3.1233 \(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{12}} \, dx\)

Optimal. Leaf size=118 \[ \frac {16 \left (a+b x+c x^2\right )^{7/2}}{693 d^{12} \left (b^2-4 a c\right )^3 (b+2 c x)^7}+\frac {8 \left (a+b x+c x^2\right )^{7/2}}{99 d^{12} \left (b^2-4 a c\right )^2 (b+2 c x)^9}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}} \]

[Out]

2/11*(c*x^2+b*x+a)^(7/2)/(-4*a*c+b^2)/d^12/(2*c*x+b)^11+8/99*(c*x^2+b*x+a)^(7/2)/(-4*a*c+b^2)^2/d^12/(2*c*x+b)
^9+16/693*(c*x^2+b*x+a)^(7/2)/(-4*a*c+b^2)^3/d^12/(2*c*x+b)^7

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Rubi [A]  time = 0.06, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {693, 682} \[ \frac {16 \left (a+b x+c x^2\right )^{7/2}}{693 d^{12} \left (b^2-4 a c\right )^3 (b+2 c x)^7}+\frac {8 \left (a+b x+c x^2\right )^{7/2}}{99 d^{12} \left (b^2-4 a c\right )^2 (b+2 c x)^9}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]

[Out]

(2*(a + b*x + c*x^2)^(7/2))/(11*(b^2 - 4*a*c)*d^12*(b + 2*c*x)^11) + (8*(a + b*x + c*x^2)^(7/2))/(99*(b^2 - 4*
a*c)^2*d^12*(b + 2*c*x)^9) + (16*(a + b*x + c*x^2)^(7/2))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^7)

Rule 682

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> 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] && NeQ[b^2 - 4*
a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]

Rule 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> 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] + Dist[(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] && NeQ[b^2 - 4*a*
c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx &=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac {4 \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx}{11 \left (b^2-4 a c\right ) d^2}\\ &=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac {8 \left (a+b x+c x^2\right )^{7/2}}{99 \left (b^2-4 a c\right )^2 d^{12} (b+2 c x)^9}+\frac {8 \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx}{99 \left (b^2-4 a c\right )^2 d^4}\\ &=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac {8 \left (a+b x+c x^2\right )^{7/2}}{99 \left (b^2-4 a c\right )^2 d^{12} (b+2 c x)^9}+\frac {16 \left (a+b x+c x^2\right )^{7/2}}{693 \left (b^2-4 a c\right )^3 d^{12} (b+2 c x)^7}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 110, normalized size = 0.93 \[ \frac {2 (a+x (b+c x))^{7/2} \left (16 c^2 \left (63 a^2-28 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (38 c x^2-77 a\right )+64 b c^2 x \left (4 c x^2-7 a\right )+99 b^4+176 b^3 c x\right )}{693 d^{12} \left (b^2-4 a c\right )^3 (b+2 c x)^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]

[Out]

(2*(a + x*(b + c*x))^(7/2)*(99*b^4 + 176*b^3*c*x + 64*b*c^2*x*(-7*a + 4*c*x^2) + 8*b^2*c*(-77*a + 38*c*x^2) +
16*c^2*(63*a^2 - 28*a*c*x^2 + 8*c^2*x^4)))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^11)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^12,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 4.90, size = 2371, normalized size = 20.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^12,x, algorithm="giac")

[Out]

1/5544*(29568*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^16*c^(17/2) + 236544*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*
b*c^8 + 868560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*b^2*c^(15/2) + 73920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^14*a*c^(17/2) + 1940400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*b^3*c^7 + 517440*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^13*a*b*c^8 + 2953104*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*b^4*c^(13/2) + 1600368*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^12*a*b^2*c^(15/2) + 162624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a^2*c^(17/2) + 3256176*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^11*b^5*c^6 + 2875488*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a*b^3*c^7 + 975744*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^2*b*c^8 + 2709168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^6*c^(11/2)
+ 3307920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a*b^4*c^(13/2) + 2583504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^10*a^2*b^2*c^(15/2) + 133056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^3*c^(17/2) + 1755600*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^9*b^7*c^5 + 2513280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^5*c^6 + 3973200*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^9*a^2*b^3*c^7 + 665280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*b*c^8 + 910800*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^8*b^8*c^(9/2) + 1227600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^6*c^(11/2) + 394
4160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*b^4*c^(13/2) + 1401840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^
3*b^2*c^(15/2) + 95040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^4*c^(17/2) + 387024*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^7*b^9*c^4 + 319968*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^7*c^5 + 2670624*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^7*a^2*b^5*c^6 + 1615680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b^3*c^7 + 380160*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^7*a^4*b*c^8 + 136488*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^10*c^(7/2) - 20592*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^6*a*b^8*c^(9/2) + 1284624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^6*c^(11/2) +
1092960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*b^4*c^(13/2) + 641520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*
a^4*b^2*c^(15/2) + 19008*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^5*c^(17/2) + 39864*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^5*b^11*c^3 - 58080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^9*c^4 + 460944*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*a^2*b^7*c^5 + 418176*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^5*c^6 + 594000*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^5*a^4*b^3*c^7 + 57024*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^5*b*c^8 + 9460*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*b^12*c^(5/2) - 27720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^10*c^(7/2) + 132000*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^8*c^(9/2) + 64240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^6*c^(1
1/2) + 330000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*b^4*c^(13/2) + 66000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^4*a^5*b^2*c^(15/2) + 3520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^6*c^(17/2) + 1760*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*b^13*c^2 - 7920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^11*c^3 + 31680*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^3*a^2*b^9*c^4 - 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^7*c^5 + 113520*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^3*a^4*b^5*c^6 + 36960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*b^3*c^7 + 7040*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*a^6*b*c^8 + 242*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^14*c^(3/2) - 1496*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^2*a*b^12*c^(5/2) + 6072*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^10*c^(7/2) - 8
800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^8*c^(9/2) + 24640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^
6*c^(11/2) + 8976*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^4*c^(13/2) + 6512*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*a^6*b^2*c^(15/2) - 704*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^7*c^(17/2) + 22*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*b^15*c - 176*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^13*c^2 + 792*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a^2*b^11*c^3 - 1760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^9*c^4 + 3520*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*a^4*b^7*c^5 + 2992*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b^3*c^7 - 704*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a^7*b*c^8 + b^16*sqrt(c) - 10*a*b^14*c^(3/2) + 52*a^2*b^12*c^(5/2) - 152*a^3*b^10*c^(7/2) + 320*a^4*b^8*c
^(9/2) - 320*a^5*b^6*c^(11/2) + 640*a^6*b^4*c^(13/2) - 304*a^7*b^2*c^(15/2) + 64*a^8*c^(17/2))/((2*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^11*c^4*d^12)

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maple [A]  time = 0.05, size = 133, normalized size = 1.13 \[ -\frac {2 \left (128 c^{4} x^{4}+256 b \,c^{3} x^{3}-448 a \,c^{3} x^{2}+304 x^{2} b^{2} c^{2}-448 a b \,c^{2} x +176 x \,b^{3} c +1008 a^{2} c^{2}-616 a \,b^{2} c +99 b^{4}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{693 \left (2 c x +b \right )^{11} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) d^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^12,x)

[Out]

-2/693*(128*c^4*x^4+256*b*c^3*x^3-448*a*c^3*x^2+304*b^2*c^2*x^2-448*a*b*c^2*x+176*b^3*c*x+1008*a^2*c^2-616*a*b
^2*c+99*b^4)*(c*x^2+b*x+a)^(7/2)/(2*c*x+b)^11/d^12/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [B]  time = 15.67, size = 9995, normalized size = 84.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x)

[Out]

(((b*((b*((b*((b*((b*((4*c^4*(40*a*c - 3*b^2))/(231*d^12*(4*a*c - b^2)^3*(96*a*c^3 - 24*b^2*c^2)) - (8*b^2*c^4
)/(231*d^12*(4*a*c - b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (10*b*c^3*(120*a*c - 23*b^2))/(693*d^12*(4*a*c
- b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (5168*a^2*c^4 + 58*b^4*c^2 - 1384*a*b^2*c^3)/(693*d^12*(4*a*c - b^
2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (b*c*(115*b^4 + 2584*a^2*c^2 - 1092*a*b^2*c))/(231*d^12*(4*a*c - b^2)^
3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (133*b^6 + 4704*a^3*c^3 + 348*a^2*b^2*c^2 - 906*a*b^4*c)/(693*d^12*(4*a*c
- b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (133*a*b^5 - 1118*a^2*b^3*c + 2352*a^3*b*c^2)/(693*d^12*(4*a*c -
b^2)^3*(96*a*c^3 - 24*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^6 + (((b*((b*(b^2/(6160*c*d^12*(4*a*c -
b^2)^5) - (32*a*c^3 - 5*b^2*c^2)/(9240*c^3*d^12*(4*a*c - b^2)^5)))/(2*c) - (7*b^3*c - 32*a*b*c^2)/(9240*c^3*d^
12*(4*a*c - b^2)^5)))/(2*c) - (6*b^4 + 126*a^2*c^2 - 55*a*b^2*c)/(9240*c^3*d^12*(4*a*c - b^2)^5))*(a + b*x + c
*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*(b^2/(5280*c*d^12*(4*a*c - b^2)^5) - (76*a*c^3 - 13*b^2*c^2)/(15840*c^3*d^
12*(4*a*c - b^2)^5)))/(2*c) - (17*b^3*c - 76*a*b*c^2)/(15840*c^3*d^12*(4*a*c - b^2)^5)))/(2*c) + (23*b^4 + 296
*a^2*c^2 - 167*a*b^2*c)/(15840*c^3*d^12*(4*a*c - b^2)^5))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((5*
a*c - b^2)/(1155*c*d^12*(4*a*c - b^2)^5) - b^2/(9240*c*d^12*(4*a*c - b^2)^5)))/(2*c) - (b*(30*a*c - 7*b^2))/(6
930*c^2*d^12*(4*a*c - b^2)^5)))/(2*c) + (147*b^4 + 2584*a^2*c^2 - 1232*a*b^2*c)/(55440*c^3*d^12*(4*a*c - b^2)^
5))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) - (((b*((b*((b*((b*((368*a*c^4 - 32*b^2*c^3)/(198*d^12*(4*a*c - b^2)^
2*(112*a*c^3 - 28*b^2*c^2)) - (10*b^2*c^3)/(99*d^12*(4*a*c - b^2)^2*(112*a*c^3 - 28*b^2*c^2))))/(2*c) - (8*b*c
^2*(46*a*c - 9*b^2))/(99*d^12*(4*a*c - b^2)^2*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (39*b^4*c + 2592*a^2*c^3 - 7
44*a*b^2*c^2)/(198*d^12*(4*a*c - b^2)^2*(112*a*c^3 - 28*b^2*c^2))))/(2*c) - (119*b^5 + 2592*a^2*b*c^2 - 1112*a
*b^3*c)/(198*d^12*(4*a*c - b^2)^2*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (119*a*b^4 + 2240*a^3*c^2 - 1032*a^2*b^2
*c)/(198*d^12*(4*a*c - b^2)^2*(112*a*c^3 - 28*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^7 + (((b*((b*((b
*((b*((b^2*c)/(4158*d^12*(4*a*c - b^2)^6) - (608*a*c^4 - 92*b^2*c^3)/(83160*c^2*d^12*(4*a*c - b^2)^6)))/(2*c)
+ (b*(152*a*c - 33*b^2))/(10395*d^12*(4*a*c - b^2)^6)))/(2*c) - (365*b^4*c + 9248*a^2*c^3 - 3712*a*b^2*c^2)/(8
3160*c^2*d^12*(4*a*c - b^2)^6)))/(2*c) + (505*b^5 + 9248*a^2*b*c^2 - 4320*a*b^3*c)/(83160*c^2*d^12*(4*a*c - b^
2)^6)))/(2*c) - (505*a*b^4 + 8656*a^3*c^2 - 4180*a^2*b^2*c)/(83160*c^2*d^12*(4*a*c - b^2)^6))*(a + b*x + c*x^2
)^(1/2))/(b + 2*c*x) + (((b*((b*((4*c*(7*a*c - b^2))/(99*d^12*(4*a*c - b^2)^2*(80*a*c^3 - 20*b^2*c^2)) - (b^2*
c)/(66*d^12*(4*a*c - b^2)^2*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (2*b*(14*a*c - 3*b^2))/(99*d^12*(4*a*c - b^2)^2
*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (23*b^4 + 576*a^2*c^2 - 232*a*b^2*c)/(792*c*d^12*(4*a*c - b^2)^2*(80*a*c^3
- 20*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^5 - (((b*((b*((c*(10*a*c - b^2))/(231*d^12*(4*a*c - b^2)
^3*(48*a*c^3 - 12*b^2*c^2)) - (b^2*c)/(308*d^12*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (2*b*(5*a*c
- b^2))/(231*d^12*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (24*a^2*c^2 - 3*b^4 + 8*a*b^2*c)/(1848*c
*d^12*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^3 - (((32*a*c - 7*b^2)/(1
4784*c^3*d^12*(4*a*c - b^2)^4) - b^2/(14784*c^3*d^12*(4*a*c - b^2)^4))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) -
(((56*a*c - 13*b^2)/(15840*c^3*d^12*(4*a*c - b^2)^4) - b^2/(15840*c^3*d^12*(4*a*c - b^2)^4))*(a + b*x + c*x^2)
^(1/2))/(b + 2*c*x) - (((170*a*c - 43*b^2)/(11088*c^3*d^12*(4*a*c - b^2)^4) + b^2/(22176*c^3*d^12*(4*a*c - b^2
)^4))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) - (((b*((b*(b^2/(5280*c*d^12*(4*a*c - b^2)^5) - (276*a*c^3 - 51*b^2
*c^2)/(47520*c^3*d^12*(4*a*c - b^2)^5)))/(2*c) + (b*(92*a*c - 21*b^2))/(15840*c^2*d^12*(4*a*c - b^2)^5)))/(2*c
) - (133*b^4 + 2392*a^2*c^2 - 1127*a*b^2*c)/(47520*c^3*d^12*(4*a*c - b^2)^5))*(a + b*x + c*x^2)^(1/2))/(b + 2*
c*x) + (((b*((b*((b*((b*((b*((8*c^4*(46*a*c - b^2))/(99*d^12*(4*a*c - b^2)^2*(128*a*c^3 - 32*b^2*c^2)) - (8*b^
2*c^4)/(33*d^12*(4*a*c - b^2)^2*(128*a*c^3 - 32*b^2*c^2))))/(2*c) - (40*b*c^3*(23*a*c - 4*b^2))/(99*d^12*(4*a*
c - b^2)^2*(128*a*c^3 - 32*b^2*c^2))))/(2*c) - (66*b^4*c^2 - 5184*a^2*c^4 + 752*a*b^2*c^3)/(198*d^12*(4*a*c -
b^2)^2*(128*a*c^3 - 32*b^2*c^2))))/(2*c) - (277*b^5*c - 2968*a*b^3*c^2 + 7776*a^2*b*c^3)/(198*d^12*(4*a*c - b^
2)^2*(128*a*c^3 - 32*b^2*c^2))))/(2*c) + (119*b^6 + 4480*a^3*c^3 + 528*a^2*b^2*c^2 - 874*a*b^4*c)/(198*d^12*(4
*a*c - b^2)^2*(128*a*c^3 - 32*b^2*c^2))))/(2*c) - (119*a*b^5 - 1032*a^2*b^3*c + 2240*a^3*b*c^2)/(198*d^12*(4*a
*c - b^2)^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*(
10*a*c - b^2))/(495*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2)) - (8*b^2*c^4)/(1155*d^12*(4*a*c - b^2)^4*(64
*a*c^3 - 16*b^2*c^2))))/(2*c) - (8*b*c^3*(5*a*c - b^2))/(99*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2))))/(2
*c) + (7568*a^2*c^4 + 158*b^4*c^2 - 2384*a*b^2*c^3)/(3465*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2))))/(2*c
) - (545*b^5*c - 4976*a*b^3*c^2 + 11352*a^2*b*c^3)/(3465*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2))))/(2*c)
+ (203*b^6 + 7024*a^3*c^3 + 408*a^2*b^2*c^2 - 1346*a*b^4*c)/(3465*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2
))))/(2*c) - (203*a*b^5 - 1688*a^2*b^3*c + 3512*a^3*b*c^2)/(3465*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2))
)*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^4 + (((b*((b*((b*((b*((b*((4*c^4*(152*a*c - 17*b^2))/(10395*d^12*(4*a*c
- b^2)^5*(32*a*c^3 - 8*b^2*c^2)) - (8*b^2*c^4)/(3465*d^12*(4*a*c - b^2)^5*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (
2*b*c^3*(152*a*c - 31*b^2))/(2079*d^12*(4*a*c - b^2)^5*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (18496*a^2*c^4 + 466*
b^4*c^2 - 6208*a*b^2*c^3)/(20790*d^12*(4*a*c - b^2)^5*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (1375*b^5*c - 12352*a*
b^3*c^2 + 27744*a^2*b*c^3)/(20790*d^12*(4*a*c - b^2)^5*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (505*b^6 + 17312*a^3*
c^3 + 888*a^2*b^2*c^2 - 3310*a*b^4*c)/(20790*d^12*(4*a*c - b^2)^5*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (505*a*b^5
- 4180*a^2*b^3*c + 8656*a^3*b*c^2)/(20790*d^12*(4*a*c - b^2)^5*(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*((2*c^3*(56*a*c + b^2))/(11*d^12*(4*a*c - b^2)*(144*a*c^3 - 36*b^2*c^2))
- (10*b^2*c^3)/(11*d^12*(4*a*c - b^2)*(144*a*c^3 - 36*b^2*c^2))))/(2*c) - (4*b*c^2*(56*a*c - 9*b^2))/(11*d^12*
(4*a*c - b^2)*(144*a*c^3 - 36*b^2*c^2))))/(2*c) + (576*a^2*c^3 - 33*b^4*c + 48*a*b^2*c^2)/(22*d^12*(4*a*c - b^
2)*(144*a*c^3 - 36*b^2*c^2))))/(2*c) - (11*b^5 + 576*a^2*b*c^2 - 176*a*b^3*c)/(22*d^12*(4*a*c - b^2)*(144*a*c^
3 - 36*b^2*c^2))))/(2*c) + (11*a*b^4 + 368*a^3*c^2 - 132*a^2*b^2*c)/(22*d^12*(4*a*c - b^2)*(144*a*c^3 - 36*b^2
*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^9 - (((b*((b*((b*((b*((24*c^3*(a*c + b^2))/(d^12*(176*a*c^3 - 44*
b^2*c^2)) - (10*b^2*c^3)/(d^12*(176*a*c^3 - 44*b^2*c^2))))/(2*c) - (4*c^2*(2*b^3 + 12*a*b*c))/(d^12*(176*a*c^3
- 44*b^2*c^2))))/(2*c) + (24*a*c^2*(a*c + b^2))/(d^12*(176*a*c^3 - 44*b^2*c^2))))/(2*c) - (24*a^2*b*c^2)/(d^1
2*(176*a*c^3 - 44*b^2*c^2))))/(2*c) + (8*a^3*c^2)/(d^12*(176*a*c^3 - 44*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b
+ 2*c*x)^11 - (((b*((b*((b*((40*a*c^3)/(77*d^12*(4*a*c - b^2)^2*(96*a*c^3 - 24*b^2*c^2)) - (4*b^2*c^2)/(77*d^
12*(4*a*c - b^2)^2*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (10*b*c*(6*a*c - b^2))/(77*d^12*(4*a*c - b^2)^2*(96*a*c^
3 - 24*b^2*c^2))))/(2*c) + (48*a^2*c^3 - 22*b^4*c + 96*a*b^2*c^2)/(308*c*d^12*(4*a*c - b^2)^2*(96*a*c^3 - 24*b
^2*c^2))))/(2*c) - (24*a^2*b*c^2 - 3*b^5 + 8*a*b^3*c)/(308*c*d^12*(4*a*c - b^2)^2*(96*a*c^3 - 24*b^2*c^2)))*(a
+ b*x + c*x^2)^(1/2))/(b + 2*c*x)^6 - (((b*((b*((3*b^2*c)/(22*d^12*(4*a*c - b^2)*(112*a*c^3 - 28*b^2*c^2)) -
(3*c*(4*a*c + b^2))/(22*d^12*(4*a*c - b^2)*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (b*(12*a*c - b^2))/(22*d^12*(4*
a*c - b^2)*(112*a*c^3 - 28*b^2*c^2))))/(2*c) - (b^4 + 48*a^2*c^2 - 12*a*b^2*c)/(88*c*d^12*(4*a*c - b^2)*(112*a
*c^3 - 28*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^7 + (((b*(b^2/(1386*d^12*(4*a*c - b^2)^3*(32*a*c^3 -
8*b^2*c^2)) + (340*a*c^2 - 88*b^2*c)/(2772*c*d^12*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (43*b^3 -
170*a*b*c)/(2772*c*d^12*(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^2/(924*d^12*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2)) - (64*a*c^2 - 10*b^2*c)/(3696*c*d^12*(4*a*c - b^2)^3*
(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (7*b^3 - 32*a*b*c)/(3696*c*d^12*(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 - (((6*a*c - b^2)/(308*c*d^12*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2)) - b
^2/(616*c*d^12*(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*((b*((b
*((b*((c^3*(64*a*c - b^2))/(231*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 20*b^2*c^2)) - (5*b^2*c^3)/(231*d^12*(4*a*c -
b^2)^3*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (2*b*c^2*(64*a*c - 11*b^2))/(231*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 2
0*b^2*c^2))))/(2*c) - (36*b^4*c^2 - 2016*a^2*c^4 + 240*a*b^2*c^3)/(1848*c*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 20*
b^2*c^2))))/(2*c) - (b*(17*b^4 + 504*a^2*c^2 - 188*a*b^2*c))/(462*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 20*b^2*c^2)
)))/(2*c) + (15*b^6 + 576*a^3*c^3 + 72*a^2*b^2*c^2 - 112*a*b^4*c)/(1848*c*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 20*
b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^5 + (((b*((b*((b*((b*((2*c^3*(23*a*c - 2*b^2))/(495*d^12*(4*a*
c - b^2)^4*(48*a*c^3 - 12*b^2*c^2)) - (b^2*c^3)/(198*d^12*(4*a*c - b^2)^4*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (
2*b*c^2*(46*a*c - 9*b^2))/(495*d^12*(4*a*c - b^2)^4*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (32*a^2*c^4 - 121*b^4*c
^2 + 536*a*b^2*c^3)/(3960*c*d^12*(4*a*c - b^2)^4*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (b*(32*a^2*c^2 - 41*b^4 +
168*a*b^2*c))/(3960*d^12*(4*a*c - b^2)^4*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (69*b^6 - 4736*a^3*c^3 + 3560*a^2*
b^2*c^2 - 869*a*b^4*c)/(3960*c*d^12*(4*a*c - b^2)^4*(48*a*c^3 - 12*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*
c*x)^3 + (((b*((b*((b*((b*((b*((28*c^4*(8*a*c + b^2))/(11*d^12*(4*a*c - b^2)*(160*a*c^3 - 40*b^2*c^2)) - (24*b
^2*c^4)/(11*d^12*(4*a*c - b^2)*(160*a*c^3 - 40*b^2*c^2))))/(2*c) - (70*b*c^3*(8*a*c - b^2))/(11*d^12*(4*a*c -
b^2)*(160*a*c^3 - 40*b^2*c^2))))/(2*c) + (1152*a^2*c^4 - 138*b^4*c^2 + 544*a*b^2*c^3)/(22*d^12*(4*a*c - b^2)*(
160*a*c^3 - 40*b^2*c^2))))/(2*c) + (b*c*(11*b^4 - 1728*a^2*c^2 + 304*a*b^2*c))/(22*d^12*(4*a*c - b^2)*(160*a*c
^3 - 40*b^2*c^2))))/(2*c) + (11*b^6 + 736*a^3*c^3 + 312*a^2*b^2*c^2 - 154*a*b^4*c)/(22*d^12*(4*a*c - b^2)*(160
*a*c^3 - 40*b^2*c^2))))/(2*c) - (11*a*b^5 - 132*a^2*b^3*c + 368*a^3*b*c^2)/(22*d^12*(4*a*c - b^2)*(160*a*c^3 -
40*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^10 - (((b*((b*((b*((b*((b*((8*a*c^5)/(55*d^12*(4*a*c - b^2
)^4*(64*a*c^3 - 16*b^2*c^2)) - (4*b^2*c^4)/(385*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (2*b*c
^3*(6*a*c - b^2))/(33*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (32*b^4*c^3 - 2288*a^2*c^5 + 304
*a*b^2*c^4)/(2310*c*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (4*b*c*(5*b^4 + 143*a^2*c^2 - 54*a
*b^2*c))/(385*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (22*b^6*c + 3888*a^3*c^4 - 24*a*b^4*c^2
- 1200*a^2*b^2*c^3)/(2310*c*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (15*b^7 - 1944*a^3*b*c^3 +
1172*a^2*b^3*c^2 - 232*a*b^5*c)/(2310*c*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2)))*(a + b*x + c*x^2)^(1/2
))/(b + 2*c*x)^4 - (((b*((b*((b^2*c)/(462*d^12*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2)) - (92*a*c^3 - 17*b^2*c
^2)/(1386*c*d^12*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (b*(92*a*c - 21*b^2))/(1386*d^12*(4*a*c -
b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (35*b^4 + 648*a^2*c^2 - 301*a*b^2*c)/(1386*c*d^12*(4*a*c - b^2)^3*(4
8*a*c^3 - 12*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^3 - (((b*((b*((b*((b*((10*c^3*(8*a*c - b^2))/(231
*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 20*b^2*c^2)) - (10*b^2*c^3)/(693*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 20*b^2*c^2
))))/(2*c) - (20*b*c^2*(24*a*c - 5*b^2))/(693*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (79*b^4*
c + 2584*a^2*c^3 - 932*a*b^2*c^2)/(693*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (133*b^5 + 2584
*a^2*b*c^2 - 1172*a*b^3*c)/(693*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (133*a*b^4 + 2352*a^3*
c^2 - 1118*a^2*b^2*c)/(693*d^12*(4*a*c - b^2)^3*(80*a*c^3 - 20*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)
^5 - (((b*((b*((b*((b*((8*c^3*(7*a*c - b^2))/(693*d^12*(4*a*c - b^2)^4*(48*a*c^3 - 12*b^2*c^2)) - (2*b^2*c^3)/
(693*d^12*(4*a*c - b^2)^4*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (8*b*c^2*(14*a*c - 3*b^2))/(693*d^12*(4*a*c - b^2
)^4*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (139*b^4*c + 3784*a^2*c^3 - 1472*a*b^2*c^2)/(3465*d^12*(4*a*c - b^2)^4*
(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (203*b^5 + 3784*a^2*b*c^2 - 1752*a*b^3*c)/(3465*d^12*(4*a*c - b^2)^4*(48*a*
c^3 - 12*b^2*c^2))))/(2*c) + (203*a*b^4 + 3512*a^3*c^2 - 1688*a^2*b^2*c)/(3465*d^12*(4*a*c - b^2)^4*(48*a*c^3
- 12*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^3 + (((b*((b*((b*((4*b^2*c^2)/(495*d^12*(4*a*c - b^2)^3*(
64*a*c^3 - 16*b^2*c^2)) - (152*a*c^4 - 18*b^2*c^3)/(990*c*d^12*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2*c^2))))/(2*c
) + (b*c*(228*a*c - 47*b^2))/(990*d^12*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (63*b^4*c + 592*a^2*
c^3 - 410*a*b^2*c^2)/(990*c*d^12*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (23*b^5 + 296*a^2*b*c^2 -
167*a*b^3*c)/(990*c*d^12*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^4 + ((
(b*((b*((b*((b^2*c^2)/(495*d^12*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2)) - (552*a*c^4 - 78*b^2*c^3)/(11880*c*d^
12*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (b*c*(276*a*c - 59*b^2))/(3960*d^12*(4*a*c - b^2)^4*(32*a
*c^3 - 8*b^2*c^2))))/(2*c) - (203*b^4*c + 4784*a^2*c^3 - 1978*a*b^2*c^2)/(11880*c*d^12*(4*a*c - b^2)^4*(32*a*c
^3 - 8*b^2*c^2))))/(2*c) + (133*b^5 + 2392*a^2*b*c^2 - 1127*a*b^3*c)/(11880*c*d^12*(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*((c*(14*a*c - b^2))/(3080*d^12*(4*a*c
- b^2)^6) - (b^2*c)/(3696*d^12*(4*a*c - b^2)^6)))/(2*c) - (b*(21*a*c - 4*b^2))/(2310*d^12*(4*a*c - b^2)^6)))/(
2*c) + (1144*a^2*c^4 + 16*b^4*c^2 - 320*a*b^2*c^3)/(36960*c^3*d^12*(4*a*c - b^2)^6)))/(2*c) - (b*(13*b^4 + 286
*a^2*c^2 - 122*a*b^2*c))/(9240*c^2*d^12*(4*a*c - b^2)^6)))/(2*c) - (15*b^6 - 1944*a^3*c^3 + 1172*a^2*b^2*c^2 -
232*a*b^4*c)/(36960*c^3*d^12*(4*a*c - b^2)^6))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((b*((b*((c*(1
0*a*c - b^2))/(1848*d^12*(4*a*c - b^2)^6) - (b^2*c)/(3696*d^12*(4*a*c - b^2)^6)))/(2*c) - (b*(5*a*c - b^2))/(4
62*d^12*(4*a*c - b^2)^6)))/(2*c) + (3632*a^2*c^4 + 92*b^4*c^2 - 1216*a*b^2*c^3)/(73920*c^3*d^12*(4*a*c - b^2)^
6)))/(2*c) - (b*(45*b^4 + 908*a^2*c^2 - 404*a*b^2*c))/(18480*c^2*d^12*(4*a*c - b^2)^6)))/(2*c) + (9*b^6 + 2672
*a^3*c^3 - 1096*a^2*b^2*c^2 + 72*a*b^4*c)/(73920*c^3*d^12*(4*a*c - b^2)^6))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*
x) - (((b*((b*((b*((b*((b*((2*c^4*(25*a*c - b^2))/(1155*d^12*(4*a*c - b^2)^5*(32*a*c^3 - 8*b^2*c^2)) - (b^2*c^
4)/(385*d^12*(4*a*c - b^2)^5*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (b*c^3*(50*a*c - 9*b^2))/(462*d^12*(4*a*c - b^2
)^5*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (7264*a^2*c^5 + 24*b^4*c^3 - 1632*a*b^2*c^4)/(18480*c*d^12*(4*a*c - b^2)
^5*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (b*c*(113*b^4 + 2724*a^2*c^2 - 1112*a*b^2*c))/(4620*d^12*(4*a*c - b^2)^5*
(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (198*b^6*c + 5344*a^3*c^4 - 1472*a*b^4*c^2 + 1440*a^2*b^2*c^3)/(18480*c*d^12
*(4*a*c - b^2)^5*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (9*b^7 + 2672*a^3*b*c^3 - 1096*a^2*b^3*c^2 + 72*a*b^5*c)/(1
8480*c*d^12*(4*a*c - b^2)^5*(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*((b*((c^5*(100*a*c + 3*b^2))/(1155*d^12*(4*a*c - b^2)^5*(48*a*c^3 - 12*b^2*c^2)) - (b^2*c^5)/(165*d^12*(4
*a*c - b^2)^5*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (b*c^4*(300*a*c - 47*b^2))/(1155*d^12*(4*a*c - b^2)^5*(48*a*c
^3 - 12*b^2*c^2))))/(2*c) - (156*b^4*c^4 - 7264*a^2*c^6 + 632*a*b^2*c^5)/(9240*c*d^12*(4*a*c - b^2)^5*(48*a*c^
3 - 12*b^2*c^2))))/(2*c) - (2*b*c^2*(29*b^4 + 908*a^2*c^2 - 329*a*b^2*c))/(1155*d^12*(4*a*c - b^2)^5*(48*a*c^3
- 12*b^2*c^2))))/(2*c) + (26080*a^3*c^5 + 100*b^6*c^2 + 192*a*b^4*c^3 - 8664*a^2*b^2*c^4)/(9240*c*d^12*(4*a*c
- b^2)^5*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (216*b^7*c - 3224*a*b^5*c^2 - 26080*a^3*b*c^4 + 15928*a^2*b^3*c^3
)/(9240*c*d^12*(4*a*c - b^2)^5*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (45*b^8 + 31104*a^4*c^4 + 7228*a^2*b^4*c^2 -
24584*a^3*b^2*c^3 - 936*a*b^6*c)/(9240*c*d^12*(4*a*c - b^2)^5*(48*a*c^3 - 12*b^2*c^2)))*(a + b*x + c*x^2)^(1/
2))/(b + 2*c*x)^3 + (a + b*x + c*x^2)^(1/2)/(2464*c^3*d^12*(4*a*c - b^2)^3*(b + 2*c*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx}{d^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**12,x)

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920
*b**8*c**4*x**4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8
+ 112640*b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(b**2
*x**2*sqrt(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**
4*x**4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*
b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(c**2*x**4*sqr
t(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 +
25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9
*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(2*a*b*x*sqrt(a + b*x +
c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*
c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 675
84*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)
/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*c**5*x**
5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 67584*b**2*
c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**12
+ 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*c**5*x**5 + 5913
6*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 67584*b**2*c**10*x*
*10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x))/d**12

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