\(\int \frac {(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^8} \, dx\) [174]

Optimal result

Integrand size = 26, antiderivative size = 79 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}+\frac {4 \left (a+b x+c x^2\right )^{5/2}}{35 \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^5} \] Output:

2/7*(c*x^2+b*x+a)^(5/2)/(-4*a*c+b^2)/d^8/(2*c*x+b)^7+4/35*(c*x^2+b*x+a)^(5 
/2)/(-4*a*c+b^2)^2/d^8/(2*c*x+b)^5
 

Mathematica [A] (verified)

Time = 10.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {2 (a+x (b+c x))^{5/2} \left (7 b^2+8 b c x+4 c \left (-5 a+2 c x^2\right )\right )}{35 \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^7} \] Input:

Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^8,x]
 

Output:

(2*(a + x*(b + c*x))^(5/2)*(7*b^2 + 8*b*c*x + 4*c*(-5*a + 2*c*x^2)))/(35*( 
b^2 - 4*a*c)^2*d^8*(b + 2*c*x)^7)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1117, 27, 1106}

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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {2 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{d^6 (b+2 c x)^6}dx}{7 d^2 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(b+2 c x)^6}dx}{7 d^8 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7}\)

\(\Big \downarrow \) 1106

\(\displaystyle \frac {4 \left (a+b x+c x^2\right )^{5/2}}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^5}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7}\)

Input:

Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^8,x]
 

Output:

(2*(a + b*x + c*x^2)^(5/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7) + (4*(a + 
b*x + c*x^2)^(5/2))/(35*(b^2 - 4*a*c)^2*d^8*(b + 2*c*x)^5)
 

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 1106

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[2*c*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)* 
(b^2 - 4*a*c))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 
0] && EqQ[m + 2*p + 3, 0]
 

rule 1117

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m 
+ 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* 
c)))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & 
& (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) 
/2])
 

Maple [A] (verified)

Time = 1.96 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89

method	result	size
gosper	\(-\frac {2 \left (-8 c^{2} x^{2}-8 c b x +20 a c -7 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{35 \left (2 c x +b \right )^{7} d^{8} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\)	\(70\)
orering	\(-\frac {2 \left (-8 c^{2} x^{2}-8 c b x +20 a c -7 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (2 c x +b \right )}{35 \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \left (2 c d x +b d \right )^{8}}\)	\(76\)
default	\(\frac {-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{7 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{7}}+\frac {32 c^{3} \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{35 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{5}}}{256 d^{8} c^{8}}\)	\(122\)
trager	\(-\frac {2 \left (-8 c^{4} x^{6}-24 b \,c^{3} x^{5}+4 a \,c^{3} x^{4}-31 b^{2} c^{2} x^{4}+8 a b \,c^{2} x^{3}-22 b^{3} c \,x^{3}+32 a^{2} c^{2} x^{2}-10 a \,b^{2} c \,x^{2}-7 b^{4} x^{2}+32 a^{2} b c x -14 a \,b^{3} x +20 c \,a^{3}-7 a^{2} b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{35 d^{8} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \left (2 c x +b \right )^{7}}\)	\(162\)

Input:

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^8,x,method=_RETURNVERBOSE)
 

Output:

-2/35*(-8*c^2*x^2-8*b*c*x+20*a*c-7*b^2)*(c*x^2+b*x+a)^(5/2)/(2*c*x+b)^7/d^ 
8/(16*a^2*c^2-8*a*b^2*c+b^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (71) = 142\).

Time = 5.34 (sec) , antiderivative size = 398, normalized size of antiderivative = 5.04 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {2 \, {\left (8 \, c^{4} x^{6} + 24 \, b c^{3} x^{5} + {\left (31 \, b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 7 \, a^{2} b^{2} - 20 \, a^{3} c + 2 \, {\left (11 \, b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (7 \, b^{4} + 10 \, a b^{2} c - 32 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (7 \, a b^{3} - 16 \, a^{2} b c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{35 \, {\left (128 \, {\left (b^{4} c^{7} - 8 \, a b^{2} c^{8} + 16 \, a^{2} c^{9}\right )} d^{8} x^{7} + 448 \, {\left (b^{5} c^{6} - 8 \, a b^{3} c^{7} + 16 \, a^{2} b c^{8}\right )} d^{8} x^{6} + 672 \, {\left (b^{6} c^{5} - 8 \, a b^{4} c^{6} + 16 \, a^{2} b^{2} c^{7}\right )} d^{8} x^{5} + 560 \, {\left (b^{7} c^{4} - 8 \, a b^{5} c^{5} + 16 \, a^{2} b^{3} c^{6}\right )} d^{8} x^{4} + 280 \, {\left (b^{8} c^{3} - 8 \, a b^{6} c^{4} + 16 \, a^{2} b^{4} c^{5}\right )} d^{8} x^{3} + 84 \, {\left (b^{9} c^{2} - 8 \, a b^{7} c^{3} + 16 \, a^{2} b^{5} c^{4}\right )} d^{8} x^{2} + 14 \, {\left (b^{10} c - 8 \, a b^{8} c^{2} + 16 \, a^{2} b^{6} c^{3}\right )} d^{8} x + {\left (b^{11} - 8 \, a b^{9} c + 16 \, a^{2} b^{7} c^{2}\right )} d^{8}\right )}} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^8,x, algorithm="fricas")
 

Output:

2/35*(8*c^4*x^6 + 24*b*c^3*x^5 + (31*b^2*c^2 - 4*a*c^3)*x^4 + 7*a^2*b^2 - 
20*a^3*c + 2*(11*b^3*c - 4*a*b*c^2)*x^3 + (7*b^4 + 10*a*b^2*c - 32*a^2*c^2 
)*x^2 + 2*(7*a*b^3 - 16*a^2*b*c)*x)*sqrt(c*x^2 + b*x + a)/(128*(b^4*c^7 - 
8*a*b^2*c^8 + 16*a^2*c^9)*d^8*x^7 + 448*(b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b* 
c^8)*d^8*x^6 + 672*(b^6*c^5 - 8*a*b^4*c^6 + 16*a^2*b^2*c^7)*d^8*x^5 + 560* 
(b^7*c^4 - 8*a*b^5*c^5 + 16*a^2*b^3*c^6)*d^8*x^4 + 280*(b^8*c^3 - 8*a*b^6* 
c^4 + 16*a^2*b^4*c^5)*d^8*x^3 + 84*(b^9*c^2 - 8*a*b^7*c^3 + 16*a^2*b^5*c^4 
)*d^8*x^2 + 14*(b^10*c - 8*a*b^8*c^2 + 16*a^2*b^6*c^3)*d^8*x + (b^11 - 8*a 
*b^9*c + 16*a^2*b^7*c^2)*d^8)
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx}{d^{8}} \] Input:

integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**8,x)
 

Output:

(Integral(a*sqrt(a + b*x + c*x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x** 
2 + 448*b**5*c**3*x**3 + 1120*b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792* 
b**2*c**6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x**8), x) + Integral(b*x*sqrt 
(a + b*x + c*x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c** 
3*x**3 + 1120*b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 
 1024*b*c**7*x**7 + 256*c**8*x**8), x) + Integral(c*x**2*sqrt(a + b*x + c* 
x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120 
*b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c**7* 
x**7 + 256*c**8*x**8), x))/d**8
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^8,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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (71) = 142\).

Time = 0.53 (sec) , antiderivative size = 997, normalized size of antiderivative = 12.62 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^8,x, algorithm="giac")
 

Output:

1/280*(560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*c^5 + 2800*(sqrt(c)*x - 
sqrt(c*x^2 + b*x + a))^9*b*c^(9/2) + 6160*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
a))^8*b^2*c^4 + 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^5 + 7840*(sq 
rt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^3*c^(7/2) + 2240*(sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))^7*a*b*c^(9/2) + 6440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6 
*b^4*c^3 + 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^4 + 1120*(sq 
rt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^5 + 3640*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^5*b^5*c^(5/2) + 2240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b 
^3*c^(7/2) + 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b*c^(9/2) + 14 
84*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^6*c^2 + 392*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^4*a*b^4*c^3 + 4032*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4* 
a^2*b^2*c^4 + 224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*c^5 + 448*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^3*b^7*c^(3/2) - 336*(sqrt(c)*x - sqrt(c*x^ 
2 + b*x + a))^3*a*b^5*c^(5/2) + 2464*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3 
*a^2*b^3*c^(7/2) + 448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b*c^(9/2) 
 + 98*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^8*c - 224*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^2*a*b^6*c^2 + 840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2* 
a^2*b^4*c^3 + 224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^2*c^4 + 112* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*c^5 + 14*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))*b^9*sqrt(c) - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^7...
 

Mupad [B] (verification not implemented)

Time = 6.23 (sec) , antiderivative size = 1814, normalized size of antiderivative = 22.96 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\text {Too large to display} \] Input:

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^8,x)
 

Output:

(((9*a*c - 2*b^2)/(70*c^2*d^8*(4*a*c - b^2)^3) - b^2/(280*c^2*d^8*(4*a*c - 
 b^2)^3))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) - (((b*((b*((16*c^2*(7*a*c 
- b^2))/(35*d^8*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2)) - (6*b^2*c^2)/(35 
*d^8*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (4*c*(6*b^3 - 28*a 
*b*c))/(35*d^8*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (4*c*(6* 
a*b^2 - 26*a^2*c))/(35*d^8*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2)))*(a + 
b*x + c*x^2)^(1/2))/(b + 2*c*x)^3 - (((b*((b*((6*c^2*(12*a*c - b^2))/(7*d^ 
8*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2)) - (6*b^2*c^2)/(7*d^8*(4*a*c - b^2 
)*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (2*c*(7*b^3 - 36*a*b*c))/(7*d^8*(4*a* 
c - b^2)*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (2*c*(7*a*b^2 - 32*a^2*c))/(7* 
d^8*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 
2*c*x)^5 - ((b^2/(14*d^8*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2)) - (8*a*c - 
 b^2)/(14*d^8*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2)))*(a + b*x + c*x^2)^(1 
/2))/(b + 2*c*x)^3 + (((b*((b*(b^2/(70*d^8*(4*a*c - b^2)^4) - (68*a*c^2 - 
11*b^2*c)/(210*c*d^8*(4*a*c - b^2)^4)))/(2*c) - (15*b^3 - 68*a*b*c)/(210*c 
*d^8*(4*a*c - b^2)^4)))/(2*c) + (15*a*b^2 - 64*a^2*c)/(210*c*d^8*(4*a*c - 
b^2)^4))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((b*((16*c^3*(14* 
a*c - b^2))/(35*d^8*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2)) - (16*b^2*c^3 
)/(35*d^8*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (16*b*c^2*(21 
*a*c - 4*b^2))/(35*d^8*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2))))/(2*c)...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 642, normalized size of antiderivative = 8.13 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx=\frac {-320 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{4}+112 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c^{3}-512 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{4} x -512 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{5} x^{2}+224 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} c^{3} x +160 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{4} x^{2}-128 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{5} x^{3}-64 \sqrt {c \,x^{2}+b x +a}\, a \,c^{6} x^{4}+112 \sqrt {c \,x^{2}+b x +a}\, b^{4} c^{3} x^{2}+352 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{4} x^{3}+496 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{5} x^{4}+384 \sqrt {c \,x^{2}+b x +a}\, b \,c^{6} x^{5}+128 \sqrt {c \,x^{2}+b x +a}\, c^{7} x^{6}-\sqrt {c}\, b^{7}-14 \sqrt {c}\, b^{6} c x -84 \sqrt {c}\, b^{5} c^{2} x^{2}-280 \sqrt {c}\, b^{4} c^{3} x^{3}-560 \sqrt {c}\, b^{3} c^{4} x^{4}-672 \sqrt {c}\, b^{2} c^{5} x^{5}-448 \sqrt {c}\, b \,c^{6} x^{6}-128 \sqrt {c}\, c^{7} x^{7}}{280 c^{3} d^{8} \left (2048 a^{2} c^{9} x^{7}-1024 a \,b^{2} c^{8} x^{7}+128 b^{4} c^{7} x^{7}+7168 a^{2} b \,c^{8} x^{6}-3584 a \,b^{3} c^{7} x^{6}+448 b^{5} c^{6} x^{6}+10752 a^{2} b^{2} c^{7} x^{5}-5376 a \,b^{4} c^{6} x^{5}+672 b^{6} c^{5} x^{5}+8960 a^{2} b^{3} c^{6} x^{4}-4480 a \,b^{5} c^{5} x^{4}+560 b^{7} c^{4} x^{4}+4480 a^{2} b^{4} c^{5} x^{3}-2240 a \,b^{6} c^{4} x^{3}+280 b^{8} c^{3} x^{3}+1344 a^{2} b^{5} c^{4} x^{2}-672 a \,b^{7} c^{3} x^{2}+84 b^{9} c^{2} x^{2}+224 a^{2} b^{6} c^{3} x -112 a \,b^{8} c^{2} x +14 b^{10} c x +16 a^{2} b^{7} c^{2}-8 a \,b^{9} c +b^{11}\right )} \] Input:

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^8,x)
 

Output:

( - 320*sqrt(a + b*x + c*x**2)*a**3*c**4 + 112*sqrt(a + b*x + c*x**2)*a**2 
*b**2*c**3 - 512*sqrt(a + b*x + c*x**2)*a**2*b*c**4*x - 512*sqrt(a + b*x + 
 c*x**2)*a**2*c**5*x**2 + 224*sqrt(a + b*x + c*x**2)*a*b**3*c**3*x + 160*s 
qrt(a + b*x + c*x**2)*a*b**2*c**4*x**2 - 128*sqrt(a + b*x + c*x**2)*a*b*c* 
*5*x**3 - 64*sqrt(a + b*x + c*x**2)*a*c**6*x**4 + 112*sqrt(a + b*x + c*x** 
2)*b**4*c**3*x**2 + 352*sqrt(a + b*x + c*x**2)*b**3*c**4*x**3 + 496*sqrt(a 
 + b*x + c*x**2)*b**2*c**5*x**4 + 384*sqrt(a + b*x + c*x**2)*b*c**6*x**5 + 
 128*sqrt(a + b*x + c*x**2)*c**7*x**6 - sqrt(c)*b**7 - 14*sqrt(c)*b**6*c*x 
 - 84*sqrt(c)*b**5*c**2*x**2 - 280*sqrt(c)*b**4*c**3*x**3 - 560*sqrt(c)*b* 
*3*c**4*x**4 - 672*sqrt(c)*b**2*c**5*x**5 - 448*sqrt(c)*b*c**6*x**6 - 128* 
sqrt(c)*c**7*x**7)/(280*c**3*d**8*(16*a**2*b**7*c**2 + 224*a**2*b**6*c**3* 
x + 1344*a**2*b**5*c**4*x**2 + 4480*a**2*b**4*c**5*x**3 + 8960*a**2*b**3*c 
**6*x**4 + 10752*a**2*b**2*c**7*x**5 + 7168*a**2*b*c**8*x**6 + 2048*a**2*c 
**9*x**7 - 8*a*b**9*c - 112*a*b**8*c**2*x - 672*a*b**7*c**3*x**2 - 2240*a* 
b**6*c**4*x**3 - 4480*a*b**5*c**5*x**4 - 5376*a*b**4*c**6*x**5 - 3584*a*b* 
*3*c**7*x**6 - 1024*a*b**2*c**8*x**7 + b**11 + 14*b**10*c*x + 84*b**9*c**2 
*x**2 + 280*b**8*c**3*x**3 + 560*b**7*c**4*x**4 + 672*b**6*c**5*x**5 + 448 
*b**5*c**6*x**6 + 128*b**4*c**7*x**7))