∖int A+Bx____________ x7∕2∖left(a+bx+cx2∖right)∖,dx

3.1015 \(\int \frac{A+B x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=307 \[ -\frac{2 \left (-a A c-a b B+A b^2\right )}{a^3 \sqrt{x}}-\frac{\sqrt{2} \sqrt{c} \left (-\frac{a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt{b^2-4 a c}}-a A c-a b B+A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a^3 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} \left (\frac{a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt{b^2-4 a c}}-a A c-a b B+A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a^3 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 (A b-a B)}{3 a^2 x^{3/2}}-\frac{2 A}{5 a x^{5/2}} \]

[Out]

(-2*A)/(5*a*x^(5/2)) + (2*(A*b - a*B))/(3*a^2*x^(3/2)) - (2*(A*b^2 - a*b*B - a*A

*c))/(a^3*Sqrt[x]) - (Sqrt[2]*Sqrt[c]*(A*b^2 - a*b*B - a*A*c - (a*B*(b^2 - 2*a*c

) - A*(b^3 - 3*a*b*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[

b - Sqrt[b^2 - 4*a*c]]])/(a^3*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*Sqrt[c]*(A

*b^2 - a*b*B - a*A*c + (a*B*(b^2 - 2*a*c) - A*(b^3 - 3*a*b*c))/Sqrt[b^2 - 4*a*c]

)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^3*Sqrt[b + S

qrt[b^2 - 4*a*c]])

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Rubi [A] time = 4.07047, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{2 \left (-a A c-a b B+A b^2\right )}{a^3 \sqrt{x}}-\frac{\sqrt{2} \sqrt{c} \left (-\frac{a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt{b^2-4 a c}}-a A c-a b B+A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a^3 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} \left (\frac{a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt{b^2-4 a c}}-a A c-a b B+A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a^3 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 (A b-a B)}{3 a^2 x^{3/2}}-\frac{2 A}{5 a x^{5/2}} \]

Antiderivative was successfully veriﬁed.

[In] Int[(A + B*x)/(x^(7/2)*(a + b*x + c*x^2)),x]

[Out]

(-2*A)/(5*a*x^(5/2)) + (2*(A*b - a*B))/(3*a^2*x^(3/2)) - (2*(A*b^2 - a*b*B - a*A

*c))/(a^3*Sqrt[x]) - (Sqrt[2]*Sqrt[c]*(A*b^2 - a*b*B - a*A*c - (a*B*(b^2 - 2*a*c

) - A*(b^3 - 3*a*b*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[

b - Sqrt[b^2 - 4*a*c]]])/(a^3*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*Sqrt[c]*(A

*b^2 - a*b*B - a*A*c + (a*B*(b^2 - 2*a*c) - A*(b^3 - 3*a*b*c))/Sqrt[b^2 - 4*a*c]

)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^3*Sqrt[b + S

qrt[b^2 - 4*a*c]])

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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((B*x+A)/x**(7/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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Mathematica [A] time = 0.940001, size = 356, normalized size = 1.16 \[ \frac{-\frac{6 a^2 A}{x^{5/2}}+\frac{30 \left (a A c+a b B-A b^2\right )}{\sqrt{x}}+\frac{15 \sqrt{2} \sqrt{c} \left (a B \left (b \sqrt{b^2-4 a c}-2 a c+b^2\right )-A \left (b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}-3 a b c+b^3\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{15 \sqrt{2} \sqrt{c} \left (A \left (-b^2 \sqrt{b^2-4 a c}+a c \sqrt{b^2-4 a c}-3 a b c+b^3\right )+a B \left (b \sqrt{b^2-4 a c}+2 a c-b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{10 a (A b-a B)}{x^{3/2}}}{15 a^3} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(A + B*x)/(x^(7/2)*(a + b*x + c*x^2)),x]

[Out]

((-6*a^2*A)/x^(5/2) + (10*a*(A*b - a*B))/x^(3/2) + (30*(-(A*b^2) + a*b*B + a*A*c

))/Sqrt[x] + (15*Sqrt[2]*Sqrt[c]*(a*B*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c]) - A*(b

^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*S

qrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b

^2 - 4*a*c]]) + (15*Sqrt[2]*Sqrt[c]*(a*B*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c]) +

A*(b^3 - 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[

2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sq

rt[b^2 - 4*a*c]]))/(15*a^3)

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Maple [B] time = 0.051, size = 913, normalized size = 3. \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((B*x+A)/x^(7/2)/(c*x^2+b*x+a),x)

[Out]

-2/5*A/a/x^(5/2)+2/3/x^(3/2)/a^2*A*b-2/3*B/a/x^(3/2)+2*A*c/a^2/x^(1/2)-2/a^3/x^(

1/2)*b^2*A+2/a^2/x^(1/2)*b*B-1/a^2*c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)

*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A+1/a^3*c*2^(1/2)/

((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1

/2))*c)^(1/2))*b^2*A-3/a^2*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2

))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b+1/a

^3*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1

/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3-1/a^2*c*2^(1/2)/((-b+(-4*a*

c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/

2))*b*B+2/a*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arc

tanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B-1/a^2*c/(-4*a*c+b^2)

^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+

(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2*B+1/a^2*c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)

^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A-1/a^3*c*2^(1

/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(

1/2))*c)^(1/2))*b^2*A-3/a^2*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2

))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b+1/a^3

*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*

2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3+1/a^2*c*2^(1/2)/((b+(-4*a*c+b^2)

^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*B+

2/a*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(

1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B-1/a^2*c/(-4*a*c+b^2)^(1/2)*2^(1

/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(

1/2))*c)^(1/2))*b^2*B

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{2 \,{\left (\frac{3 \, A a^{3}}{x^{\frac{5}{2}}} + 15 \,{\left ({\left (b^{3} - 2 \, a b c\right )} A -{\left (a b^{2} - a^{2} c\right )} B\right )} \sqrt{x} - \frac{15 \,{\left (B a^{2} b -{\left (a b^{2} - a^{2} c\right )} A\right )}}{\sqrt{x}} + \frac{5 \,{\left (B a^{3} - A a^{2} b\right )}}{x^{\frac{3}{2}}}\right )}}{15 \, a^{4}} - \int -\frac{{\left ({\left (b^{3} c - 2 \, a b c^{2}\right )} A -{\left (a b^{2} c - a^{2} c^{2}\right )} B\right )} x^{\frac{3}{2}} +{\left ({\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} A -{\left (a b^{3} - 2 \, a^{2} b c\right )} B\right )} \sqrt{x}}{a^{4} c x^{2} + a^{4} b x + a^{5}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((B*x + A)/((c*x^2 + b*x + a)*x^(7/2)),x, algorithm="maxima")

[Out]

-2/15*(3*A*a^3/x^(5/2) + 15*((b^3 - 2*a*b*c)*A - (a*b^2 - a^2*c)*B)*sqrt(x) - 15

*(B*a^2*b - (a*b^2 - a^2*c)*A)/sqrt(x) + 5*(B*a^3 - A*a^2*b)/x^(3/2))/a^4 - inte

grate(-(((b^3*c - 2*a*b*c^2)*A - (a*b^2*c - a^2*c^2)*B)*x^(3/2) + ((b^4 - 3*a*b^

2*c + a^2*c^2)*A - (a*b^3 - 2*a^2*b*c)*B)*sqrt(x))/(a^4*c*x^2 + a^4*b*x + a^5),

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Fricas [A] time = 3.13694, size = 10753, normalized size = 35.03 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((B*x + A)/((c*x^2 + b*x + a)*x^(7/2)),x, algorithm="fricas")

[Out]

1/30*(15*sqrt(2)*a^3*x^(5/2)*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*B

*a^4 - 7*A^2*a^3*b)*c^3 + (5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 -

(5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2*a*b^5)*c + (a^7*b^2 - 4*a^8*c)*sqrt((B^4

*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^

4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*

A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3

*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 + 31*A^4

*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A

^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^

2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^7

*b^2 - 4*a^8*c))*log(sqrt(2)*(B^3*a^3*b^8 - 3*A*B^2*a^2*b^9 + 3*A^2*B*a*b^10 - A

^3*b^11 - 4*(A^2*B*a^6 - 2*A^3*a^5*b)*c^5 + (4*B^3*a^7 - 32*A*B^2*a^6*b + 77*A^2

*B*a^5*b^2 - 54*A^3*a^4*b^3)*c^4 - (17*B^3*a^6*b^2 - 92*A*B^2*a^5*b^3 + 151*A^2*

B*a^4*b^4 - 77*A^3*a^3*b^5)*c^3 + (20*B^3*a^5*b^4 - 81*A*B^2*a^4*b^5 + 105*A^2*B

*a^3*b^6 - 44*A^3*a^2*b^7)*c^2 - (8*B^3*a^4*b^6 - 27*A*B^2*a^3*b^7 + 30*A^2*B*a^

2*b^8 - 11*A^3*a*b^9)*c - (B*a^8*b^5 - A*a^7*b^6 + 8*A*a^10*c^3 + 6*(2*B*a^10*b

- 3*A*a^9*b^2)*c^2 - (7*B*a^9*b^3 - 8*A*a^8*b^4)*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*

a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*(A^2*

B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^

2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A

*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11

*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7 + 37*A

^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*

A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))*sqrt(-(B^2*a^2*b^5 - 2*

A*B*a*b^6 + A^2*b^7 + (4*A*B*a^4 - 7*A^2*a^3*b)*c^3 + (5*B^2*a^4*b - 18*A*B*a^3*

b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2*a*b^5)*c + (

a^7*b^2 - 4*a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*

A^3*B*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a

^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3

+ 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4

- 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 +

132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^

6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/

(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c)) + 4*(A^4*a^3*c^7 + (5*A^3*B*a^3*b -

6*A^4*a^2*b^2)*c^6 - (B^4*a^5 - 7*A*B^3*a^4*b + 9*A^2*B^2*a^3*b^2 + 2*A^3*B*a^2

*b^3 - 5*A^4*a*b^4)*c^5 + (3*B^4*a^4*b^2 - 11*A*B^3*a^3*b^3 + 12*A^2*B^2*a^2*b^4

- 3*A^3*B*a*b^5 - A^4*b^6)*c^4 - (B^4*a^3*b^4 - 3*A*B^3*a^2*b^5 + 3*A^2*B^2*a*b

^6 - A^3*B*b^7)*c^3)*sqrt(x)) - 15*sqrt(2)*a^3*x^(5/2)*sqrt(-(B^2*a^2*b^5 - 2*A*

B*a*b^6 + A^2*b^7 + (4*A*B*a^4 - 7*A^2*a^3*b)*c^3 + (5*B^2*a^4*b - 18*A*B*a^3*b^

2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2*a*b^5)*c + (a^

7*b^2 - 4*a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^

3*B*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5

*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 +

46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 -

80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 1

32*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6

- 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a

^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c))*log(-sqrt(2)*(B^3*a^3*b^8 - 3*A*B^2*a

^2*b^9 + 3*A^2*B*a*b^10 - A^3*b^11 - 4*(A^2*B*a^6 - 2*A^3*a^5*b)*c^5 + (4*B^3*a^

7 - 32*A*B^2*a^6*b + 77*A^2*B*a^5*b^2 - 54*A^3*a^4*b^3)*c^4 - (17*B^3*a^6*b^2 -

92*A*B^2*a^5*b^3 + 151*A^2*B*a^4*b^4 - 77*A^3*a^3*b^5)*c^3 + (20*B^3*a^5*b^4 - 8

1*A*B^2*a^4*b^5 + 105*A^2*B*a^3*b^6 - 44*A^3*a^2*b^7)*c^2 - (8*B^3*a^4*b^6 - 27*

A*B^2*a^3*b^7 + 30*A^2*B*a^2*b^8 - 11*A^3*a*b^9)*c - (B*a^8*b^5 - A*a^7*b^6 + 8*

A*a^10*c^3 + 6*(2*B*a^10*b - 3*A*a^9*b^2)*c^2 - (7*B*a^9*b^3 - 8*A*a^8*b^4)*c)*s

qrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b

^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a

^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^

4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5

+ 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6

- 116*A^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7

+ 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c

)))*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*B*a^4 - 7*A^2*a^3*b)*c^3 +

(5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a

^2*b^4 + 7*A^2*a*b^5)*c + (a^7*b^2 - 4*a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^

9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^

7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*

a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a

^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a

^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a^2

*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*

a^2*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c)) + 4*(A^4

*a^3*c^7 + (5*A^3*B*a^3*b - 6*A^4*a^2*b^2)*c^6 - (B^4*a^5 - 7*A*B^3*a^4*b + 9*A^

2*B^2*a^3*b^2 + 2*A^3*B*a^2*b^3 - 5*A^4*a*b^4)*c^5 + (3*B^4*a^4*b^2 - 11*A*B^3*a

^3*b^3 + 12*A^2*B^2*a^2*b^4 - 3*A^3*B*a*b^5 - A^4*b^6)*c^4 - (B^4*a^3*b^4 - 3*A*

B^3*a^2*b^5 + 3*A^2*B^2*a*b^6 - A^3*B*b^7)*c^3)*sqrt(x)) + 15*sqrt(2)*a^3*x^(5/2

)*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*B*a^4 - 7*A^2*a^3*b)*c^3 + (

5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (5*B^2*a^3*b^3 - 12*A*B*a^2

*b^4 + 7*A^2*a*b^5)*c - (a^7*b^2 - 4*a^8*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9

+ 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7

- 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^

6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6

*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6

*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a^2*b

^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^

2*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c))*log(sqrt(2

)*(B^3*a^3*b^8 - 3*A*B^2*a^2*b^9 + 3*A^2*B*a*b^10 - A^3*b^11 - 4*(A^2*B*a^6 - 2*

A^3*a^5*b)*c^5 + (4*B^3*a^7 - 32*A*B^2*a^6*b + 77*A^2*B*a^5*b^2 - 54*A^3*a^4*b^3

)*c^4 - (17*B^3*a^6*b^2 - 92*A*B^2*a^5*b^3 + 151*A^2*B*a^4*b^4 - 77*A^3*a^3*b^5)

*c^3 + (20*B^3*a^5*b^4 - 81*A*B^2*a^4*b^5 + 105*A^2*B*a^3*b^6 - 44*A^3*a^2*b^7)*

c^2 - (8*B^3*a^4*b^6 - 27*A*B^2*a^3*b^7 + 30*A^2*B*a^2*b^8 - 11*A^3*a*b^9)*c + (

B*a^8*b^5 - A*a^7*b^6 + 8*A*a^10*c^3 + 6*(2*B*a^10*b - 3*A*a^9*b^2)*c^2 - (7*B*a

^9*b^3 - 8*A*a^8*b^4)*c)*sqrt((B^4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^1

0 - 4*A^3*B*a*b^11 + A^4*b^12 + A^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6

*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^

5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a

^5*b^4 - 80*A^3*B*a^4*b^5 + 31*A^4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5

*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*

a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*b^1

0)*c)/(a^14*b^2 - 4*a^15*c)))*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*

B*a^4 - 7*A^2*a^3*b)*c^3 + (5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 -

(5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2*a*b^5)*c - (a^7*b^2 - 4*a^8*c)*sqrt((B^

4*a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A

^4*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12

*A*B^3*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(

3*B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 + 31*A^

4*a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*

A^3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A

^2*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^

7*b^2 - 4*a^8*c)) + 4*(A^4*a^3*c^7 + (5*A^3*B*a^3*b - 6*A^4*a^2*b^2)*c^6 - (B^4*

a^5 - 7*A*B^3*a^4*b + 9*A^2*B^2*a^3*b^2 + 2*A^3*B*a^2*b^3 - 5*A^4*a*b^4)*c^5 + (

3*B^4*a^4*b^2 - 11*A*B^3*a^3*b^3 + 12*A^2*B^2*a^2*b^4 - 3*A^3*B*a*b^5 - A^4*b^6)

*c^4 - (B^4*a^3*b^4 - 3*A*B^3*a^2*b^5 + 3*A^2*B^2*a*b^6 - A^3*B*b^7)*c^3)*sqrt(x

)) - 15*sqrt(2)*a^3*x^(5/2)*sqrt(-(B^2*a^2*b^5 - 2*A*B*a*b^6 + A^2*b^7 + (4*A*B*

a^4 - 7*A^2*a^3*b)*c^3 + (5*B^2*a^4*b - 18*A*B*a^3*b^2 + 14*A^2*a^2*b^3)*c^2 - (

5*B^2*a^3*b^3 - 12*A*B*a^2*b^4 + 7*A^2*a*b^5)*c - (a^7*b^2 - 4*a^8*c)*sqrt((B^4*

a^4*b^8 - 4*A*B^3*a^3*b^9 + 6*A^2*B^2*a^2*b^10 - 4*A^3*B*a*b^11 + A^4*b^12 + A^4

*a^6*c^6 - 2*(A^2*B^2*a^7 - 6*A^3*B*a^6*b + 6*A^4*a^5*b^2)*c^5 + (B^4*a^8 - 12*A

*B^3*a^7*b + 54*A^2*B^2*a^6*b^2 - 88*A^3*B*a^5*b^3 + 46*A^4*a^4*b^4)*c^4 - 2*(3*

B^4*a^7*b^2 - 26*A*B^3*a^6*b^3 + 72*A^2*B^2*a^5*b^4 - 80*A^3*B*a^4*b^5 + 31*A^4*

a^3*b^6)*c^3 + (11*B^4*a^6*b^4 - 64*A*B^3*a^5*b^5 + 132*A^2*B^2*a^4*b^6 - 116*A^

3*B*a^3*b^7 + 37*A^4*a^2*b^8)*c^2 - 2*(3*B^4*a^5*b^6 - 14*A*B^3*a^4*b^7 + 24*A^2

*B^2*a^3*b^8 - 18*A^3*B*a^2*b^9 + 5*A^4*a*b^10)*c)/(a^14*b^2 - 4*a^15*c)))/(a^7*

b^2 - 4*a^8*c))*log(-sqrt(2)*(B^3*a^3*b^8 - 3*A*B^2*a^2*b^9 + 3*A^2*B*a*b^10 - A

^3*b^11 - 4*(A^2*B*a^6 - 2*A^3*a^5*b)*c^5 + (4*B^3*a^7 - 32*A*B^2*a^6*b + 77*A^2

*B*a^5*b^2 - 54*A^3*a^4*b^3)*c^4 - (17*B^3*a^6*b^2 - 92*A*B^2*a^5*b^3 + 151*A^2*

B*a^4*b^4 - 77*A^3*a^3*b^5)*c^3 + (20*B^3*a^5*b^4 - 81*A*B^2*a^4*b^5 + 105*A^2*B

*a^3*b^6 - 44*A^3*a^2*b^7)*c^2 - (8*B^3*a^4*b^6 - 27*A*B^2*a^3*b^7 + 30*A^2*B*a^

2*b^8 - 11*A^3*a*b^9)*c + (B*a^8*b^5 - A*a^7*b^6 + 8*A*a^10*c^3 + 6*(2*B*a^10*b