\(\int \frac {x^6 (c+d x)}{(a+b x^2)^{9/2}} \, dx\) [1320]

Optimal result

Integrand size = 20, antiderivative size = 119 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {a^2 (a d-b c x)}{7 b^4 \left (a+b x^2\right )^{7/2}}-\frac {3 a (7 a d-5 b c x)}{35 b^4 \left (a+b x^2\right )^{5/2}}+\frac {7 a d-3 b c x}{7 b^4 \left (a+b x^2\right )^{3/2}}-\frac {7 a d-b c x}{7 a b^4 \sqrt {a+b x^2}} \] Output:

1/7*a^2*(-b*c*x+a*d)/b^4/(b*x^2+a)^(7/2)-3/35*a*(-5*b*c*x+7*a*d)/b^4/(b*x^ 
2+a)^(5/2)+1/7*(-3*b*c*x+7*a*d)/b^4/(b*x^2+a)^(3/2)-1/7*(-b*c*x+7*a*d)/a/b 
^4/(b*x^2+a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.58 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-16 a^4 d-56 a^3 b d x^2-70 a^2 b^2 d x^4-35 a b^3 d x^6+5 b^4 c x^7}{35 a b^4 \left (a+b x^2\right )^{7/2}} \] Input:

Integrate[(x^6*(c + d*x))/(a + b*x^2)^(9/2),x]
 

Output:

(-16*a^4*d - 56*a^3*b*d*x^2 - 70*a^2*b^2*d*x^4 - 35*a*b^3*d*x^6 + 5*b^4*c* 
x^7)/(35*a*b^4*(a + b*x^2)^(7/2))
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {530, 25, 2345, 27, 2345, 27, 453}

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.

\(\displaystyle \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 530

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2345

\(\displaystyle \frac {-\frac {\int \frac {5 \left (\frac {2 c a^3}{b^3}+\frac {14 d x a^3}{b^3}-\frac {7 d x^3 a^2}{b^2}-\frac {7 c x^2 a^2}{b^2}\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a}-\frac {3 a^2 (7 a d-5 b c x)}{5 b^4 \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {a^2 (a d-b c x)}{7 b^4 \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\frac {2 c a^3}{b^3}+\frac {14 d x a^3}{b^3}-\frac {7 d x^3 a^2}{b^2}-\frac {7 c x^2 a^2}{b^2}}{\left (b x^2+a\right )^{5/2}}dx}{a}-\frac {3 a^2 (7 a d-5 b c x)}{5 b^4 \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {a^2 (a d-b c x)}{7 b^4 \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 2345

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 453

\(\displaystyle \frac {a^2 (a d-b c x)}{7 b^4 \left (a+b x^2\right )^{7/2}}+\frac {-\frac {3 a^2 (7 a d-5 b c x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {\frac {a (7 a d-b c x)}{b^4 \sqrt {a+b x^2}}-\frac {a^2 (7 a d-3 b c x)}{b^4 \left (a+b x^2\right )^{3/2}}}{a}}{7 a}\)

Input:

Int[(x^6*(c + d*x))/(a + b*x^2)^(9/2),x]
 

Output:

(a^2*(a*d - b*c*x))/(7*b^4*(a + b*x^2)^(7/2)) + ((-3*a^2*(7*a*d - 5*b*c*x) 
)/(5*b^4*(a + b*x^2)^(5/2)) - (-((a^2*(7*a*d - 3*b*c*x))/(b^4*(a + b*x^2)^ 
(3/2))) + (a*(7*a*d - b*c*x))/(b^4*Sqrt[a + b*x^2]))/a)/(7*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 453

Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-(a* 
d - b*c*x)/(a*b*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b, c, d}, x]
 

rule 530

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb 
ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co 
eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po 
lynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x 
)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(a 
 + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; 
FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 
1] && IntegerQ[2*p]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.55

method	result	size
gosper	\(-\frac {-5 b^{4} c \,x^{7}+35 a \,b^{3} d \,x^{6}+70 a^{2} b^{2} d \,x^{4}+56 a^{3} b d \,x^{2}+16 a^{4} d}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,b^{4}}\)	\(66\)
trager	\(-\frac {-5 b^{4} c \,x^{7}+35 a \,b^{3} d \,x^{6}+70 a^{2} b^{2} d \,x^{4}+56 a^{3} b d \,x^{2}+16 a^{4} d}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,b^{4}}\)	\(66\)
orering	\(-\frac {-5 b^{4} c \,x^{7}+35 a \,b^{3} d \,x^{6}+70 a^{2} b^{2} d \,x^{4}+56 a^{3} b d \,x^{2}+16 a^{4} d}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,b^{4}}\)	\(66\)
default	\(c \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+d \left (-\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {6 a \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )}{b}\right )\)	\(230\)

Input:

int(x^6*(d*x+c)/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

-1/35*(-5*b^4*c*x^7+35*a*b^3*d*x^6+70*a^2*b^2*d*x^4+56*a^3*b*d*x^2+16*a^4* 
d)/(b*x^2+a)^(7/2)/a/b^4
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (5 \, b^{4} c x^{7} - 35 \, a b^{3} d x^{6} - 70 \, a^{2} b^{2} d x^{4} - 56 \, a^{3} b d x^{2} - 16 \, a^{4} d\right )} \sqrt {b x^{2} + a}}{35 \, {\left (a b^{8} x^{8} + 4 \, a^{2} b^{7} x^{6} + 6 \, a^{3} b^{6} x^{4} + 4 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}} \] Input:

integrate(x^6*(d*x+c)/(b*x^2+a)^(9/2),x, algorithm="fricas")
 

Output:

1/35*(5*b^4*c*x^7 - 35*a*b^3*d*x^6 - 70*a^2*b^2*d*x^4 - 56*a^3*b*d*x^2 - 1 
6*a^4*d)*sqrt(b*x^2 + a)/(a*b^8*x^8 + 4*a^2*b^7*x^6 + 6*a^3*b^6*x^4 + 4*a^ 
4*b^5*x^2 + a^5*b^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (107) = 214\).

Time = 22.20 (sec) , antiderivative size = 464, normalized size of antiderivative = 3.90 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {c x^{7}}{7 a^{\frac {9}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 21 a^{\frac {7}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 21 a^{\frac {5}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 7 a^{\frac {3}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} + d \left (\begin {cases} - \frac {16 a^{3}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} - \frac {56 a^{2} b x^{2}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} - \frac {70 a b^{2} x^{4}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} - \frac {35 b^{3} x^{6}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) \] Input:

integrate(x**6*(d*x+c)/(b*x**2+a)**(9/2),x)
 

Output:

c*x**7/(7*a**(9/2)*sqrt(1 + b*x**2/a) + 21*a**(7/2)*b*x**2*sqrt(1 + b*x**2 
/a) + 21*a**(5/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 7*a**(3/2)*b**3*x**6*sqrt 
(1 + b*x**2/a)) + d*Piecewise((-16*a**3/(35*a**3*b**4*sqrt(a + b*x**2) + 1 
05*a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35 
*b**7*x**6*sqrt(a + b*x**2)) - 56*a**2*b*x**2/(35*a**3*b**4*sqrt(a + b*x** 
2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2 
) + 35*b**7*x**6*sqrt(a + b*x**2)) - 70*a*b**2*x**4/(35*a**3*b**4*sqrt(a + 
 b*x**2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + 
b*x**2) + 35*b**7*x**6*sqrt(a + b*x**2)) - 35*b**3*x**6/(35*a**3*b**4*sqrt 
(a + b*x**2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt( 
a + b*x**2) + 35*b**7*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**8/(8*a**(9/2) 
), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.55 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {d x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {c x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {2 \, a d x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {5 \, a c x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {8 \, a^{2} d x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {c x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {c x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, a c x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, a^{2} c x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {16 \, a^{3} d}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} \] Input:

integrate(x^6*(d*x+c)/(b*x^2+a)^(9/2),x, algorithm="maxima")
 

Output:

-d*x^6/((b*x^2 + a)^(7/2)*b) - 1/2*c*x^5/((b*x^2 + a)^(7/2)*b) - 2*a*d*x^4 
/((b*x^2 + a)^(7/2)*b^2) - 5/8*a*c*x^3/((b*x^2 + a)^(7/2)*b^2) - 8/5*a^2*d 
*x^2/((b*x^2 + a)^(7/2)*b^3) + 1/14*c*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*c*x/ 
(sqrt(b*x^2 + a)*a*b^3) + 3/56*a*c*x/((b*x^2 + a)^(5/2)*b^3) - 15/56*a^2*c 
*x/((b*x^2 + a)^(7/2)*b^3) - 16/35*a^3*d/((b*x^2 + a)^(7/2)*b^4)
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.55 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (5 \, {\left ({\left (\frac {c x}{a} - \frac {7 \, d}{b}\right )} x^{2} - \frac {14 \, a d}{b^{2}}\right )} x^{2} - \frac {56 \, a^{2} d}{b^{3}}\right )} x^{2} - \frac {16 \, a^{3} d}{b^{4}}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \] Input:

integrate(x^6*(d*x+c)/(b*x^2+a)^(9/2),x, algorithm="giac")
 

Output:

1/35*((5*((c*x/a - 7*d/b)*x^2 - 14*a*d/b^2)*x^2 - 56*a^2*d/b^3)*x^2 - 16*a 
^3*d/b^4)/(b*x^2 + a)^(7/2)
 

Mupad [B] (verification not implemented)

Time = 8.82 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {a\,d}{b^4}-\frac {3\,c\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{3/2}}-\frac {\frac {3\,a^2\,d}{5\,b^4}-\frac {3\,a\,c\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{5/2}}+\frac {\frac {a^3\,d}{7\,b^4}-\frac {a^2\,c\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {d}{b^4}-\frac {c\,x}{7\,a\,b^3}}{\sqrt {b\,x^2+a}} \] Input:

int((x^6*(c + d*x))/(a + b*x^2)^(9/2),x)
 

Output:

((a*d)/b^4 - (3*c*x)/(7*b^3))/(a + b*x^2)^(3/2) - ((3*a^2*d)/(5*b^4) - (3* 
a*c*x)/(7*b^3))/(a + b*x^2)^(5/2) + ((a^3*d)/(7*b^4) - (a^2*c*x)/(7*b^3))/ 
(a + b*x^2)^(7/2) - (d/b^4 - (c*x)/(7*a*b^3))/(a + b*x^2)^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.64 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-16 \sqrt {b \,x^{2}+a}\, a^{4} d -56 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{2}-70 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{4}-35 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,x^{6}+5 \sqrt {b \,x^{2}+a}\, b^{4} c \,x^{7}+5 \sqrt {b}\, a^{4} c +20 \sqrt {b}\, a^{3} b c \,x^{2}+30 \sqrt {b}\, a^{2} b^{2} c \,x^{4}+20 \sqrt {b}\, a \,b^{3} c \,x^{6}+5 \sqrt {b}\, b^{4} c \,x^{8}}{35 a \,b^{4} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:

int(x^6*(d*x+c)/(b*x^2+a)^(9/2),x)
 

Output:

( - 16*sqrt(a + b*x**2)*a**4*d - 56*sqrt(a + b*x**2)*a**3*b*d*x**2 - 70*sq 
rt(a + b*x**2)*a**2*b**2*d*x**4 - 35*sqrt(a + b*x**2)*a*b**3*d*x**6 + 5*sq 
rt(a + b*x**2)*b**4*c*x**7 + 5*sqrt(b)*a**4*c + 20*sqrt(b)*a**3*b*c*x**2 + 
 30*sqrt(b)*a**2*b**2*c*x**4 + 20*sqrt(b)*a*b**3*c*x**6 + 5*sqrt(b)*b**4*c 
*x**8)/(35*a*b**4*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x**4 + 4*a*b**3*x**6 
 + b**4*x**8))