\(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{12}} \, dx\) [193]

Optimal result

Integrand size = 26, antiderivative size = 118 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac {8 \left (a+b x+c x^2\right )^{7/2}}{99 \left (b^2-4 a c\right )^2 d^{12} (b+2 c x)^9}+\frac {16 \left (a+b x+c x^2\right )^{7/2}}{693 \left (b^2-4 a c\right )^3 d^{12} (b+2 c x)^7} \] Output:

2/11*(c*x^2+b*x+a)^(7/2)/(-4*a*c+b^2)/d^12/(2*c*x+b)^11+8/99*(c*x^2+b*x+a) 
^(7/2)/(-4*a*c+b^2)^2/d^12/(2*c*x+b)^9+16/693*(c*x^2+b*x+a)^(7/2)/(-4*a*c+ 
b^2)^3/d^12/(2*c*x+b)^7
 

Mathematica [A] (verified)

Time = 10.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\frac {2 (a+x (b+c x))^{7/2} \left (99 b^4+176 b^3 c x+64 b c^2 x \left (-7 a+4 c x^2\right )+8 b^2 c \left (-77 a+38 c x^2\right )+16 c^2 \left (63 a^2-28 a c x^2+8 c^2 x^4\right )\right )}{693 \left (b^2-4 a c\right )^3 d^{12} (b+2 c x)^{11}} \] Input:

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]
 

Output:

(2*(a + x*(b + c*x))^(7/2)*(99*b^4 + 176*b^3*c*x + 64*b*c^2*x*(-7*a + 4*c* 
x^2) + 8*b^2*c*(-77*a + 38*c*x^2) + 16*c^2*(63*a^2 - 28*a*c*x^2 + 8*c^2*x^ 
4)))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^11)
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1117, 27, 1117, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {4 \int \frac {\left (c x^2+b x+a\right )^{5/2}}{d^{10} (b+2 c x)^{10}}dx}{11 d^2 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 \int \frac {\left (c x^2+b x+a\right )^{5/2}}{(b+2 c x)^{10}}dx}{11 d^{12} \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}}\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {\left (c x^2+b x+a\right )^{5/2}}{(b+2 c x)^8}dx}{9 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 \left (b^2-4 a c\right ) (b+2 c x)^9}\right )}{11 d^{12} \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}}\)

\(\Big \downarrow \) 1106

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}}+\frac {4 \left (\frac {4 \left (a+b x+c x^2\right )^{7/2}}{63 \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 \left (b^2-4 a c\right ) (b+2 c x)^9}\right )}{11 d^{12} \left (b^2-4 a c\right )}\)

Input:

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]
 

Output:

(2*(a + b*x + c*x^2)^(7/2))/(11*(b^2 - 4*a*c)*d^12*(b + 2*c*x)^11) + (4*(( 
2*(a + b*x + c*x^2)^(7/2))/(9*(b^2 - 4*a*c)*(b + 2*c*x)^9) + (4*(a + b*x + 
 c*x^2)^(7/2))/(63*(b^2 - 4*a*c)^2*(b + 2*c*x)^7)))/(11*(b^2 - 4*a*c)*d^12 
)
 

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 = 58.77 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.13

method	result	size
gosper	\(-\frac {2 \left (128 c^{4} x^{4}+256 b \,c^{3} x^{3}-448 a \,c^{3} x^{2}+304 b^{2} c^{2} x^{2}-448 a b \,c^{2} x +176 b^{3} c x +1008 a^{2} c^{2}-616 c a \,b^{2}+99 b^{4}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{693 \left (2 c x +b \right )^{11} d^{12} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\)	\(133\)
orering	\(-\frac {2 \left (128 c^{4} x^{4}+256 b \,c^{3} x^{3}-448 a \,c^{3} x^{2}+304 b^{2} c^{2} x^{2}-448 a b \,c^{2} x +176 b^{3} c x +1008 a^{2} c^{2}-616 c a \,b^{2}+99 b^{4}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (2 c x +b \right )}{693 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (2 c d x +b d \right )^{12}}\)	\(139\)
default	\(\frac {-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{11 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{11}}-\frac {16 c^{2} \left (-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{9 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{9}}+\frac {32 c^{3} \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{63 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{7}}\right )}{11 \left (4 a c -b^{2}\right )}}{4096 d^{12} c^{12}}\)	\(195\)
trager	\(-\frac {2 \left (128 c^{7} x^{10}+640 b \,c^{6} x^{9}-64 a \,c^{6} x^{8}+1456 c^{5} b^{2} x^{8}-256 c^{5} b a \,x^{7}+1984 b^{3} c^{4} x^{7}+48 a^{2} c^{5} x^{6}-472 a \,b^{2} c^{4} x^{6}+1795 b^{4} c^{3} x^{6}+144 a^{2} b \,c^{4} x^{5}-520 a \,b^{3} c^{3} x^{5}+1129 b^{5} c^{2} x^{5}+1808 a^{3} c^{4} x^{4}-1176 a^{2} b^{2} c^{3} x^{4}-31 a \,b^{4} c^{2} x^{4}+473 b^{6} c \,x^{4}+3616 a^{3} b \,c^{3} x^{3}-2592 a^{2} b^{3} c^{2} x^{3}+506 a \,b^{5} c \,x^{3}+99 b^{7} x^{3}+2576 a^{4} c^{3} x^{2}+136 a^{3} b^{2} c^{2} x^{2}-1023 a^{2} b^{4} c \,x^{2}+297 a \,b^{6} x^{2}+2576 c^{2} a^{4} b x -1672 a^{3} b^{3} c x +297 a^{2} b^{5} x +1008 a^{5} c^{2}-616 a^{4} b^{2} c +99 a^{3} b^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{693 d^{12} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (2 c x +b \right )^{11}}\)	\(378\)

Input:

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^12,x,method=_RETURNVERBOSE)
 

Output:

-2/693*(128*c^4*x^4+256*b*c^3*x^3-448*a*c^3*x^2+304*b^2*c^2*x^2-448*a*b*c^ 
2*x+176*b^3*c*x+1008*a^2*c^2-616*a*b^2*c+99*b^4)*(c*x^2+b*x+a)^(7/2)/(2*c* 
x+b)^11/d^12/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 851 vs. \(2 (106) = 212\).

Time = 49.50 (sec) , antiderivative size = 851, normalized size of antiderivative = 7.21 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^12,x, algorithm="fricas")
 

Output:

2/693*(128*c^7*x^10 + 640*b*c^6*x^9 + 16*(91*b^2*c^5 - 4*a*c^6)*x^8 + 64*( 
31*b^3*c^4 - 4*a*b*c^5)*x^7 + 99*a^3*b^4 - 616*a^4*b^2*c + 1008*a^5*c^2 + 
(1795*b^4*c^3 - 472*a*b^2*c^4 + 48*a^2*c^5)*x^6 + (1129*b^5*c^2 - 520*a*b^ 
3*c^3 + 144*a^2*b*c^4)*x^5 + (473*b^6*c - 31*a*b^4*c^2 - 1176*a^2*b^2*c^3 
+ 1808*a^3*c^4)*x^4 + (99*b^7 + 506*a*b^5*c - 2592*a^2*b^3*c^2 + 3616*a^3* 
b*c^3)*x^3 + (297*a*b^6 - 1023*a^2*b^4*c + 136*a^3*b^2*c^2 + 2576*a^4*c^3) 
*x^2 + (297*a^2*b^5 - 1672*a^3*b^3*c + 2576*a^4*b*c^2)*x)*sqrt(c*x^2 + b*x 
 + a)/(2048*(b^6*c^11 - 12*a*b^4*c^12 + 48*a^2*b^2*c^13 - 64*a^3*c^14)*d^1 
2*x^11 + 11264*(b^7*c^10 - 12*a*b^5*c^11 + 48*a^2*b^3*c^12 - 64*a^3*b*c^13 
)*d^12*x^10 + 28160*(b^8*c^9 - 12*a*b^6*c^10 + 48*a^2*b^4*c^11 - 64*a^3*b^ 
2*c^12)*d^12*x^9 + 42240*(b^9*c^8 - 12*a*b^7*c^9 + 48*a^2*b^5*c^10 - 64*a^ 
3*b^3*c^11)*d^12*x^8 + 42240*(b^10*c^7 - 12*a*b^8*c^8 + 48*a^2*b^6*c^9 - 6 
4*a^3*b^4*c^10)*d^12*x^7 + 29568*(b^11*c^6 - 12*a*b^9*c^7 + 48*a^2*b^7*c^8 
 - 64*a^3*b^5*c^9)*d^12*x^6 + 14784*(b^12*c^5 - 12*a*b^10*c^6 + 48*a^2*b^8 
*c^7 - 64*a^3*b^6*c^8)*d^12*x^5 + 5280*(b^13*c^4 - 12*a*b^11*c^5 + 48*a^2* 
b^9*c^6 - 64*a^3*b^7*c^7)*d^12*x^4 + 1320*(b^14*c^3 - 12*a*b^12*c^4 + 48*a 
^2*b^10*c^5 - 64*a^3*b^8*c^6)*d^12*x^3 + 220*(b^15*c^2 - 12*a*b^13*c^3 + 4 
8*a^2*b^11*c^4 - 64*a^3*b^9*c^5)*d^12*x^2 + 22*(b^16*c - 12*a*b^14*c^2 + 4 
8*a^2*b^12*c^3 - 64*a^3*b^10*c^4)*d^12*x + (b^17 - 12*a*b^15*c + 48*a^2*b^ 
13*c^2 - 64*a^3*b^11*c^3)*d^12)
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx =\text {Too large to display} \] Input:

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**12,x)
 

Output:

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c* 
*2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*c**5*x**5 
 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 
112640*b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 409 
6*c**12*x**12), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b**12 + 24 
*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x* 
*4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 
 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 
24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(c**2*x**4*sqrt(a + 
b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3 
*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 
+ 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 
67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Int 
egral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c** 
2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*c**5*x**5 
+ 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 1 
12640*b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096 
*c**12*x**12), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**12 + 24 
*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x* 
*4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^12,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. 2365 vs. \(2 (106) = 212\).

Time = 1.49 (sec) , antiderivative size = 2365, normalized size of antiderivative = 20.04 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^12,x, algorithm="giac")
 

Output:

1/5544*(29568*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^16*c^8 + 236544*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^15*b*c^(15/2) + 868560*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^14*b^2*c^7 + 73920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*a*c 
^8 + 1940400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*b^3*c^(13/2) + 517440* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*a*b*c^(15/2) + 2953104*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x + a))^12*b^4*c^6 + 1600368*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^12*a*b^2*c^7 + 162624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a^2*c^ 
8 + 3256176*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b^5*c^(11/2) + 2875488* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a*b^3*c^(13/2) + 975744*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^11*a^2*b*c^(15/2) + 2709168*(sqrt(c)*x - sqrt(c*x 
^2 + b*x + a))^10*b^6*c^5 + 3307920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10 
*a*b^4*c^6 + 2583504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^2*b^2*c^7 + 
133056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^3*c^8 + 1755600*(sqrt(c)*x 
 - sqrt(c*x^2 + b*x + a))^9*b^7*c^(9/2) + 2513280*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^9*a*b^5*c^(11/2) + 3973200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
^9*a^2*b^3*c^(13/2) + 665280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*b*c 
^(15/2) + 910800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^8*c^4 + 1227600*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^6*c^5 + 3944160*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^8*a^2*b^4*c^6 + 1401840*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))^8*a^3*b^2*c^7 + 95040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^4*c^...
 

Mupad [B] (verification not implemented)

Time = 15.90 (sec) , antiderivative size = 9995, normalized size of antiderivative = 84.70 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\text {Too large to display} \] Input:

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x)
 

Output:

(((b*((b*((b*((b*((b*((4*c^4*(40*a*c - 3*b^2))/(231*d^12*(4*a*c - b^2)^3*( 
96*a*c^3 - 24*b^2*c^2)) - (8*b^2*c^4)/(231*d^12*(4*a*c - b^2)^3*(96*a*c^3 
- 24*b^2*c^2))))/(2*c) - (10*b*c^3*(120*a*c - 23*b^2))/(693*d^12*(4*a*c - 
b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (5168*a^2*c^4 + 58*b^4*c^2 - 138 
4*a*b^2*c^3)/(693*d^12*(4*a*c - b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - 
(b*c*(115*b^4 + 2584*a^2*c^2 - 1092*a*b^2*c))/(231*d^12*(4*a*c - b^2)^3*(9 
6*a*c^3 - 24*b^2*c^2))))/(2*c) + (133*b^6 + 4704*a^3*c^3 + 348*a^2*b^2*c^2 
 - 906*a*b^4*c)/(693*d^12*(4*a*c - b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) 
 - (133*a*b^5 - 1118*a^2*b^3*c + 2352*a^3*b*c^2)/(693*d^12*(4*a*c - b^2)^3 
*(96*a*c^3 - 24*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^6 + (((b*( 
(b*(b^2/(6160*c*d^12*(4*a*c - b^2)^5) - (32*a*c^3 - 5*b^2*c^2)/(9240*c^3*d 
^12*(4*a*c - b^2)^5)))/(2*c) - (7*b^3*c - 32*a*b*c^2)/(9240*c^3*d^12*(4*a* 
c - b^2)^5)))/(2*c) - (6*b^4 + 126*a^2*c^2 - 55*a*b^2*c)/(9240*c^3*d^12*(4 
*a*c - b^2)^5))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*(b^2/(5280 
*c*d^12*(4*a*c - b^2)^5) - (76*a*c^3 - 13*b^2*c^2)/(15840*c^3*d^12*(4*a*c 
- b^2)^5)))/(2*c) - (17*b^3*c - 76*a*b*c^2)/(15840*c^3*d^12*(4*a*c - b^2)^ 
5)))/(2*c) + (23*b^4 + 296*a^2*c^2 - 167*a*b^2*c)/(15840*c^3*d^12*(4*a*c - 
 b^2)^5))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((5*a*c - b^2)/( 
1155*c*d^12*(4*a*c - b^2)^5) - b^2/(9240*c*d^12*(4*a*c - b^2)^5)))/(2*c) - 
 (b*(30*a*c - 7*b^2))/(6930*c^2*d^12*(4*a*c - b^2)^5)))/(2*c) + (147*b^...
 

Reduce [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 1401, normalized size of antiderivative = 11.87 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^12,x)
 

Output:

( - 16128*sqrt(a + b*x + c*x**2)*a**5*c**6 + 9856*sqrt(a + b*x + c*x**2)*a 
**4*b**2*c**5 - 41216*sqrt(a + b*x + c*x**2)*a**4*b*c**6*x - 41216*sqrt(a 
+ b*x + c*x**2)*a**4*c**7*x**2 - 1584*sqrt(a + b*x + c*x**2)*a**3*b**4*c** 
4 + 26752*sqrt(a + b*x + c*x**2)*a**3*b**3*c**5*x - 2176*sqrt(a + b*x + c* 
x**2)*a**3*b**2*c**6*x**2 - 57856*sqrt(a + b*x + c*x**2)*a**3*b*c**7*x**3 
- 28928*sqrt(a + b*x + c*x**2)*a**3*c**8*x**4 - 4752*sqrt(a + b*x + c*x**2 
)*a**2*b**5*c**4*x + 16368*sqrt(a + b*x + c*x**2)*a**2*b**4*c**5*x**2 + 41 
472*sqrt(a + b*x + c*x**2)*a**2*b**3*c**6*x**3 + 18816*sqrt(a + b*x + c*x* 
*2)*a**2*b**2*c**7*x**4 - 2304*sqrt(a + b*x + c*x**2)*a**2*b*c**8*x**5 - 7 
68*sqrt(a + b*x + c*x**2)*a**2*c**9*x**6 - 4752*sqrt(a + b*x + c*x**2)*a*b 
**6*c**4*x**2 - 8096*sqrt(a + b*x + c*x**2)*a*b**5*c**5*x**3 + 496*sqrt(a 
+ b*x + c*x**2)*a*b**4*c**6*x**4 + 8320*sqrt(a + b*x + c*x**2)*a*b**3*c**7 
*x**5 + 7552*sqrt(a + b*x + c*x**2)*a*b**2*c**8*x**6 + 4096*sqrt(a + b*x + 
 c*x**2)*a*b*c**9*x**7 + 1024*sqrt(a + b*x + c*x**2)*a*c**10*x**8 - 1584*s 
qrt(a + b*x + c*x**2)*b**7*c**4*x**3 - 7568*sqrt(a + b*x + c*x**2)*b**6*c* 
*5*x**4 - 18064*sqrt(a + b*x + c*x**2)*b**5*c**6*x**5 - 28720*sqrt(a + b*x 
 + c*x**2)*b**4*c**7*x**6 - 31744*sqrt(a + b*x + c*x**2)*b**3*c**8*x**7 - 
23296*sqrt(a + b*x + c*x**2)*b**2*c**9*x**8 - 10240*sqrt(a + b*x + c*x**2) 
*b*c**10*x**9 - 2048*sqrt(a + b*x + c*x**2)*c**11*x**10 + sqrt(c)*b**11 + 
22*sqrt(c)*b**10*c*x + 220*sqrt(c)*b**9*c**2*x**2 + 1320*sqrt(c)*b**8*c...