Integrand size = 26, antiderivative size = 79 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac {4 \left (a+b x+c x^2\right )^{7/2}}{63 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)^7} \]
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
Time = 10.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\frac {2 (a+x (b+c x))^{7/2} \left (9 b^2+8 b c x+4 c \left (-7 a+2 c x^2\right )\right )}{63 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)^9} \]
(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)
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 |
\(\displaystyle \frac {4 \left (a+b x+c x^2\right )^{7/2}}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9}\) |
(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)
3.13.31.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
Time = 24.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(-\frac {2 \left (-8 c^{2} x^{2}-8 b c x +28 a c -9 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{63 \left (2 c x +b \right )^{9} d^{10} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(70\) |
default | \(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{9 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{9}}+\frac {32 c^{3} \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{63 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{7}}}{1024 d^{10} c^{10}}\) | \(122\) |
trager | \(-\frac {2 \left (-8 c^{5} x^{8}-32 b \,c^{4} x^{7}+4 a \,c^{4} x^{6}-57 b^{2} c^{3} x^{6}+12 a b \,c^{3} x^{5}-59 c^{2} x^{5} b^{3}+60 a^{2} c^{3} x^{4}-15 a \,b^{2} c^{2} x^{4}-35 b^{4} c \,x^{4}+120 a^{2} b \,c^{2} x^{3}-50 a \,b^{3} c \,x^{3}-9 x^{3} b^{5}+76 a^{3} c^{2} x^{2}+33 c \,x^{2} a^{2} b^{2}-27 b^{4} x^{2} a +76 a^{3} b c x -27 a^{2} b^{3} x +28 a^{4} c -9 a^{3} b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{63 d^{10} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \left (2 c x +b \right )^{9}}\) | \(231\) |
-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)
Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (71) = 142\).
Time = 18.08 (sec) , antiderivative size = 530, normalized size of antiderivative = 6.71 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\frac {2 \, {\left (8 \, c^{5} x^{8} + 32 \, b c^{4} x^{7} + {\left (57 \, b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} + {\left (59 \, b^{3} c^{2} - 12 \, a b c^{3}\right )} x^{5} + 9 \, a^{3} b^{2} - 28 \, a^{4} c + 5 \, {\left (7 \, b^{4} c + 3 \, a b^{2} c^{2} - 12 \, a^{2} c^{3}\right )} x^{4} + {\left (9 \, b^{5} + 50 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} x^{3} + {\left (27 \, a b^{4} - 33 \, a^{2} b^{2} c - 76 \, a^{3} c^{2}\right )} x^{2} + {\left (27 \, a^{2} b^{3} - 76 \, a^{3} b c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{63 \, {\left (512 \, {\left (b^{4} c^{9} - 8 \, a b^{2} c^{10} + 16 \, a^{2} c^{11}\right )} d^{10} x^{9} + 2304 \, {\left (b^{5} c^{8} - 8 \, a b^{3} c^{9} + 16 \, a^{2} b c^{10}\right )} d^{10} x^{8} + 4608 \, {\left (b^{6} c^{7} - 8 \, a b^{4} c^{8} + 16 \, a^{2} b^{2} c^{9}\right )} d^{10} x^{7} + 5376 \, {\left (b^{7} c^{6} - 8 \, a b^{5} c^{7} + 16 \, a^{2} b^{3} c^{8}\right )} d^{10} x^{6} + 4032 \, {\left (b^{8} c^{5} - 8 \, a b^{6} c^{6} + 16 \, a^{2} b^{4} c^{7}\right )} d^{10} x^{5} + 2016 \, {\left (b^{9} c^{4} - 8 \, a b^{7} c^{5} + 16 \, a^{2} b^{5} c^{6}\right )} d^{10} x^{4} + 672 \, {\left (b^{10} c^{3} - 8 \, a b^{8} c^{4} + 16 \, a^{2} b^{6} c^{5}\right )} d^{10} x^{3} + 144 \, {\left (b^{11} c^{2} - 8 \, a b^{9} c^{3} + 16 \, a^{2} b^{7} c^{4}\right )} d^{10} x^{2} + 18 \, {\left (b^{12} c - 8 \, a b^{10} c^{2} + 16 \, a^{2} b^{8} c^{3}\right )} d^{10} x + {\left (b^{13} - 8 \, a b^{11} c + 16 \, a^{2} b^{9} c^{2}\right )} d^{10}\right )}} \]
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)
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx}{d^{10}} \]
(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...
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 1843 vs. \(2 (71) = 142\).
Time = 1.15 (sec) , antiderivative size = 1843, normalized size of antiderivative = 23.33 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\text {Too large to display} \]
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...
Time = 16.22 (sec) , antiderivative size = 5511, normalized size of antiderivative = 69.76 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx=\text {Too large to display} \]
(((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*...