\(\int \frac {(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^9} \, dx\) [175]

Optimal result

Integrand size = 26, antiderivative size = 207 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=-\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{512 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^4}+\frac {3 \sqrt {a+b x+c x^2}}{1024 c^2 \left (b^2-4 a c\right )^2 d^9 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac {3 \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2048 c^{5/2} \left (b^2-4 a c\right )^{5/2} d^9} \] Output:

-1/128*(c*x^2+b*x+a)^(1/2)/c^2/d^9/(2*c*x+b)^6+1/512*(c*x^2+b*x+a)^(1/2)/c 
^2/(-4*a*c+b^2)/d^9/(2*c*x+b)^4+3/1024*(c*x^2+b*x+a)^(1/2)/c^2/(-4*a*c+b^2 
)^2/d^9/(2*c*x+b)^2-1/16*(c*x^2+b*x+a)^(3/2)/c/d^9/(2*c*x+b)^8+3/2048*arct 
an(2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(5/2)/(-4*a*c+b^2)^ 
(5/2)/d^9
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.30 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\frac {2 (a+x (b+c x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},5,\frac {7}{2},\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{5 \left (b^2-4 a c\right )^5 d^9} \] Input:

Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9,x]
 

Output:

(2*(a + x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, 5, 7/2, (4*c*(a + x*(b + 
 c*x)))/(-b^2 + 4*a*c)])/(5*(b^2 - 4*a*c)^5*d^9)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1108, 27, 1108, 1117, 1117, 1112, 218}

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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx\)

\(\Big \downarrow \) 1108

\(\displaystyle \frac {3 \int \frac {\sqrt {c x^2+b x+a}}{d^7 (b+2 c x)^7}dx}{32 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \int \frac {\sqrt {c x^2+b x+a}}{(b+2 c x)^7}dx}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}\)

\(\Big \downarrow \) 1108

\(\displaystyle \frac {3 \left (\frac {\int \frac {1}{(b+2 c x)^5 \sqrt {c x^2+b x+a}}dx}{24 c}-\frac {\sqrt {a+b x+c x^2}}{12 c (b+2 c x)^6}\right )}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {3 \left (\frac {\frac {3 \int \frac {1}{(b+2 c x)^3 \sqrt {c x^2+b x+a}}dx}{4 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (b+2 c x)^4}}{24 c}-\frac {\sqrt {a+b x+c x^2}}{12 c (b+2 c x)^6}\right )}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {3 \left (\frac {\frac {3 \left (\frac {\int \frac {1}{(b+2 c x) \sqrt {c x^2+b x+a}}dx}{2 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}\right )}{4 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (b+2 c x)^4}}{24 c}-\frac {\sqrt {a+b x+c x^2}}{12 c (b+2 c x)^6}\right )}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}\)

\(\Big \downarrow \) 1112

\(\displaystyle \frac {3 \left (\frac {\frac {3 \left (\frac {2 c \int \frac {1}{8 \left (c x^2+b x+a\right ) c^2+2 \left (b^2-4 a c\right ) c}d\sqrt {c x^2+b x+a}}{b^2-4 a c}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}\right )}{4 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (b+2 c x)^4}}{24 c}-\frac {\sqrt {a+b x+c x^2}}{12 c (b+2 c x)^6}\right )}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {3 \left (\frac {\frac {3 \left (\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {c} \left (b^2-4 a c\right )^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}\right )}{4 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (b+2 c x)^4}}{24 c}-\frac {\sqrt {a+b x+c x^2}}{12 c (b+2 c x)^6}\right )}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}\)

Input:

Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9,x]
 

Output:

-1/16*(a + b*x + c*x^2)^(3/2)/(c*d^9*(b + 2*c*x)^8) + (3*(-1/12*Sqrt[a + b 
*x + c*x^2]/(c*(b + 2*c*x)^6) + (Sqrt[a + b*x + c*x^2]/(2*(b^2 - 4*a*c)*(b 
 + 2*c*x)^4) + (3*(Sqrt[a + b*x + c*x^2]/((b^2 - 4*a*c)*(b + 2*c*x)^2) + A 
rcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]]/(2*Sqrt[c]*(b^2 
 - 4*a*c)^(3/2))))/(4*(b^2 - 4*a*c)))/(24*c)))/(32*c*d^9)
 

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1108

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

rule 1112

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symb 
ol] :> Simp[4*c   Subst[Int[1/(b^2*e - 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a 
+ b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 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 = 1.09 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(-\frac {\frac {3 \left (2 c x +b \right )^{8} \operatorname {arctanh}\left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, c}{\sqrt {4 a \,c^{2}-b^{2} c}}\right )}{2048}+\sqrt {c \,x^{2}+b x +a}\, \left (\frac {c^{2} x^{2}}{2}+\left (\frac {b x}{2}+a \right ) c -\frac {b^{2}}{8}\right ) \left (-\frac {3 c^{4} x^{4}}{8}+x^{2} \left (-\frac {3 b x}{4}+a \right ) c^{3}+\left (-\frac {13}{16} b^{2} x^{2}+a b x +a^{2}\right ) c^{2}-\frac {\left (\frac {7 b x}{4}+a \right ) b^{2} c}{4}-\frac {3 b^{4}}{128}\right ) \sqrt {4 a \,c^{2}-b^{2} c}}{16 \sqrt {4 a \,c^{2}-b^{2} c}\, \left (2 c x +b \right )^{8} \left (a c -\frac {b^{2}}{4}\right )^{2} d^{9} c^{2}}\)	\(194\)
default	\(\frac {-\frac {c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{2 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{8}}-\frac {3 c^{2} \left (-\frac {2 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6}}-\frac {2 c^{2} \left (-\frac {c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}+\frac {c^{2} \left (-\frac {2 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {6 c^{2} \left (\frac {\left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\right )}{2 \left (4 a c -b^{2}\right )}}{512 d^{9} c^{9}}\)	\(488\)

Input:

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^9,x,method=_RETURNVERBOSE)
 

Output:

-1/16/(4*a*c^2-b^2*c)^(1/2)*(3/2048*(2*c*x+b)^8*arctanh(2*(c*x^2+b*x+a)^(1 
/2)*c/(4*a*c^2-b^2*c)^(1/2))+(c*x^2+b*x+a)^(1/2)*(1/2*c^2*x^2+(1/2*b*x+a)* 
c-1/8*b^2)*(-3/8*c^4*x^4+x^2*(-3/4*b*x+a)*c^3+(-13/16*b^2*x^2+a*b*x+a^2)*c 
^2-1/4*(7/4*b*x+a)*b^2*c-3/128*b^4)*(4*a*c^2-b^2*c)^(1/2))/(2*c*x+b)^8/(a* 
c-1/4*b^2)^2/d^9/c^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 801 vs. \(2 (179) = 358\).

Time = 9.79 (sec) , antiderivative size = 1639, normalized size of antiderivative = 7.92 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^9,x, algorithm="fricas")
 

Output:

[-1/4096*(3*(256*c^8*x^8 + 1024*b*c^7*x^7 + 1792*b^2*c^6*x^6 + 1792*b^3*c^ 
5*x^5 + 1120*b^4*c^4*x^4 + 448*b^5*c^3*x^3 + 112*b^6*c^2*x^2 + 16*b^7*c*x 
+ b^8)*sqrt(-b^2*c + 4*a*c^2)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4* 
sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a))/(4*c^2*x^2 + 4*b*c*x + b^2)) 
 + 4*(3*b^8*c - 4*a*b^6*c^2 - 416*a^2*b^4*c^3 + 2560*a^3*b^2*c^4 - 4096*a^ 
4*c^5 - 192*(b^2*c^7 - 4*a*c^8)*x^6 - 576*(b^3*c^6 - 4*a*b*c^7)*x^5 - 16*( 
47*b^4*c^5 - 196*a*b^2*c^6 + 32*a^2*c^7)*x^4 - 32*(17*b^5*c^4 - 76*a*b^3*c 
^5 + 32*a^2*b*c^6)*x^3 - 12*(11*b^6*c^3 + 4*a*b^4*c^4 - 320*a^2*b^2*c^5 + 
512*a^3*c^6)*x^2 + 4*(11*b^7*c^2 - 220*a*b^5*c^3 + 1088*a^2*b^3*c^4 - 1536 
*a^3*b*c^5)*x)*sqrt(c*x^2 + b*x + a))/(256*(b^6*c^11 - 12*a*b^4*c^12 + 48* 
a^2*b^2*c^13 - 64*a^3*c^14)*d^9*x^8 + 1024*(b^7*c^10 - 12*a*b^5*c^11 + 48* 
a^2*b^3*c^12 - 64*a^3*b*c^13)*d^9*x^7 + 1792*(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^9*x^6 + 1792*(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^9*x^5 + 1120*(b^10*c^7 - 12*a*b^8*c^8 
 + 48*a^2*b^6*c^9 - 64*a^3*b^4*c^10)*d^9*x^4 + 448*(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^9*x^3 + 112*(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^9*x^2 + 16*(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^9*x + (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^9), -1/2048*(3*(256*c^8*x^8 + 1024*b*c 
^7*x^7 + 1792*b^2*c^6*x^6 + 1792*b^3*c^5*x^5 + 1120*b^4*c^4*x^4 + 448*b...
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx}{d^{9}} \] Input:

integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**9,x)
 

Output:

(Integral(a*sqrt(a + b*x + c*x**2)/(b**9 + 18*b**8*c*x + 144*b**7*c**2*x** 
2 + 672*b**6*c**3*x**3 + 2016*b**5*c**4*x**4 + 4032*b**4*c**5*x**5 + 5376* 
b**3*c**6*x**6 + 4608*b**2*c**7*x**7 + 2304*b*c**8*x**8 + 512*c**9*x**9), 
x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**9 + 18*b**8*c*x + 144*b**7*c* 
*2*x**2 + 672*b**6*c**3*x**3 + 2016*b**5*c**4*x**4 + 4032*b**4*c**5*x**5 + 
 5376*b**3*c**6*x**6 + 4608*b**2*c**7*x**7 + 2304*b*c**8*x**8 + 512*c**9*x 
**9), x) + Integral(c*x**2*sqrt(a + b*x + c*x**2)/(b**9 + 18*b**8*c*x + 14 
4*b**7*c**2*x**2 + 672*b**6*c**3*x**3 + 2016*b**5*c**4*x**4 + 4032*b**4*c* 
*5*x**5 + 5376*b**3*c**6*x**6 + 4608*b**2*c**7*x**7 + 2304*b*c**8*x**8 + 5 
12*c**9*x**9), x))/d**9
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^9,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. 2225 vs. \(2 (179) = 358\).

Time = 0.62 (sec) , antiderivative size = 2225, normalized size of antiderivative = 10.75 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^9,x, algorithm="giac")
 

Output:

3/1024*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt( 
b^2*c - 4*a*c^2))/((b^4*c^2*d^9 - 8*a*b^2*c^3*d^9 + 16*a^2*c^4*d^9)*sqrt(b 
^2*c - 4*a*c^2)) - 1/1024*(384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*c^(1 
5/2) + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*b*c^7 + 10816*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^13*b^2*c^(13/2) - 2944*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^13*a*c^(15/2) + 26624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*b 
^3*c^6 - 19136*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a*b*c^7 + 44448*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^11*b^4*c^(11/2) - 36096*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^11*a*b^2*c^(13/2) - 42624*(sqrt(c)*x - sqrt(c*x^2 + b*x 
+ a))^11*a^2*c^(15/2) + 47696*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^5*c 
^5 + 11968*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a*b^3*c^6 - 234432*(sqrt 
(c)*x - sqrt(c*x^2 + b*x + a))^10*a^2*b*c^7 + 27632*(sqrt(c)*x - sqrt(c*x^ 
2 + b*x + a))^9*b^6*c^(9/2) + 145376*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9 
*a*b^4*c^(11/2) - 521664*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b^2*c^( 
13/2) - 85888*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*c^(15/2) + 248688* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^5*c^5 - 589248*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^8*a^2*b^3*c^6 - 386496*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
a))^8*a^3*b*c^7 - 13816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^8*c^(7/2) 
+ 221056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^6*c^(9/2) - 331584*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^4*c^(11/2) - 687104*(sqrt(c)*x ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^9} \,d x \] Input:

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9,x)
 

Output:

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9, x)
 

Reduce [B] (verification not implemented)

Time = 10.04 (sec) , antiderivative size = 2202, normalized size of antiderivative = 10.64 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^9,x)
 

Output:

(3*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt( 
a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**8 + 48*sqrt(c)*sqrt( 
4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2 
) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**7*c*x + 336*sqrt(c)*sqrt(4*a*c - b** 
2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c 
*x)/sqrt(4*a*c - b**2))*b**6*c**2*x**2 + 1344*sqrt(c)*sqrt(4*a*c - b**2)*l 
og(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/ 
sqrt(4*a*c - b**2))*b**5*c**3*x**3 + 3360*sqrt(c)*sqrt(4*a*c - b**2)*log(( 
 - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt 
(4*a*c - b**2))*b**4*c**4*x**4 + 5376*sqrt(c)*sqrt(4*a*c - b**2)*log(( - s 
qrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a 
*c - b**2))*b**3*c**5*x**5 + 5376*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt( 
4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - 
 b**2))*b**2*c**6*x**6 + 3072*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a* 
c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b** 
2))*b*c**7*x**7 + 768*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2 
) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*c**8 
*x**8 - 3*sqrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*s 
qrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**8 - 48*sqrt(c)*s 
qrt(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*...