3.2229 \(\int \frac {f+g x}{(d+e x)^2 (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=283 \[ \frac {128 c^2 (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^6 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 c (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (-7 b e g+4 c d g+10 c e f)}{35 e^2 (d+e x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

16/105*c*(-7*b*e*g+4*c*d*g+10*c*e*f)*(2*c*x+b)/e/(-b*e+2*c*d)^4/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)-2/7*(-d
*g+e*f)/e^2/(-b*e+2*c*d)/(e*x+d)^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)-2/35*(-7*b*e*g+4*c*d*g+10*c*e*f)/e^2
/(-b*e+2*c*d)^2/(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)+128/105*c^2*(-7*b*e*g+4*c*d*g+10*c*e*f)*(2*c*x+
b)/e/(-b*e+2*c*d)^6/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)

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Rubi [A]  time = 0.40, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {792, 658, 614, 613} \[ \frac {128 c^2 (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^6 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 c (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (-7 b e g+4 c d g+10 c e f)}{35 e^2 (d+e x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)/((d + e*x)^2*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]

[Out]

(16*c*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x))/(105*e*(2*c*d - b*e)^4*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2)) - (2*(e*f - d*g))/(7*e^2*(2*c*d - b*e)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(
10*c*e*f + 4*c*d*g - 7*b*e*g))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))
+ (128*c^2*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x))/(105*e*(2*c*d - b*e)^6*Sqrt[d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2])

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 658

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rubi steps

\begin {align*} \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(10 c e f+4 c d g-7 b e g) \int \frac {1}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{7 e (2 c d-b e)}\\ &=-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(8 c (10 c e f+4 c d g-7 b e g)) \int \frac {1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{35 e (2 c d-b e)^2}\\ &=\frac {16 c (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {\left (64 c^2 (10 c e f+4 c d g-7 b e g)\right ) \int \frac {1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{105 e (2 c d-b e)^4}\\ &=\frac {16 c (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {128 c^2 (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^6 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 468, normalized size = 1.65 \[ \frac {-6 b^5 e^5 (2 d g+5 e f+7 e g x)+4 b^4 c e^4 \left (43 d^2 g+2 d e (45 f+73 g x)+e^2 x (15 f+28 g x)\right )-16 b^3 c^2 e^3 \left (88 d^3 g+d^2 e (115 f+293 g x)+2 d e^2 x (25 f+86 g x)+2 e^3 x^2 (5 f+21 g x)\right )+96 b^2 c^3 e^2 \left (17 d^4 g+20 d^3 e (3 f+2 g x)+d^2 e^2 x (65 f-54 g x)+40 d e^3 x^2 (f-2 g x)+2 e^4 x^3 (5 f-14 g x)\right )+32 b c^4 e \left (6 d^5 g-39 d^4 e (5 f-g x)+12 d^3 e^2 x (24 g x-5 f)+4 d^2 e^3 x^2 (75 f+43 g x)+8 d e^4 x^3 (45 f-8 g x)+8 e^5 x^4 (15 f-7 g x)\right )-64 c^5 \left (9 d^6 g-6 d^5 e (5 f-3 g x)+3 d^4 e^2 x (15 f+16 g x)+8 d^3 e^3 x^2 (15 f+g x)+4 d^2 e^4 x^3 (5 f-8 g x)-16 d e^5 x^4 (5 f+g x)-40 e^6 f x^5\right )}{105 e^2 (d+e x)^3 (b e-2 c d)^6 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)/((d + e*x)^2*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]

[Out]

(-6*b^5*e^5*(5*e*f + 2*d*g + 7*e*g*x) + 96*b^2*c^3*e^2*(17*d^4*g + d^2*e^2*x*(65*f - 54*g*x) + 2*e^4*x^3*(5*f
- 14*g*x) + 40*d*e^3*x^2*(f - 2*g*x) + 20*d^3*e*(3*f + 2*g*x)) - 64*c^5*(9*d^6*g - 40*e^6*f*x^5 + 4*d^2*e^4*x^
3*(5*f - 8*g*x) - 6*d^5*e*(5*f - 3*g*x) - 16*d*e^5*x^4*(5*f + g*x) + 8*d^3*e^3*x^2*(15*f + g*x) + 3*d^4*e^2*x*
(15*f + 16*g*x)) + 32*b*c^4*e*(6*d^5*g + 8*d*e^4*x^3*(45*f - 8*g*x) + 8*e^5*x^4*(15*f - 7*g*x) - 39*d^4*e*(5*f
- g*x) + 12*d^3*e^2*x*(-5*f + 24*g*x) + 4*d^2*e^3*x^2*(75*f + 43*g*x)) + 4*b^4*c*e^4*(43*d^2*g + e^2*x*(15*f
+ 28*g*x) + 2*d*e*(45*f + 73*g*x)) - 16*b^3*c^2*e^3*(88*d^3*g + 2*e^3*x^2*(5*f + 21*g*x) + 2*d*e^2*x*(25*f + 8
6*g*x) + d^2*e*(115*f + 293*g*x)))/(105*e^2*(-2*c*d + b*e)^6*(d + e*x)^3*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)
*(-(b*e) + c*(d - e*x))])

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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((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x

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maple [B]  time = 0.05, size = 782, normalized size = 2.76 \[ -\frac {2 \left (c e x +b e -c d \right ) \left (896 b \,c^{4} e^{6} g \,x^{5}-512 c^{5} d \,e^{5} g \,x^{5}-1280 c^{5} e^{6} f \,x^{5}+1344 b^{2} c^{3} e^{6} g \,x^{4}+1024 b \,c^{4} d \,e^{5} g \,x^{4}-1920 b \,c^{4} e^{6} f \,x^{4}-1024 c^{5} d^{2} e^{4} g \,x^{4}-2560 c^{5} d \,e^{5} f \,x^{4}+336 b^{3} c^{2} e^{6} g \,x^{3}+3840 b^{2} c^{3} d \,e^{5} g \,x^{3}-480 b^{2} c^{3} e^{6} f \,x^{3}-2752 b \,c^{4} d^{2} e^{4} g \,x^{3}-5760 b \,c^{4} d \,e^{5} f \,x^{3}+256 c^{5} d^{3} e^{3} g \,x^{3}+640 c^{5} d^{2} e^{4} f \,x^{3}-56 b^{4} c \,e^{6} g \,x^{2}+1376 b^{3} c^{2} d \,e^{5} g \,x^{2}+80 b^{3} c^{2} e^{6} f \,x^{2}+2592 b^{2} c^{3} d^{2} e^{4} g \,x^{2}-1920 b^{2} c^{3} d \,e^{5} f \,x^{2}-4608 b \,c^{4} d^{3} e^{3} g \,x^{2}-4800 b \,c^{4} d^{2} e^{4} f \,x^{2}+1536 c^{5} d^{4} e^{2} g \,x^{2}+3840 c^{5} d^{3} e^{3} f \,x^{2}+21 b^{5} e^{6} g x -292 b^{4} c d \,e^{5} g x -30 b^{4} c \,e^{6} f x +2344 b^{3} c^{2} d^{2} e^{4} g x +400 b^{3} c^{2} d \,e^{5} f x -1920 b^{2} c^{3} d^{3} e^{3} g x -3120 b^{2} c^{3} d^{2} e^{4} f x -624 b \,c^{4} d^{4} e^{2} g x +960 b \,c^{4} d^{3} e^{3} f x +576 c^{5} d^{5} e g x +1440 c^{5} d^{4} e^{2} f x +6 b^{5} d \,e^{5} g +15 b^{5} e^{6} f -86 b^{4} c \,d^{2} e^{4} g -180 b^{4} c d \,e^{5} f +704 b^{3} c^{2} d^{3} e^{3} g +920 b^{3} c^{2} d^{2} e^{4} f -816 b^{2} c^{3} d^{4} e^{2} g -2880 b^{2} c^{3} d^{3} e^{3} f -96 b \,c^{4} d^{5} e g +3120 b \,c^{4} d^{4} e^{2} f +288 c^{5} d^{6} g -960 c^{5} d^{5} e f \right )}{105 \left (e x +d \right ) \left (b^{6} e^{6}-12 b^{5} c d \,e^{5}+60 b^{4} c^{2} d^{2} e^{4}-160 b^{3} c^{3} d^{3} e^{3}+240 b^{2} c^{4} d^{4} e^{2}-192 b \,c^{5} d^{5} e +64 c^{6} d^{6}\right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

-2/105*(c*e*x+b*e-c*d)*(896*b*c^4*e^6*g*x^5-512*c^5*d*e^5*g*x^5-1280*c^5*e^6*f*x^5+1344*b^2*c^3*e^6*g*x^4+1024
*b*c^4*d*e^5*g*x^4-1920*b*c^4*e^6*f*x^4-1024*c^5*d^2*e^4*g*x^4-2560*c^5*d*e^5*f*x^4+336*b^3*c^2*e^6*g*x^3+3840
*b^2*c^3*d*e^5*g*x^3-480*b^2*c^3*e^6*f*x^3-2752*b*c^4*d^2*e^4*g*x^3-5760*b*c^4*d*e^5*f*x^3+256*c^5*d^3*e^3*g*x
^3+640*c^5*d^2*e^4*f*x^3-56*b^4*c*e^6*g*x^2+1376*b^3*c^2*d*e^5*g*x^2+80*b^3*c^2*e^6*f*x^2+2592*b^2*c^3*d^2*e^4
*g*x^2-1920*b^2*c^3*d*e^5*f*x^2-4608*b*c^4*d^3*e^3*g*x^2-4800*b*c^4*d^2*e^4*f*x^2+1536*c^5*d^4*e^2*g*x^2+3840*
c^5*d^3*e^3*f*x^2+21*b^5*e^6*g*x-292*b^4*c*d*e^5*g*x-30*b^4*c*e^6*f*x+2344*b^3*c^2*d^2*e^4*g*x+400*b^3*c^2*d*e
^5*f*x-1920*b^2*c^3*d^3*e^3*g*x-3120*b^2*c^3*d^2*e^4*f*x-624*b*c^4*d^4*e^2*g*x+960*b*c^4*d^3*e^3*f*x+576*c^5*d
^5*e*g*x+1440*c^5*d^4*e^2*f*x+6*b^5*d*e^5*g+15*b^5*e^6*f-86*b^4*c*d^2*e^4*g-180*b^4*c*d*e^5*f+704*b^3*c^2*d^3*
e^3*g+920*b^3*c^2*d^2*e^4*f-816*b^2*c^3*d^4*e^2*g-2880*b^2*c^3*d^3*e^3*f-96*b*c^4*d^5*e*g+3120*b*c^4*d^4*e^2*f
+288*c^5*d^6*g-960*c^5*d^5*e*f)/(e*x+d)/(b^6*e^6-12*b^5*c*d*e^5+60*b^4*c^2*d^2*e^4-160*b^3*c^3*d^3*e^3+240*b^2
*c^4*d^4*e^2-192*b*c^5*d^5*e+64*c^6*d^6)/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)

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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((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),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(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?

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mupad [B]  time = 10.35, size = 11539, normalized size = 40.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)/((d + e*x)^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2)),x)

[Out]

((800*c^6*d^4*g + 558*b^3*c^3*e^4*f - 222*b^4*c^2*e^4*g - 4192*c^6*d^3*e*f + 384*b*c^5*d^3*e*g + 6624*b*c^5*d^
2*e^2*f - 3392*b^2*c^4*d*e^3*f + 1248*b^3*c^3*d*e^3*g - 1984*b^2*c^4*d^2*e^2*g)/(105*e^2*(b*e - 2*c*d)^8) - x*
((b*((b*((8*c^5*e*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^8) + (16*b*c^5*e^2*g)/(105*(b*e - 2*c*d)^8
)))/c - (4*c^4*(40*c^2*d^2*g - 33*b^2*e^2*g + 62*b*c*e^2*f - 88*c^2*d*e*f + 16*b*c*d*e*g))/(105*(b*e - 2*c*d)^
8) + (16*c^5*g*(c*d^2 - b*d*e))/(105*(b*e - 2*c*d)^8)))/c + (44*b^2*c^4*e^4*f - 30*b^3*c^3*e^4*g - 672*c^6*d^2
*e^2*f + 224*c^6*d^3*e*g + 320*b*c^5*d*e^3*f + 160*b*c^5*d^2*e^2*g - 128*b^2*c^4*d*e^3*g)/(105*e^2*(b*e - 2*c*
d)^8) + ((c*d^2 - b*d*e)*((8*c^5*e*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^8) + (16*b*c^5*e^2*g)/(10
5*(b*e - 2*c*d)^8)))/(c*e^2)) + ((c*d^2 - b*d*e)*((b*((8*c^5*e*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*
d)^8) + (16*b*c^5*e^2*g)/(105*(b*e - 2*c*d)^8)))/c - (4*c^4*(40*c^2*d^2*g - 33*b^2*e^2*g + 62*b*c*e^2*f - 88*c
^2*d*e*f + 16*b*c*d*e*g))/(105*(b*e - 2*c*d)^8) + (16*c^5*g*(c*d^2 - b*d*e))/(105*(b*e - 2*c*d)^8)))/(c*e^2))/
(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + (((8*c^2*g*(3*b*e - 4*c*d))/(105*e^2*(b*e - 2*c*d)^6) - (16*c^3*
d*g)/(105*e^2*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((8*c^4*(2*c
*d*g - 7*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^8) + (16*c^5*d*g)/(105*(b*e - 2*c*d)^8)))/e + (76*b^2*c^3*e^2*g
- 176*c^5*d^2*g + 304*c^5*d*e*f - 200*b*c^4*e^2*f + 8*b*c^4*d*e*g)/(105*e*(b*e - 2*c*d)^8)))/e - (2*b*c^2*(13*
b^2*e^2*g - 44*c^2*d^2*g - 44*b*c*e^2*f + 76*c^2*d*e*f + 4*b*c*d*e*g))/(105*e*(b*e - 2*c*d)^8))*(c*d^2 - c*e^2
*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((4*b*c*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (8*c^2*d*g)/
(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((276*
b^2*c^3*e^3*f - 352*c^5*d^3*g - 88*b^3*c^2*e^3*g + 608*c^5*d^2*e*f - 832*b*c^4*d*e^2*f + 288*b*c^4*d^2*e*g + 1
04*b^2*c^3*d*e^2*g)/(105*e^2*(b*e - 2*c*d)^8) + (d*((d*((16*c^4*(4*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c
*d)^8) + (16*c^5*d*g)/(105*(b*e - 2*c*d)^8)))/e - (272*b*c^4*e^3*f - 148*b^2*c^3*e^3*g - 448*c^5*d*e^2*f + 128
*c^5*d^2*e*g + 160*b*c^4*d*e^2*g)/(105*e^2*(b*e - 2*c*d)^8)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/
(d + e*x) + (((d*((d*((2*c^2*e^3*(5*b*e*g + 2*c*d*g - 6*c*e*f))/(7*(b*e - 2*c*d)^3*(5*b^3*e^6 - 40*c^3*d^3*e^3
+ 60*b*c^2*d^2*e^4 - 30*b^2*c*d*e^5)) - (4*c^3*d*e^3*g)/(7*(b*e - 2*c*d)^3*(5*b^3*e^6 - 40*c^3*d^3*e^3 + 60*b
*c^2*d^2*e^4 - 30*b^2*c*d*e^5))))/e - (e*(8*b^2*c*e^3*g - 38*b*c^2*e^3*f + 52*c^3*d*e^2*f - 48*c^3*d^2*e*g + 2
6*b*c^2*d*e^2*g))/(7*(b*e - 2*c*d)^3*(5*b^3*e^6 - 40*c^3*d^3*e^3 + 60*b*c^2*d^2*e^4 - 30*b^2*c*d*e^5))))/e - (
e*(16*c^3*d^3*g - 16*b^3*e^3*g + 40*b^2*c*e^3*f + 96*c^3*d^2*e*f - 122*b*c^2*d*e^2*f - 48*b*c^2*d^2*e*g + 48*b
^2*c*d*e^2*g))/(7*(b*e - 2*c*d)^3*(5*b^3*e^6 - 40*c^3*d^3*e^3 + 60*b*c^2*d^2*e^4 - 30*b^2*c*d*e^5)))*(c*d^2 -
c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((8*c*g*(2*b*e - 3*c*d))/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2
*c*d)^4) - (8*c^2*d*g)/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)
)/(d + e*x)^2 - (((d*((d*((24*c^3*e^2*(b*g - c*f))/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^6) - (8*c^4*d*e*g)/(3
5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^6)))/e - (26*b^2*c^2*e^2*g - 96*c^4*d^2*g + 128*c^4*d*e*f - 88*b*c^3*e^2*f
+ 32*b*c^3*d*e*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^6)))/e + (2*b*c*(4*b^2*e^2*g - 24*c^2*d^2*g - 19*b*c*
e^2*f + 32*c^2*d*e*f + 8*b*c*d*e*g))/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*
e^2*x)^(1/2))/(d + e*x)^2 + (((2*e^2*f)/(7*b^3*e^6 - 56*c^3*d^3*e^3 + 84*b*c^2*d^2*e^4 - 42*b^2*c*d*e^5) - (2*
d*e*g)/(7*b^3*e^6 - 56*c^3*d^3*e^3 + 84*b*c^2*d^2*e^4 - 42*b^2*c*d*e^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)
^(1/2))/(d + e*x)^4 - (((2*b*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (4*c*d*g)/(7*e*(5*b*e^2 - 10*c*d*e)
*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 + ((x*(((e*(b*e - c*d) + c*d*e)*((
(e*(b*e - c*d) + c*d*e)*(((e*(b*e - c*d) + c*d*e)*((2*c^2*(88*c^6*d^2*e^3*g - 82*b^2*c^4*e^5*g + 136*b*c^5*e^5
*f - 176*c^6*d*e^4*f + 32*b*c^5*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - ((e*
(b*e - c*d) + c*d*e)*((16*c^7*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 -
4*b*c^2*d*e)) + (16*c^7*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*
d*e)) - (8*b*c^7*e^4*g)/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (8*b*c^6*e^3*(
2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*d*e^2*g*(b
*e - c*d))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (d*(b*e - c*d)*((16*c^7*e^3
*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*e^2*g*(e
*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^7*e^4*g)/(105*(b*e
- 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(78*b^2*c^4*e^5*f - 40*b^3*c^3*e^5*g - 3
60*c^6*d^2*e^3*f + 96*c^6*d^3*e^2*g + 96*b*c^5*d*e^4*f + 168*b*c^5*d^2*e^3*g - 84*b^2*c^4*d*e^4*g))/(105*e*(b*
e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (b*c*(88*c^6*d^2*e^3*g - 82*b^2*c^4*e^5*g + 136*b*c^5*e^
5*f - 176*c^6*d*e^4*f + 32*b*c^5*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*
e^2) - (2*c^2*(296*b^3*c^3*e^5*f - 116*b^4*c^2*e^5*g - 2272*c^6*d^3*e^2*f + 480*c^6*d^4*e*g + 3768*b*c^5*d^2*e
^3*f - 1932*b^2*c^4*d*e^4*f + 80*b*c^5*d^3*e^2*g + 712*b^3*c^3*d*e^4*g - 1086*b^2*c^4*d^2*e^3*g))/(105*e*(b*e
- 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (d*(b*e - c*d)*((2*c^2*(88*c^6*d^2*e^3*g - 82*b^2*c^4*e^5*
g + 136*b*c^5*e^5*f - 176*c^6*d*e^4*f + 32*b*c^5*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b
*c^2*d*e)) - ((e*(b*e - c*d) + c*d*e)*((16*c^7*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*
d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2
*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^7*e^4*g)/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2)
+ (8*b*c^6*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (
16*c^7*d*e^2*g*(b*e - c*d))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (b*c*(78*b
^2*c^4*e^5*f - 40*b^3*c^3*e^5*g - 360*c^6*d^2*e^3*f + 96*c^6*d^3*e^2*g + 96*b*c^5*d*e^4*f + 168*b*c^5*d^2*e^3*
g - 84*b^2*c^4*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (2*c^2*(832
*c^6*d^5*g + 332*b^4*c^2*e^5*f - 176*b^5*c*e^5*g + 2880*c^6*d^4*e*f - 3280*b*c^5*d^4*e*g - 6896*b*c^5*d^3*e^2*
f - 2360*b^3*c^3*d*e^4*f + 1312*b^4*c^2*d*e^4*g + 6114*b^2*c^4*d^2*e^3*f + 5024*b^2*c^4*d^3*e^2*g - 3712*b^3*c
^3*d^2*e^3*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (d*(b*e - c*d)*(((e*(b*e - c*d)
+ c*d*e)*((2*c^2*(88*c^6*d^2*e^3*g - 82*b^2*c^4*e^5*g + 136*b*c^5*e^5*f - 176*c^6*d*e^4*f + 32*b*c^5*d*e^4*g)
)/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - ((e*(b*e - c*d) + c*d*e)*((16*c^7*e^3*(2*c*d
*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*e^2*g*(e*(b*e -
c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^7*e^4*g)/(105*(b*e - 2*c*
d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (8*b*c^6*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e
- 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*d*e^2*g*(b*e - c*d))/(105*(b*e - 2*c*d)^6*(4*c^3*d
^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (d*(b*e - c*d)*((16*c^7*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*
e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d
)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^7*e^4*g)/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b
*c^2*d*e))))/(c*e^2) - (2*c^2*(78*b^2*c^4*e^5*f - 40*b^3*c^3*e^5*g - 360*c^6*d^2*e^3*f + 96*c^6*d^3*e^2*g + 96
*b*c^5*d*e^4*f + 168*b*c^5*d^2*e^3*g - 84*b^2*c^4*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*
b*c^2*d*e)) - (b*c*(88*c^6*d^2*e^3*g - 82*b^2*c^4*e^5*g + 136*b*c^5*e^5*f - 176*c^6*d*e^4*f + 32*b*c^5*d*e^4*g
))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (b*c*(296*b^3*c^3*e^5*f - 116*b^4
*c^2*e^5*g - 2272*c^6*d^3*e^2*f + 480*c^6*d^4*e*g + 3768*b*c^5*d^2*e^3*f - 1932*b^2*c^4*d*e^4*f + 80*b*c^5*d^3
*e^2*g + 712*b^3*c^3*d*e^4*g - 1086*b^2*c^4*d^2*e^3*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^
2*d*e))) + (d*(b*e - c*d)*(((e*(b*e - c*d) + c*d*e)*(((e*(b*e - c*d) + c*d*e)*((2*c^2*(88*c^6*d^2*e^3*g - 82*b
^2*c^4*e^5*g + 136*b*c^5*e^5*f - 176*c^6*d*e^4*f + 32*b*c^5*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*
c*e^2 - 4*b*c^2*d*e)) - ((e*(b*e - c*d) + c*d*e)*((16*c^7*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d
)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^
3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^7*e^4*g)/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)
)))/(c*e^2) + (8*b*c^6*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^
2*d*e)) + (16*c^7*d*e^2*g*(b*e - c*d))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) +
(d*(b*e - c*d)*((16*c^7*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*
c^2*d*e)) + (16*c^7*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))
- (8*b*c^7*e^4*g)/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(78*b^2*c^4*
e^5*f - 40*b^3*c^3*e^5*g - 360*c^6*d^2*e^3*f + 96*c^6*d^3*e^2*g + 96*b*c^5*d*e^4*f + 168*b*c^5*d^2*e^3*g - 84*
b^2*c^4*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (b*c*(88*c^6*d^2*e^3*g - 82*
b^2*c^4*e^5*g + 136*b*c^5*e^5*f - 176*c^6*d*e^4*f + 32*b*c^5*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2
*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(296*b^3*c^3*e^5*f - 116*b^4*c^2*e^5*g - 2272*c^6*d^3*e^2*f + 480*c^
6*d^4*e*g + 3768*b*c^5*d^2*e^3*f - 1932*b^2*c^4*d*e^4*f + 80*b*c^5*d^3*e^2*g + 712*b^3*c^3*d*e^4*g - 1086*b^2*
c^4*d^2*e^3*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (d*(b*e - c*d)*((2*c^2*(88*c^6
*d^2*e^3*g - 82*b^2*c^4*e^5*g + 136*b*c^5*e^5*f - 176*c^6*d*e^4*f + 32*b*c^5*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*
(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - ((e*(b*e - c*d) + c*d*e)*((16*c^7*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/
(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e
- 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^7*e^4*g)/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e
^2 - 4*b*c^2*d*e))))/(c*e^2) + (8*b*c^6*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b
^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*d*e^2*g*(b*e - c*d))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*
d*e))))/(c*e^2) + (b*c*(78*b^2*c^4*e^5*f - 40*b^3*c^3*e^5*g - 360*c^6*d^2*e^3*f + 96*c^6*d^3*e^2*g + 96*b*c^5*
d*e^4*f + 168*b*c^5*d^2*e^3*g - 84*b^2*c^4*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d
*e))))/(c*e^2) + (b*c*(832*c^6*d^5*g + 332*b^4*c^2*e^5*f - 176*b^5*c*e^5*g + 2880*c^6*d^4*e*f - 3280*b*c^5*d^4
*e*g - 6896*b*c^5*d^3*e^2*f - 2360*b^3*c^3*d*e^4*f + 1312*b^4*c^2*d*e^4*g + 6114*b^2*c^4*d^2*e^3*f + 5024*b^2*
c^4*d^3*e^2*g - 3712*b^3*c^3*d^2*e^3*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)))*(c*d^2
- c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/((d + e*x)^2*(b*e - c*d + c*e*x)^2) + ((x*(((e*(b*e - c*d) + c*d*e)*(((
e*(b*e - c*d) + c*d*e)*(((e*(b*e - c*d) + c*d*e)*((16*c^7*e^2*(28*c^2*d^2*g - 45*b^2*e^2*g + 68*b*c*e^2*f - 88
*c^2*d*e*f + 32*b*c*d*e*g))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - ((e*(b*e - c*d) + c*
d*e)*((64*c^8*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) +
(64*c^8*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (32*b*c^
8*e^4*g)/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (32*b*c^7*e^3*(2*c*d*g - 5*b*
e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (64*c^8*d*e^2*g*(b*e - c*d))/(10
5*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (d*(b*e - c*d)*((64*c^8*e^3*(2*c*d*g - 5*
b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (64*c^8*e^2*g*(e*(b*e - c*d) +
c*d*e))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (32*b*c^8*e^4*g)/(105*(b*e - 2*c*d)^8*(
4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(408*b^2*c^5*e^5*f - 288*b^3*c^4*e^5*g - 1056*c^7*d^2
*e^3*f + 640*c^7*d^3*e^2*g - 96*b*c^6*d^2*e^3*g + 240*b^2*c^5*d*e^4*g))/(105*e*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^
2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^6*e^2*(28*c^2*d^2*g - 45*b^2*e^2*g + 68*b*c*e^2*f - 88*c^2*d*e*f + 32*b*c*d*e
*g))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(736*b^3*c^4*e^5*f - 264*b
^4*c^3*e^5*g - 6272*c^7*d^3*e^2*f + 640*c^7*d^4*e*g + 10464*b*c^6*d^2*e^3*f - 5232*b^2*c^5*d*e^4*f + 1216*b*c^
6*d^3*e^2*g + 1952*b^3*c^4*d*e^4*g - 3480*b^2*c^5*d^2*e^3*g))/(105*e*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 -
4*b*c^2*d*e)) - (d*(b*e - c*d)*((16*c^7*e^2*(28*c^2*d^2*g - 45*b^2*e^2*g + 68*b*c*e^2*f - 88*c^2*d*e*f + 32*b*
c*d*e*g))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - ((e*(b*e - c*d) + c*d*e)*((64*c^8*e^3*
(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (64*c^8*e^2*g*(e*
(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (32*b*c^8*e^4*g)/(105*(b*e
- 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (32*b*c^7*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(1
05*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (64*c^8*d*e^2*g*(b*e - c*d))/(105*(b*e - 2*c*d)^8*
(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (b*c*(408*b^2*c^5*e^5*f - 288*b^3*c^4*e^5*g - 1056*c^7*d^2*
e^3*f + 640*c^7*d^3*e^2*g - 96*b*c^6*d^2*e^3*g + 240*b^2*c^5*d*e^4*g))/(105*e*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2
*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(7104*c^7*d^5*g + 948*b^4*c^3*e^5*f - 732*b^5*c^2*e^5*g + 21696*c^7*
d^4*e*f - 28928*b*c^6*d^4*e*g - 40256*b*c^6*d^3*e^2*f - 8320*b^3*c^4*d*e^4*f + 6636*b^4*c^3*d*e^4*g + 27576*b^
2*c^5*d^2*e^3*f + 38688*b^2*c^5*d^3*e^2*g - 23360*b^3*c^4*d^2*e^3*g))/(105*e*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*
c*e^2 - 4*b*c^2*d*e)) - (d*(b*e - c*d)*(((e*(b*e - c*d) + c*d*e)*((16*c^7*e^2*(28*c^2*d^2*g - 45*b^2*e^2*g + 6
8*b*c*e^2*f - 88*c^2*d*e*f + 32*b*c*d*e*g))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - ((e*
(b*e - c*d) + c*d*e)*((64*c^8*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 -
4*b*c^2*d*e)) + (64*c^8*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*
d*e)) - (32*b*c^8*e^4*g)/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (32*b*c^7*e^3
*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (64*c^8*d*e^2*g*
(b*e - c*d))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (d*(b*e - c*d)*((64*c^8*e
^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (64*c^8*e^2*g*
(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (32*b*c^8*e^4*g)/(105*(
b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(408*b^2*c^5*e^5*f - 288*b^3*c^4*e^5*
g - 1056*c^7*d^2*e^3*f + 640*c^7*d^3*e^2*g - 96*b*c^6*d^2*e^3*g + 240*b^2*c^5*d*e^4*g))/(105*e*(b*e - 2*c*d)^8
*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^6*e^2*(28*c^2*d^2*g - 45*b^2*e^2*g + 68*b*c*e^2*f - 88*c^2*d*
e*f + 32*b*c*d*e*g))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (b*c*(736*b^3*c^4
*e^5*f - 264*b^4*c^3*e^5*g - 6272*c^7*d^3*e^2*f + 640*c^7*d^4*e*g + 10464*b*c^6*d^2*e^3*f - 5232*b^2*c^5*d*e^4
*f + 1216*b*c^6*d^3*e^2*g + 1952*b^3*c^4*d*e^4*g - 3480*b^2*c^5*d^2*e^3*g))/(105*e*(b*e - 2*c*d)^8*(4*c^3*d^2
+ b^2*c*e^2 - 4*b*c^2*d*e))) + (d*(b*e - c*d)*(((e*(b*e - c*d) + c*d*e)*(((e*(b*e - c*d) + c*d*e)*((16*c^7*e^2
*(28*c^2*d^2*g - 45*b^2*e^2*g + 68*b*c*e^2*f - 88*c^2*d*e*f + 32*b*c*d*e*g))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 +
b^2*c*e^2 - 4*b*c^2*d*e)) - ((e*(b*e - c*d) + c*d*e)*((64*c^8*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e -
2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (64*c^8*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^8*
(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (32*b*c^8*e^4*g)/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^
2*d*e))))/(c*e^2) + (32*b*c^7*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 -
4*b*c^2*d*e)) + (64*c^8*d*e^2*g*(b*e - c*d))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c
*e^2) + (d*(b*e - c*d)*((64*c^8*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2
- 4*b*c^2*d*e)) + (64*c^8*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^
2*d*e)) - (32*b*c^8*e^4*g)/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(408
*b^2*c^5*e^5*f - 288*b^3*c^4*e^5*g - 1056*c^7*d^2*e^3*f + 640*c^7*d^3*e^2*g - 96*b*c^6*d^2*e^3*g + 240*b^2*c^5
*d*e^4*g))/(105*e*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^6*e^2*(28*c^2*d^2*g - 45*b^2
*e^2*g + 68*b*c*e^2*f - 88*c^2*d*e*f + 32*b*c*d*e*g))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*
e))))/(c*e^2) - (2*c^2*(736*b^3*c^4*e^5*f - 264*b^4*c^3*e^5*g - 6272*c^7*d^3*e^2*f + 640*c^7*d^4*e*g + 10464*b
*c^6*d^2*e^3*f - 5232*b^2*c^5*d*e^4*f + 1216*b*c^6*d^3*e^2*g + 1952*b^3*c^4*d*e^4*g - 3480*b^2*c^5*d^2*e^3*g))
/(105*e*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (d*(b*e - c*d)*((16*c^7*e^2*(28*c^2*d^2*g - 4
5*b^2*e^2*g + 68*b*c*e^2*f - 88*c^2*d*e*f + 32*b*c*d*e*g))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c
^2*d*e)) - ((e*(b*e - c*d) + c*d*e)*((64*c^8*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^
2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (64*c^8*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c
*e^2 - 4*b*c^2*d*e)) - (32*b*c^8*e^4*g)/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2)
+ (32*b*c^7*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (
64*c^8*d*e^2*g*(b*e - c*d))/(105*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (b*c*(408*
b^2*c^5*e^5*f - 288*b^3*c^4*e^5*g - 1056*c^7*d^2*e^3*f + 640*c^7*d^3*e^2*g - 96*b*c^6*d^2*e^3*g + 240*b^2*c^5*
d*e^4*g))/(105*e*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (b*c*(7104*c^7*d^5*g + 948
*b^4*c^3*e^5*f - 732*b^5*c^2*e^5*g + 21696*c^7*d^4*e*f - 28928*b*c^6*d^4*e*g - 40256*b*c^6*d^3*e^2*f - 8320*b^
3*c^4*d*e^4*f + 6636*b^4*c^3*d*e^4*g + 27576*b^2*c^5*d^2*e^3*f + 38688*b^2*c^5*d^3*e^2*g - 23360*b^3*c^4*d^2*e
^3*g))/(105*e*(b*e - 2*c*d)^8*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1
/2))/((d + e*x)*(b*e - c*d + c*e*x))

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sympy [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((g*x+f)/(e*x+d)**2/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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