\(\int \frac {(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^{10}} \, dx\) [176]

Optimal result

Integrand size = 26, antiderivative size = 118 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{10}} \, dx=\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac {8 \left (a+b x+c x^2\right )^{5/2}}{63 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)^7}+\frac {16 \left (a+b x+c x^2\right )^{5/2}}{315 \left (b^2-4 a c\right )^3 d^{10} (b+2 c x)^5} \] Output:

2/9*(c*x^2+b*x+a)^(5/2)/(-4*a*c+b^2)/d^10/(2*c*x+b)^9+8/63*(c*x^2+b*x+a)^( 
5/2)/(-4*a*c+b^2)^2/d^10/(2*c*x+b)^7+16/315*(c*x^2+b*x+a)^(5/2)/(-4*a*c+b^ 
2)^3/d^10/(2*c*x+b)^5
 

Mathematica [A] (verified)

Time = 10.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{10}} \, dx=\frac {2 (a+x (b+c x))^{5/2} \left (63 b^4+144 b^3 c x+64 b c^2 x \left (-5 a+4 c x^2\right )+8 b^2 c \left (-45 a+34 c x^2\right )+16 c^2 \left (35 a^2-20 a c x^2+8 c^2 x^4\right )\right )}{315 \left (b^2-4 a c\right )^3 d^{10} (b+2 c x)^9} \] Input:

Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^10,x]
 

Output:

(2*(a + x*(b + c*x))^(5/2)*(63*b^4 + 144*b^3*c*x + 64*b*c^2*x*(-5*a + 4*c* 
x^2) + 8*b^2*c*(-45*a + 34*c*x^2) + 16*c^2*(35*a^2 - 20*a*c*x^2 + 8*c^2*x^ 
4)))/(315*(b^2 - 4*a*c)^3*d^10*(b + 2*c*x)^9)
 

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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{10}} \, dx\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {4 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{d^8 (b+2 c x)^8}dx}{9 d^2 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(b+2 c x)^8}dx}{9 d^{10} \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9}\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(b+2 c x)^6}dx}{7 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) (b+2 c x)^7}\right )}{9 d^{10} \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9}\)

\(\Big \downarrow \) 1106

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9}+\frac {4 \left (\frac {4 \left (a+b x+c x^2\right )^{5/2}}{35 \left (b^2-4 a c\right )^2 (b+2 c x)^5}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) (b+2 c x)^7}\right )}{9 d^{10} \left (b^2-4 a c\right )}\)

Input:

Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^10,x]
 

Output:

(2*(a + b*x + c*x^2)^(5/2))/(9*(b^2 - 4*a*c)*d^10*(b + 2*c*x)^9) + (4*((2* 
(a + b*x + c*x^2)^(5/2))/(7*(b^2 - 4*a*c)*(b + 2*c*x)^7) + (4*(a + b*x + c 
*x^2)^(5/2))/(35*(b^2 - 4*a*c)^2*(b + 2*c*x)^5)))/(9*(b^2 - 4*a*c)*d^10)
 

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 = 2.86 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.13

method	result	size
gosper	\(-\frac {2 \left (128 c^{4} x^{4}+256 b \,c^{3} x^{3}-320 a \,c^{3} x^{2}+272 b^{2} c^{2} x^{2}-320 a b \,c^{2} x +144 b^{3} c x +560 a^{2} c^{2}-360 c a \,b^{2}+63 b^{4}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{315 \left (2 c x +b \right )^{9} d^{10} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\)	\(133\)
orering	\(-\frac {2 \left (128 c^{4} x^{4}+256 b \,c^{3} x^{3}-320 a \,c^{3} x^{2}+272 b^{2} c^{2} x^{2}-320 a b \,c^{2} x +144 b^{3} c x +560 a^{2} c^{2}-360 c a \,b^{2}+63 b^{4}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (2 c x +b \right )}{315 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (2 c d x +b d \right )^{10}}\)	\(139\)
default	\(\frac {-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{9 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{9}}-\frac {16 c^{2} \left (-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{7 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{7}}+\frac {32 c^{3} \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{35 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{5}}\right )}{9 \left (4 a c -b^{2}\right )}}{1024 d^{10} c^{10}}\)	\(195\)
trager	\(-\frac {2 \left (128 c^{6} x^{8}+512 b \,c^{5} x^{7}-64 a \,c^{5} x^{6}+912 b^{2} c^{4} x^{6}-192 x^{5} a b \,c^{4}+944 b^{3} c^{3} x^{5}+48 a^{2} c^{4} x^{4}-264 a \,b^{2} c^{3} x^{4}+623 b^{4} c^{2} x^{4}+96 a^{2} b \,c^{3} x^{3}-208 a \,b^{3} c^{2} x^{3}+270 b^{5} c \,x^{3}+800 a^{3} c^{3} x^{2}-528 a^{2} b^{2} c^{2} x^{2}+54 a \,b^{4} c \,x^{2}+63 b^{6} x^{2}+800 a^{3} b \,c^{2} x -576 a^{2} b^{3} c x +126 a \,b^{5} x +560 a^{4} c^{2}-360 a^{3} b^{2} c +63 a^{2} b^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{315 d^{10} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (2 c x +b \right )^{9}}\)	\(281\)

Input:

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^10,x,method=_RETURNVERBOSE)
 

Output:

-2/315*(128*c^4*x^4+256*b*c^3*x^3-320*a*c^3*x^2+272*b^2*c^2*x^2-320*a*b*c^ 
2*x+144*b^3*c*x+560*a^2*c^2-360*a*b^2*c+63*b^4)*(c*x^2+b*x+a)^(5/2)/(2*c*x 
+b)^9/d^10/(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. 676 vs. \(2 (106) = 212\).

Time = 15.05 (sec) , antiderivative size = 676, normalized size of antiderivative = 5.73 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{10}} \, dx=\frac {2 \, {\left (128 \, c^{6} x^{8} + 512 \, b c^{5} x^{7} + 16 \, {\left (57 \, b^{2} c^{4} - 4 \, a c^{5}\right )} x^{6} + 63 \, a^{2} b^{4} - 360 \, a^{3} b^{2} c + 560 \, a^{4} c^{2} + 16 \, {\left (59 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} x^{5} + {\left (623 \, b^{4} c^{2} - 264 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (135 \, b^{5} c - 104 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x^{3} + {\left (63 \, b^{6} + 54 \, a b^{4} c - 528 \, a^{2} b^{2} c^{2} + 800 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (63 \, a b^{5} - 288 \, a^{2} b^{3} c + 400 \, a^{3} b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{315 \, {\left (512 \, {\left (b^{6} c^{9} - 12 \, a b^{4} c^{10} + 48 \, a^{2} b^{2} c^{11} - 64 \, a^{3} c^{12}\right )} d^{10} x^{9} + 2304 \, {\left (b^{7} c^{8} - 12 \, a b^{5} c^{9} + 48 \, a^{2} b^{3} c^{10} - 64 \, a^{3} b c^{11}\right )} d^{10} x^{8} + 4608 \, {\left (b^{8} c^{7} - 12 \, a b^{6} c^{8} + 48 \, a^{2} b^{4} c^{9} - 64 \, a^{3} b^{2} c^{10}\right )} d^{10} x^{7} + 5376 \, {\left (b^{9} c^{6} - 12 \, a b^{7} c^{7} + 48 \, a^{2} b^{5} c^{8} - 64 \, a^{3} b^{3} c^{9}\right )} d^{10} x^{6} + 4032 \, {\left (b^{10} c^{5} - 12 \, a b^{8} c^{6} + 48 \, a^{2} b^{6} c^{7} - 64 \, a^{3} b^{4} c^{8}\right )} d^{10} x^{5} + 2016 \, {\left (b^{11} c^{4} - 12 \, a b^{9} c^{5} + 48 \, a^{2} b^{7} c^{6} - 64 \, a^{3} b^{5} c^{7}\right )} d^{10} x^{4} + 672 \, {\left (b^{12} c^{3} - 12 \, a b^{10} c^{4} + 48 \, a^{2} b^{8} c^{5} - 64 \, a^{3} b^{6} c^{6}\right )} d^{10} x^{3} + 144 \, {\left (b^{13} c^{2} - 12 \, a b^{11} c^{3} + 48 \, a^{2} b^{9} c^{4} - 64 \, a^{3} b^{7} c^{5}\right )} d^{10} x^{2} + 18 \, {\left (b^{14} c - 12 \, a b^{12} c^{2} + 48 \, a^{2} b^{10} c^{3} - 64 \, a^{3} b^{8} c^{4}\right )} d^{10} x + {\left (b^{15} - 12 \, a b^{13} c + 48 \, a^{2} b^{11} c^{2} - 64 \, a^{3} b^{9} c^{3}\right )} d^{10}\right )}} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^10,x, algorithm="fricas")
 

Output:

2/315*(128*c^6*x^8 + 512*b*c^5*x^7 + 16*(57*b^2*c^4 - 4*a*c^5)*x^6 + 63*a^ 
2*b^4 - 360*a^3*b^2*c + 560*a^4*c^2 + 16*(59*b^3*c^3 - 12*a*b*c^4)*x^5 + ( 
623*b^4*c^2 - 264*a*b^2*c^3 + 48*a^2*c^4)*x^4 + 2*(135*b^5*c - 104*a*b^3*c 
^2 + 48*a^2*b*c^3)*x^3 + (63*b^6 + 54*a*b^4*c - 528*a^2*b^2*c^2 + 800*a^3* 
c^3)*x^2 + 2*(63*a*b^5 - 288*a^2*b^3*c + 400*a^3*b*c^2)*x)*sqrt(c*x^2 + b* 
x + a)/(512*(b^6*c^9 - 12*a*b^4*c^10 + 48*a^2*b^2*c^11 - 64*a^3*c^12)*d^10 
*x^9 + 2304*(b^7*c^8 - 12*a*b^5*c^9 + 48*a^2*b^3*c^10 - 64*a^3*b*c^11)*d^1 
0*x^8 + 4608*(b^8*c^7 - 12*a*b^6*c^8 + 48*a^2*b^4*c^9 - 64*a^3*b^2*c^10)*d 
^10*x^7 + 5376*(b^9*c^6 - 12*a*b^7*c^7 + 48*a^2*b^5*c^8 - 64*a^3*b^3*c^9)* 
d^10*x^6 + 4032*(b^10*c^5 - 12*a*b^8*c^6 + 48*a^2*b^6*c^7 - 64*a^3*b^4*c^8 
)*d^10*x^5 + 2016*(b^11*c^4 - 12*a*b^9*c^5 + 48*a^2*b^7*c^6 - 64*a^3*b^5*c 
^7)*d^10*x^4 + 672*(b^12*c^3 - 12*a*b^10*c^4 + 48*a^2*b^8*c^5 - 64*a^3*b^6 
*c^6)*d^10*x^3 + 144*(b^13*c^2 - 12*a*b^11*c^3 + 48*a^2*b^9*c^4 - 64*a^3*b 
^7*c^5)*d^10*x^2 + 18*(b^14*c - 12*a*b^12*c^2 + 48*a^2*b^10*c^3 - 64*a^3*b 
^8*c^4)*d^10*x + (b^15 - 12*a*b^13*c + 48*a^2*b^11*c^2 - 64*a^3*b^9*c^3)*d 
^10)
 

Sympy [F]

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

integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**10,x)
 

Output:

(Integral(a*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 + 1344 
0*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*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) + Integ 
ral(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))/d**10
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{10}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(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. 1387 vs. \(2 (106) = 212\).

Time = 0.75 (sec) , antiderivative size = 1387, normalized size of antiderivative = 11.75 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{10}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^10,x, algorithm="giac")
 

Output:

1/630*(3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*c^6 + 20160*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^11*b*c^(11/2) + 54180*(sqrt(c)*x - sqrt(c*x^2 + b 
*x + a))^10*b^2*c^5 + 5040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a*c^6 + 
86100*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c^(9/2) + 25200*(sqrt(c)*x 
 - sqrt(c*x^2 + b*x + a))^9*a*b*c^(11/2) + 90216*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^8*b^4*c^4 + 53172*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^2*c 
^5 + 7056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*c^6 + 66024*(sqrt(c)*x 
 - sqrt(c*x^2 + b*x + a))^7*b^5*c^(7/2) + 61488*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))^7*a*b^3*c^(9/2) + 28224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^ 
2*b*c^(11/2) + 35028*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^6*c^3 + 41832 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^4*c^4 + 47880*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^6*a^2*b^2*c^5 + 2016*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) 
)^6*a^3*c^6 + 13860*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^7*c^(5/2) + 16 
128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^5*c^(7/2) + 44856*(sqrt(c)*x 
 - sqrt(c*x^2 + b*x + a))^5*a^2*b^3*c^(9/2) + 6048*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^5*a^3*b*c^(11/2) + 4176*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4 
*b^8*c^2 + 2484*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^6*c^3 + 25416*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^4*c^4 + 6984*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^4*a^3*b^2*c^5 + 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4 
*a^4*c^6 + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^9*c^(3/2) - 576*...
 

Mupad [B] (verification not implemented)

Time = 7.74 (sec) , antiderivative size = 3111, normalized size of antiderivative = 26.36 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{10}} \, dx=\text {Too large to display} \] Input:

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^10,x)
 

Output:

(((b*((b*((b*((4*c^3*(44*a*c - b^2))/(9*d^10*(4*a*c - b^2)*(128*a*c^3 - 32 
*b^2*c^2)) - (16*b^2*c^3)/(9*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2))) 
)/(2*c) + (2*c*(23*b^3*c - 132*a*b*c^2))/(9*d^10*(4*a*c - b^2)*(128*a*c^3 
- 32*b^2*c^2))))/(2*c) + (2*c*(80*a^2*c^2 - 9*b^4 + 26*a*b^2*c))/(9*d^10*( 
4*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2))))/(2*c) + (2*c*(9*a*b^3 - 40*a^2*b* 
c))/(9*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^2/(18*d^10*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2)) 
- (8*a*c - b^2)/(18*d^10*(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/(126*d^10*(4*a*c - b^2)^2*(48*a*c^3 
- 12*b^2*c^2)) - (22*a*c - 5*b^2)/(63*d^10*(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*((c*(6*a*c - b^2) 
)/(30*d^10*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2)) - (b^2*c)/(90*d^10*(4*a 
*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (2*b^3 - 9*a*b*c)/(90*d^10*( 
4*a*c - b^2)^3*(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*((4*c*(104*a*c^3 - 16*b^2*c^2))/(945*d^10*(4*a*c - b^2 
)^4*(32*a*c^3 - 8*b^2*c^2)) - (16*b^2*c^3)/(945*d^10*(4*a*c - b^2)^4*(32*a 
*c^3 - 8*b^2*c^2))))/(2*c) - (8*b*c^2*(78*a*c - 17*b^2))/(945*d^10*(4*a*c 
- b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (4*c*(100*a^2*c^2 - 12*b^4 + 28 
*a*b^2*c))/(945*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (8* 
a*b*c*(25*a*c - 6*b^2))/(945*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2...
 

Reduce [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 1083, normalized size of antiderivative = 9.18 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{10}} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^10,x)
 

Output:

( - 2240*sqrt(a + b*x + c*x**2)*a**4*c**5 + 1440*sqrt(a + b*x + c*x**2)*a* 
*3*b**2*c**4 - 3200*sqrt(a + b*x + c*x**2)*a**3*b*c**5*x - 3200*sqrt(a + b 
*x + c*x**2)*a**3*c**6*x**2 - 252*sqrt(a + b*x + c*x**2)*a**2*b**4*c**3 + 
2304*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 - 384*sqrt(a + b*x + c*x**2)*a**2*b*c**6*x**3 - 192*s 
qrt(a + b*x + c*x**2)*a**2*c**7*x**4 - 504*sqrt(a + b*x + c*x**2)*a*b**5*c 
**3*x - 216*sqrt(a + b*x + c*x**2)*a*b**4*c**4*x**2 + 832*sqrt(a + b*x + c 
*x**2)*a*b**3*c**5*x**3 + 1056*sqrt(a + b*x + c*x**2)*a*b**2*c**6*x**4 + 7 
68*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 - 252*sqrt(a + b*x + c*x**2)*b**6*c**3*x**2 - 1080*sqrt(a + b*x + c 
*x**2)*b**5*c**4*x**3 - 2492*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 + 144*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 + 4608*sqrt(c)*b**2*c 
**7*x**7 + 2304*sqrt(c)*b*c**8*x**8 + 512*sqrt(c)*c**9*x**9)/(630*c**3*d** 
10*(64*a**3*b**9*c**3 + 1152*a**3*b**8*c**4*x + 9216*a**3*b**7*c**5*x**2 + 
 43008*a**3*b**6*c**6*x**3 + 129024*a**3*b**5*c**7*x**4 + 258048*a**3*b**4 
*c**8*x**5 + 344064*a**3*b**3*c**9*x**6 + 294912*a**3*b**2*c**10*x**7 +...