3.20.85 \(\int \frac {f+g x}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=285 \[ -\frac {16 c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x) (2 c d-b e)^4}-\frac {8 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x)^2 (2 c d-b e)^3}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{35 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (d+e x)^4 (2 c d-b e)} \]

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Rubi [A]  time = 0.44, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {792, 658, 650} \begin {gather*} -\frac {16 c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x) (2 c d-b e)^4}-\frac {8 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x)^2 (2 c d-b e)^3}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{35 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (d+e x)^4 (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(f + g*x)/((d + e*x)^4*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]

[Out]

(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(7*e^2*(2*c*d - b*e)*(d + e*x)^4) - (2*(6*c*e*f + 8
*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^3) - (8*c*(6*c*
e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^3*(d + e*x)^2) - (1
6*c^2*(6*c*e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^4*(d + e
*x))

Rule 650

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))/((p + 1)*(2*c*d - b*e)), 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] && EqQ[m + 2*p + 2, 0]

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

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Mathematica [A]  time = 0.14, size = 247, normalized size = 0.87 \begin {gather*} \frac {2 (b e-c d+c e x) \left (-3 b^3 e^3 (2 d g+5 e f+7 e g x)+2 b^2 c e^2 \left (23 d^2 g+d e (54 f+82 g x)+e^2 x (9 f+14 g x)\right )-4 b c^2 e \left (36 d^3 g+d^2 e (69 f+131 g x)+2 d e^2 x (15 f+32 g x)+2 e^3 x^2 (3 f+7 g x)\right )+8 c^3 \left (13 d^4 g+4 d^3 e (9 f+13 g x)+d^2 e^2 x (39 f+32 g x)+8 d e^3 x^2 (3 f+g x)+6 e^4 f x^3\right )\right )}{105 e^2 (d+e x)^3 (b e-2 c d)^4 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)/((d + e*x)^4*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]

[Out]

(2*(-(c*d) + b*e + c*e*x)*(-3*b^3*e^3*(5*e*f + 2*d*g + 7*e*g*x) + 8*c^3*(13*d^4*g + 6*e^4*f*x^3 + 8*d*e^3*x^2*
(3*f + g*x) + 4*d^3*e*(9*f + 13*g*x) + d^2*e^2*x*(39*f + 32*g*x)) + 2*b^2*c*e^2*(23*d^2*g + e^2*x*(9*f + 14*g*
x) + d*e*(54*f + 82*g*x)) - 4*b*c^2*e*(36*d^3*g + 2*e^3*x^2*(3*f + 7*g*x) + 2*d*e^2*x*(15*f + 32*g*x) + d^2*e*
(69*f + 131*g*x))))/(105*e^2*(-2*c*d + b*e)^4*(d + e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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IntegrateAlgebraic [B]  time = 43.86, size = 8629, normalized size = 30.28 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(f + g*x)/((d + e*x)^4*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]

[Out]

Result too large to show

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fricas [B]  time = 59.83, size = 606, normalized size = 2.13 \begin {gather*} -\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (8 \, {\left (6 \, c^{3} e^{4} f + {\left (8 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} g\right )} x^{3} + 4 \, {\left (6 \, {\left (8 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} f + {\left (64 \, c^{3} d^{2} e^{2} - 64 \, b c^{2} d e^{3} + 7 \, b^{2} c e^{4}\right )} g\right )} x^{2} + 3 \, {\left (96 \, c^{3} d^{3} e - 92 \, b c^{2} d^{2} e^{2} + 36 \, b^{2} c d e^{3} - 5 \, b^{3} e^{4}\right )} f + 2 \, {\left (52 \, c^{3} d^{4} - 72 \, b c^{2} d^{3} e + 23 \, b^{2} c d^{2} e^{2} - 3 \, b^{3} d e^{3}\right )} g + {\left (6 \, {\left (52 \, c^{3} d^{2} e^{2} - 20 \, b c^{2} d e^{3} + 3 \, b^{2} c e^{4}\right )} f + {\left (416 \, c^{3} d^{3} e - 524 \, b c^{2} d^{2} e^{2} + 164 \, b^{2} c d e^{3} - 21 \, b^{3} e^{4}\right )} g\right )} x\right )}}{105 \, {\left (16 \, c^{4} d^{8} e^{2} - 32 \, b c^{3} d^{7} e^{3} + 24 \, b^{2} c^{2} d^{6} e^{4} - 8 \, b^{3} c d^{5} e^{5} + b^{4} d^{4} e^{6} + {\left (16 \, c^{4} d^{4} e^{6} - 32 \, b c^{3} d^{3} e^{7} + 24 \, b^{2} c^{2} d^{2} e^{8} - 8 \, b^{3} c d e^{9} + b^{4} e^{10}\right )} x^{4} + 4 \, {\left (16 \, c^{4} d^{5} e^{5} - 32 \, b c^{3} d^{4} e^{6} + 24 \, b^{2} c^{2} d^{3} e^{7} - 8 \, b^{3} c d^{2} e^{8} + b^{4} d e^{9}\right )} x^{3} + 6 \, {\left (16 \, c^{4} d^{6} e^{4} - 32 \, b c^{3} d^{5} e^{5} + 24 \, b^{2} c^{2} d^{4} e^{6} - 8 \, b^{3} c d^{3} e^{7} + b^{4} d^{2} e^{8}\right )} x^{2} + 4 \, {\left (16 \, c^{4} d^{7} e^{3} - 32 \, b c^{3} d^{6} e^{4} + 24 \, b^{2} c^{2} d^{5} e^{5} - 8 \, b^{3} c d^{4} e^{6} + b^{4} d^{3} e^{7}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

-2/105*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(8*(6*c^3*e^4*f + (8*c^3*d*e^3 - 7*b*c^2*e^4)*g)*x^3 + 4*(6*
(8*c^3*d*e^3 - b*c^2*e^4)*f + (64*c^3*d^2*e^2 - 64*b*c^2*d*e^3 + 7*b^2*c*e^4)*g)*x^2 + 3*(96*c^3*d^3*e - 92*b*
c^2*d^2*e^2 + 36*b^2*c*d*e^3 - 5*b^3*e^4)*f + 2*(52*c^3*d^4 - 72*b*c^2*d^3*e + 23*b^2*c*d^2*e^2 - 3*b^3*d*e^3)
*g + (6*(52*c^3*d^2*e^2 - 20*b*c^2*d*e^3 + 3*b^2*c*e^4)*f + (416*c^3*d^3*e - 524*b*c^2*d^2*e^2 + 164*b^2*c*d*e
^3 - 21*b^3*e^4)*g)*x)/(16*c^4*d^8*e^2 - 32*b*c^3*d^7*e^3 + 24*b^2*c^2*d^6*e^4 - 8*b^3*c*d^5*e^5 + b^4*d^4*e^6
+ (16*c^4*d^4*e^6 - 32*b*c^3*d^3*e^7 + 24*b^2*c^2*d^2*e^8 - 8*b^3*c*d*e^9 + b^4*e^10)*x^4 + 4*(16*c^4*d^5*e^5
- 32*b*c^3*d^4*e^6 + 24*b^2*c^2*d^3*e^7 - 8*b^3*c*d^2*e^8 + b^4*d*e^9)*x^3 + 6*(16*c^4*d^6*e^4 - 32*b*c^3*d^5
*e^5 + 24*b^2*c^2*d^4*e^6 - 8*b^3*c*d^3*e^7 + b^4*d^2*e^8)*x^2 + 4*(16*c^4*d^7*e^3 - 32*b*c^3*d^6*e^4 + 24*b^2
*c^2*d^5*e^5 - 8*b^3*c*d^4*e^6 + b^4*d^3*e^7)*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((-73728*exp(2)^2*(sqrt(-c*exp(2)*x^2-
b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^5*g*b^3*d*exp(1)^7+12288*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)
*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^5*g*b^3*d*exp(1)^5+98304*c*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2
-b*d*exp(1))-sqrt(-c*exp(2))*x)^5*g*b^2*d^2*exp(1)^8+319488*c*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*
d*exp(1))-sqrt(-c*exp(2))*x)^5*g*b^2*d^2*exp(1)^6-49152*c*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*ex
p(1))-sqrt(-c*exp(2))*x)^5*g*b^2*d^2*exp(1)^4-196608*c^2*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1
))-sqrt(-c*exp(2))*x)^5*g*b*d^3*exp(1)^7-638976*c^2*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-
sqrt(-c*exp(2))*x)^5*g*b*d^3*exp(1)^5+98304*c^2*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt
(-c*exp(2))*x)^5*g*b*d^3*exp(1)^3+98304*c^3*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*ex
p(2))*x)^5*g*d^4*exp(1)^6+393216*c^3*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))
*x)^5*g*d^4*exp(1)^4+61440*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^5*b^3*
f*exp(1)^6-147456*c*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^5*b^2*f*d*exp
(1)^7-221184*c*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^5*b^2*f*d*exp(1)^5
+442368*c^2*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^5*b*f*d^2*exp(1)^6+29
4912*c^2*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^5*b*f*d^2*exp(1)^4-29491
2*c^3*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^5*f*d^3*exp(1)^5-196608*c^3
*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^5*f*d^3*exp(1)^3+368640*exp(2)^2
*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*g*b^3*d^2*exp(1)^6-6144
0*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*g*b^3*d^2*exp
(1)^4-491520*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*g*
b^2*d^3*exp(1)^7-1597440*c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*e
xp(2))*x)^4*g*b^2*d^3*exp(1)^5+245760*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(
1))-sqrt(-c*exp(2))*x)^4*g*b^2*d^3*exp(1)^3+983040*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c
*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*g*b*d^4*exp(1)^6+3194880*c^2*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^
2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*g*b*d^4*exp(1)^4-491520*c^2*exp(2)^3*sqrt(-c*exp(2))*(sqrt
(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*g*b*d^4*exp(1)^2-491520*c^3*exp(2)*sqrt(-c*ex
p(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*g*d^5*exp(1)^5-1966080*c^3*exp(2)^
2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*g*d^5*exp(1)^3-307200*
exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*b^3*f*d*exp(1)^
5+737280*c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*b^2*
f*d^2*exp(1)^6+1105920*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp
(2))*x)^4*b^2*f*d^2*exp(1)^4-2211840*c^2*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp
(1))-sqrt(-c*exp(2))*x)^4*b*f*d^3*exp(1)^5-1474560*c^2*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x
+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*b*f*d^3*exp(1)^3+1474560*c^3*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*
x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*f*d^4*exp(1)^4+983040*c^3*exp(2)^3*sqrt(-c*exp(2))*(sqrt
(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^4*f*d^4*exp(1)^2-196608*exp(2)^2*(sqrt(-c*exp(2
)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^4*d^2*exp(1)^8+229376*exp(2)^3*(sqrt(-c*exp(2)*x^2
-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^4*d^2*exp(1)^6-32768*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp
(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^4*d^2*exp(1)^4+1572864*c*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2
)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^3*d^3*exp(1)^7-1261568*c*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*
x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^3*d^3*exp(1)^5+303104*c*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c
*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^3*d^3*exp(1)^3-5111808*c^2*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c
*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^2*d^4*exp(1)^6+2113536*c^2*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c
*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^2*d^4*exp(1)^4-688128*c^2*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*
d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^2*d^4*exp(1)^2+6291456*c^3*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*
d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b*d^5*exp(1)^5+1015808*c^3*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^
2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b*d^5*exp(1)^3+65536*c^3*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*
d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b*d^5*exp(1)-2555904*c^4*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*ex
p(1))-sqrt(-c*exp(2))*x)^3*g*d^6*exp(1)^4-2097152*c^4*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1)
)-sqrt(-c*exp(2))*x)^3*g*d^6*exp(1)^2-262144*c^4*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqr
t(-c*exp(2))*x)^3*g*d^6+163840*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*
b^4*f*d*exp(1)^7-163840*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b^4*f*d
*exp(1)^5-393216*c*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b^3*f*d^2*ex
p(1)^8-360448*c*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b^3*f*d^2*exp(1
)^6+139264*c*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b^3*f*d^2*exp(1)^4
+1572864*c^2*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b^2*f*d^3*exp(1)^7
+1277952*c^2*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b^2*f*d^3*exp(1)^5
+835584*c^2*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b^2*f*d^3*exp(1)^3-
1966080*c^3*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b*f*d^4*exp(1)^6-37
68320*c^3*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b*f*d^4*exp(1)^4-1638
400*c^3*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b*f*d^4*exp(1)^2+786432
*c^4*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*f*d^5*exp(1)^5+2686976*c^4
*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*f*d^5*exp(1)^3+1441792*c^4*exp
(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*f*d^5*exp(1)+196608*exp(2)*sqrt(-c
*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^4*d^3*exp(1)^9-98304*exp(2)
^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^4*d^3*exp(1)^5-98
304*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^4*d^3*e
xp(1)^3-1376256*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2
*g*b^3*d^4*exp(1)^8-589824*c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c
*exp(2))*x)^2*g*b^3*d^4*exp(1)^6+1130496*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*e
xp(1))-sqrt(-c*exp(2))*x)^2*g*b^3*d^4*exp(1)^4+221184*c*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*
x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^3*d^4*exp(1)^2+2949120*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)
*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^2*d^5*exp(1)^7+4521984*c^2*exp(2)^2*sqrt(-c*exp(2))
*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^2*d^5*exp(1)^5-3883008*c^2*exp(2)^3
*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^2*d^5*exp(1)^3+9830
4*c^2*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^2*d^5
*exp(1)-2555904*c^3*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)
^2*g*b*d^6*exp(1)^6-7274496*c^3*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt
(-c*exp(2))*x)^2*g*b*d^6*exp(1)^4+2064384*c^3*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*
d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b*d^6*exp(1)^2+393216*c^3*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(
2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b*d^6+786432*c^4*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*ex
p(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*d^7*exp(1)^5+3342336*c^4*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp
(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*d^7*exp(1)^3+786432*c^4*exp(2)^3*sqrt(-c*exp(2))*(
sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*d^7*exp(1)-491520*exp(2)^3*sqrt(-c*exp(
2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*b^4*f*d^2*exp(1)^6+491520*exp(2)^4*s
qrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*b^4*f*d^2*exp(1)^4+393216
*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*b^3*f*d^3*exp(
1)^9+2260992*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*
b^3*f*d^3*exp(1)^5-2039808*c*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c
*exp(2))*x)^2*b^3*f*d^3*exp(1)^3-1179648*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*e
xp(1))-sqrt(-c*exp(2))*x)^2*b^2*f*d^4*exp(1)^8-1179648*c^2*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(
2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*b^2*f*d^4*exp(1)^6-4423680*c^2*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*e
xp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*b^2*f*d^4*exp(1)^4+3096576*c^2*exp(2)^4*sqrt(-c*ex
p(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*b^2*f*d^4*exp(1)^2+1179648*c^3*exp
(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*b*f*d^5*exp(1)^7+235
9296*c^3*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*b*f*d^
5*exp(1)^5+5996544*c^3*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2
))*x)^2*b*f*d^5*exp(1)^3-2162688*c^3*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))
-sqrt(-c*exp(2))*x)^2*b*f*d^5*exp(1)-393216*c^4*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*
d*exp(1))-sqrt(-c*exp(2))*x)^2*f*d^6*exp(1)^6-1179648*c^4*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2
)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*f*d^6*exp(1)^4-3342336*c^4*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-c*exp(2)
*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*f*d^6*exp(1)^2-122880*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*ex
p(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^5*d^3*exp(1)^9+233472*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+
c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^5*d^3*exp(1)^7-98304*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*
d*exp(1))-sqrt(-c*exp(2))*x)*g*b^5*d^3*exp(1)^5-12288*exp(2)^5*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1)
)-sqrt(-c*exp(2))*x)*g*b^5*d^3*exp(1)^3-98304*c*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-
c*exp(2))*x)*g*b^4*d^4*exp(1)^10+1499136*c*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*e
xp(2))*x)*g*b^4*d^4*exp(1)^8-2236416*c*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2
))*x)*g*b^4*d^4*exp(1)^6+860160*c*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)
*g*b^4*d^4*exp(1)^4-24576*c*exp(2)^5*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^4
*d^4*exp(1)^2+393216*c^2*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^3*d^5*
exp(1)^9-5922816*c^2*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^3*d^5*ex
p(1)^7+7311360*c^2*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^3*d^5*exp(
1)^5-2236416*c^2*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^3*d^5*exp(1)
^3+147456*c^2*exp(2)^5*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^3*d^5*exp(1)-58
9824*c^3*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^2*d^6*exp(1)^8+1000243
2*c^3*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^2*d^6*exp(1)^6-8454144*
c^3*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^2*d^6*exp(1)^4+688128*c^3
*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^2*d^6*exp(1)^2+196608*c^3*ex
p(2)^5*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^2*d^6+393216*c^4*exp(2)*(sqrt(-
c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b*d^7*exp(1)^7-7618560*c^4*exp(2)^2*(sqrt(-c*ex
p(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b*d^7*exp(1)^5+2752512*c^4*exp(2)^3*(sqrt(-c*exp(2)
*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b*d^7*exp(1)^3+786432*c^4*exp(2)^4*(sqrt(-c*exp(2)*x^2-
b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b*d^7*exp(1)-98304*c^5*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x
+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*d^8*exp(1)^6+2162688*c^5*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2
-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*d^8*exp(1)^4+393216*c^5*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*ex
p(1))-sqrt(-c*exp(2))*x)*g*d^8*exp(1)^2+135168*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(
-c*exp(2))*x)*b^5*f*d^2*exp(1)^8-270336*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(
2))*x)*b^5*f*d^2*exp(1)^6+135168*exp(2)^5*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*
b^5*f*d^2*exp(1)^4+147456*c*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^4*f
*d^3*exp(1)^9-1523712*c*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^4*f*d^3
*exp(1)^7+2113536*c*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^4*f*d^3*exp
(1)^5-737280*c*exp(2)^5*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^4*f*d^3*exp(1)^3
+49152*c^2*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^3*f*d^4*exp(1)^8+441
1392*c^2*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^3*f*d^4*exp(1)^6-55296
00*c^2*exp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^3*f*d^4*exp(1)^4+1376256
*c^2*exp(2)^5*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^3*f*d^4*exp(1)^2-1032192*c
^3*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^2*f*d^5*exp(1)^7-6758400*c^3
*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^2*f*d^5*exp(1)^5+7028736*c^3*e
xp(2)^4*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^2*f*d^5*exp(1)^3-1081344*c^3*exp
(2)^5*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^2*f*d^5*exp(1)+1327104*c^4*exp(2)^
2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b*f*d^6*exp(1)^6+5701632*c^4*exp(2)^3*(s
qrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b*f*d^6*exp(1)^4-3342336*c^4*exp(2)^4*(sqrt(
-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b*f*d^6*exp(1)^2-491520*c^5*exp(2)^2*(sqrt(-c*ex
p(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*f*d^7*exp(1)^5-1966080*c^5*exp(2)^3*(sqrt(-c*exp(2)*x
^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*f*d^7*exp(1)^3+196608*exp(2)*sqrt(-c*exp(2))*g*b^5*d^4*exp(
1)^10-466944*exp(2)^2*sqrt(-c*exp(2))*g*b^5*d^4*exp(1)^8+356352*exp(2)^3*sqrt(-c*exp(2))*g*b^5*d^4*exp(1)^6-98
304*exp(2)^4*sqrt(-c*exp(2))*g*b^5*d^4*exp(1)^4+12288*exp(2)^5*sqrt(-c*exp(2))*g*b^5*d^4*exp(1)^2-1212416*c*ex
p(2)*sqrt(-c*exp(2))*g*b^4*d^5*exp(1)^9+2564096*c*exp(2)^2*sqrt(-c*exp(2))*g*b^4*d^5*exp(1)^7-1695744*c*exp(2)
^3*sqrt(-c*exp(2))*g*b^4*d^5*exp(1)^5+385024*c*exp(2)^4*sqrt(-c*exp(2))*g*b^4*d^5*exp(1)^3-40960*c*exp(2)^5*sq
rt(-c*exp(2))*g*b^4*d^5*exp(1)+2883584*c^2*exp(2)*sqrt(-c*exp(2))*g*b^3*d^6*exp(1)^8-4825088*c^2*exp(2)^2*sqrt
(-c*exp(2))*g*b^3*d^6*exp(1)^6+2125824*c^2*exp(2)^3*sqrt(-c*exp(2))*g*b^3*d^6*exp(1)^4-90112*c^2*exp(2)^4*sqrt
(-c*exp(2))*g*b^3*d^6*exp(1)^2-32768*c^2*exp(2)^5*sqrt(-c*exp(2))*g*b^3*d^6-3342336*c^3*exp(2)*sqrt(-c*exp(2))
*g*b^2*d^7*exp(1)^7+3760128*c^3*exp(2)^2*sqrt(-c*exp(2))*g*b^2*d^7*exp(1)^5-589824*c^3*exp(2)^3*sqrt(-c*exp(2)
)*g*b^2*d^7*exp(1)^3-196608*c^3*exp(2)^4*sqrt(-c*exp(2))*g*b^2*d^7*exp(1)+1900544*c^4*exp(2)*sqrt(-c*exp(2))*g
*b*d^8*exp(1)^6-966656*c^4*exp(2)^2*sqrt(-c*exp(2))*g*b*d^8*exp(1)^4-196608*c^4*exp(2)^3*sqrt(-c*exp(2))*g*b*d
^8*exp(1)^2-425984*c^5*exp(2)*sqrt(-c*exp(2))*g*d^9*exp(1)^5-65536*c^5*exp(2)^2*sqrt(-c*exp(2))*g*d^9*exp(1)^3
-196608*exp(2)^2*sqrt(-c*exp(2))*b^5*f*d^3*exp(1)^9+454656*exp(2)^3*sqrt(-c*exp(2))*b^5*f*d^3*exp(1)^7-319488*
exp(2)^4*sqrt(-c*exp(2))*b^5*f*d^3*exp(1)^5+61440*exp(2)^5*sqrt(-c*exp(2))*b^5*f*d^3*exp(1)^3+131072*c*exp(2)*
sqrt(-c*exp(2))*b^4*f*d^4*exp(1)^10+704512*c*exp(2)^2*sqrt(-c*exp(2))*b^4*f*d^4*exp(1)^8-1818624*c*exp(2)^3*sq
rt(-c*exp(2))*b^4*f*d^4*exp(1)^6+1163264*c*exp(2)^4*sqrt(-c*exp(2))*b^4*f*d^4*exp(1)^4-180224*c*exp(2)^5*sqrt(
-c*exp(2))*b^4*f*d^4*exp(1)^2-524288*c^2*exp(2)*sqrt(-c*exp(2))*b^3*f*d^5*exp(1)^9-1294336*c^2*exp(2)^2*sqrt(-
c*exp(2))*b^3*f*d^5*exp(1)^7+3256320*c^2*exp(2)^3*sqrt(-c*exp(2))*b^3*f*d^5*exp(1)^5-1679360*c^2*exp(2)^4*sqrt
(-c*exp(2))*b^3*f*d^5*exp(1)^3+180224*c^2*exp(2)^5*sqrt(-c*exp(2))*b^3*f*d^5*exp(1)+786432*c^3*exp(2)*sqrt(-c*
exp(2))*b^2*f*d^6*exp(1)^8+1622016*c^3*exp(2)^2*sqrt(-c*exp(2))*b^2*f*d^6*exp(1)^6-2875392*c^3*exp(2)^3*sqrt(-
c*exp(2))*b^2*f*d^6*exp(1)^4+835584*c^3*exp(2)^4*sqrt(-c*exp(2))*b^2*f*d^6*exp(1)^2-524288*c^4*exp(2)*sqrt(-c*
exp(2))*b*f*d^7*exp(1)^7-1196032*c^4*exp(2)^2*sqrt(-c*exp(2))*b*f*d^7*exp(1)^5+983040*c^4*exp(2)^3*sqrt(-c*exp
(2))*b*f*d^7*exp(1)^3+131072*c^5*exp(2)*sqrt(-c*exp(2))*f*d^8*exp(1)^6+360448*c^5*exp(2)^2*sqrt(-c*exp(2))*f*d
^8*exp(1)^4)/(-196608*b^3*d^3*exp(1)^10+589824*exp(2)*b^3*d^3*exp(1)^8-589824*exp(2)^2*b^3*d^3*exp(1)^6+196608
*exp(2)^3*b^3*d^3*exp(1)^4+589824*c*b^2*d^4*exp(1)^9-1769472*c*exp(2)*b^2*d^4*exp(1)^7+1769472*c*exp(2)^2*b^2*
d^4*exp(1)^5-589824*c*exp(2)^3*b^2*d^4*exp(1)^3-589824*c^2*b*d^5*exp(1)^8+1769472*c^2*exp(2)*b*d^5*exp(1)^6-17
69472*c^2*exp(2)^2*b*d^5*exp(1)^4+589824*c^2*exp(2)^3*b*d^5*exp(1)^2+196608*c^3*d^6*exp(1)^7-589824*c^3*exp(2)
*d^6*exp(1)^5+589824*c^3*exp(2)^2*d^6*exp(1)^3-196608*c^3*exp(2)^3*d^6*exp(1))/((sqrt(-c*exp(2)*x^2-b*exp(2)*x
+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*exp(1)-2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(
1))-sqrt(-c*exp(2))*x)*d+b*d*exp(1)^2-exp(2)*b*d-c*d^2*exp(1))^3+(12*exp(2)^2*g*b^3*d*exp(1)^4-2*exp(2)^3*g*b^
3*d*exp(1)^2-16*c*exp(2)*g*b^2*d^2*exp(1)^5-52*c*exp(2)^2*g*b^2*d^2*exp(1)^3+8*c*exp(2)^3*g*b^2*d^2*exp(1)+32*
c^2*exp(2)*g*b*d^3*exp(1)^4+104*c^2*exp(2)^2*g*b*d^3*exp(1)^2-16*c^2*exp(2)^3*g*b*d^3-16*c^3*exp(2)*g*d^4*exp(
1)^3-64*c^3*exp(2)^2*g*d^4*exp(1)-10*exp(2)^3*b^3*f*exp(1)^3+24*c*exp(2)^2*b^2*f*d*exp(1)^4+36*c*exp(2)^3*b^2*
f*d*exp(1)^2-72*c^2*exp(2)^2*b*f*d^2*exp(1)^3-48*c^2*exp(2)^3*b*f*d^2*exp(1)+48*c^3*exp(2)^2*f*d^3*exp(1)^2+32
*c^3*exp(2)^3*f*d^3)/32/(b^3*d^3*exp(1)^9-3*exp(2)*b^3*d^3*exp(1)^7+3*exp(2)^2*b^3*d^3*exp(1)^5-exp(2)^3*b^3*d
^3*exp(1)^3-3*c*b^2*d^4*exp(1)^8+9*c*exp(2)*b^2*d^4*exp(1)^6-9*c*exp(2)^2*b^2*d^4*exp(1)^4+3*c*exp(2)^3*b^2*d^
4*exp(1)^2+3*c^2*b*d^5*exp(1)^7-9*c^2*exp(2)*b*d^5*exp(1)^5+9*c^2*exp(2)^2*b*d^5*exp(1)^3-3*c^2*exp(2)^3*b*d^5
*exp(1)-c^3*d^6*exp(1)^6+3*c^3*exp(2)*d^6*exp(1)^4-3*c^3*exp(2)^2*d^6*exp(1)^2+c^3*exp(2)^3*d^6)/sqrt(b*d*exp(
1)^3-c*d^2*exp(1)^2+c*d^2*exp(2)-b*d*exp(1)*exp(2))*atan((-d*sqrt(-c*exp(2))+(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*
d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*exp(1))/sqrt(b*d*exp(1)^3-c*d^2*exp(1)^2+c*d^2*exp(2)-b*d*exp(1)*exp(2))))

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maple [A]  time = 0.05, size = 382, normalized size = 1.34 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (56 b \,c^{2} e^{4} g \,x^{3}-64 c^{3} d \,e^{3} g \,x^{3}-48 c^{3} e^{4} f \,x^{3}-28 b^{2} c \,e^{4} g \,x^{2}+256 b \,c^{2} d \,e^{3} g \,x^{2}+24 b \,c^{2} e^{4} f \,x^{2}-256 c^{3} d^{2} e^{2} g \,x^{2}-192 c^{3} d \,e^{3} f \,x^{2}+21 b^{3} e^{4} g x -164 b^{2} c d \,e^{3} g x -18 b^{2} c \,e^{4} f x +524 b \,c^{2} d^{2} e^{2} g x +120 b \,c^{2} d \,e^{3} f x -416 c^{3} d^{3} e g x -312 c^{3} d^{2} e^{2} f x +6 b^{3} d \,e^{3} g +15 b^{3} e^{4} f -46 b^{2} c \,d^{2} e^{2} g -108 b^{2} c d \,e^{3} f +144 b \,c^{2} d^{3} e g +276 b \,c^{2} d^{2} e^{2} f -104 c^{3} d^{4} g -288 c^{3} d^{3} e f \right )}{105 \left (e x +d \right )^{3} \left (b^{4} e^{4}-8 b^{3} c d \,e^{3}+24 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)/(e*x+d)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-2/105*(c*e*x+b*e-c*d)*(56*b*c^2*e^4*g*x^3-64*c^3*d*e^3*g*x^3-48*c^3*e^4*f*x^3-28*b^2*c*e^4*g*x^2+256*b*c^2*d*
e^3*g*x^2+24*b*c^2*e^4*f*x^2-256*c^3*d^2*e^2*g*x^2-192*c^3*d*e^3*f*x^2+21*b^3*e^4*g*x-164*b^2*c*d*e^3*g*x-18*b
^2*c*e^4*f*x+524*b*c^2*d^2*e^2*g*x+120*b*c^2*d*e^3*f*x-416*c^3*d^3*e*g*x-312*c^3*d^2*e^2*f*x+6*b^3*d*e^3*g+15*
b^3*e^4*f-46*b^2*c*d^2*e^2*g-108*b^2*c*d*e^3*f+144*b*c^2*d^3*e*g+276*b*c^2*d^2*e^2*f-104*c^3*d^4*g-288*c^3*d^3
*e*f)/(e*x+d)^3/e^2/(b^4*e^4-8*b^3*c*d*e^3+24*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d^4)/(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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 = 4.75, size = 624, normalized size = 2.19 \begin {gather*} \frac {\left (\frac {40\,c^2\,d\,g+48\,c^2\,e\,f-40\,b\,c\,e\,g}{35\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}-\frac {8\,c^2\,d\,g}{35\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {8\,c\,g\,\left (2\,b\,e-3\,c\,d\right )}{35\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}-\frac {8\,c^2\,d\,g}{35\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {2\,b\,g}{7\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}-\frac {4\,c\,d\,g}{7\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {16\,c\,d\,g-16\,b\,e\,g+12\,c\,e\,f}{7\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}+\frac {4\,c\,d\,g}{7\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {2\,f}{7\,b\,e^2-14\,c\,d\,e}-\frac {2\,d\,g}{e\,\left (7\,b\,e^2-14\,c\,d\,e\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^4}-\frac {\left (\frac {112\,c^3\,d\,g+96\,c^3\,e\,f-112\,b\,c^2\,e\,g}{105\,e^2\,{\left (b\,e-2\,c\,d\right )}^4}+\frac {16\,c^3\,d\,g}{105\,e^2\,{\left (b\,e-2\,c\,d\right )}^4}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)/((d + e*x)^4*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)),x)

[Out]

(((40*c^2*d*g + 48*c^2*e*f - 40*b*c*e*g)/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) - (8*c^2*d*g)/(35*e*(3*b*e
^2 - 6*c*d*e)*(b*e - 2*c*d)^2))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((8*c*g*(2*b*e - 3
*c*d))/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) - (8*c^2*d*g)/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2))*(c
*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((2*b*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)) - (4
*c*d*g)/(7*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (
((16*c*d*g - 16*b*e*g + 12*c*e*f)/(7*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)) + (4*c*d*g)/(7*e*(5*b*e^2 - 10*c*d*
e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 + (((2*f)/(7*b*e^2 - 14*c*d*e) - (
2*d*g)/(e*(7*b*e^2 - 14*c*d*e)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^4 - (((112*c^3*d*g + 9
6*c^3*e*f - 112*b*c^2*e*g)/(105*e^2*(b*e - 2*c*d)^4) + (16*c^3*d*g)/(105*e^2*(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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f + g x}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)**4/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

Integral((f + g*x)/(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)**4), x)

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