3.20.48 \(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx\)
Optimal. Leaf size=137 \[ -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-5 b e g+8 c d g+2 c e f)}{15 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 e^2 (d+e x)^4 (2 c d-b e)} \]
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Rubi [A] time = 0.21, antiderivative size = 137, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used =
{792, 650} \begin {gather*} -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-5 b e g+8 c d g+2 c e f)}{15 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 e^2 (d+e x)^4 (2 c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^4,x]
[Out]
(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(5*e^2*(2*c*d - b*e)*(d + e*x)^4) - (2*(2*c*e*f +
8*c*d*g - 5*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(15*e^2*(2*c*d - b*e)^2*(d + e*x)^3)
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 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) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 e^2 (2 c d-b e) (d+e x)^4}+\frac {(2 c e f+8 c d g-5 b e g) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx}{5 e (2 c d-b e)}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 e^2 (2 c d-b e) (d+e x)^4}-\frac {2 (2 c e f+8 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15 e^2 (2 c d-b e)^2 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 102, normalized size = 0.74 \begin {gather*} \frac {2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)} \left (2 c \left (d^2 g+4 d e (f+g x)+e^2 f x\right )-b e (2 d g+3 e f+5 e g x)\right )}{15 e^2 (d+e x)^3 (b e-2 c d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^4,x]
[Out]
(2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-(b*e*(3*e*f + 2*d*g + 5*e*g*x)) + 2*c*(d^2*
g + e^2*f*x + 4*d*e*(f + g*x))))/(15*e^2*(-2*c*d + b*e)^2*(d + e*x)^3)
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IntegrateAlgebraic [B] time = 26.20, size = 5369, normalized size = 39.19 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^4,x]
[Out]
Result too large to show
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fricas [B] time = 7.01, size = 307, normalized size = 2.24 \begin {gather*} \frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c^{2} e^{3} f + {\left (8 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} g\right )} x^{2} - {\left (8 \, c^{2} d^{2} e - 11 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f - 2 \, {\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2}\right )} g + {\left ({\left (6 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (6 \, c^{2} d^{2} e - 11 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )} g\right )} x\right )}}{15 \, {\left (4 \, c^{2} d^{5} e^{2} - 4 \, b c d^{4} e^{3} + b^{2} d^{3} e^{4} + {\left (4 \, c^{2} d^{2} e^{5} - 4 \, b c d e^{6} + b^{2} e^{7}\right )} x^{3} + 3 \, {\left (4 \, c^{2} d^{3} e^{4} - 4 \, b c d^{2} e^{5} + b^{2} d e^{6}\right )} x^{2} + 3 \, {\left (4 \, c^{2} d^{4} e^{3} - 4 \, b c d^{3} e^{4} + b^{2} d^{2} e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4,x, algorithm="fricas")
[Out]
2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((2*c^2*e^3*f + (8*c^2*d*e^2 - 5*b*c*e^3)*g)*x^2 - (8*c^2*d^2*
e - 11*b*c*d*e^2 + 3*b^2*e^3)*f - 2*(c^2*d^3 - 2*b*c*d^2*e + b^2*d*e^2)*g + ((6*c^2*d*e^2 - b*c*e^3)*f - (6*c^
2*d^2*e - 11*b*c*d*e^2 + 5*b^2*e^3)*g)*x)/(4*c^2*d^5*e^2 - 4*b*c*d^4*e^3 + b^2*d^3*e^4 + (4*c^2*d^2*e^5 - 4*b*
c*d*e^6 + b^2*e^7)*x^3 + 3*(4*c^2*d^3*e^4 - 4*b*c*d^2*e^5 + b^2*d*e^6)*x^2 + 3*(4*c^2*d^4*e^3 - 4*b*c*d^3*e^4
+ b^2*d^2*e^5)*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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((100663296*exp(2)^2*(sqrt(-b*d*exp(1)
-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*b^3*d*exp(1)^7-50331648*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*
exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*b^3*d*exp(1)^5+402653184*c*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(
2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*b^2*d^2*exp(1)^8-1509949440*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp
(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*b^2*d^2*exp(1)^6+805306368*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp
(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*b^2*d^2*exp(1)^4-805306368*c^2*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp
(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*b*d^3*exp(1)^7+3019898880*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*ex
p(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*b*d^3*exp(1)^5-1610612736*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*e
xp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*b*d^3*exp(1)^3+402653184*c^3*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp
(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*d^4*exp(1)^6-1610612736*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(
2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*d^4*exp(1)^4+805306368*c^3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)
+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*d^4*exp(1)^2-50331648*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-
c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b^3*f*exp(1)^6+201326592*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^
2*exp(2))-sqrt(-c*exp(2))*x)^5*b^2*f*d*exp(1)^7+100663296*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*
exp(2))-sqrt(-c*exp(2))*x)^5*b^2*f*d*exp(1)^5-603979776*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*
exp(2))-sqrt(-c*exp(2))*x)^5*b*f*d^2*exp(1)^6+402653184*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*
exp(2))-sqrt(-c*exp(2))*x)^5*f*d^3*exp(1)^5-805306368*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^
2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*b^3*d^2*exp(1)^8+1107296256*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-
b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*b^3*d^2*exp(1)^6-553648128*exp(2)^3*sqrt(-c*exp(2))*(sqr
t(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*b^3*d^2*exp(1)^4+2013265920*c*exp(2)*sqrt(
-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*b^2*d^3*exp(1)^7-503316480*
c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*b^2*d^3*exp
(1)^5-1610612736*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x
)^4*g*b*d^4*exp(1)^6-3825205248*c^2*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-
sqrt(-c*exp(2))*x)^4*g*b*d^4*exp(1)^4+2415919104*c^2*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d
^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*b*d^4*exp(1)^2+402653184*c^3*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-
b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*d^5*exp(1)^5+3221225472*c^3*exp(2)^2*sqrt(-c*exp(2))*(sq
rt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*d^5*exp(1)^3-1610612736*c^3*exp(2)^3*sqrt
(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*d^5*exp(1)+251658240*exp(2
)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b^3*f*d*exp(1)^5-805
306368*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b^2*f*d^
2*exp(1)^8+603979776*c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2
))*x)^4*b^2*f*d^2*exp(1)^6-1308622848*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(
2))-sqrt(-c*exp(2))*x)^4*b^2*f*d^2*exp(1)^4+1610612736*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)
+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b*f*d^3*exp(1)^7-201326592*c^2*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*e
xp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b*f*d^3*exp(1)^5+1610612736*c^2*exp(2)^3*sqrt(-c*exp
(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b*f*d^3*exp(1)^3-805306368*c^3*exp(
2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*f*d^4*exp(1)^6-402653
184*c^3*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*f*d^4*e
xp(1)^4-805306368*c^3*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2)
)*x)^4*f*d^4*exp(1)^2+134217728*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3
*g*b^4*d^2*exp(1)^6-134217728*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*g
*b^4*d^2*exp(1)^4-2415919104*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*
g*b^3*d^3*exp(1)^7+2885681152*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3
*g*b^3*d^3*exp(1)^5-973078528*c*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3
*g*b^3*d^3*exp(1)^3+8858370048*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x
)^3*g*b^2*d^4*exp(1)^6-8254390272*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2)
)*x)^3*g*b^2*d^4*exp(1)^4+2415919104*c^2*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp
(2))*x)^3*g*b^2*d^4*exp(1)^2-10468982784*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c
*exp(2))*x)^3*g*b*d^5*exp(1)^5+4160749568*c^3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-
c*exp(2))*x)^3*g*b*d^5*exp(1)^3+268435456*c^3*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-
c*exp(2))*x)^3*g*b*d^5*exp(1)+4026531840*c^4*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c
*exp(2))*x)^3*g*d^6*exp(1)^4+1073741824*c^4*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*
exp(2))*x)^3*g*d^6*exp(1)^2-1073741824*c^4*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*e
xp(2))*x)^3*g*d^6-134217728*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b^4
*f*d*exp(1)^7+134217728*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b^4*f*d
*exp(1)^5-805306368*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b^3*f*d^2
*exp(1)^8+2550136832*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b^3*f*d^
2*exp(1)^6-1241513984*c*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b^3*f*d
^2*exp(1)^4-4429185024*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b^2*
f*d^3*exp(1)^5+1409286144*c^2*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b
^2*f*d^3*exp(1)^3+2415919104*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^
3*b*f*d^4*exp(1)^6+3892314112*c^3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)
^3*b*f*d^4*exp(1)^4-268435456*c^3*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)
^3*b*f*d^4*exp(1)^2-1610612736*c^4*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x
)^3*f*d^5*exp(1)^5-1879048192*c^4*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)
^3*f*d^5*exp(1)^3-536870912*c^4*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3
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)*x)^2*g*b^4*d^3*exp(1)^9+1610612736*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))
-sqrt(-c*exp(2))*x)^2*g*b^4*d^3*exp(1)^7-1207959552*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^
2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*b^4*d^3*exp(1)^5+402653184*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b
*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*b^4*d^3*exp(1)^3+4026531840*c*exp(2)*sqrt(-c*exp(2))*(sqr
t(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*b^3*d^4*exp(1)^8-4831838208*c*exp(2)^2*sqr
t(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*b^3*d^4*exp(1)^6+14092861
44*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*b^3*d^4*
exp(1)^4-100663296*c*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))
*x)^2*g*b^3*d^4*exp(1)^2-7247757312*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2)
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c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*b^2*d^5*exp(1)^5+5838471168*c^2*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d
*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*b^2*d^5*exp(1)^3-2415919104*c^2*exp(2)^4*sqrt(-c
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+6442450944*c^3*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2
*g*b*d^6*exp(1)^4-7650410496*c^3*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqr
t(-c*exp(2))*x)^2*g*b*d^6*exp(1)^2+1610612736*c^3*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-
c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*b*d^6-1610612736*c^4*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+
c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*d^7*exp(1)^5-4026531840*c^4*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp
(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*d^7*exp(1)^3+1610612736*c^4*exp(2)^3*sqrt(-c*exp(2))
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xp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^4*f*d^2*exp(1)
^6+402653184*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^
4*f*d^2*exp(1)^4-2415919104*c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-
c*exp(2))*x)^2*b^3*f*d^3*exp(1)^7+2013265920*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x
^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^3*f*d^3*exp(1)^5-100663296*c*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*
exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^3*f*d^3*exp(1)^3+5637144576*c^2*exp(2)^2*sqrt(-c*exp(2))*(sq
rt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*f*d^4*exp(1)^6-2013265920*c^2*exp(2)^3*
sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*f*d^4*exp(1)^4-60397
9776*c^2*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*f*
d^4*exp(1)^2-7247757312*c^3*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*
exp(2))*x)^2*b*f*d^5*exp(1)^5+402653184*c^3*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*
exp(2))-sqrt(-c*exp(2))*x)^2*b*f*d^5*exp(1)^3+805306368*c^3*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp
(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*f*d^5*exp(1)+3221225472*c^4*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d
*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*f*d^6*exp(1)^4+805306368*c^4*exp(2)^3*sqrt(-c*exp(
2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*f*d^6*exp(1)^2-100663296*exp(2)^2*(s
qrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^5*d^3*exp(1)^9+251658240*exp(2)^3*(sqrt(
-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^5*d^3*exp(1)^7-201326592*exp(2)^4*(sqrt(-b*d
*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^5*d^3*exp(1)^5+50331648*exp(2)^5*(sqrt(-b*d*exp(
1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^5*d^3*exp(1)^3-402653184*c*exp(2)*(sqrt(-b*d*exp(1)-b
*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^4*d^4*exp(1)^10+100663296*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*
x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^4*d^4*exp(1)^8+905969664*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*
exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^4*d^4*exp(1)^6-905969664*c*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*ex
p(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^4*d^4*exp(1)^4+301989888*c*exp(2)^5*(sqrt(-b*d*exp(1)-b*x*exp(
2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^4*d^4*exp(1)^2+1610612736*c^2*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2
)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^3*d^5*exp(1)^9+2717908992*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(
2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^3*d^5*exp(1)^7-8606711808*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp
(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^3*d^5*exp(1)^5+5939134464*c^2*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*ex
p(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^3*d^5*exp(1)^3-1409286144*c^2*exp(2)^5*(sqrt(-b*d*exp(1)-b*x*e
xp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^3*d^5*exp(1)-2415919104*c^3*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(
2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^2*d^6*exp(1)^8-7751073792*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp
(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^2*d^6*exp(1)^6+14294188032*c^3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*e
xp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^2*d^6*exp(1)^4-6442450944*c^3*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*
exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^2*d^6*exp(1)^2+805306368*c^3*exp(2)^5*(sqrt(-b*d*exp(1)-b*x*
exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b^2*d^6+1610612736*c^4*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d
^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b*d^7*exp(1)^7+7449083904*c^4*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^
2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b*d^7*exp(1)^5-7650410496*c^4*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2
-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b*d^7*exp(1)^3+1610612736*c^4*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-
c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*b*d^7*exp(1)-402653184*c^5*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*
exp(2))-sqrt(-c*exp(2))*x)*g*d^8*exp(1)^6-2415919104*c^5*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp
(2))-sqrt(-c*exp(2))*x)*g*d^8*exp(1)^4+805306368*c^5*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))
-sqrt(-c*exp(2))*x)*g*d^8*exp(1)^2+50331648*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*
exp(2))*x)*b^5*f*d^2*exp(1)^8-100663296*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(
2))*x)*b^5*f*d^2*exp(1)^6+50331648*exp(2)^5*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x
)*b^5*f*d^2*exp(1)^4+603979776*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*
b^4*f*d^3*exp(1)^9-603979776*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^
4*f*d^3*exp(1)^7-201326592*c*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*
f*d^3*exp(1)^5+201326592*c*exp(2)^5*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*f*
d^3*exp(1)^3-3019898880*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*f
*d^4*exp(1)^8+3070230528*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*
f*d^4*exp(1)^6+100663296*c^2*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*
f*d^4*exp(1)^4-402653184*c^2*exp(2)^5*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*
f*d^4*exp(1)^2+5435817984*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2
*f*d^5*exp(1)^7-3724541952*c^3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^
2*f*d^5*exp(1)^5-603979776*c^3*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^
2*f*d^5*exp(1)^3+402653184*c^3*exp(2)^5*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^
2*f*d^5*exp(1)-4227858432*c^4*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*f
*d^6*exp(1)^6+402653184*c^4*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*f*d
^6*exp(1)^4+805306368*c^4*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*f*d^6
*exp(1)^2+1207959552*c^5*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*f*d^7*ex
p(1)^5+805306368*c^5*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*f*d^7*exp(1)
^3+100663296*exp(2)^2*sqrt(-c*exp(2))*g*b^5*d^4*exp(1)^8-251658240*exp(2)^3*sqrt(-c*exp(2))*g*b^5*d^4*exp(1)^6
+201326592*exp(2)^4*sqrt(-c*exp(2))*g*b^5*d^4*exp(1)^4-50331648*exp(2)^5*sqrt(-c*exp(2))*g*b^5*d^4*exp(1)^2+67
1088640*c*exp(2)*sqrt(-c*exp(2))*g*b^4*d^5*exp(1)^9-1979711488*c*exp(2)^2*sqrt(-c*exp(2))*g*b^4*d^5*exp(1)^7+2
315255808*c*exp(2)^3*sqrt(-c*exp(2))*g*b^4*d^5*exp(1)^5-1241513984*c*exp(2)^4*sqrt(-c*exp(2))*g*b^4*d^5*exp(1)
^3+234881024*c*exp(2)^5*sqrt(-c*exp(2))*g*b^4*d^5*exp(1)-2684354560*c^2*exp(2)*sqrt(-c*exp(2))*g*b^3*d^6*exp(1
)^8+5603590144*c^2*exp(2)^2*sqrt(-c*exp(2))*g*b^3*d^6*exp(1)^6-4278190080*c^2*exp(2)^3*sqrt(-c*exp(2))*g*b^3*d
^6*exp(1)^4+1442840576*c^2*exp(2)^4*sqrt(-c*exp(2))*g*b^3*d^6*exp(1)^2-134217728*c^2*exp(2)^5*sqrt(-c*exp(2))*
g*b^3*d^6+4026531840*c^3*exp(2)*sqrt(-c*exp(2))*g*b^2*d^7*exp(1)^7-5939134464*c^3*exp(2)^2*sqrt(-c*exp(2))*g*b
^2*d^7*exp(1)^5+2617245696*c^3*exp(2)^3*sqrt(-c*exp(2))*g*b^2*d^7*exp(1)^3-402653184*c^3*exp(2)^4*sqrt(-c*exp(
2))*g*b^2*d^7*exp(1)-2684354560*c^4*exp(2)*sqrt(-c*exp(2))*g*b*d^8*exp(1)^6+2483027968*c^4*exp(2)^2*sqrt(-c*ex
p(2))*g*b*d^8*exp(1)^4-402653184*c^4*exp(2)^3*sqrt(-c*exp(2))*g*b*d^8*exp(1)^2+671088640*c^5*exp(2)*sqrt(-c*ex
p(2))*g*d^9*exp(1)^5-268435456*c^5*exp(2)^2*sqrt(-c*exp(2))*g*d^9*exp(1)^3-50331648*exp(2)^3*sqrt(-c*exp(2))*b
^5*f*d^3*exp(1)^7+100663296*exp(2)^4*sqrt(-c*exp(2))*b^5*f*d^3*exp(1)^5-50331648*exp(2)^5*sqrt(-c*exp(2))*b^5*
f*d^3*exp(1)^3-268435456*c*exp(2)*sqrt(-c*exp(2))*b^4*f*d^4*exp(1)^10+469762048*c*exp(2)^2*sqrt(-c*exp(2))*b^4
*f*d^4*exp(1)^8-201326592*c*exp(2)^3*sqrt(-c*exp(2))*b^4*f*d^4*exp(1)^6-67108864*c*exp(2)^4*sqrt(-c*exp(2))*b^
4*f*d^4*exp(1)^4+67108864*c*exp(2)^5*sqrt(-c*exp(2))*b^4*f*d^4*exp(1)^2+1073741824*c^2*exp(2)*sqrt(-c*exp(2))*
b^3*f*d^5*exp(1)^9-1275068416*c^2*exp(2)^2*sqrt(-c*exp(2))*b^3*f*d^5*exp(1)^7+150994944*c^2*exp(2)^3*sqrt(-c*e
xp(2))*b^3*f*d^5*exp(1)^5+167772160*c^2*exp(2)^4*sqrt(-c*exp(2))*b^3*f*d^5*exp(1)^3-67108864*c^2*exp(2)^5*sqrt
(-c*exp(2))*b^3*f*d^5*exp(1)-1610612736*c^3*exp(2)*sqrt(-c*exp(2))*b^2*f*d^6*exp(1)^8+1006632960*c^3*exp(2)^2*
sqrt(-c*exp(2))*b^2*f*d^6*exp(1)^6+503316480*c^3*exp(2)^3*sqrt(-c*exp(2))*b^2*f*d^6*exp(1)^4-201326592*c^3*exp
(2)^4*sqrt(-c*exp(2))*b^2*f*d^6*exp(1)^2+1073741824*c^4*exp(2)*sqrt(-c*exp(2))*b*f*d^7*exp(1)^7-67108864*c^4*e
xp(2)^2*sqrt(-c*exp(2))*b*f*d^7*exp(1)^5-402653184*c^4*exp(2)^3*sqrt(-c*exp(2))*b*f*d^7*exp(1)^3-268435456*c^5
*exp(2)*sqrt(-c*exp(2))*f*d^8*exp(1)^6-134217728*c^5*exp(2)^2*sqrt(-c*exp(2))*f*d^8*exp(1)^4)/(805306368*b^2*d
^2*exp(1)^9-1610612736*exp(2)*b^2*d^2*exp(1)^7+805306368*exp(2)^2*b^2*d^2*exp(1)^5-1610612736*c*b*d^3*exp(1)^8
+3221225472*c*exp(2)*b*d^3*exp(1)^6-1610612736*c*exp(2)^2*b*d^3*exp(1)^4+805306368*c^2*d^4*exp(1)^7-1610612736
*c^2*exp(2)*d^4*exp(1)^5+805306368*c^2*exp(2)^2*d^4*exp(1)^3)/((sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2)
)-sqrt(-c*exp(2))*x)^2*exp(1)-2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2
))*x)*d+b*d*exp(1)^2-exp(2)*b*d-c*d^2*exp(1))^3+(4*exp(2)^2*g*b^3*d*exp(1)^2-2*exp(2)^3*g*b^3*d-16*c*exp(2)*g*
b^2*d^2*exp(1)^3+4*c*exp(2)^2*g*b^2*d^2*exp(1)+32*c^2*exp(2)*g*b*d^3*exp(1)^2-8*c^2*exp(2)^2*g*b*d^3-16*c^3*ex
p(2)*g*d^4*exp(1)-2*exp(2)^3*b^3*f*exp(1)+8*c*exp(2)^2*b^2*f*d*exp(1)^2+4*c*exp(2)^3*b^2*f*d-24*c^2*exp(2)^2*b
*f*d^2*exp(1)+16*c^3*exp(2)^2*f*d^3)/32/(b^2*d^2*exp(1)^6-2*exp(2)*b^2*d^2*exp(1)^4+exp(2)^2*b^2*d^2*exp(1)^2-
2*c*b*d^3*exp(1)^5+4*c*exp(2)*b*d^3*exp(1)^3-2*c*exp(2)^2*b*d^3*exp(1)+c^2*d^4*exp(1)^4-2*c^2*exp(2)*d^4*exp(1
)^2+c^2*exp(2)^2*d^4)/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(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-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 = 0.07, size = 128, normalized size = 0.93 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (5 b \,e^{2} g x -8 c d e g x -2 c \,e^{2} f x +2 b d e g +3 b \,e^{2} f -2 c \,d^{2} g -8 c d e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{15 \left (e x +d \right )^{3} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4,x)
[Out]
-2/15*(c*e*x+b*e-c*d)*(5*b*e^2*g*x-8*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g+3*b*e^2*f-2*c*d^2*g-8*c*d*e*f)*(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^3/e^2/(b^2*e^2-4*b*c*d*e+4*c^2*d^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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4,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 = 4.29, size = 1022, normalized size = 7.46 \begin {gather*} \frac {\left (\frac {20\,g\,b^2\,c\,e^2-64\,g\,b\,c^2\,d\,e+12\,f\,b\,c^2\,e^2+48\,g\,c^3\,d^2-16\,f\,c^3\,d\,e}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {d\,\left (\frac {4\,c^2\,\left (7\,b\,e\,g-12\,c\,d\,g+2\,c\,e\,f\right )}{15\,e\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {8\,c^3\,d\,g}{15\,e\,{\left (b\,e-2\,c\,d\right )}^3}\right )}{e}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x}-\frac {\left (\frac {2\,f\,\left (b\,e-c\,d\right )}{5\,b\,e^2-10\,c\,d\,e}-\frac {d\,\left (\frac {2\,b\,e\,g-2\,c\,d\,g+2\,c\,e\,f}{5\,b\,e^2-10\,c\,d\,e}-\frac {2\,c\,d\,g}{5\,b\,e^2-10\,c\,d\,e}\right )}{e}\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 {d\,\left (\frac {4\,c^2\,e\,f-8\,c^2\,d\,g+6\,b\,c\,e\,g}{5\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}-\frac {4\,c^2\,d\,g}{5\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )}{e}-\frac {2\,b\,\left (b\,e\,g-2\,c\,d\,g+c\,e\,f\right )}{5\,\left (3\,b\,e^2-6\,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 )}^2}+\frac {\left (\frac {d\,\left (\frac {8\,c^2\,\left (6\,b\,e\,g-11\,c\,d\,g+c\,e\,f\right )}{15\,e\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {8\,c^3\,d\,g}{15\,e\,{\left (b\,e-2\,c\,d\right )}^3}\right )}{e}-\frac {8\,c\,\left (b\,e-c\,d\right )\,\left (5\,b\,e\,g-10\,c\,d\,g+c\,e\,f\right )}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x}+\frac {\left (\frac {d\,\left (\frac {4\,c\,\left (4\,b\,e\,g-7\,c\,d\,g+c\,e\,f\right )}{5\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}-\frac {4\,c^2\,d\,g}{5\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )}{e}-\frac {4\,\left (b\,e-c\,d\right )\,\left (3\,b\,e\,g-6\,c\,d\,g+c\,e\,f\right )}{5\,e\,\left (3\,b\,e^2-6\,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 )}^2}+\frac {\left (\frac {d\,\left (\frac {8\,c^3\,e\,f-24\,c^3\,d\,g+16\,b\,c^2\,e\,g}{15\,e\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {8\,c^3\,d\,g}{15\,e\,{\left (b\,e-2\,c\,d\right )}^3}\right )}{e}-\frac {2\,b\,c\,\left (3\,b\,e\,g-6\,c\,d\,g+2\,c\,e\,f\right )}{15\,e\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x}-\frac {\left (\frac {d\,\left (\frac {8\,c^3\,e\,f-64\,c^3\,d\,g+36\,b\,c^2\,e\,g}{15\,e\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {8\,c^3\,d\,g}{15\,e\,{\left (b\,e-2\,c\,d\right )}^3}\right )}{e}-\frac {4\,b\,c\,\left (4\,b\,e\,g-8\,c\,d\,g+c\,e\,f\right )}{15\,e\,{\left (b\,e-2\,c\,d\right )}^3}\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)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^4,x)
[Out]
(((48*c^3*d^2*g - 16*c^3*d*e*f + 12*b*c^2*e^2*f + 20*b^2*c*e^2*g - 64*b*c^2*d*e*g)/(15*e^2*(b*e - 2*c*d)^3) -
(d*((4*c^2*(7*b*e*g - 12*c*d*g + 2*c*e*f))/(15*e*(b*e - 2*c*d)^3) - (8*c^3*d*g)/(15*e*(b*e - 2*c*d)^3)))/e)*(c
*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((2*f*(b*e - c*d))/(5*b*e^2 - 10*c*d*e) - (d*((2*b*e*g
- 2*c*d*g + 2*c*e*f)/(5*b*e^2 - 10*c*d*e) - (2*c*d*g)/(5*b*e^2 - 10*c*d*e)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e -
b*e^2*x)^(1/2))/(d + e*x)^3 - (((d*((4*c^2*e*f - 8*c^2*d*g + 6*b*c*e*g)/(5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d))
- (4*c^2*d*g)/(5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d))))/e - (2*b*(b*e*g - 2*c*d*g + c*e*f))/(5*(3*b*e^2 - 6*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)^2 + (((d*((8*c^2*(6*b*e*g - 11*c*d*g
+ c*e*f))/(15*e*(b*e - 2*c*d)^3) - (8*c^3*d*g)/(15*e*(b*e - 2*c*d)^3)))/e - (8*c*(b*e - c*d)*(5*b*e*g - 10*c*
d*g + c*e*f))/(15*e^2*(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) + (((d*((4*c*(4
*b*e*g - 7*c*d*g + c*e*f))/(5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)) - (4*c^2*d*g)/(5*(3*b*e^2 - 6*c*d*e)*(b*e - 2
*c*d))))/e - (4*(b*e - c*d)*(3*b*e*g - 6*c*d*g + c*e*f))/(5*e*(3*b*e^2 - 6*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)^2 + (((d*((8*c^3*e*f - 24*c^3*d*g + 16*b*c^2*e*g)/(15*e*(b*e - 2*c*
d)^3) - (8*c^3*d*g)/(15*e*(b*e - 2*c*d)^3)))/e - (2*b*c*(3*b*e*g - 6*c*d*g + 2*c*e*f))/(15*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) - (((d*((8*c^3*e*f - 64*c^3*d*g + 36*b*c^2*e*g)/(15*e*
(b*e - 2*c*d)^3) - (8*c^3*d*g)/(15*e*(b*e - 2*c*d)^3)))/e - (4*b*c*(4*b*e*g - 8*c*d*g + c*e*f))/(15*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)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g 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)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**4,x)
[Out]
Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**4, x)
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