\(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{10}} \, dx\) [191]

Optimal result

Integrand size = 26, antiderivative size = 79 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac {4 \left (a+b x+c x^2\right )^{7/2}}{63 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)^7} \] Output:

2/9*(c*x^2+b*x+a)^(7/2)/(-4*a*c+b^2)/d^10/(2*c*x+b)^9+4/63*(c*x^2+b*x+a)^( 
7/2)/(-4*a*c+b^2)^2/d^10/(2*c*x+b)^7
 

Mathematica [A] (verified)

Time = 10.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\frac {2 (a+x (b+c x))^{7/2} \left (9 b^2+8 b c x+4 c \left (-7 a+2 c x^2\right )\right )}{63 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)^9} \] Input:

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^10,x]
 

Output:

(2*(a + x*(b + c*x))^(7/2)*(9*b^2 + 8*b*c*x + 4*c*(-7*a + 2*c*x^2)))/(63*( 
b^2 - 4*a*c)^2*d^10*(b + 2*c*x)^9)
 

Rubi [A] (verified)

Time = 0.21 (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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {2 \int \frac {\left (c x^2+b x+a\right )^{5/2}}{d^8 (b+2 c x)^8}dx}{9 d^2 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {\left (c x^2+b x+a\right )^{5/2}}{(b+2 c x)^8}dx}{9 d^{10} \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9}\)

\(\Big \downarrow \) 1106

\(\displaystyle \frac {4 \left (a+b x+c x^2\right )^{7/2}}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9}\)

Input:

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^10,x]
 

Output:

(2*(a + b*x + c*x^2)^(7/2))/(9*(b^2 - 4*a*c)*d^10*(b + 2*c*x)^9) + (4*(a + 
 b*x + c*x^2)^(7/2))/(63*(b^2 - 4*a*c)^2*d^10*(b + 2*c*x)^7)
 

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 = 19.79 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89

method	result	size
gosper	\(-\frac {2 \left (-8 c^{2} x^{2}-8 c b x +28 a c -9 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{63 \left (2 c x +b \right )^{9} d^{10} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\)	\(70\)
orering	\(-\frac {2 \left (-8 c^{2} x^{2}-8 c b x +28 a c -9 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (2 c x +b \right )}{63 \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \left (2 c d x +b d \right )^{10}}\)	\(76\)
default	\(\frac {-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{9 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{9}}+\frac {32 c^{3} \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{63 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{7}}}{1024 d^{10} c^{10}}\)	\(122\)
trager	\(-\frac {2 \left (-8 c^{5} x^{8}-32 b \,c^{4} x^{7}+4 a \,c^{4} x^{6}-57 c^{3} b^{2} x^{6}+12 a b \,c^{3} x^{5}-59 b^{3} c^{2} x^{5}+60 a^{2} c^{3} x^{4}-15 a \,b^{2} c^{2} x^{4}-35 x^{4} b^{4} c +120 a^{2} b \,c^{2} x^{3}-50 a \,b^{3} c \,x^{3}-9 b^{5} x^{3}+76 c^{2} a^{3} x^{2}+33 a^{2} b^{2} c \,x^{2}-27 a \,b^{4} x^{2}+76 a^{3} b c x -27 a^{2} b^{3} x +28 c \,a^{4}-9 a^{3} b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{63 d^{10} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \left (2 c x +b \right )^{9}}\)	\(231\)

Input:

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^10,x,method=_RETURNVERBOSE)
 

Output:

-2/63*(-8*c^2*x^2-8*b*c*x+28*a*c-9*b^2)*(c*x^2+b*x+a)^(7/2)/(2*c*x+b)^9/d^ 
10/(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. 530 vs. \(2 (71) = 142\).

Time = 15.88 (sec) , antiderivative size = 530, normalized size of antiderivative = 6.71 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\frac {2 \, {\left (8 \, c^{5} x^{8} + 32 \, b c^{4} x^{7} + {\left (57 \, b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} + {\left (59 \, b^{3} c^{2} - 12 \, a b c^{3}\right )} x^{5} + 9 \, a^{3} b^{2} - 28 \, a^{4} c + 5 \, {\left (7 \, b^{4} c + 3 \, a b^{2} c^{2} - 12 \, a^{2} c^{3}\right )} x^{4} + {\left (9 \, b^{5} + 50 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} x^{3} + {\left (27 \, a b^{4} - 33 \, a^{2} b^{2} c - 76 \, a^{3} c^{2}\right )} x^{2} + {\left (27 \, a^{2} b^{3} - 76 \, a^{3} b c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{63 \, {\left (512 \, {\left (b^{4} c^{9} - 8 \, a b^{2} c^{10} + 16 \, a^{2} c^{11}\right )} d^{10} x^{9} + 2304 \, {\left (b^{5} c^{8} - 8 \, a b^{3} c^{9} + 16 \, a^{2} b c^{10}\right )} d^{10} x^{8} + 4608 \, {\left (b^{6} c^{7} - 8 \, a b^{4} c^{8} + 16 \, a^{2} b^{2} c^{9}\right )} d^{10} x^{7} + 5376 \, {\left (b^{7} c^{6} - 8 \, a b^{5} c^{7} + 16 \, a^{2} b^{3} c^{8}\right )} d^{10} x^{6} + 4032 \, {\left (b^{8} c^{5} - 8 \, a b^{6} c^{6} + 16 \, a^{2} b^{4} c^{7}\right )} d^{10} x^{5} + 2016 \, {\left (b^{9} c^{4} - 8 \, a b^{7} c^{5} + 16 \, a^{2} b^{5} c^{6}\right )} d^{10} x^{4} + 672 \, {\left (b^{10} c^{3} - 8 \, a b^{8} c^{4} + 16 \, a^{2} b^{6} c^{5}\right )} d^{10} x^{3} + 144 \, {\left (b^{11} c^{2} - 8 \, a b^{9} c^{3} + 16 \, a^{2} b^{7} c^{4}\right )} d^{10} x^{2} + 18 \, {\left (b^{12} c - 8 \, a b^{10} c^{2} + 16 \, a^{2} b^{8} c^{3}\right )} d^{10} x + {\left (b^{13} - 8 \, a b^{11} c + 16 \, a^{2} b^{9} c^{2}\right )} d^{10}\right )}} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^10,x, algorithm="fricas")
 

Output:

2/63*(8*c^5*x^8 + 32*b*c^4*x^7 + (57*b^2*c^3 - 4*a*c^4)*x^6 + (59*b^3*c^2 
- 12*a*b*c^3)*x^5 + 9*a^3*b^2 - 28*a^4*c + 5*(7*b^4*c + 3*a*b^2*c^2 - 12*a 
^2*c^3)*x^4 + (9*b^5 + 50*a*b^3*c - 120*a^2*b*c^2)*x^3 + (27*a*b^4 - 33*a^ 
2*b^2*c - 76*a^3*c^2)*x^2 + (27*a^2*b^3 - 76*a^3*b*c)*x)*sqrt(c*x^2 + b*x 
+ a)/(512*(b^4*c^9 - 8*a*b^2*c^10 + 16*a^2*c^11)*d^10*x^9 + 2304*(b^5*c^8 
- 8*a*b^3*c^9 + 16*a^2*b*c^10)*d^10*x^8 + 4608*(b^6*c^7 - 8*a*b^4*c^8 + 16 
*a^2*b^2*c^9)*d^10*x^7 + 5376*(b^7*c^6 - 8*a*b^5*c^7 + 16*a^2*b^3*c^8)*d^1 
0*x^6 + 4032*(b^8*c^5 - 8*a*b^6*c^6 + 16*a^2*b^4*c^7)*d^10*x^5 + 2016*(b^9 
*c^4 - 8*a*b^7*c^5 + 16*a^2*b^5*c^6)*d^10*x^4 + 672*(b^10*c^3 - 8*a*b^8*c^ 
4 + 16*a^2*b^6*c^5)*d^10*x^3 + 144*(b^11*c^2 - 8*a*b^9*c^3 + 16*a^2*b^7*c^ 
4)*d^10*x^2 + 18*(b^12*c - 8*a*b^10*c^2 + 16*a^2*b^8*c^3)*d^10*x + (b^13 - 
 8*a*b^11*c + 16*a^2*b^9*c^2)*d^10)
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx}{d^{10}} \] Input:

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**10,x)
 

Output:

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**10 + 20*b**9*c*x + 180*b**8*c**2 
*x**2 + 960*b**7*c**3*x**3 + 3360*b**6*c**4*x**4 + 8064*b**5*c**5*x**5 + 1 
3440*b**4*c**6*x**6 + 15360*b**3*c**7*x**7 + 11520*b**2*c**8*x**8 + 5120*b 
*c**9*x**9 + 1024*c**10*x**10), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x 
**2)/(b**10 + 20*b**9*c*x + 180*b**8*c**2*x**2 + 960*b**7*c**3*x**3 + 3360 
*b**6*c**4*x**4 + 8064*b**5*c**5*x**5 + 13440*b**4*c**6*x**6 + 15360*b**3* 
c**7*x**7 + 11520*b**2*c**8*x**8 + 5120*b*c**9*x**9 + 1024*c**10*x**10), x 
) + Integral(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**10 + 20*b**9*c*x + 180*b 
**8*c**2*x**2 + 960*b**7*c**3*x**3 + 3360*b**6*c**4*x**4 + 8064*b**5*c**5* 
x**5 + 13440*b**4*c**6*x**6 + 15360*b**3*c**7*x**7 + 11520*b**2*c**8*x**8 
+ 5120*b*c**9*x**9 + 1024*c**10*x**10), x) + Integral(2*a*b*x*sqrt(a + b*x 
 + c*x**2)/(b**10 + 20*b**9*c*x + 180*b**8*c**2*x**2 + 960*b**7*c**3*x**3 
+ 3360*b**6*c**4*x**4 + 8064*b**5*c**5*x**5 + 13440*b**4*c**6*x**6 + 15360 
*b**3*c**7*x**7 + 11520*b**2*c**8*x**8 + 5120*b*c**9*x**9 + 1024*c**10*x** 
10), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**10 + 20*b**9*c*x 
+ 180*b**8*c**2*x**2 + 960*b**7*c**3*x**3 + 3360*b**6*c**4*x**4 + 8064*b** 
5*c**5*x**5 + 13440*b**4*c**6*x**6 + 15360*b**3*c**7*x**7 + 11520*b**2*c** 
8*x**8 + 5120*b*c**9*x**9 + 1024*c**10*x**10), x) + Integral(2*b*c*x**3*sq 
rt(a + b*x + c*x**2)/(b**10 + 20*b**9*c*x + 180*b**8*c**2*x**2 + 960*b**7* 
c**3*x**3 + 3360*b**6*c**4*x**4 + 8064*b**5*c**5*x**5 + 13440*b**4*c**6...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^10,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. 1843 vs. \(2 (71) = 142\).

Time = 1.25 (sec) , antiderivative size = 1843, normalized size of antiderivative = 23.33 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^10,x, algorithm="giac")
 

Output:

1/2016*(4032*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*c^7 + 28224*(sqrt(c)*x 
 - sqrt(c*x^2 + b*x + a))^13*b*c^(13/2) + 90048*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))^12*b^2*c^6 + 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a*c^7 + 
 173376*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b^3*c^(11/2) + 40320*(sqrt( 
c)*x - sqrt(c*x^2 + b*x + a))^11*a*b*c^(13/2) + 225792*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^10*b^4*c^5 + 100800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^1 
0*a*b^2*c^6 + 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^2*c^7 + 21235 
2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^5*c^(9/2) + 134400*(sqrt(c)*x - 
sqrt(c*x^2 + b*x + a))^9*a*b^3*c^(11/2) + 100800*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^9*a^2*b*c^(13/2) + 151200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8 
*b^6*c^4 + 96768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^4*c^5 + 217728* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*b^2*c^6 + 12096*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))^8*a^3*c^7 + 84672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
7*b^7*c^(7/2) + 24192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^5*c^(9/2) 
+ 266112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^3*c^(11/2) + 48384*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b*c^(13/2) + 38304*(sqrt(c)*x - sq 
rt(c*x^2 + b*x + a))^6*b^8*c^3 - 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
^6*a*b^6*c^4 + 205632*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^4*c^5 + 
72576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*b^2*c^6 + 12096*(sqrt(c)*x 
 - sqrt(c*x^2 + b*x + a))^6*a^4*c^7 + 14112*(sqrt(c)*x - sqrt(c*x^2 + b...
 

Mupad [B] (verification not implemented)

Time = 9.74 (sec) , antiderivative size = 5511, normalized size of antiderivative = 69.76 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\text {Too large to display} \] Input:

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^10,x)
 

Output:

(((b*((b*((b*((b*((b*((4*c^4*(16*a*c + 3*b^2))/(3*d^10*(4*a*c - b^2)*(128* 
a*c^3 - 32*b^2*c^2)) - (8*b^2*c^4)/(3*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b 
^2*c^2))))/(2*c) - (10*b*c^3*(48*a*c - 5*b^2))/(9*d^10*(4*a*c - b^2)*(128* 
a*c^3 - 32*b^2*c^2))))/(2*c) + (960*a^2*c^4 - 110*b^4*c^2 + 480*a*b^2*c^3) 
/(18*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2))))/(2*c) + (b*c*(3*b^4 - 
480*a^2*c^2 + 80*a*b^2*c))/(6*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2)) 
))/(2*c) + (9*b^6 + 608*a^3*c^3 + 264*a^2*b^2*c^2 - 126*a*b^4*c)/(18*d^10* 
(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2))))/(2*c) - (9*a*b^5 - 108*a^2*b^3*c 
 + 304*a^3*b*c^2)/(18*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2)))*(a + b 
*x + c*x^2)^(1/2))/(b + 2*c*x)^8 + (((b*((b*((b*((b*((b*((8*c^4*(18*a*c - 
b^2))/(315*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2)) - (8*b^2*c^4)/(315 
*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (8*b*c^3*(27*a*c - 
 5*b^2))/(189*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (4496 
*a^2*c^4 + 46*b^4*c^2 - 1168*a*b^2*c^3)/(945*d^10*(4*a*c - b^2)^4*(32*a*c^ 
3 - 8*b^2*c^2))))/(2*c) - (b*c*(99*b^4 + 2248*a^2*c^2 - 944*a*b^2*c))/(315 
*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (115*b^6 + 4080*a^ 
3*c^3 + 312*a^2*b^2*c^2 - 786*a*b^4*c)/(945*d^10*(4*a*c - b^2)^4*(32*a*c^3 
 - 8*b^2*c^2))))/(2*c) - (115*a*b^5 - 968*a^2*b^3*c + 2040*a^3*b*c^2)/(945 
*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b 
 + 2*c*x)^2 + (((b*((b*((b*((b*((b*((8*c^4*(38*a*c + b^2))/(63*d^10*(4*...
 

Reduce [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 882, normalized size of antiderivative = 11.16 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^10,x)
 

Output:

( - 1792*sqrt(a + b*x + c*x**2)*a**4*c**5 + 576*sqrt(a + b*x + c*x**2)*a** 
3*b**2*c**4 - 4864*sqrt(a + b*x + c*x**2)*a**3*b*c**5*x - 4864*sqrt(a + b* 
x + c*x**2)*a**3*c**6*x**2 + 1728*sqrt(a + b*x + c*x**2)*a**2*b**3*c**4*x 
- 2112*sqrt(a + b*x + c*x**2)*a**2*b**2*c**5*x**2 - 7680*sqrt(a + b*x + c* 
x**2)*a**2*b*c**6*x**3 - 3840*sqrt(a + b*x + c*x**2)*a**2*c**7*x**4 + 1728 
*sqrt(a + b*x + c*x**2)*a*b**4*c**4*x**2 + 3200*sqrt(a + b*x + c*x**2)*a*b 
**3*c**5*x**3 + 960*sqrt(a + b*x + c*x**2)*a*b**2*c**6*x**4 - 768*sqrt(a + 
 b*x + c*x**2)*a*b*c**7*x**5 - 256*sqrt(a + b*x + c*x**2)*a*c**8*x**6 + 57 
6*sqrt(a + b*x + c*x**2)*b**5*c**4*x**3 + 2240*sqrt(a + b*x + c*x**2)*b**4 
*c**5*x**4 + 3776*sqrt(a + b*x + c*x**2)*b**3*c**6*x**5 + 3648*sqrt(a + b* 
x + c*x**2)*b**2*c**7*x**6 + 2048*sqrt(a + b*x + c*x**2)*b*c**8*x**7 + 512 
*sqrt(a + b*x + c*x**2)*c**9*x**8 - sqrt(c)*b**9 - 18*sqrt(c)*b**8*c*x - 1 
44*sqrt(c)*b**7*c**2*x**2 - 672*sqrt(c)*b**6*c**3*x**3 - 2016*sqrt(c)*b**5 
*c**4*x**4 - 4032*sqrt(c)*b**4*c**5*x**5 - 5376*sqrt(c)*b**3*c**6*x**6 - 4 
608*sqrt(c)*b**2*c**7*x**7 - 2304*sqrt(c)*b*c**8*x**8 - 512*sqrt(c)*c**9*x 
**9)/(2016*c**4*d**10*(16*a**2*b**9*c**2 + 288*a**2*b**8*c**3*x + 2304*a** 
2*b**7*c**4*x**2 + 10752*a**2*b**6*c**5*x**3 + 32256*a**2*b**5*c**6*x**4 + 
 64512*a**2*b**4*c**7*x**5 + 86016*a**2*b**3*c**8*x**6 + 73728*a**2*b**2*c 
**9*x**7 + 36864*a**2*b*c**10*x**8 + 8192*a**2*c**11*x**9 - 8*a*b**11*c - 
144*a*b**10*c**2*x - 1152*a*b**9*c**3*x**2 - 5376*a*b**8*c**4*x**3 - 16...