\(\int \frac {(c+d x)^3 (a+b x^2)^{3/2}}{x^{10}} \, dx\) [1074]

Optimal result

Integrand size = 22, antiderivative size = 271 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^{10}} \, dx=\frac {d \left (9 b c^2-8 a d^2\right ) \sqrt {a+b x^2}}{48 x^6}+\frac {7 b d \left (9 b c^2-8 a d^2\right ) \sqrt {a+b x^2}}{192 a x^4}+\frac {b^2 d \left (9 b c^2-8 a d^2\right ) \sqrt {a+b x^2}}{128 a^2 x^2}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}-\frac {3 c^2 d \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac {c \left (4 b c^2-27 a d^2\right ) \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac {2 b c \left (4 b c^2-27 a d^2\right ) \left (a+b x^2\right )^{5/2}}{315 a^3 x^5}-\frac {b^3 d \left (9 b c^2-8 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}} \] Output:

1/48*d*(-8*a*d^2+9*b*c^2)*(b*x^2+a)^(1/2)/x^6+7/192*b*d*(-8*a*d^2+9*b*c^2) 
*(b*x^2+a)^(1/2)/a/x^4+1/128*b^2*d*(-8*a*d^2+9*b*c^2)*(b*x^2+a)^(1/2)/a^2/ 
x^2-1/9*c^3*(b*x^2+a)^(5/2)/a/x^9-3/8*c^2*d*(b*x^2+a)^(5/2)/a/x^8+1/63*c*( 
-27*a*d^2+4*b*c^2)*(b*x^2+a)^(5/2)/a^2/x^7-2/315*b*c*(-27*a*d^2+4*b*c^2)*( 
b*x^2+a)^(5/2)/a^3/x^5-1/128*b^3*d*(-8*a*d^2+9*b*c^2)*arctanh((b*x^2+a)^(1 
/2)/a^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 1.68 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^{10}} \, dx=-\frac {\sqrt {a+b x^2} \left (1024 b^4 c^3 x^8-a b^3 c x^6 \left (512 c^2+2835 c d x+6912 d^2 x^2\right )+80 a^4 \left (56 c^3+189 c^2 d x+216 c d^2 x^2+84 d^3 x^3\right )+6 a^2 b^2 x^4 \left (64 c^3+315 c^2 d x+576 c d^2 x^2+420 d^3 x^3\right )+8 a^3 b x^2 \left (800 c^3+2835 c^2 d x+3456 c d^2 x^2+1470 d^3 x^3\right )\right )}{40320 a^3 x^9}+\frac {b^3 d \left (9 b c^2-8 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{64 a^{5/2}} \] Input:

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

Output:

-1/40320*(Sqrt[a + b*x^2]*(1024*b^4*c^3*x^8 - a*b^3*c*x^6*(512*c^2 + 2835* 
c*d*x + 6912*d^2*x^2) + 80*a^4*(56*c^3 + 189*c^2*d*x + 216*c*d^2*x^2 + 84* 
d^3*x^3) + 6*a^2*b^2*x^4*(64*c^3 + 315*c^2*d*x + 576*c*d^2*x^2 + 420*d^3*x 
^3) + 8*a^3*b*x^2*(800*c^3 + 2835*c^2*d*x + 3456*c*d^2*x^2 + 1470*d^3*x^3) 
))/(a^3*x^9) + (b^3*d*(9*b*c^2 - 8*a*d^2)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b* 
x^2])/Sqrt[a]])/(64*a^(5/2))
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.96, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {540, 25, 2338, 27, 539, 25, 539, 27, 534, 243, 51, 51, 73, 221}

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.

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

\(\Big \downarrow \) 540

\(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (9 a x^2 d^3+27 a c^2 d-c \left (4 b c^2-27 a d^2\right ) x\right )}{x^9}dx}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (9 a x^2 d^3+27 a c^2 d-c \left (4 b c^2-27 a d^2\right ) x\right )}{x^9}dx}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {-\frac {\int \frac {a \left (8 c \left (4 b c^2-27 a d^2\right )+9 d \left (9 b c^2-8 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^8}dx}{8 a}-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{8} \int \frac {\left (8 c \left (4 b c^2-27 a d^2\right )+9 d \left (9 b c^2-8 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^8}dx-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{8} \left (\frac {\int -\frac {\left (63 a d \left (9 b c^2-8 a d^2\right )-16 b c \left (4 b c^2-27 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^7}dx}{7 a}+\frac {8 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{7 a x^7}\right )-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{7 a x^7}-\frac {\int \frac {\left (63 a d \left (9 b c^2-8 a d^2\right )-16 b c \left (4 b c^2-27 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^7}dx}{7 a}\right )-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{7 a x^7}-\frac {-\frac {\int \frac {3 a b \left (32 c \left (4 b c^2-27 a d^2\right )+21 d \left (9 b c^2-8 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^6}dx}{6 a}-\frac {21 d \left (a+b x^2\right )^{5/2} \left (9 b c^2-8 a d^2\right )}{2 x^6}}{7 a}\right )-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{7 a x^7}-\frac {-\frac {1}{2} b \int \frac {\left (32 c \left (4 b c^2-27 a d^2\right )+21 d \left (9 b c^2-8 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^6}dx-\frac {21 d \left (a+b x^2\right )^{5/2} \left (9 b c^2-8 a d^2\right )}{2 x^6}}{7 a}\right )-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{7 a x^7}-\frac {-\frac {1}{2} b \left (21 d \left (9 b c^2-8 a d^2\right ) \int \frac {\left (b x^2+a\right )^{3/2}}{x^5}dx-\frac {32 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{5 a x^5}\right )-\frac {21 d \left (a+b x^2\right )^{5/2} \left (9 b c^2-8 a d^2\right )}{2 x^6}}{7 a}\right )-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{7 a x^7}-\frac {-\frac {1}{2} b \left (\frac {21}{2} d \left (9 b c^2-8 a d^2\right ) \int \frac {\left (b x^2+a\right )^{3/2}}{x^6}dx^2-\frac {32 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{5 a x^5}\right )-\frac {21 d \left (a+b x^2\right )^{5/2} \left (9 b c^2-8 a d^2\right )}{2 x^6}}{7 a}\right )-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{7 a x^7}-\frac {-\frac {1}{2} b \left (\frac {21}{2} d \left (9 b c^2-8 a d^2\right ) \left (\frac {3}{4} b \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {32 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{5 a x^5}\right )-\frac {21 d \left (a+b x^2\right )^{5/2} \left (9 b c^2-8 a d^2\right )}{2 x^6}}{7 a}\right )-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{7 a x^7}-\frac {-\frac {1}{2} b \left (\frac {21}{2} d \left (9 b c^2-8 a d^2\right ) \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {32 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{5 a x^5}\right )-\frac {21 d \left (a+b x^2\right )^{5/2} \left (9 b c^2-8 a d^2\right )}{2 x^6}}{7 a}\right )-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{7 a x^7}-\frac {-\frac {1}{2} b \left (\frac {21}{2} d \left (9 b c^2-8 a d^2\right ) \left (\frac {3}{4} b \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {32 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{5 a x^5}\right )-\frac {21 d \left (a+b x^2\right )^{5/2} \left (9 b c^2-8 a d^2\right )}{2 x^6}}{7 a}\right )-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{7 a x^7}-\frac {-\frac {1}{2} b \left (\frac {21}{2} d \left (\frac {3}{4} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right ) \left (9 b c^2-8 a d^2\right )-\frac {32 c \left (a+b x^2\right )^{5/2} \left (4 b c^2-27 a d^2\right )}{5 a x^5}\right )-\frac {21 d \left (a+b x^2\right )^{5/2} \left (9 b c^2-8 a d^2\right )}{2 x^6}}{7 a}\right )-\frac {27 c^2 d \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

Input:

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

Output:

-1/9*(c^3*(a + b*x^2)^(5/2))/(a*x^9) + ((-27*c^2*d*(a + b*x^2)^(5/2))/(8*x 
^8) + ((8*c*(4*b*c^2 - 27*a*d^2)*(a + b*x^2)^(5/2))/(7*a*x^7) - ((-21*d*(9 
*b*c^2 - 8*a*d^2)*(a + b*x^2)^(5/2))/(2*x^6) - (b*((-32*c*(4*b*c^2 - 27*a* 
d^2)*(a + b*x^2)^(5/2))/(5*a*x^5) + (21*d*(9*b*c^2 - 8*a*d^2)*(-1/2*(a + b 
*x^2)^(3/2)/x^4 + (3*b*(-(Sqrt[a + b*x^2]/x^2) - (b*ArcTanh[Sqrt[a + b*x^2 
]/Sqrt[a]])/Sqrt[a]))/4))/2))/2)/(7*a))/8)/(9*a)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 534

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

rule 539

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) 
   Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] 
/; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.97

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-6912 a \,b^{3} c \,d^{2} x^{8}+1024 b^{4} c^{3} x^{8}+2520 a^{2} b^{2} d^{3} x^{7}-2835 a \,b^{3} c^{2} d \,x^{7}+3456 a^{2} b^{2} c \,d^{2} x^{6}-512 a \,b^{3} c^{3} x^{6}+11760 a^{3} b \,d^{3} x^{5}+1890 a^{2} b^{2} c^{2} d \,x^{5}+27648 a^{3} b c \,d^{2} x^{4}+384 a^{2} b^{2} c^{3} x^{4}+6720 a^{4} d^{3} x^{3}+22680 a^{3} b \,c^{2} d \,x^{3}+17280 a^{4} c \,d^{2} x^{2}+6400 a^{3} b \,c^{3} x^{2}+15120 a^{4} c^{2} d x +4480 a^{4} c^{3}\right )}{40320 x^{9} a^{3}}+\frac {\left (8 a \,d^{2}-9 b \,c^{2}\right ) b^{3} d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{128 a^{\frac {5}{2}}}\)	\(263\)
default	\(c^{3} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 a \,x^{9}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )+d^{3} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+3 c \,d^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )+3 c^{2} d \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 a \,x^{8}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )\)	\(390\)

Input:

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

Output:

-1/40320*(b*x^2+a)^(1/2)*(-6912*a*b^3*c*d^2*x^8+1024*b^4*c^3*x^8+2520*a^2* 
b^2*d^3*x^7-2835*a*b^3*c^2*d*x^7+3456*a^2*b^2*c*d^2*x^6-512*a*b^3*c^3*x^6+ 
11760*a^3*b*d^3*x^5+1890*a^2*b^2*c^2*d*x^5+27648*a^3*b*c*d^2*x^4+384*a^2*b 
^2*c^3*x^4+6720*a^4*d^3*x^3+22680*a^3*b*c^2*d*x^3+17280*a^4*c*d^2*x^2+6400 
*a^3*b*c^3*x^2+15120*a^4*c^2*d*x+4480*a^4*c^3)/x^9/a^3+1/128*(8*a*d^2-9*b* 
c^2)/a^(5/2)*b^3*d*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.00 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^{10}} \, dx=\left [-\frac {315 \, {\left (9 \, b^{4} c^{2} d - 8 \, a b^{3} d^{3}\right )} \sqrt {a} x^{9} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (256 \, {\left (4 \, b^{4} c^{3} - 27 \, a b^{3} c d^{2}\right )} x^{8} + 15120 \, a^{4} c^{2} d x - 315 \, {\left (9 \, a b^{3} c^{2} d - 8 \, a^{2} b^{2} d^{3}\right )} x^{7} + 4480 \, a^{4} c^{3} - 128 \, {\left (4 \, a b^{3} c^{3} - 27 \, a^{2} b^{2} c d^{2}\right )} x^{6} + 210 \, {\left (9 \, a^{2} b^{2} c^{2} d + 56 \, a^{3} b d^{3}\right )} x^{5} + 384 \, {\left (a^{2} b^{2} c^{3} + 72 \, a^{3} b c d^{2}\right )} x^{4} + 840 \, {\left (27 \, a^{3} b c^{2} d + 8 \, a^{4} d^{3}\right )} x^{3} + 640 \, {\left (10 \, a^{3} b c^{3} + 27 \, a^{4} c d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{80640 \, a^{3} x^{9}}, \frac {315 \, {\left (9 \, b^{4} c^{2} d - 8 \, a b^{3} d^{3}\right )} \sqrt {-a} x^{9} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (256 \, {\left (4 \, b^{4} c^{3} - 27 \, a b^{3} c d^{2}\right )} x^{8} + 15120 \, a^{4} c^{2} d x - 315 \, {\left (9 \, a b^{3} c^{2} d - 8 \, a^{2} b^{2} d^{3}\right )} x^{7} + 4480 \, a^{4} c^{3} - 128 \, {\left (4 \, a b^{3} c^{3} - 27 \, a^{2} b^{2} c d^{2}\right )} x^{6} + 210 \, {\left (9 \, a^{2} b^{2} c^{2} d + 56 \, a^{3} b d^{3}\right )} x^{5} + 384 \, {\left (a^{2} b^{2} c^{3} + 72 \, a^{3} b c d^{2}\right )} x^{4} + 840 \, {\left (27 \, a^{3} b c^{2} d + 8 \, a^{4} d^{3}\right )} x^{3} + 640 \, {\left (10 \, a^{3} b c^{3} + 27 \, a^{4} c d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{40320 \, a^{3} x^{9}}\right ] \] Input:

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

Output:

[-1/80640*(315*(9*b^4*c^2*d - 8*a*b^3*d^3)*sqrt(a)*x^9*log(-(b*x^2 + 2*sqr 
t(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(256*(4*b^4*c^3 - 27*a*b^3*c*d^2)*x^8 
 + 15120*a^4*c^2*d*x - 315*(9*a*b^3*c^2*d - 8*a^2*b^2*d^3)*x^7 + 4480*a^4* 
c^3 - 128*(4*a*b^3*c^3 - 27*a^2*b^2*c*d^2)*x^6 + 210*(9*a^2*b^2*c^2*d + 56 
*a^3*b*d^3)*x^5 + 384*(a^2*b^2*c^3 + 72*a^3*b*c*d^2)*x^4 + 840*(27*a^3*b*c 
^2*d + 8*a^4*d^3)*x^3 + 640*(10*a^3*b*c^3 + 27*a^4*c*d^2)*x^2)*sqrt(b*x^2 
+ a))/(a^3*x^9), 1/40320*(315*(9*b^4*c^2*d - 8*a*b^3*d^3)*sqrt(-a)*x^9*arc 
tan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (256*(4*b^4*c^3 - 27*a*b^3*c*d^2)*x^8 + 
15120*a^4*c^2*d*x - 315*(9*a*b^3*c^2*d - 8*a^2*b^2*d^3)*x^7 + 4480*a^4*c^3 
 - 128*(4*a*b^3*c^3 - 27*a^2*b^2*c*d^2)*x^6 + 210*(9*a^2*b^2*c^2*d + 56*a^ 
3*b*d^3)*x^5 + 384*(a^2*b^2*c^3 + 72*a^3*b*c*d^2)*x^4 + 840*(27*a^3*b*c^2* 
d + 8*a^4*d^3)*x^3 + 640*(10*a^3*b*c^3 + 27*a^4*c*d^2)*x^2)*sqrt(b*x^2 + a 
))/(a^3*x^9)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1785 vs. \(2 (260) = 520\).

Time = 36.20 (sec) , antiderivative size = 1785, normalized size of antiderivative = 6.59 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^{10}} \, dx=\text {Too large to display} \] Input:

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

Output:

-35*a**8*b**(19/2)*c**3*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a** 
6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 110*a**7*b* 
*(21/2)*c**3*x**2*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**1 
0*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 114*a**6*b**(23/2 
)*c**3*x**4*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**1 
0 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 45*a**6*b**(9/2)*c*d**2 
*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3* 
b**6*x**10) - 40*a**5*b**(25/2)*c**3*x**6*sqrt(a/(b*x**2) + 1)/(315*a**7*b 
**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x* 
*14) - 15*a**5*b**(11/2)*c**3*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 2 
10*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 99*a**5*b**(11/2)*c*d**2*x**2*s 
qrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b* 
*6*x**10) + 5*a**4*b**(27/2)*c**3*x**8*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9 
*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14 
) - 33*a**4*b**(13/2)*c**3*x**2*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 
 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 51*a**4*b**(13/2)*c*d**2*x**4 
*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3* 
b**6*x**10) + 30*a**3*b**(29/2)*c**3*x**10*sqrt(a/(b*x**2) + 1)/(315*a**7* 
b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x 
**14) - 17*a**3*b**(15/2)*c**3*x**4*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.45 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^{10}} \, dx=-\frac {9 \, b^{4} c^{2} d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {5}{2}}} + \frac {b^{3} d^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4} c^{2} d}{128 \, a^{4}} + \frac {9 \, \sqrt {b x^{2} + a} b^{4} c^{2} d}{128 \, a^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} d^{3}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} b^{3} d^{3}}{16 \, a^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3} c^{2} d}{128 \, a^{4} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} d^{3}}{48 \, a^{3} x^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} c^{2} d}{64 \, a^{3} x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b d^{3}}{24 \, a^{2} x^{4}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} c^{3}}{315 \, a^{3} x^{5}} + \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b c d^{2}}{35 \, a^{2} x^{5}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b c^{2} d}{16 \, a^{2} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} d^{3}}{6 \, a x^{6}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b c^{3}}{63 \, a^{2} x^{7}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c d^{2}}{7 \, a x^{7}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2} d}{8 \, a x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{3}}{9 \, a x^{9}} \] Input:

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

Output:

-9/128*b^4*c^2*d*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + 1/16*b^3*d^3*arcs 
inh(a/(sqrt(a*b)*abs(x)))/a^(3/2) + 3/128*(b*x^2 + a)^(3/2)*b^4*c^2*d/a^4 
+ 9/128*sqrt(b*x^2 + a)*b^4*c^2*d/a^3 - 1/48*(b*x^2 + a)^(3/2)*b^3*d^3/a^3 
 - 1/16*sqrt(b*x^2 + a)*b^3*d^3/a^2 - 3/128*(b*x^2 + a)^(5/2)*b^3*c^2*d/(a 
^4*x^2) + 1/48*(b*x^2 + a)^(5/2)*b^2*d^3/(a^3*x^2) - 3/64*(b*x^2 + a)^(5/2 
)*b^2*c^2*d/(a^3*x^4) + 1/24*(b*x^2 + a)^(5/2)*b*d^3/(a^2*x^4) - 8/315*(b* 
x^2 + a)^(5/2)*b^2*c^3/(a^3*x^5) + 6/35*(b*x^2 + a)^(5/2)*b*c*d^2/(a^2*x^5 
) + 3/16*(b*x^2 + a)^(5/2)*b*c^2*d/(a^2*x^6) - 1/6*(b*x^2 + a)^(5/2)*d^3/( 
a*x^6) + 4/63*(b*x^2 + a)^(5/2)*b*c^3/(a^2*x^7) - 3/7*(b*x^2 + a)^(5/2)*c* 
d^2/(a*x^7) - 3/8*(b*x^2 + a)^(5/2)*c^2*d/(a*x^8) - 1/9*(b*x^2 + a)^(5/2)* 
c^3/(a*x^9)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (235) = 470\).

Time = 0.15 (sec) , antiderivative size = 983, normalized size of antiderivative = 3.63 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^{10}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/64*(9*b^4*c^2*d - 8*a*b^3*d^3)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqr 
t(-a))/(sqrt(-a)*a^2) - 1/20160*(2835*(sqrt(b)*x - sqrt(b*x^2 + a))^17*b^4 
*c^2*d - 2520*(sqrt(b)*x - sqrt(b*x^2 + a))^17*a*b^3*d^3 - 24570*(sqrt(b)* 
x - sqrt(b*x^2 + a))^15*a*b^4*c^2*d - 31920*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
15*a^2*b^3*d^3 - 241920*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^2*b^(7/2)*c*d^2 
 - 292950*(sqrt(b)*x - sqrt(b*x^2 + a))^13*a^2*b^4*c^2*d + 45360*(sqrt(b)* 
x - sqrt(b*x^2 + a))^13*a^3*b^3*d^3 - 215040*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^12*a^2*b^(9/2)*c^3 + 241920*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^3*b^(7/2)* 
c*d^2 - 319410*(sqrt(b)*x - sqrt(b*x^2 + a))^11*a^3*b^4*c^2*d + 15120*(sqr 
t(b)*x - sqrt(b*x^2 + a))^11*a^4*b^3*d^3 - 322560*(sqrt(b)*x - sqrt(b*x^2 
+ a))^10*a^3*b^(9/2)*c^3 - 241920*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^4*b^( 
7/2)*c*d^2 - 451584*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^4*b^(9/2)*c^3 + 6289 
92*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^5*b^(7/2)*c*d^2 + 319410*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^7*a^5*b^4*c^2*d - 15120*(sqrt(b)*x - sqrt(b*x^2 + a))^7* 
a^6*b^3*d^3 - 129024*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^5*b^(9/2)*c^3 - 338 
688*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^6*b^(7/2)*c*d^2 + 292950*(sqrt(b)*x 
- sqrt(b*x^2 + a))^5*a^6*b^4*c^2*d - 45360*(sqrt(b)*x - sqrt(b*x^2 + a))^5 
*a^7*b^3*d^3 - 36864*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^6*b^(9/2)*c^3 + 691 
2*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^7*b^(7/2)*c*d^2 + 24570*(sqrt(b)*x - s 
qrt(b*x^2 + a))^3*a^7*b^4*c^2*d + 31920*(sqrt(b)*x - sqrt(b*x^2 + a))^3...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 7.33 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.87 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^{10}} \, dx=\frac {-4480 \sqrt {b \,x^{2}+a}\, a^{4} c^{3}-15120 \sqrt {b \,x^{2}+a}\, a^{4} c^{2} d x -17280 \sqrt {b \,x^{2}+a}\, a^{4} c \,d^{2} x^{2}-6720 \sqrt {b \,x^{2}+a}\, a^{4} d^{3} x^{3}-6400 \sqrt {b \,x^{2}+a}\, a^{3} b \,c^{3} x^{2}-22680 \sqrt {b \,x^{2}+a}\, a^{3} b \,c^{2} d \,x^{3}-27648 \sqrt {b \,x^{2}+a}\, a^{3} b c \,d^{2} x^{4}-11760 \sqrt {b \,x^{2}+a}\, a^{3} b \,d^{3} x^{5}-384 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c^{3} x^{4}-1890 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c^{2} d \,x^{5}-3456 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,d^{2} x^{6}-2520 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d^{3} x^{7}+512 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{3} x^{6}+2835 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{2} d \,x^{7}+6912 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,d^{2} x^{8}-1024 \sqrt {b \,x^{2}+a}\, b^{4} c^{3} x^{8}-2520 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d^{3} x^{9}+2835 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c^{2} d \,x^{9}+2520 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d^{3} x^{9}-2835 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c^{2} d \,x^{9}-6912 \sqrt {b}\, a \,b^{3} c \,d^{2} x^{9}+1024 \sqrt {b}\, b^{4} c^{3} x^{9}}{40320 a^{3} x^{9}} \] Input:

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

Output:

( - 4480*sqrt(a + b*x**2)*a**4*c**3 - 15120*sqrt(a + b*x**2)*a**4*c**2*d*x 
 - 17280*sqrt(a + b*x**2)*a**4*c*d**2*x**2 - 6720*sqrt(a + b*x**2)*a**4*d* 
*3*x**3 - 6400*sqrt(a + b*x**2)*a**3*b*c**3*x**2 - 22680*sqrt(a + b*x**2)* 
a**3*b*c**2*d*x**3 - 27648*sqrt(a + b*x**2)*a**3*b*c*d**2*x**4 - 11760*sqr 
t(a + b*x**2)*a**3*b*d**3*x**5 - 384*sqrt(a + b*x**2)*a**2*b**2*c**3*x**4 
- 1890*sqrt(a + b*x**2)*a**2*b**2*c**2*d*x**5 - 3456*sqrt(a + b*x**2)*a**2 
*b**2*c*d**2*x**6 - 2520*sqrt(a + b*x**2)*a**2*b**2*d**3*x**7 + 512*sqrt(a 
 + b*x**2)*a*b**3*c**3*x**6 + 2835*sqrt(a + b*x**2)*a*b**3*c**2*d*x**7 + 6 
912*sqrt(a + b*x**2)*a*b**3*c*d**2*x**8 - 1024*sqrt(a + b*x**2)*b**4*c**3* 
x**8 - 2520*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))* 
a*b**3*d**3*x**9 + 2835*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)* 
x)/sqrt(a))*b**4*c**2*d*x**9 + 2520*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a 
) + sqrt(b)*x)/sqrt(a))*a*b**3*d**3*x**9 - 2835*sqrt(a)*log((sqrt(a + b*x* 
*2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**4*c**2*d*x**9 - 6912*sqrt(b)*a*b**3 
*c*d**2*x**9 + 1024*sqrt(b)*b**4*c**3*x**9)/(40320*a**3*x**9)