (\frac {\left (-\frac {16 h x}{5}+g \right ) b}{3}+a h \right ) e f}{5}+\frac {8 b^{2} e^{2} h}{175}\right ) b f \,d^{2}+\frac {63 c^{4} \left (\left (\frac {\left (-\frac {55 h x}{21}+g \right ) b}{3}+a h \right ) f +\frac {26 e h b}{63}\right ) b^{2} f^{2} d}{5}-\frac {33 b^{3} c^{5} f^{3} h}{5}\right )}{4}}{d^{6} \sqrt {\left (c f -d e \right ) d}\, \left (x d +c \right )^{2} f^{2}}\)	\(758\)
risch	\(\text {Expression too large to display}\)	\(1049\)
derivativedivides	\(\text {Expression too large to display}\)	\(1324\)
default	\(\text {Expression too large to display}\)	\(1324\)

Input:

int((b*x+a)^3*(f*x+e)^(3/2)*(h*x+g)/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

15/4/((c*f-d*e)*d)^(1/2)*(-(a*d-b*c)*(d*x+c)^2*f^2*(1/5*(-a^2*f^2*g-4*a*e* 
(a*h+3*b*g)*f-8*b*e^2*(a*h+b*g))*d^3+c*(a*(a*h+14/5*b*g)*f^2+36/5*(a*h+7/9 
*b*g)*b*e*f+16/5*b^2*e^2*h)*d^2-6*c^2*((a*h+7/10*b*g)*f+8/5*e*h*b)*b*f*d+3 
3/5*b^2*c^3*f^2*h)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))+((c*f-d*e)* 
d)^(1/2)*(f*x+e)^(1/2)*((-1/3*(-8/25*x^3*(5/7*h*x+g)*b^3-8/5*a*x^2*(3/5*h* 
x+g)*b^2-24/5*a^2*x*(1/3*h*x+g)*b+a^3*(-8/5*h*x+g))*x*f^3-2/15*(-8/5*x^3*( 
4/7*h*x+g)*b^3-16*a*x^2*(3/10*h*x+g)*b^2+6*a^2*x*(-8/3*h*x+g)*b+a^3*(2*h*x 
+g))*e*f^2+8/25*x^2*(1/3*(1/7*h*x+g)*b+a*h)*b^2*e^2*f-16/525*b^3*e^3*h*x^2 
)*d^5-2/15*c*((12/5*x^3*(11/21*h*x+g)*b^3+28*a*x^2*(9/35*h*x+g)*b^2-75/2*( 
-56/75*h*x+g)*a^2*x*b+3/2*a^3*(-25/3*h*x+g))*f^3+(64/5*x^2*(13/56*h*x+g)*b 
^3-44*a*x*(-48/55*h*x+g)*b^2+3*a^2*(-44/3*h*x+g)*b+h*a^3)*e*f^2-24/5*x*(1/ 
3*(-19/14*h*x+g)*b+a*h)*b^2*e^2*f+16/35*b^3*e^3*h*x)*d^4+c^2*((56/25*x^2*( 
11/49*h*x+g)*b^3-35/3*a*x*(-72/125*h*x+g)*b^2+3*a^2*(-35/9*h*x+g)*b+h*a^3) 
*f^3+10/3*b*(-182/125*x*(-408/637*h*x+g)*b^2+a*(-546/125*h*x+g)*b+a^2*h)*e 
*f^2+8/25*(1/3*(-41/7*h*x+g)*b+a*h)*b^2*e^2*f-16/525*b^3*e^3*h)*d^3-7*c^3* 
(((88/175*h*x^2-g*x)*b^2+a*(-3*h*x+g)*b+a^2*h)*f^2+6/5*b*(1/3*(-16/5*h*x+g 
)*b+a*h)*e*f+8/175*b^2*e^2*h)*b*f*d^2+63/5*c^4*((1/3*(-55/21*h*x+g)*b+a*h) 
*f+26/63*e*h*b)*b^2*f^2*d-33/5*b^3*c^5*f^3*h))/d^6/(d*x+c)^2/f^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2275 vs. \(2 (462) = 924\).

Time = 0.30 (sec) , antiderivative size = 4563, normalized size of antiderivative = 9.24 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^3*(f*x+e)^(3/2)*(h*x+g)/(d*x+c)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**3*(f*x+e)**(3/2)*(h*x+g)/(d*x+c)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^3*(f*x+e)^(3/2)*(h*x+g)/(d*x+c)^3,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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1617 vs. \(2 (462) = 924\).

Time = 0.19 (sec) , antiderivative size = 1617, normalized size of antiderivative = 3.27 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^3*(f*x+e)^(3/2)*(h*x+g)/(d*x+c)^3,x, algorithm="giac")
 

Output:

-3/4*(8*b^3*c*d^3*e^2*g - 8*a*b^2*d^4*e^2*g - 28*b^3*c^2*d^2*e*f*g + 40*a* 
b^2*c*d^3*e*f*g - 12*a^2*b*d^4*e*f*g + 21*b^3*c^3*d*f^2*g - 35*a*b^2*c^2*d 
^2*f^2*g + 15*a^2*b*c*d^3*f^2*g - a^3*d^4*f^2*g - 16*b^3*c^2*d^2*e^2*h + 2 
4*a*b^2*c*d^3*e^2*h - 8*a^2*b*d^4*e^2*h + 48*b^3*c^3*d*e*f*h - 84*a*b^2*c^ 
2*d^2*e*f*h + 40*a^2*b*c*d^3*e*f*h - 4*a^3*d^4*e*f*h - 33*b^3*c^4*f^2*h + 
63*a*b^2*c^3*d*f^2*h - 35*a^2*b*c^2*d^2*f^2*h + 5*a^3*c*d^3*f^2*h)*arctan( 
sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f)*d^6) - 1/4*(12 
*(f*x + e)^(3/2)*b^3*c^2*d^3*e*f*g - 24*(f*x + e)^(3/2)*a*b^2*c*d^4*e*f*g 
+ 12*(f*x + e)^(3/2)*a^2*b*d^5*e*f*g - 12*sqrt(f*x + e)*b^3*c^2*d^3*e^2*f* 
g + 24*sqrt(f*x + e)*a*b^2*c*d^4*e^2*f*g - 12*sqrt(f*x + e)*a^2*b*d^5*e^2* 
f*g - 17*(f*x + e)^(3/2)*b^3*c^3*d^2*f^2*g + 39*(f*x + e)^(3/2)*a*b^2*c^2* 
d^3*f^2*g - 27*(f*x + e)^(3/2)*a^2*b*c*d^4*f^2*g + 5*(f*x + e)^(3/2)*a^3*d 
^5*f^2*g + 27*sqrt(f*x + e)*b^3*c^3*d^2*e*f^2*g - 57*sqrt(f*x + e)*a*b^2*c 
^2*d^3*e*f^2*g + 33*sqrt(f*x + e)*a^2*b*c*d^4*e*f^2*g - 3*sqrt(f*x + e)*a^ 
3*d^5*e*f^2*g - 15*sqrt(f*x + e)*b^3*c^4*d*f^3*g + 33*sqrt(f*x + e)*a*b^2* 
c^3*d^2*f^3*g - 21*sqrt(f*x + e)*a^2*b*c^2*d^3*f^3*g + 3*sqrt(f*x + e)*a^3 
*c*d^4*f^3*g - 16*(f*x + e)^(3/2)*b^3*c^3*d^2*e*f*h + 36*(f*x + e)^(3/2)*a 
*b^2*c^2*d^3*e*f*h - 24*(f*x + e)^(3/2)*a^2*b*c*d^4*e*f*h + 4*(f*x + e)^(3 
/2)*a^3*d^5*e*f*h + 16*sqrt(f*x + e)*b^3*c^3*d^2*e^2*f*h - 36*sqrt(f*x + e 
)*a*b^2*c^2*d^3*e^2*f*h + 24*sqrt(f*x + e)*a^2*b*c*d^4*e^2*f*h - 4*sqrt...
 

Mupad [B] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 1753, normalized size of antiderivative = 3.55 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

int(((e + f*x)^(3/2)*(g + h*x)*(a + b*x)^3)/(c + d*x)^3,x)
 

Output:

(e + f*x)^(5/2)*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(5*d^3*f^2) - (6*b^ 
3*h*(c*f - d*e))/(5*d^4*f^2)) - ((e + f*x)^(3/2)*((5*a^3*d^5*f^2*g)/4 + a^ 
3*d^5*e*f*h - (9*a^3*c*d^4*f^2*h)/4 + (21*b^3*c^4*d*f^2*h)/4 - (17*b^3*c^3 
*d^2*f^2*g)/4 + (39*a*b^2*c^2*d^3*f^2*g)/4 - (51*a*b^2*c^3*d^2*f^2*h)/4 + 
(39*a^2*b*c^2*d^3*f^2*h)/4 + 3*a^2*b*d^5*e*f*g - (27*a^2*b*c*d^4*f^2*g)/4 
+ 3*b^3*c^2*d^3*e*f*g - 4*b^3*c^3*d^2*e*f*h + 9*a*b^2*c^2*d^3*e*f*h - 6*a* 
b^2*c*d^4*e*f*g - 6*a^2*b*c*d^4*e*f*h) - (e + f*x)^(1/2)*((15*b^3*c^4*d*f^ 
3*g)/4 - (3*a^3*c*d^4*f^3*g)/4 - (19*b^3*c^5*f^3*h)/4 + (3*a^3*d^5*e*f^2*g 
)/4 + a^3*d^5*e^2*f*h + (7*a^3*c^2*d^3*f^3*h)/4 - (33*a*b^2*c^3*d^2*f^3*g) 
/4 + (21*a^2*b*c^2*d^3*f^3*g)/4 - (33*a^2*b*c^3*d^2*f^3*h)/4 + 3*b^3*c^2*d 
^3*e^2*f*g - (27*b^3*c^3*d^2*e*f^2*g)/4 - 4*b^3*c^3*d^2*e^2*f*h + (45*a*b^ 
2*c^4*d*f^3*h)/4 + 3*a^2*b*d^5*e^2*f*g - (11*a^3*c*d^4*e*f^2*h)/4 + (35*b^ 
3*c^4*d*e*f^2*h)/4 - 6*a*b^2*c*d^4*e^2*f*g - (33*a^2*b*c*d^4*e*f^2*g)/4 - 
6*a^2*b*c*d^4*e^2*f*h + (57*a*b^2*c^2*d^3*e*f^2*g)/4 + 9*a*b^2*c^2*d^3*e^2 
*f*h - (81*a*b^2*c^3*d^2*e*f^2*h)/4 + (57*a^2*b*c^2*d^3*e*f^2*h)/4))/(d^8* 
(e + f*x)^2 - (e + f*x)*(2*d^8*e - 2*c*d^7*f) + d^8*e^2 + c^2*d^6*f^2 - 2* 
c*d^7*e*f) - (e + f*x)^(3/2)*((((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(d^3 
*f^2) - (6*b^3*h*(c*f - d*e))/(d^4*f^2))*(c*f - d*e))/d - (2*b*(a*f - b*e) 
*(a*f*h - 2*b*e*h + b*f*g))/(d^3*f^2) + (2*b^3*h*(c*f - d*e)^2)/(d^5*f^2)) 
 + (e + f*x)^(1/2)*((3*(c*f - d*e)*((3*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^...
 

Reduce [B] (verification not implemented)

Time = 1.09 (sec) , antiderivative size = 5520, normalized size of antiderivative = 11.17 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^3*(f*x+e)^(3/2)*(h*x+g)/(d*x+c)^3,x)
 

Output:

( - 525*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - 
 d*e)))*a**3*c**3*d**3*f**4*h + 420*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + 
 f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c**2*d**4*e*f**3*h + 105*sqrt(d)* 
sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c** 
2*d**4*f**4*g - 1050*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt( 
d)*sqrt(c*f - d*e)))*a**3*c**2*d**4*f**4*h*x + 840*sqrt(d)*sqrt(c*f - d*e) 
*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c*d**5*e*f**3*h*x 
+ 210*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d 
*e)))*a**3*c*d**5*f**4*g*x - 525*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f* 
x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c*d**5*f**4*h*x**2 + 420*sqrt(d)*sqr 
t(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*d**6*e 
*f**3*h*x**2 + 105*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d) 
*sqrt(c*f - d*e)))*a**3*d**6*f**4*g*x**2 + 3675*sqrt(d)*sqrt(c*f - d*e)*at 
an((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**4*d**2*f**4*h - 
4200*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d* 
e)))*a**2*b*c**3*d**3*e*f**3*h - 1575*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e 
 + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**3*d**3*f**4*g + 7350*sqrt( 
d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2* 
b*c**3*d**3*f**4*h*x + 840*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/ 
(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**2*d**4*e**2*f**2*h + 1260*sqrt(d)*...