3.14.64 \(\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=430 \[ -\frac {e \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^4}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (33 a e+2 b d)+35 b^2 e^2+8 c^2 d^2\right )}{240 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{40 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}+\frac {e \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{9/2}}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )} \]

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Rubi [A]  time = 0.65, antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {834, 806, 720, 724, 206} \begin {gather*} -\frac {e \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^4}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (33 a e+2 b d)+35 b^2 e^2+8 c^2 d^2\right )}{240 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{40 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}+\frac {e \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{9/2}}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^6,x]

[Out]

-((b^2 - 4*a*c)*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x
+ c*x^2])/(128*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^2) + ((2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(5*(c*d^2 - b*
d*e + a*e^2)*(d + e*x)^5) + ((8*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(2*b*d + 5*a*e))*(a + b*x + c*x^2)^(3/2))/(40*(c*d
^2 - b*d*e + a*e^2)^2*(d + e*x)^4) + ((2*c*d - b*e)*(8*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(2*b*d + 33*a*e))*(a + b*x
+ c*x^2)^(3/2))/(240*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^3) + ((b^2 - 4*a*c)^2*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*
c*e*(6*b*d + a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2
])])/(256*(c*d^2 - b*d*e + a*e^2)^(9/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 720

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

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), 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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx &=\frac {(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {\int \frac {\left (\frac {1}{2} \left (-4 b c d+7 b^2 e-20 a c e\right )-2 c (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx}{5 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {\left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}+\frac {\int \frac {\left (\frac {1}{4} \left (-64 b^2 c d e+224 a c^2 d e+35 b^3 e^2+4 b c \left (2 c d^2-33 a e^2\right )\right )+\frac {1}{2} c \left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx}{20 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {\left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}+\frac {(2 c d-b e) \left (8 c^2 d^2+35 b^2 e^2-4 c e (2 b d+33 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}-\frac {\left (\left (b^2-4 a c\right ) e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx}{32 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {\left (b^2-4 a c\right ) e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}+\frac {(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {\left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}+\frac {(2 c d-b e) \left (8 c^2 d^2+35 b^2 e^2-4 c e (2 b d+33 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}+\frac {\left (\left (b^2-4 a c\right )^2 e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{256 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {\left (b^2-4 a c\right ) e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}+\frac {(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {\left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}+\frac {(2 c d-b e) \left (8 c^2 d^2+35 b^2 e^2-4 c e (2 b d+33 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}-\frac {\left (\left (b^2-4 a c\right )^2 e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{128 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {\left (b^2-4 a c\right ) e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}+\frac {(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {\left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}+\frac {(2 c d-b e) \left (8 c^2 d^2+35 b^2 e^2-4 c e (2 b d+33 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}+\frac {\left (b^2-4 a c\right )^2 e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c d^2-b d e+a e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 3.11, size = 401, normalized size = 0.93 \begin {gather*} \frac {\frac {(a+x (b+c x))^{3/2} (2 c d-b e) \left (-4 c e (33 a e+2 b d)+35 b^2 e^2+8 c^2 d^2\right )}{48 (d+e x)^3 \left (e (a e-b d)+c d^2\right )^2}+\frac {(a+x (b+c x))^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{8 (d+e x)^4 \left (e (a e-b d)+c d^2\right )}-\frac {5 e \left (b^2-4 a c\right ) \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (\left (b^2-4 a c\right ) (d+e x)^2 \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )+2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2} (-2 a e+b (d-e x)+2 c d x)\right )}{256 (d+e x)^2 \left (e (a e-b d)+c d^2\right )^{7/2}}+\frac {(a+x (b+c x))^{3/2} (2 c d-b e)}{(d+e x)^5}}{5 \left (e (a e-b d)+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^6,x]

[Out]

(((2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(d + e*x)^5 + ((8*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(2*b*d + 5*a*e))*(a + x
*(b + c*x))^(3/2))/(8*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^4) + ((2*c*d - b*e)*(8*c^2*d^2 + 35*b^2*e^2 - 4*c*e
*(2*b*d + 33*a*e))*(a + x*(b + c*x))^(3/2))/(48*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)^3) - (5*(b^2 - 4*a*c)*e
*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]*(-2*a*
e + 2*c*d*x + b*(d - e*x)) + (b^2 - 4*a*c)*(d + e*x)^2*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^
2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(256*(c*d^2 + e*(-(b*d) + a*e))^(7/2)*(d + e*x)^2))/(5*(c*d^2
+ e*(-(b*d) + a*e)))

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IntegrateAlgebraic [F]  time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^6,x]

[Out]

$Aborted

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fricas [B]  time = 86.81, size = 6858, normalized size = 15.95

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

[-1/7680*(15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^7*e - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6*e^2
+ (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^5*e^3 + (24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*e
^6 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^7 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e^8)*
x^5 + 5*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3*e^5 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2*e^6 + (
7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d*e^7)*x^4 + 10*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^
4*e^4 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3*e^5 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*
d^2*e^6)*x^3 + 10*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5*e^3 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d
^4*e^4 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^3*e^5)*x^2 + 5*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*
a^2*c^4)*d^6*e^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5*e^3 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 6
4*a^3*c^3)*d^4*e^4)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2
- 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*
c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 4*(1280*a*c^5*
d^9 - 384*a^5*b*e^9 - 40*(9*b^3*c^3 + 100*a*b*c^4)*d^8*e + 16*(45*b^4*c^2 + 485*a*b^2*c^3 - 92*a^2*c^4)*d^7*e^
2 - (465*b^5*c + 9520*a*b^3*c^2 - 272*a^2*b*c^3)*d^6*e^3 + (105*b^6 + 5690*a*b^4*c + 8496*a^2*b^2*c^2 - 3616*a
^3*c^3)*d^5*e^4 - 5*(263*a*b^5 + 1984*a^2*b^3*c - 48*a^3*b*c^2)*d^4*e^5 + 2*(1657*a^2*b^4 + 3024*a^3*b^2*c - 5
28*a^4*c^2)*d^3*e^6 - 8*(449*a^3*b^3 + 132*a^4*b*c)*d^2*e^7 + 48*(39*a^4*b^2 - 4*a^5*c)*d*e^8 + (128*c^6*d^7*e
^2 - 448*b*c^5*d^6*e^3 + 32*(13*b^2*c^4 + 32*a*c^5)*d^5*e^4 + 80*(b^3*c^3 - 32*a*b*c^4)*d^4*e^5 - 2*(233*b^4*c
^2 - 1704*a*b^2*c^3 + 848*a^2*c^4)*d^3*e^6 + (395*b^5*c - 2552*a*b^3*c^2 + 2544*a^2*b*c^3)*d^2*e^7 - (105*b^6
- 470*a*b^4*c - 672*a^2*b^2*c^2 + 2592*a^3*c^3)*d*e^8 + (105*a*b^5 - 760*a^2*b^3*c + 1296*a^3*b*c^2)*e^9)*x^4
+ 2*(320*c^6*d^8*e - 1152*b*c^5*d^7*e^2 + 128*(9*b^2*c^4 + 20*a*c^5)*d^6*e^3 + 32*(3*b^3*c^3 - 208*a*b*c^4)*d^
5*e^4 - 5*(219*b^4*c^2 - 1688*a*b^2*c^3 + 560*a^2*c^4)*d^4*e^5 + 4*(231*b^5*c - 1448*a*b^3*c^2 + 976*a^2*b*c^3
)*d^3*e^6 - (245*b^6 - 958*a*b^4*c - 2304*a^2*b^2*c^2 + 5280*a^3*c^3)*d^2*e^7 + 8*(35*a*b^5 - 236*a^2*b^3*c +
336*a^3*b*c^2)*d*e^8 - (35*a^2*b^4 - 216*a^3*b^2*c + 240*a^4*c^2)*e^9)*x^3 + 2*(640*c^6*d^9 - 2400*b*c^5*d^8*e
+ 24*(111*b^2*c^4 + 212*a*c^5)*d^7*e^2 - 12*(17*b^3*c^3 + 1164*a*b*c^4)*d^6*e^3 - 3*(649*b^4*c^2 - 5696*a*b^2
*c^3 + 752*a^2*c^4)*d^5*e^4 + 15*(113*b^5*c - 700*a*b^3*c^2 + 96*a^2*b*c^3)*d^4*e^5 - 8*(56*b^6 - 171*a*b^4*c
- 855*a^2*b^2*c^2 + 892*a^3*c^3)*d^3*e^6 + 3*(203*a*b^5 - 1220*a^2*b^3*c + 928*a^3*b*c^2)*d^2*e^7 - 3*(63*a^2*
b^4 - 352*a^3*b^2*c + 144*a^4*c^2)*d*e^8 + 4*(7*a^3*b^3 - 36*a^4*b*c)*e^9)*x^2 + 2*(640*b*c^5*d^9 - 440*(7*b^2
*c^4 - 4*a*c^5)*d^8*e + 8*(665*b^3*c^3 - 244*a*b*c^4)*d^7*e^2 - (4565*b^4*c^2 + 1128*a*b^2*c^3 + 4880*a^2*c^4)
*d^6*e^3 + 8*(260*b^5*c + 337*a*b^3*c^2 + 1548*a^2*b*c^3)*d^5*e^4 - 5*(79*b^6 + 454*a*b^4*c + 1632*a^2*b^2*c^2
+ 1760*a^3*c^3)*d^4*e^5 + 4*(171*a*b^5 + 622*a^2*b^3*c + 2616*a^3*b*c^2)*d^3*e^6 - 3*(139*a^2*b^4 + 1272*a^3*
b^2*c + 880*a^4*c^2)*d^2*e^7 + 8*(19*a^3*b^3 + 276*a^4*b*c)*d*e^8 - 24*(a^4*b^2 + 20*a^5*c)*e^9)*x)*sqrt(c*x^2
+ b*x + a))/(c^5*d^15 - 5*b*c^4*d^14*e - 5*a^4*b*d^6*e^9 + a^5*d^5*e^10 + 5*(2*b^2*c^3 + a*c^4)*d^13*e^2 - 10
*(b^3*c^2 + 2*a*b*c^3)*d^12*e^3 + 5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^11*e^4 - (b^5 + 20*a*b^3*c + 30*a^2*b*
c^2)*d^10*e^5 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^9*e^6 - 10*(a^2*b^3 + 2*a^3*b*c)*d^8*e^7 + 5*(2*a^3*b^2
+ a^4*c)*d^7*e^8 + (c^5*d^10*e^5 - 5*b*c^4*d^9*e^6 - 5*a^4*b*d*e^14 + a^5*e^15 + 5*(2*b^2*c^3 + a*c^4)*d^8*e^7
- 10*(b^3*c^2 + 2*a*b*c^3)*d^7*e^8 + 5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^6*e^9 - (b^5 + 20*a*b^3*c + 30*a^2
*b*c^2)*d^5*e^10 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^4*e^11 - 10*(a^2*b^3 + 2*a^3*b*c)*d^3*e^12 + 5*(2*a^3
*b^2 + a^4*c)*d^2*e^13)*x^5 + 5*(c^5*d^11*e^4 - 5*b*c^4*d^10*e^5 - 5*a^4*b*d^2*e^13 + a^5*d*e^14 + 5*(2*b^2*c^
3 + a*c^4)*d^9*e^6 - 10*(b^3*c^2 + 2*a*b*c^3)*d^8*e^7 + 5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^7*e^8 - (b^5 + 2
0*a*b^3*c + 30*a^2*b*c^2)*d^6*e^9 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^5*e^10 - 10*(a^2*b^3 + 2*a^3*b*c)*d^
4*e^11 + 5*(2*a^3*b^2 + a^4*c)*d^3*e^12)*x^4 + 10*(c^5*d^12*e^3 - 5*b*c^4*d^11*e^4 - 5*a^4*b*d^3*e^12 + a^5*d^
2*e^13 + 5*(2*b^2*c^3 + a*c^4)*d^10*e^5 - 10*(b^3*c^2 + 2*a*b*c^3)*d^9*e^6 + 5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^
3)*d^8*e^7 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^7*e^8 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^6*e^9 - 10*(a^2
*b^3 + 2*a^3*b*c)*d^5*e^10 + 5*(2*a^3*b^2 + a^4*c)*d^4*e^11)*x^3 + 10*(c^5*d^13*e^2 - 5*b*c^4*d^12*e^3 - 5*a^4
*b*d^4*e^11 + a^5*d^3*e^12 + 5*(2*b^2*c^3 + a*c^4)*d^11*e^4 - 10*(b^3*c^2 + 2*a*b*c^3)*d^10*e^5 + 5*(b^4*c + 6
*a*b^2*c^2 + 2*a^2*c^3)*d^9*e^6 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^8*e^7 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c
^2)*d^7*e^8 - 10*(a^2*b^3 + 2*a^3*b*c)*d^6*e^9 + 5*(2*a^3*b^2 + a^4*c)*d^5*e^10)*x^2 + 5*(c^5*d^14*e - 5*b*c^4
*d^13*e^2 - 5*a^4*b*d^5*e^10 + a^5*d^4*e^11 + 5*(2*b^2*c^3 + a*c^4)*d^12*e^3 - 10*(b^3*c^2 + 2*a*b*c^3)*d^11*e
^4 + 5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^10*e^5 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^9*e^6 + 5*(a*b^4 + 6*a
^2*b^2*c + 2*a^3*c^2)*d^8*e^7 - 10*(a^2*b^3 + 2*a^3*b*c)*d^7*e^8 + 5*(2*a^3*b^2 + a^4*c)*d^6*e^9)*x), 1/3840*(
15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^7*e - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6*e^2 + (7*b^6 -
60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^5*e^3 + (24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*e^6 - 24*(b
^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^7 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e^8)*x^5 + 5*(2
4*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3*e^5 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2*e^6 + (7*b^6 - 60
*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d*e^7)*x^4 + 10*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4*e^4 - 24
*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3*e^5 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^2*e^6)*x
^3 + 10*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5*e^3 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4*e^4 + (
7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^3*e^5)*x^2 + 5*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d
^6*e^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5*e^3 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)
*d^4*e^4)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d
- 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d
*e + a*b*e^2)*x)) + 2*(1280*a*c^5*d^9 - 384*a^5*b*e^9 - 40*(9*b^3*c^3 + 100*a*b*c^4)*d^8*e + 16*(45*b^4*c^2 +
485*a*b^2*c^3 - 92*a^2*c^4)*d^7*e^2 - (465*b^5*c + 9520*a*b^3*c^2 - 272*a^2*b*c^3)*d^6*e^3 + (105*b^6 + 5690*a
*b^4*c + 8496*a^2*b^2*c^2 - 3616*a^3*c^3)*d^5*e^4 - 5*(263*a*b^5 + 1984*a^2*b^3*c - 48*a^3*b*c^2)*d^4*e^5 + 2*
(1657*a^2*b^4 + 3024*a^3*b^2*c - 528*a^4*c^2)*d^3*e^6 - 8*(449*a^3*b^3 + 132*a^4*b*c)*d^2*e^7 + 48*(39*a^4*b^2
- 4*a^5*c)*d*e^8 + (128*c^6*d^7*e^2 - 448*b*c^5*d^6*e^3 + 32*(13*b^2*c^4 + 32*a*c^5)*d^5*e^4 + 80*(b^3*c^3 -
32*a*b*c^4)*d^4*e^5 - 2*(233*b^4*c^2 - 1704*a*b^2*c^3 + 848*a^2*c^4)*d^3*e^6 + (395*b^5*c - 2552*a*b^3*c^2 + 2
544*a^2*b*c^3)*d^2*e^7 - (105*b^6 - 470*a*b^4*c - 672*a^2*b^2*c^2 + 2592*a^3*c^3)*d*e^8 + (105*a*b^5 - 760*a^2
*b^3*c + 1296*a^3*b*c^2)*e^9)*x^4 + 2*(320*c^6*d^8*e - 1152*b*c^5*d^7*e^2 + 128*(9*b^2*c^4 + 20*a*c^5)*d^6*e^3
+ 32*(3*b^3*c^3 - 208*a*b*c^4)*d^5*e^4 - 5*(219*b^4*c^2 - 1688*a*b^2*c^3 + 560*a^2*c^4)*d^4*e^5 + 4*(231*b^5*
c - 1448*a*b^3*c^2 + 976*a^2*b*c^3)*d^3*e^6 - (245*b^6 - 958*a*b^4*c - 2304*a^2*b^2*c^2 + 5280*a^3*c^3)*d^2*e^
7 + 8*(35*a*b^5 - 236*a^2*b^3*c + 336*a^3*b*c^2)*d*e^8 - (35*a^2*b^4 - 216*a^3*b^2*c + 240*a^4*c^2)*e^9)*x^3 +
2*(640*c^6*d^9 - 2400*b*c^5*d^8*e + 24*(111*b^2*c^4 + 212*a*c^5)*d^7*e^2 - 12*(17*b^3*c^3 + 1164*a*b*c^4)*d^6
*e^3 - 3*(649*b^4*c^2 - 5696*a*b^2*c^3 + 752*a^2*c^4)*d^5*e^4 + 15*(113*b^5*c - 700*a*b^3*c^2 + 96*a^2*b*c^3)*
d^4*e^5 - 8*(56*b^6 - 171*a*b^4*c - 855*a^2*b^2*c^2 + 892*a^3*c^3)*d^3*e^6 + 3*(203*a*b^5 - 1220*a^2*b^3*c + 9
28*a^3*b*c^2)*d^2*e^7 - 3*(63*a^2*b^4 - 352*a^3*b^2*c + 144*a^4*c^2)*d*e^8 + 4*(7*a^3*b^3 - 36*a^4*b*c)*e^9)*x
^2 + 2*(640*b*c^5*d^9 - 440*(7*b^2*c^4 - 4*a*c^5)*d^8*e + 8*(665*b^3*c^3 - 244*a*b*c^4)*d^7*e^2 - (4565*b^4*c^
2 + 1128*a*b^2*c^3 + 4880*a^2*c^4)*d^6*e^3 + 8*(260*b^5*c + 337*a*b^3*c^2 + 1548*a^2*b*c^3)*d^5*e^4 - 5*(79*b^
6 + 454*a*b^4*c + 1632*a^2*b^2*c^2 + 1760*a^3*c^3)*d^4*e^5 + 4*(171*a*b^5 + 622*a^2*b^3*c + 2616*a^3*b*c^2)*d^
3*e^6 - 3*(139*a^2*b^4 + 1272*a^3*b^2*c + 880*a^4*c^2)*d^2*e^7 + 8*(19*a^3*b^3 + 276*a^4*b*c)*d*e^8 - 24*(a^4*
b^2 + 20*a^5*c)*e^9)*x)*sqrt(c*x^2 + b*x + a))/(c^5*d^15 - 5*b*c^4*d^14*e - 5*a^4*b*d^6*e^9 + a^5*d^5*e^10 + 5
*(2*b^2*c^3 + a*c^4)*d^13*e^2 - 10*(b^3*c^2 + 2*a*b*c^3)*d^12*e^3 + 5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^11*e
^4 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^10*e^5 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^9*e^6 - 10*(a^2*b^3 +
2*a^3*b*c)*d^8*e^7 + 5*(2*a^3*b^2 + a^4*c)*d^7*e^8 + (c^5*d^10*e^5 - 5*b*c^4*d^9*e^6 - 5*a^4*b*d*e^14 + a^5*e^
15 + 5*(2*b^2*c^3 + a*c^4)*d^8*e^7 - 10*(b^3*c^2 + 2*a*b*c^3)*d^7*e^8 + 5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^
6*e^9 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^5*e^10 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^4*e^11 - 10*(a^2*b^
3 + 2*a^3*b*c)*d^3*e^12 + 5*(2*a^3*b^2 + a^4*c)*d^2*e^13)*x^5 + 5*(c^5*d^11*e^4 - 5*b*c^4*d^10*e^5 - 5*a^4*b*d
^2*e^13 + a^5*d*e^14 + 5*(2*b^2*c^3 + a*c^4)*d^9*e^6 - 10*(b^3*c^2 + 2*a*b*c^3)*d^8*e^7 + 5*(b^4*c + 6*a*b^2*c
^2 + 2*a^2*c^3)*d^7*e^8 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^6*e^9 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^5*
e^10 - 10*(a^2*b^3 + 2*a^3*b*c)*d^4*e^11 + 5*(2*a^3*b^2 + a^4*c)*d^3*e^12)*x^4 + 10*(c^5*d^12*e^3 - 5*b*c^4*d^
11*e^4 - 5*a^4*b*d^3*e^12 + a^5*d^2*e^13 + 5*(2*b^2*c^3 + a*c^4)*d^10*e^5 - 10*(b^3*c^2 + 2*a*b*c^3)*d^9*e^6 +
5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^8*e^7 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^7*e^8 + 5*(a*b^4 + 6*a^2*b^
2*c + 2*a^3*c^2)*d^6*e^9 - 10*(a^2*b^3 + 2*a^3*b*c)*d^5*e^10 + 5*(2*a^3*b^2 + a^4*c)*d^4*e^11)*x^3 + 10*(c^5*d
^13*e^2 - 5*b*c^4*d^12*e^3 - 5*a^4*b*d^4*e^11 + a^5*d^3*e^12 + 5*(2*b^2*c^3 + a*c^4)*d^11*e^4 - 10*(b^3*c^2 +
2*a*b*c^3)*d^10*e^5 + 5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^9*e^6 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^8*e^7
+ 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^7*e^8 - 10*(a^2*b^3 + 2*a^3*b*c)*d^6*e^9 + 5*(2*a^3*b^2 + a^4*c)*d^5*e
^10)*x^2 + 5*(c^5*d^14*e - 5*b*c^4*d^13*e^2 - 5*a^4*b*d^5*e^10 + a^5*d^4*e^11 + 5*(2*b^2*c^3 + a*c^4)*d^12*e^3
- 10*(b^3*c^2 + 2*a*b*c^3)*d^11*e^4 + 5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^10*e^5 - (b^5 + 20*a*b^3*c + 30*a
^2*b*c^2)*d^9*e^6 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^8*e^7 - 10*(a^2*b^3 + 2*a^3*b*c)*d^7*e^8 + 5*(2*a^3*
b^2 + a^4*c)*d^6*e^9)*x)]

________________________________________________________________________________________

giac [B]  time = 73.16, size = 10103, normalized size = 23.50

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

1/128*(24*b^4*c^2*d^2*e - 192*a*b^2*c^3*d^2*e + 384*a^2*c^4*d^2*e - 24*b^5*c*d*e^2 + 192*a*b^3*c^2*d*e^2 - 384
*a^2*b*c^3*d*e^2 + 7*b^6*e^3 - 60*a*b^4*c*e^3 + 144*a^2*b^2*c^2*e^3 - 64*a^3*c^3*e^3)*arctan(-((sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^
2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*
e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*sq
rt(-c*d^2 + b*d*e - a*e^2)) + 1/1920*(10240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*c^(15/2)*d^9*e + 4096*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^5*c^8*d^10 + 10240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*c^7*d^8*e^2 + 10240*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^7*d^9*e + 10240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(15/2)*d^10 -
10240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b*c^(13/2)*d^8*e^2 - 10240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*
b^2*c^(13/2)*d^9*e - 10240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(15/2)*d^9*e + 10240*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^3*b^2*c^7*d^10 - 40960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b*c^6*d^7*e^3 - 40448*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*b^2*c^6*d^8*e^2 + 8192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^7*d^8*e^2 - 20480*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c^6*d^9*e - 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^7*d^9*
e + 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(13/2)*d^10 - 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6
*b^2*c^(11/2)*d^7*e^3 + 40960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*c^(13/2)*d^7*e^3 - 29440*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^4*b^3*c^(11/2)*d^8*e^2 + 66560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(13/2)*d^8*e^2
- 12160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*c^(11/2)*d^9*e - 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
2*a*b^2*c^(13/2)*d^9*e + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c^6*d^10 + 61440*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^7*b^2*c^5*d^6*e^4 + 40960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*c^6*d^6*e^4 - 22528*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*b^3*c^5*d^7*e^3 + 90112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c^6*d^7*e^3 - 512
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*c^5*d^8*e^2 + 107520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c
^6*d^8*e^2 + 10240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^7*d^8*e^2 - 3200*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*b^5*c^5*d^9*e - 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c^6*d^9*e + 128*b^5*c^(11/2)*d^10 + 14336
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^3*c^(9/2)*d^6*e^4 + 13440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*
c^(9/2)*d^7*e^3 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(11/2)*d^7*e^3 - 71680*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^4*a^2*c^(13/2)*d^7*e^3 + 2240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^5*c^(9/2)*d^8*e^2 + 7
1680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3*c^(11/2)*d^8*e^2 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
2*a^2*b*c^(13/2)*d^8*e^2 - 320*b^6*c^(9/2)*d^9*e - 640*a*b^4*c^(11/2)*d^9*e - 40960*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^7*b^3*c^4*d^5*e^5 - 122880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b*c^5*d^5*e^5 + 133280*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*b^4*c^4*d^6*e^4 - 48384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c^5*d^6*e^4 -
218624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^6*d^6*e^4 + 16640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5
*c^4*d^7*e^3 - 71680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*c^5*d^7*e^3 - 184320*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*a^2*b*c^6*d^7*e^3 + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^6*c^4*d^8*e^2 + 21760*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a*b^4*c^5*d^8*e^2 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c^6*d^8*e^2 - 131
120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^4*c^(7/2)*d^5*e^5 - 97920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*
b^2*c^(9/2)*d^5*e^5 - 234240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^(11/2)*d^5*e^5 + 53200*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^4*b^5*c^(7/2)*d^6*e^4 + 40320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^3*c^(9/2)*d^6*e^
4 - 295680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b*c^(11/2)*d^6*e^4 + 7520*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*b^6*c^(7/2)*d^7*e^3 - 58240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^(9/2)*d^7*e^3 - 161280*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^(11/2)*d^7*e^3 - 10240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(
13/2)*d^7*e^3 + 96*b^7*c^(7/2)*d^8*e^2 + 2496*a*b^5*c^(9/2)*d^8*e^2 + 1280*a^2*b^3*c^(11/2)*d^8*e^2 - 1040*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^4*c^3*d^4*e^6 + 213120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^2*c^4*d^
4*e^6 - 119040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*c^5*d^4*e^6 - 155248*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^5*b^5*c^3*d^5*e^5 - 33920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^3*c^4*d^5*e^5 - 46848*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^5*a^2*b*c^5*d^5*e^5 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^6*c^3*d^6*e^4 + 93440*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^4*c^4*d^6*e^4 - 42240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*c^
5*d^6*e^4 + 92160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*c^6*d^6*e^4 + 1760*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*b^7*c^3*d^7*e^3 - 17280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c^4*d^7*e^3 - 58880*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*a^2*b^3*c^5*d^7*e^3 - 10240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c^6*d^7*e^3 - 3240*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^8*b^4*c^(5/2)*d^3*e^7 + 25920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^2*c^(7
/2)*d^3*e^7 - 51840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*c^(9/2)*d^3*e^7 + 37440*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^6*b^5*c^(5/2)*d^4*e^6 + 273920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^3*c^(7/2)*d^4*e^6 + 16896
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b*c^(9/2)*d^4*e^6 - 85120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^
6*c^(5/2)*d^5*e^5 - 74000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^4*c^(7/2)*d^5*e^5 + 90240*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^4*a^2*b^2*c^(9/2)*d^5*e^5 + 392960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*c^(11/2)*d^5*
e^5 - 7480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^7*c^(5/2)*d^6*e^4 + 52800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^2*a*b^5*c^(7/2)*d^6*e^4 + 80000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^3*c^(9/2)*d^6*e^4 + 174080*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b*c^(11/2)*d^6*e^4 + 176*b^8*c^(5/2)*d^7*e^3 - 1824*a*b^6*c^(7/2)*d^7*e^
3 - 7680*a^2*b^4*c^(9/2)*d^7*e^3 - 2560*a^3*b^2*c^(11/2)*d^7*e^3 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b
^4*c^2*d^2*e^8 + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^2*c^3*d^2*e^8 - 5760*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^9*a^2*c^4*d^2*e^8 + 9600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^5*c^2*d^3*e^7 - 117760*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^7*a*b^3*c^3*d^3*e^7 + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b*c^4*d^3*e^7 +
63700*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^6*c^2*d^4*e^6 + 236480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*
b^4*c^3*d^4*e^6 - 39360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^2*c^4*d^4*e^6 + 468480*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^5*a^3*c^5*d^4*e^6 - 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^7*c^2*d^5*e^5 - 55200*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^5*c^3*d^5*e^5 - 152320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^3*c^4*
d^5*e^5 + 509440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b*c^5*d^5*e^5 - 2540*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*b^8*c^2*d^6*e^4 + 13360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^6*c^3*d^6*e^4 + 33120*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))*a^2*b^4*c^4*d^6*e^4 + 102400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^2*c^5*d^6*e^4 + 2560*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*c^6*d^6*e^4 + 3240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^5*c^(3/2)*d
^2*e^8 - 25920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^3*c^(5/2)*d^2*e^8 + 51840*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^8*a^2*b*c^(7/2)*d^2*e^8 + 6370*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^6*c^(3/2)*d^3*e^7 - 159000*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^4*c^(5/2)*d^3*e^7 - 324000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^
2*c^(7/2)*d^3*e^7 + 278400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*c^(9/2)*d^3*e^7 + 39170*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^4*b^7*c^(3/2)*d^4*e^6 + 195720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^5*c^(5/2)*d^4*e^6
- 202400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^3*c^(7/2)*d^4*e^6 + 188800*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*a^3*b*c^(9/2)*d^4*e^6 - 490*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^8*c^(3/2)*d^5*e^5 - 8720*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^2*a*b^6*c^(5/2)*d^5*e^5 - 188880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^4*c^(7
/2)*d^5*e^5 + 191360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^2*c^(9/2)*d^5*e^5 - 70400*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^2*a^4*c^(11/2)*d^5*e^5 - 290*b^9*c^(3/2)*d^6*e^4 + 1448*a*b^7*c^(5/2)*d^6*e^4 + 2928*a^2*b^5*
c^(7/2)*d^6*e^4 + 19200*a^3*b^3*c^(9/2)*d^6*e^4 + 1280*a^4*b*c^(11/2)*d^6*e^4 + 360*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^9*b^5*c*d*e^9 - 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c^2*d*e^9 + 5760*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^9*a^2*b*c^3*d*e^9 - 1610*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^6*c*d^2*e^8 + 16440*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^7*a*b^4*c^2*d^2*e^8 + 7200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^2*c^3*d^2*
e^8 + 97920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*c^4*d^2*e^8 - 5426*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5
*b^7*c*d^3*e^7 - 140616*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^5*c^2*d^3*e^7 - 246880*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*a^2*b^3*c^3*d^3*e^7 - 101760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b*c^4*d^3*e^7 + 10930*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^8*c*d^4*e^6 + 84200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^6*c^2*d^
4*e^6 + 24240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^4*c^3*d^4*e^6 - 80640*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*a^3*b^2*c^4*d^4*e^6 - 367360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*c^5*d^4*e^6 + 690*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*b^9*c*d^5*e^5 + 1840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^7*c^2*d^5*e^5 - 61680*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^5*c^3*d^5*e^5 + 13440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^3*c^4*d^
5*e^5 - 78080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b*c^5*d^5*e^5 - 945*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
8*b^6*sqrt(c)*d*e^9 + 8100*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^4*c^(3/2)*d*e^9 - 19440*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^8*a^2*b^2*c^(5/2)*d*e^9 + 8640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^3*c^(7/2)*d*e^9 - 34
30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^7*sqrt(c)*d^2*e^8 + 17640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b
^5*c^(3/2)*d^2*e^8 + 207840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^3*c^(5/2)*d^2*e^8 - 74880*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^6*a^3*b*c^(7/2)*d^2*e^8 - 4480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^8*sqrt(c)*d^3*e
^7 - 112130*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^6*c^(3/2)*d^3*e^7 - 70200*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*a^2*b^4*c^(5/2)*d^3*e^7 + 45600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^2*c^(7/2)*d^3*e^7 - 361600
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*c^(9/2)*d^3*e^7 + 1470*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^9*sq
rt(c)*d^4*e^6 + 8780*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^7*c^(3/2)*d^4*e^6 + 86640*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2*a^2*b^5*c^(5/2)*d^4*e^6 + 50240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^3*c^(7/2)*d^4*e^6
- 375040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b*c^(9/2)*d^4*e^6 + 105*b^10*sqrt(c)*d^5*e^5 + 330*a*b^8*c
^(3/2)*d^5*e^5 - 6064*a^2*b^6*c^(5/2)*d^5*e^5 - 2160*a^3*b^4*c^(7/2)*d^5*e^5 - 21120*a^4*b^2*c^(9/2)*d^5*e^5 -
256*a^5*c^(11/2)*d^5*e^5 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^6*e^10 + 900*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^9*a*b^4*c*e^10 - 2160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b^2*c^2*e^10 + 960*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^9*a^3*c^3*e^10 - 490*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^7*d*e^9 + 2520*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^7*a*b^5*c*d*e^9 + 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^3*c^2*d*e^9 - 63360*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b*c^3*d*e^9 - 896*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^8*d^2*e^8 +
10034*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^6*c*d^2*e^8 + 203640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^
2*b^4*c^2*d^2*e^8 + 119520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^2*c^3*d^2*e^8 - 155520*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^5*a^4*c^4*d^2*e^8 - 790*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^9*d^3*e^7 - 47420*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^3*a*b^7*c*d^3*e^7 - 60720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^5*c^2*d^3*e^
7 + 76480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^3*c^3*d^3*e^7 + 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^3*a^4*b*c^4*d^3*e^7 + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^10*d^4*e^6 - 2610*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a*b^8*c*d^4*e^6 + 25060*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^6*c^2*d^4*e^6 + 60320*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a^3*b^4*c^3*d^4*e^6 - 112320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^2*c^4*d^4*e^6 +
17920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*c^5*d^4*e^6 + 3430*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^6*
sqrt(c)*d*e^9 - 29400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^4*c^(3/2)*d*e^9 - 52320*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^6*a^3*b^2*c^(5/2)*d*e^9 - 21120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^4*c^(7/2)*d*e^9 + 8960
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^7*sqrt(c)*d^2*e^8 + 130560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^
2*b^5*c^(3/2)*d^2*e^8 - 25600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^3*c^(5/2)*d^2*e^8 + 153600*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^4*a^4*b*c^(7/2)*d^2*e^8 - 8250*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^8*sqrt(c)*
d^3*e^7 - 22690*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^6*c^(3/2)*d^3*e^7 - 85480*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*a^3*b^4*c^(5/2)*d^3*e^7 + 178080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^2*c^(7/2)*d^3*e^7 +
164480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*c^(9/2)*d^3*e^7 - 420*a*b^9*sqrt(c)*d^4*e^6 + 1650*a^2*b^7*c
^(3/2)*d^4*e^6 + 10760*a^3*b^5*c^(5/2)*d^4*e^6 - 9120*a^4*b^3*c^(7/2)*d^4*e^6 + 9600*a^5*b*c^(9/2)*d^4*e^6 + 4
90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^6*e^10 - 4200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^4*c*e^1
0 + 10080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b^2*c^2*e^10 + 5760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*
a^4*c^3*e^10 + 1792*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^7*d*e^9 - 12288*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^5*a^2*b^5*c*d*e^9 - 153600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^3*c^2*d*e^9 + 92160*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^5*a^4*b*c^3*d*e^9 + 2370*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^8*d^2*e^8 + 70090*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^6*c*d^2*e^8 - 45560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^4*c^2*
d^2*e^8 + 8160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b^2*c^3*d^2*e^8 + 109440*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^3*a^5*c^4*d^2*e^8 - 420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^9*d^3*e^7 + 750*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*a^2*b^7*c*d^3*e^7 - 52040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^5*c^2*d^3*e^7 + 27040*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^3*c^3*d^3*e^7 + 128640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*b*c^4*d^3*e
^7 + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^4*b*c^(5/2)*e^10 - 4480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^4*a^2*b^6*sqrt(c)*d*e^9 - 89600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^4*c^(3/2)*d*e^9 + 30720*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^4*a^5*c^(7/2)*d*e^9 + 15930*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^7*sqrt(c)*d
^2*e^8 + 18440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^5*c^(3/2)*d^2*e^8 - 16160*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*a^4*b^3*c^(5/2)*d^2*e^8 - 82560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b*c^(7/2)*d^2*e^8 + 630
*a^2*b^8*sqrt(c)*d^3*e^7 - 5310*a^3*b^6*c^(3/2)*d^3*e^7 - 7840*a^4*b^4*c^(5/2)*d^3*e^7 + 25248*a^5*b^2*c^(7/2)
*d^3*e^7 - 1792*a^6*c^(9/2)*d^3*e^7 - 896*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^6*e^10 + 7680*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*a^3*b^4*c*e^10 + 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^4*b^2*c^2*e^10 - 23
70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^7*d*e^9 - 41640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^5*c
*d*e^9 + 37280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b^3*c^2*d*e^9 - 48000*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^3*a^5*b*c^3*d*e^9 + 630*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^8*d^2*e^8 + 6510*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*a^3*b^6*c*d^2*e^8 + 28720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^4*c^2*d^2*e^8 - 68640*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*a^5*b^2*c^3*d^2*e^8 - 46080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*c^4*d^2*e^8
+ 32000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*b^3*c^(3/2)*e^10 - 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^4*a^5*b*c^(5/2)*e^10 - 12990*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^6*sqrt(c)*d*e^9 - 1480*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*a^4*b^4*c^(3/2)*d*e^9 + 31200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^2*c^(5/2)*d
*e^9 - 17280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^6*c^(7/2)*d*e^9 - 420*a^3*b^7*sqrt(c)*d^2*e^8 + 7080*a^4*
b^5*c^(3/2)*d^2*e^8 - 3712*a^5*b^3*c^(5/2)*d^2*e^8 - 20352*a^6*b*c^(7/2)*d^2*e^8 + 790*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*a^3*b^6*e^10 + 8040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b^4*c*e^10 + 5280*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^3*a^5*b^2*c^2*e^10 - 5760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^6*c^3*e^10 - 420*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^7*d*e^9 - 8280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^5*c*d*e^9 + 4480*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*b^3*c^2*d*e^9 + 28800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b*c^3*d*
e^9 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^5*sqrt(c)*e^10 - 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^2*a^5*b^3*c^(3/2)*e^10 + 105*a^4*b^6*sqrt(c)*d*e^9 - 4740*a^5*b^4*c^(3/2)*d*e^9 + 8304*a^6*b^2*c^(5/2)*d*e^
9 + 5184*a^7*c^(7/2)*d*e^9 + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^6*e^10 + 2940*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*a^5*b^4*c*e^10 - 5520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b^2*c^2*e^10 - 960*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*a^7*c^3*e^10 + 1280*a^6*b^3*c^(3/2)*e^10 - 3072*a^7*b*c^(5/2)*e^10)/((c^4*d^8*e^3 - 4*b*
c^3*d^7*e^4 + 6*b^2*c^2*d^6*e^5 + 4*a*c^3*d^6*e^5 - 4*b^3*c*d^5*e^6 - 12*a*b*c^2*d^5*e^6 + b^4*d^4*e^7 + 12*a*
b^2*c*d^4*e^7 + 6*a^2*c^2*d^4*e^7 - 4*a*b^3*d^3*e^8 - 12*a^2*b*c*d^3*e^8 + 6*a^2*b^2*d^2*e^9 + 4*a^3*c*d^2*e^9
- 4*a^3*b*d*e^10 + a^4*e^11)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*sqrt(c)*d + b*d - a*e)^5)

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maple [B]  time = 0.09, size = 15192, normalized size = 35.33 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x)

[Out]

result too large to display

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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((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,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(a*e^2-b*d*e>0)', see `assume?`
for more details)Is a*e^2-b*d*e                            +c*d^2 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (b+2\,c\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^6,x)

[Out]

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^6, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b + 2 c x\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**6,x)

[Out]

Integral((b + 2*c*x)*sqrt(a + b*x + c*x**2)/(d + e*x)**6, x)

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