3.1457 \(\int \frac{A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx\)

Optimal. Leaf size=303 \[ -\frac{\left (-5 \sqrt{a} A \sqrt{c} e+3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\left (5 \sqrt{a} A \sqrt{c} e+3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}+\frac{x (A c d-a B e)+a (B d-A e)}{2 a \left (a-c x^2\right ) \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{e \left (5 a A e^2-6 a B d e+A c d^2\right )}{2 a \sqrt{d+e x} \left (c d^2-a e^2\right )^2} \]

[Out]

-(e*(A*c*d^2 - 6*a*B*d*e + 5*a*A*e^2))/(2*a*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]) + (
a*(B*d - A*e) + (A*c*d - a*B*e)*x)/(2*a*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a - c*x^2
)) - ((2*A*c*d + 3*a*B*e - 5*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x]
)/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)
) + ((2*A*c*d + 3*a*B*e + 5*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])
/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(1/4)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))

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Rubi [A]  time = 1.40278, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (-5 \sqrt{a} A \sqrt{c} e+3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\left (5 \sqrt{a} A \sqrt{c} e+3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}+\frac{x (A c d-a B e)+a (B d-A e)}{2 a \left (a-c x^2\right ) \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{e \left (5 a A e^2-6 a B d e+A c d^2\right )}{2 a \sqrt{d+e x} \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)^2),x]

[Out]

-(e*(A*c*d^2 - 6*a*B*d*e + 5*a*A*e^2))/(2*a*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]) + (
a*(B*d - A*e) + (A*c*d - a*B*e)*x)/(2*a*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a - c*x^2
)) - ((2*A*c*d + 3*a*B*e - 5*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x]
)/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)
) + ((2*A*c*d + 3*a*B*e + 5*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])
/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(1/4)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))

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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)/(e*x+d)**(3/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.915405, size = 342, normalized size = 1.13 \[ -\frac{\left (-5 \sqrt{a} A \sqrt{c} e+3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^2 \sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (5 \sqrt{a} A \sqrt{c} e+3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^2 \sqrt{\sqrt{a} \sqrt{c} e+c d}}+\frac{-a^2 e^2 (-4 A e+5 B d+B e x)+a c \left (A e \left (2 d^2+d e x-5 e^2 x^2\right )+B d \left (-d^2+d e x+6 e^2 x^2\right )\right )-A c^2 d^2 x (d+e x)}{2 a \left (c x^2-a\right ) \sqrt{d+e x} \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)^2),x]

[Out]

(-(A*c^2*d^2*x*(d + e*x)) - a^2*e^2*(5*B*d - 4*A*e + B*e*x) + a*c*(A*e*(2*d^2 +
d*e*x - 5*e^2*x^2) + B*d*(-d^2 + d*e*x + 6*e^2*x^2)))/(2*a*(c*d^2 - a*e^2)^2*Sqr
t[d + e*x]*(-a + c*x^2)) - ((2*A*c*d + 3*a*B*e - 5*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[
(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(4*a^(3/2)*(Sqrt[c]*d -
Sqrt[a]*e)^2*Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]) + ((2*A*c*d + 3*a*B*e + 5*Sqrt[a]*A*
Sqrt[c]*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(4*a^
(3/2)*(Sqrt[c]*d + Sqrt[a]*e)^2*Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])

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Maple [B]  time = 0.073, size = 1622, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^2,x)

[Out]

-2*e^3/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)*A+2*e^2/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)*B*d-1
/2*e^3/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*c*(e*x+d)^(3/2)*A-1/2*e/(a*e^2-c*d^2)^2
/(c*e^2*x^2-a*e^2)*c^2/a*(e*x+d)^(3/2)*A*d^2+e^2/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^
2)*c*(e*x+d)^(3/2)*B*d+3/2*e^3/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*(e*x+d)^(1/2)*A
*c*d+1/2*e/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(1/2)*A*d^3*c^2-1/2*e^4/(
a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*a*(e*x+d)^(1/2)*B-3/2*e^2/(a*e^2-c*d^2)^2/(c*e^
2*x^2-a*e^2)*(e*x+d)^(1/2)*B*c*d^2-2*e^5/(a*e^2-c*d^2)^2*a^2*c^2/(a^3*c*e^4)^(1/
2)/((a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*c*d
*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2))*A*d+1/2*e^3/(a*e^2-c*d^2)^2*a*c^3/(a^3*c*e^4
)^(1/2)/((a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((
a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2))*A*d^3+3/4*e^6/(a*e^2-c*d^2)^2*a^3*c/(a^
3*c*e^4)^(1/2)/((a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2)*arctanh(a*c*e*(e*x+d)^(
1/2)/((a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2))*B+3/4*e^4/(a*e^2-c*d^2)^2*a^2*c^
2/(a^3*c*e^4)^(1/2)/((a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2)*arctanh(a*c*e*(e*x
+d)^(1/2)/((a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2))*B*d^2+5/4*e^4/(a*e^2-c*d^2)
^2*a*c/((a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a
*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2))*A+1/4*e^2/(a*e^2-c*d^2)^2*c^2/((a*c*d*e+
(a^3*c*e^4)^(1/2))*a*c*e)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*c*d*e+(a^3*c*e^4
)^(1/2))*a*c*e)^(1/2))*A*d^2-3/2*e^3/(a*e^2-c*d^2)^2*a*c/((a*c*d*e+(a^3*c*e^4)^(
1/2))*a*c*e)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*
e)^(1/2))*B*d-2*e^5/(a*e^2-c*d^2)^2*a^2*c^2/(a^3*c*e^4)^(1/2)/((-a*c*d*e+(a^3*c*
e^4)^(1/2))*a*c*e)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c*d*e+(a^3*c*e^4)^(1/2)
)*a*c*e)^(1/2))*A*d+1/2*e^3/(a*e^2-c*d^2)^2*a*c^3/(a^3*c*e^4)^(1/2)/((-a*c*d*e+(
a^3*c*e^4)^(1/2))*a*c*e)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c*d*e+(a^3*c*e^4)
^(1/2))*a*c*e)^(1/2))*A*d^3+3/4*e^6/(a*e^2-c*d^2)^2*a^3*c/(a^3*c*e^4)^(1/2)/((-a
*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c*d*e+(a^
3*c*e^4)^(1/2))*a*c*e)^(1/2))*B+3/4*e^4/(a*e^2-c*d^2)^2*a^2*c^2/(a^3*c*e^4)^(1/2
)/((-a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c*d
*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2))*B*d^2-5/4*e^4/(a*e^2-c*d^2)^2*a*c/((-a*c*d*e
+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c*d*e+(a^3*c*e^
4)^(1/2))*a*c*e)^(1/2))*A-1/4*e^2/(a*e^2-c*d^2)^2*c^2/((-a*c*d*e+(a^3*c*e^4)^(1/
2))*a*c*e)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)
^(1/2))*A*d^2+3/2*e^3/(a*e^2-c*d^2)^2*a*c/((-a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(
1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c*d*e+(a^3*c*e^4)^(1/2))*a*c*e)^(1/2))*B*d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 - a)^2*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((c*x^2 - a)^2*(e*x + d)^(3/2)), x)

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Fricas [A]  time = 116.027, size = 16370, normalized size = 54.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 - a)^2*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

1/8*(4*B*a*c*d^3 - 8*A*a*c*d^2*e + 20*B*a^2*d*e^2 - 16*A*a^2*e^3 + 4*(A*c^2*d^2*
e - 6*B*a*c*d*e^2 + 5*A*a*c*e^3)*x^2 - (a^2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4
- (a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x + d)*sqrt((4*A^2*c^4
*d^7 + 12*A*B*a*c^3*d^6*e - 30*A*B*a^2*c^2*d^4*e^3 - 240*A*B*a^3*c*d^2*e^5 - 30*
A*B*a^4*e^7 + (9*B^2*a^2*c^2 - 35*A^2*a*c^3)*d^5*e^2 + 10*(9*B^2*a^3*c + 7*A^2*a
^2*c^2)*d^3*e^4 + 15*(3*B^2*a^4 + 7*A^2*a^3*c)*d*e^6 + (a^3*c^5*d^10 - 5*a^4*c^4
*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)
*sqrt((900*A^2*B^2*c^6*d^10*e^4 + 300*(9*A*B^3*a*c^5 - 7*A^3*B*c^6)*d^9*e^5 + 25
*(81*B^4*a^2*c^4 - 558*A^2*B^2*a*c^5 + 49*A^4*c^6)*d^8*e^6 - 240*(45*A*B^3*a^2*c
^4 - 91*A^3*B*a*c^5)*d^7*e^7 + 20*(405*B^4*a^3*c^3 + 1584*A^2*B^2*a^2*c^4 - 539*
A^4*a*c^5)*d^6*e^8 - 44280*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^9 + 6*(1485*B^4
*a^4*c^2 + 13170*A^2*B^2*a^3*c^3 + 3661*A^4*a^2*c^4)*d^4*e^10 - 48*(585*A*B^3*a^
4*c^2 + 1073*A^3*B*a^3*c^3)*d^3*e^11 + 4*(405*B^4*a^5*c + 6579*A^2*B^2*a^4*c^2 +
1925*A^4*a^3*c^3)*d^2*e^12 - 276*(9*A*B^3*a^5*c + 25*A^3*B*a^4*c^2)*d*e^13 + (8
1*B^4*a^6 + 450*A^2*B^2*a^5*c + 625*A^4*a^4*c^2)*e^14)/(a^3*c^11*d^20 - 10*a^4*c
^10*d^18*e^2 + 45*a^5*c^9*d^16*e^4 - 120*a^6*c^8*d^14*e^6 + 210*a^7*c^7*d^12*e^8
- 252*a^8*c^6*d^10*e^10 + 210*a^9*c^5*d^8*e^12 - 120*a^10*c^4*d^6*e^14 + 45*a^1
1*c^3*d^4*e^16 - 10*a^12*c^2*d^2*e^18 + a^13*c*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4
*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)
)*log(-(120*A^3*B*c^5*d^7*e^2 + 20*(27*A^2*B^2*a*c^4 - 7*A^4*c^5)*d^6*e^3 + 270*
(3*A*B^3*a^2*c^3 - 7*A^3*B*a*c^4)*d^5*e^4 + 3*(135*B^4*a^3*c^2 - 1080*A^2*B^2*a^
2*c^3 + 497*A^4*a*c^4)*d^4*e^5 - 36*(15*A*B^3*a^3*c^2 - 161*A^3*B*a^2*c^3)*d^3*e
^6 + 6*(135*B^4*a^4*c - 414*A^2*B^2*a^3*c^2 - 625*A^4*a^2*c^3)*d^2*e^7 - 6*(189*
A*B^3*a^4*c - 625*A^3*B*a^3*c^2)*d*e^8 + (81*B^4*a^5 - 625*A^4*a^3*c^2)*e^9)*sqr
t(e*x + d) + (30*A^2*B*a^2*c^5*d^8*e^3 + 35*(9*A*B^2*a^3*c^4 - A^3*a^2*c^5)*d^7*
e^4 + 15*(27*B^3*a^4*c^3 - 59*A^2*B*a^3*c^4)*d^6*e^5 - 3*(675*A*B^2*a^4*c^3 - 20
3*A^3*a^3*c^4)*d^5*e^6 + 27*(35*B^3*a^5*c^2 + 129*A^2*B*a^4*c^3)*d^4*e^7 - 3*(98
1*A*B^2*a^5*c^2 + 659*A^3*a^4*c^3)*d^3*e^8 + 3*(117*B^3*a^6*c + 827*A^2*B*a^5*c^
2)*d^2*e^9 - (531*A*B^2*a^6*c + 325*A^3*a^5*c^2)*d*e^10 + 3*(9*B^3*a^7 + 25*A^2*
B*a^6*c)*e^11 - (2*A*a^3*c^8*d^14 + 3*B*a^4*c^7*d^13*e - 19*A*a^4*c^7*d^12*e^2 -
6*B*a^5*c^6*d^11*e^3 + 60*A*a^5*c^6*d^10*e^4 - 15*B*a^6*c^5*d^9*e^5 - 85*A*a^6*
c^5*d^8*e^6 + 60*B*a^7*c^4*d^7*e^7 + 50*A*a^7*c^4*d^6*e^8 - 75*B*a^8*c^3*d^5*e^9
+ 3*A*a^8*c^3*d^4*e^10 + 42*B*a^9*c^2*d^3*e^11 - 16*A*a^9*c^2*d^2*e^12 - 9*B*a^
10*c*d*e^13 + 5*A*a^10*c*e^14)*sqrt((900*A^2*B^2*c^6*d^10*e^4 + 300*(9*A*B^3*a*c
^5 - 7*A^3*B*c^6)*d^9*e^5 + 25*(81*B^4*a^2*c^4 - 558*A^2*B^2*a*c^5 + 49*A^4*c^6)
*d^8*e^6 - 240*(45*A*B^3*a^2*c^4 - 91*A^3*B*a*c^5)*d^7*e^7 + 20*(405*B^4*a^3*c^3
+ 1584*A^2*B^2*a^2*c^4 - 539*A^4*a*c^5)*d^6*e^8 - 44280*(A*B^3*a^3*c^3 + A^3*B*
a^2*c^4)*d^5*e^9 + 6*(1485*B^4*a^4*c^2 + 13170*A^2*B^2*a^3*c^3 + 3661*A^4*a^2*c^
4)*d^4*e^10 - 48*(585*A*B^3*a^4*c^2 + 1073*A^3*B*a^3*c^3)*d^3*e^11 + 4*(405*B^4*
a^5*c + 6579*A^2*B^2*a^4*c^2 + 1925*A^4*a^3*c^3)*d^2*e^12 - 276*(9*A*B^3*a^5*c +
25*A^3*B*a^4*c^2)*d*e^13 + (81*B^4*a^6 + 450*A^2*B^2*a^5*c + 625*A^4*a^4*c^2)*e
^14)/(a^3*c^11*d^20 - 10*a^4*c^10*d^18*e^2 + 45*a^5*c^9*d^16*e^4 - 120*a^6*c^8*d
^14*e^6 + 210*a^7*c^7*d^12*e^8 - 252*a^8*c^6*d^10*e^10 + 210*a^9*c^5*d^8*e^12 -
120*a^10*c^4*d^6*e^14 + 45*a^11*c^3*d^4*e^16 - 10*a^12*c^2*d^2*e^18 + a^13*c*e^2
0)))*sqrt((4*A^2*c^4*d^7 + 12*A*B*a*c^3*d^6*e - 30*A*B*a^2*c^2*d^4*e^3 - 240*A*B
*a^3*c*d^2*e^5 - 30*A*B*a^4*e^7 + (9*B^2*a^2*c^2 - 35*A^2*a*c^3)*d^5*e^2 + 10*(9
*B^2*a^3*c + 7*A^2*a^2*c^2)*d^3*e^4 + 15*(3*B^2*a^4 + 7*A^2*a^3*c)*d*e^6 + (a^3*
c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c
*d^2*e^8 - a^8*e^10)*sqrt((900*A^2*B^2*c^6*d^10*e^4 + 300*(9*A*B^3*a*c^5 - 7*A^3
*B*c^6)*d^9*e^5 + 25*(81*B^4*a^2*c^4 - 558*A^2*B^2*a*c^5 + 49*A^4*c^6)*d^8*e^6 -
240*(45*A*B^3*a^2*c^4 - 91*A^3*B*a*c^5)*d^7*e^7 + 20*(405*B^4*a^3*c^3 + 1584*A^
2*B^2*a^2*c^4 - 539*A^4*a*c^5)*d^6*e^8 - 44280*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d
^5*e^9 + 6*(1485*B^4*a^4*c^2 + 13170*A^2*B^2*a^3*c^3 + 3661*A^4*a^2*c^4)*d^4*e^1
0 - 48*(585*A*B^3*a^4*c^2 + 1073*A^3*B*a^3*c^3)*d^3*e^11 + 4*(405*B^4*a^5*c + 65
79*A^2*B^2*a^4*c^2 + 1925*A^4*a^3*c^3)*d^2*e^12 - 276*(9*A*B^3*a^5*c + 25*A^3*B*
a^4*c^2)*d*e^13 + (81*B^4*a^6 + 450*A^2*B^2*a^5*c + 625*A^4*a^4*c^2)*e^14)/(a^3*
c^11*d^20 - 10*a^4*c^10*d^18*e^2 + 45*a^5*c^9*d^16*e^4 - 120*a^6*c^8*d^14*e^6 +
210*a^7*c^7*d^12*e^8 - 252*a^8*c^6*d^10*e^10 + 210*a^9*c^5*d^8*e^12 - 120*a^10*c
^4*d^6*e^14 + 45*a^11*c^3*d^4*e^16 - 10*a^12*c^2*d^2*e^18 + a^13*c*e^20)))/(a^3*
c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c
*d^2*e^8 - a^8*e^10))) + (a^2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4 - (a*c^3*d^4 -
2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x + d)*sqrt((4*A^2*c^4*d^7 + 12*A*B*
a*c^3*d^6*e - 30*A*B*a^2*c^2*d^4*e^3 - 240*A*B*a^3*c*d^2*e^5 - 30*A*B*a^4*e^7 +
(9*B^2*a^2*c^2 - 35*A^2*a*c^3)*d^5*e^2 + 10*(9*B^2*a^3*c + 7*A^2*a^2*c^2)*d^3*e^
4 + 15*(3*B^2*a^4 + 7*A^2*a^3*c)*d*e^6 + (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*
a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((900*A^2
*B^2*c^6*d^10*e^4 + 300*(9*A*B^3*a*c^5 - 7*A^3*B*c^6)*d^9*e^5 + 25*(81*B^4*a^2*c
^4 - 558*A^2*B^2*a*c^5 + 49*A^4*c^6)*d^8*e^6 - 240*(45*A*B^3*a^2*c^4 - 91*A^3*B*
a*c^5)*d^7*e^7 + 20*(405*B^4*a^3*c^3 + 1584*A^2*B^2*a^2*c^4 - 539*A^4*a*c^5)*d^6
*e^8 - 44280*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^9 + 6*(1485*B^4*a^4*c^2 + 131
70*A^2*B^2*a^3*c^3 + 3661*A^4*a^2*c^4)*d^4*e^10 - 48*(585*A*B^3*a^4*c^2 + 1073*A
^3*B*a^3*c^3)*d^3*e^11 + 4*(405*B^4*a^5*c + 6579*A^2*B^2*a^4*c^2 + 1925*A^4*a^3*
c^3)*d^2*e^12 - 276*(9*A*B^3*a^5*c + 25*A^3*B*a^4*c^2)*d*e^13 + (81*B^4*a^6 + 45
0*A^2*B^2*a^5*c + 625*A^4*a^4*c^2)*e^14)/(a^3*c^11*d^20 - 10*a^4*c^10*d^18*e^2 +
45*a^5*c^9*d^16*e^4 - 120*a^6*c^8*d^14*e^6 + 210*a^7*c^7*d^12*e^8 - 252*a^8*c^6
*d^10*e^10 + 210*a^9*c^5*d^8*e^12 - 120*a^10*c^4*d^6*e^14 + 45*a^11*c^3*d^4*e^16
- 10*a^12*c^2*d^2*e^18 + a^13*c*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*
a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))*log(-(120*A^
3*B*c^5*d^7*e^2 + 20*(27*A^2*B^2*a*c^4 - 7*A^4*c^5)*d^6*e^3 + 270*(3*A*B^3*a^2*c
^3 - 7*A^3*B*a*c^4)*d^5*e^4 + 3*(135*B^4*a^3*c^2 - 1080*A^2*B^2*a^2*c^3 + 497*A^
4*a*c^4)*d^4*e^5 - 36*(15*A*B^3*a^3*c^2 - 161*A^3*B*a^2*c^3)*d^3*e^6 + 6*(135*B^
4*a^4*c - 414*A^2*B^2*a^3*c^2 - 625*A^4*a^2*c^3)*d^2*e^7 - 6*(189*A*B^3*a^4*c -
625*A^3*B*a^3*c^2)*d*e^8 + (81*B^4*a^5 - 625*A^4*a^3*c^2)*e^9)*sqrt(e*x + d) - (
30*A^2*B*a^2*c^5*d^8*e^3 + 35*(9*A*B^2*a^3*c^4 - A^3*a^2*c^5)*d^7*e^4 + 15*(27*B
^3*a^4*c^3 - 59*A^2*B*a^3*c^4)*d^6*e^5 - 3*(675*A*B^2*a^4*c^3 - 203*A^3*a^3*c^4)
*d^5*e^6 + 27*(35*B^3*a^5*c^2 + 129*A^2*B*a^4*c^3)*d^4*e^7 - 3*(981*A*B^2*a^5*c^
2 + 659*A^3*a^4*c^3)*d^3*e^8 + 3*(117*B^3*a^6*c + 827*A^2*B*a^5*c^2)*d^2*e^9 - (
531*A*B^2*a^6*c + 325*A^3*a^5*c^2)*d*e^10 + 3*(9*B^3*a^7 + 25*A^2*B*a^6*c)*e^11
- (2*A*a^3*c^8*d^14 + 3*B*a^4*c^7*d^13*e - 19*A*a^4*c^7*d^12*e^2 - 6*B*a^5*c^6*d
^11*e^3 + 60*A*a^5*c^6*d^10*e^4 - 15*B*a^6*c^5*d^9*e^5 - 85*A*a^6*c^5*d^8*e^6 +
60*B*a^7*c^4*d^7*e^7 + 50*A*a^7*c^4*d^6*e^8 - 75*B*a^8*c^3*d^5*e^9 + 3*A*a^8*c^3
*d^4*e^10 + 42*B*a^9*c^2*d^3*e^11 - 16*A*a^9*c^2*d^2*e^12 - 9*B*a^10*c*d*e^13 +
5*A*a^10*c*e^14)*sqrt((900*A^2*B^2*c^6*d^10*e^4 + 300*(9*A*B^3*a*c^5 - 7*A^3*B*c
^6)*d^9*e^5 + 25*(81*B^4*a^2*c^4 - 558*A^2*B^2*a*c^5 + 49*A^4*c^6)*d^8*e^6 - 240
*(45*A*B^3*a^2*c^4 - 91*A^3*B*a*c^5)*d^7*e^7 + 20*(405*B^4*a^3*c^3 + 1584*A^2*B^
2*a^2*c^4 - 539*A^4*a*c^5)*d^6*e^8 - 44280*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e
^9 + 6*(1485*B^4*a^4*c^2 + 13170*A^2*B^2*a^3*c^3 + 3661*A^4*a^2*c^4)*d^4*e^10 -
48*(585*A*B^3*a^4*c^2 + 1073*A^3*B*a^3*c^3)*d^3*e^11 + 4*(405*B^4*a^5*c + 6579*A
^2*B^2*a^4*c^2 + 1925*A^4*a^3*c^3)*d^2*e^12 - 276*(9*A*B^3*a^5*c + 25*A^3*B*a^4*
c^2)*d*e^13 + (81*B^4*a^6 + 450*A^2*B^2*a^5*c + 625*A^4*a^4*c^2)*e^14)/(a^3*c^11
*d^20 - 10*a^4*c^10*d^18*e^2 + 45*a^5*c^9*d^16*e^4 - 120*a^6*c^8*d^14*e^6 + 210*
a^7*c^7*d^12*e^8 - 252*a^8*c^6*d^10*e^10 + 210*a^9*c^5*d^8*e^12 - 120*a^10*c^4*d
^6*e^14 + 45*a^11*c^3*d^4*e^16 - 10*a^12*c^2*d^2*e^18 + a^13*c*e^20)))*sqrt((4*A
^2*c^4*d^7 + 12*A*B*a*c^3*d^6*e - 30*A*B*a^2*c^2*d^4*e^3 - 240*A*B*a^3*c*d^2*e^5
- 30*A*B*a^4*e^7 + (9*B^2*a^2*c^2 - 35*A^2*a*c^3)*d^5*e^2 + 10*(9*B^2*a^3*c + 7
*A^2*a^2*c^2)*d^3*e^4 + 15*(3*B^2*a^4 + 7*A^2*a^3*c)*d*e^6 + (a^3*c^5*d^10 - 5*a
^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8
*e^10)*sqrt((900*A^2*B^2*c^6*d^10*e^4 + 300*(9*A*B^3*a*c^5 - 7*A^3*B*c^6)*d^9*e^
5 + 25*(81*B^4*a^2*c^4 - 558*A^2*B^2*a*c^5 + 49*A^4*c^6)*d^8*e^6 - 240*(45*A*B^3
*a^2*c^4 - 91*A^3*B*a*c^5)*d^7*e^7 + 20*(405*B^4*a^3*c^3 + 1584*A^2*B^2*a^2*c^4
- 539*A^4*a*c^5)*d^6*e^8 - 44280*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^9 + 6*(14
85*B^4*a^4*c^2 + 13170*A^2*B^2*a^3*c^3 + 3661*A^4*a^2*c^4)*d^4*e^10 - 48*(585*A*
B^3*a^4*c^2 + 1073*A^3*B*a^3*c^3)*d^3*e^11 + 4*(405*B^4*a^5*c + 6579*A^2*B^2*a^4
*c^2 + 1925*A^4*a^3*c^3)*d^2*e^12 - 276*(9*A*B^3*a^5*c + 25*A^3*B*a^4*c^2)*d*e^1
3 + (81*B^4*a^6 + 450*A^2*B^2*a^5*c + 625*A^4*a^4*c^2)*e^14)/(a^3*c^11*d^20 - 10
*a^4*c^10*d^18*e^2 + 45*a^5*c^9*d^16*e^4 - 120*a^6*c^8*d^14*e^6 + 210*a^7*c^7*d^
12*e^8 - 252*a^8*c^6*d^10*e^10 + 210*a^9*c^5*d^8*e^12 - 120*a^10*c^4*d^6*e^14 +
45*a^11*c^3*d^4*e^16 - 10*a^12*c^2*d^2*e^18 + a^13*c*e^20)))/(a^3*c^5*d^10 - 5*a
^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8
*e^10))) - (a^2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4 - (a*c^3*d^4 - 2*a^2*c^2*d^2
*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x + d)*sqrt((4*A^2*c^4*d^7 + 12*A*B*a*c^3*d^6*e -
30*A*B*a^2*c^2*d^4*e^3 - 240*A*B*a^3*c*d^2*e^5 - 30*A*B*a^4*e^7 + (9*B^2*a^2*c^2
- 35*A^2*a*c^3)*d^5*e^2 + 10*(9*B^2*a^3*c + 7*A^2*a^2*c^2)*d^3*e^4 + 15*(3*B^2*
a^4 + 7*A^2*a^3*c)*d*e^6 - (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^
4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((900*A^2*B^2*c^6*d^10*
e^4 + 300*(9*A*B^3*a*c^5 - 7*A^3*B*c^6)*d^9*e^5 + 25*(81*B^4*a^2*c^4 - 558*A^2*B
^2*a*c^5 + 49*A^4*c^6)*d^8*e^6 - 240*(45*A*B^3*a^2*c^4 - 91*A^3*B*a*c^5)*d^7*e^7
+ 20*(405*B^4*a^3*c^3 + 1584*A^2*B^2*a^2*c^4 - 539*A^4*a*c^5)*d^6*e^8 - 44280*(
A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^9 + 6*(1485*B^4*a^4*c^2 + 13170*A^2*B^2*a^3
*c^3 + 3661*A^4*a^2*c^4)*d^4*e^10 - 48*(585*A*B^3*a^4*c^2 + 1073*A^3*B*a^3*c^3)*
d^3*e^11 + 4*(405*B^4*a^5*c + 6579*A^2*B^2*a^4*c^2 + 1925*A^4*a^3*c^3)*d^2*e^12
- 276*(9*A*B^3*a^5*c + 25*A^3*B*a^4*c^2)*d*e^13 + (81*B^4*a^6 + 450*A^2*B^2*a^5*
c + 625*A^4*a^4*c^2)*e^14)/(a^3*c^11*d^20 - 10*a^4*c^10*d^18*e^2 + 45*a^5*c^9*d^
16*e^4 - 120*a^6*c^8*d^14*e^6 + 210*a^7*c^7*d^12*e^8 - 252*a^8*c^6*d^10*e^10 + 2
10*a^9*c^5*d^8*e^12 - 120*a^10*c^4*d^6*e^14 + 45*a^11*c^3*d^4*e^16 - 10*a^12*c^2
*d^2*e^18 + a^13*c*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^
4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))*log(-(120*A^3*B*c^5*d^7*e^
2 + 20*(27*A^2*B^2*a*c^4 - 7*A^4*c^5)*d^6*e^3 + 270*(3*A*B^3*a^2*c^3 - 7*A^3*B*a
*c^4)*d^5*e^4 + 3*(135*B^4*a^3*c^2 - 1080*A^2*B^2*a^2*c^3 + 497*A^4*a*c^4)*d^4*e
^5 - 36*(15*A*B^3*a^3*c^2 - 161*A^3*B*a^2*c^3)*d^3*e^6 + 6*(135*B^4*a^4*c - 414*
A^2*B^2*a^3*c^2 - 625*A^4*a^2*c^3)*d^2*e^7 - 6*(189*A*B^3*a^4*c - 625*A^3*B*a^3*
c^2)*d*e^8 + (81*B^4*a^5 - 625*A^4*a^3*c^2)*e^9)*sqrt(e*x + d) + (30*A^2*B*a^2*c
^5*d^8*e^3 + 35*(9*A*B^2*a^3*c^4 - A^3*a^2*c^5)*d^7*e^4 + 15*(27*B^3*a^4*c^3 - 5
9*A^2*B*a^3*c^4)*d^6*e^5 - 3*(675*A*B^2*a^4*c^3 - 203*A^3*a^3*c^4)*d^5*e^6 + 27*
(35*B^3*a^5*c^2 + 129*A^2*B*a^4*c^3)*d^4*e^7 - 3*(981*A*B^2*a^5*c^2 + 659*A^3*a^
4*c^3)*d^3*e^8 + 3*(117*B^3*a^6*c + 827*A^2*B*a^5*c^2)*d^2*e^9 - (531*A*B^2*a^6*
c + 325*A^3*a^5*c^2)*d*e^10 + 3*(9*B^3*a^7 + 25*A^2*B*a^6*c)*e^11 + (2*A*a^3*c^8
*d^14 + 3*B*a^4*c^7*d^13*e - 19*A*a^4*c^7*d^12*e^2 - 6*B*a^5*c^6*d^11*e^3 + 60*A
*a^5*c^6*d^10*e^4 - 15*B*a^6*c^5*d^9*e^5 - 85*A*a^6*c^5*d^8*e^6 + 60*B*a^7*c^4*d
^7*e^7 + 50*A*a^7*c^4*d^6*e^8 - 75*B*a^8*c^3*d^5*e^9 + 3*A*a^8*c^3*d^4*e^10 + 42
*B*a^9*c^2*d^3*e^11 - 16*A*a^9*c^2*d^2*e^12 - 9*B*a^10*c*d*e^13 + 5*A*a^10*c*e^1
4)*sqrt((900*A^2*B^2*c^6*d^10*e^4 + 300*(9*A*B^3*a*c^5 - 7*A^3*B*c^6)*d^9*e^5 +
25*(81*B^4*a^2*c^4 - 558*A^2*B^2*a*c^5 + 49*A^4*c^6)*d^8*e^6 - 240*(45*A*B^3*a^2
*c^4 - 91*A^3*B*a*c^5)*d^7*e^7 + 20*(405*B^4*a^3*c^3 + 1584*A^2*B^2*a^2*c^4 - 53
9*A^4*a*c^5)*d^6*e^8 - 44280*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^9 + 6*(1485*B
^4*a^4*c^2 + 13170*A^2*B^2*a^3*c^3 + 3661*A^4*a^2*c^4)*d^4*e^10 - 48*(585*A*B^3*
a^4*c^2 + 1073*A^3*B*a^3*c^3)*d^3*e^11 + 4*(405*B^4*a^5*c + 6579*A^2*B^2*a^4*c^2
+ 1925*A^4*a^3*c^3)*d^2*e^12 - 276*(9*A*B^3*a^5*c + 25*A^3*B*a^4*c^2)*d*e^13 +
(81*B^4*a^6 + 450*A^2*B^2*a^5*c + 625*A^4*a^4*c^2)*e^14)/(a^3*c^11*d^20 - 10*a^4
*c^10*d^18*e^2 + 45*a^5*c^9*d^16*e^4 - 120*a^6*c^8*d^14*e^6 + 210*a^7*c^7*d^12*e
^8 - 252*a^8*c^6*d^10*e^10 + 210*a^9*c^5*d^8*e^12 - 120*a^10*c^4*d^6*e^14 + 45*a
^11*c^3*d^4*e^16 - 10*a^12*c^2*d^2*e^18 + a^13*c*e^20)))*sqrt((4*A^2*c^4*d^7 + 1
2*A*B*a*c^3*d^6*e - 30*A*B*a^2*c^2*d^4*e^3 - 240*A*B*a^3*c*d^2*e^5 - 30*A*B*a^4*
e^7 + (9*B^2*a^2*c^2 - 35*A^2*a*c^3)*d^5*e^2 + 10*(9*B^2*a^3*c + 7*A^2*a^2*c^2)*
d^3*e^4 + 15*(3*B^2*a^4 + 7*A^2*a^3*c)*d*e^6 - (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2
+ 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((9
00*A^2*B^2*c^6*d^10*e^4 + 300*(9*A*B^3*a*c^5 - 7*A^3*B*c^6)*d^9*e^5 + 25*(81*B^4
*a^2*c^4 - 558*A^2*B^2*a*c^5 + 49*A^4*c^6)*d^8*e^6 - 240*(45*A*B^3*a^2*c^4 - 91*
A^3*B*a*c^5)*d^7*e^7 + 20*(405*B^4*a^3*c^3 + 1584*A^2*B^2*a^2*c^4 - 539*A^4*a*c^
5)*d^6*e^8 - 44280*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^9 + 6*(1485*B^4*a^4*c^2
+ 13170*A^2*B^2*a^3*c^3 + 3661*A^4*a^2*c^4)*d^4*e^10 - 48*(585*A*B^3*a^4*c^2 +
1073*A^3*B*a^3*c^3)*d^3*e^11 + 4*(405*B^4*a^5*c + 6579*A^2*B^2*a^4*c^2 + 1925*A^
4*a^3*c^3)*d^2*e^12 - 276*(9*A*B^3*a^5*c + 25*A^3*B*a^4*c^2)*d*e^13 + (81*B^4*a^
6 + 450*A^2*B^2*a^5*c + 625*A^4*a^4*c^2)*e^14)/(a^3*c^11*d^20 - 10*a^4*c^10*d^18
*e^2 + 45*a^5*c^9*d^16*e^4 - 120*a^6*c^8*d^14*e^6 + 210*a^7*c^7*d^12*e^8 - 252*a
^8*c^6*d^10*e^10 + 210*a^9*c^5*d^8*e^12 - 120*a^10*c^4*d^6*e^14 + 45*a^11*c^3*d^
4*e^16 - 10*a^12*c^2*d^2*e^18 + a^13*c*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2
+ 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))) + (a^
2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4 - (a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e
^4)*x^2)*sqrt(e*x + d)*sqrt((4*A^2*c^4*d^7 + 12*A*B*a*c^3*d^6*e - 30*A*B*a^2*c^2
*d^4*e^3 - 240*A*B*a^3*c*d^2*e^5 - 30*A*B*a^4*e^7 + (9*B^2*a^2*c^2 - 35*A^2*a*c^
3)*d^5*e^2 + 10*(9*B^2*a^3*c + 7*A^2*a^2*c^2)*d^3*e^4 + 15*(3*B^2*a^4 + 7*A^2*a^
3*c)*d*e^6 - (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2
*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((900*A^2*B^2*c^6*d^10*e^4 + 300*(9*A
*B^3*a*c^5 - 7*A^3*B*c^6)*d^9*e^5 + 25*(81*B^4*a^2*c^4 - 558*A^2*B^2*a*c^5 + 49*
A^4*c^6)*d^8*e^6 - 240*(45*A*B^3*a^2*c^4 - 91*A^3*B*a*c^5)*d^7*e^7 + 20*(405*B^4
*a^3*c^3 + 1584*A^2*B^2*a^2*c^4 - 539*A^4*a*c^5)*d^6*e^8 - 44280*(A*B^3*a^3*c^3
+ A^3*B*a^2*c^4)*d^5*e^9 + 6*(1485*B^4*a^4*c^2 + 13170*A^2*B^2*a^3*c^3 + 3661*A^
4*a^2*c^4)*d^4*e^10 - 48*(585*A*B^3*a^4*c^2 + 1073*A^3*B*a^3*c^3)*d^3*e^11 + 4*(
405*B^4*a^5*c + 6579*A^2*B^2*a^4*c^2 + 1925*A^4*a^3*c^3)*d^2*e^12 - 276*(9*A*B^3
*a^5*c + 25*A^3*B*a^4*c^2)*d*e^13 + (81*B^4*a^6 + 450*A^2*B^2*a^5*c + 625*A^4*a^
4*c^2)*e^14)/(a^3*c^11*d^20 - 10*a^4*c^10*d^18*e^2 + 45*a^5*c^9*d^16*e^4 - 120*a
^6*c^8*d^14*e^6 + 210*a^7*c^7*d^12*e^8 - 252*a^8*c^6*d^10*e^10 + 210*a^9*c^5*d^8
*e^12 - 120*a^10*c^4*d^6*e^14 + 45*a^11*c^3*d^4*e^16 - 10*a^12*c^2*d^2*e^18 + a^
13*c*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 - 10*a^6*c^2
*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))*log(-(120*A^3*B*c^5*d^7*e^2 + 20*(27*A^2
*B^2*a*c^4 - 7*A^4*c^5)*d^6*e^3 + 270*(3*A*B^3*a^2*c^3 - 7*A^3*B*a*c^4)*d^5*e^4
+ 3*(135*B^4*a^3*c^2 - 1080*A^2*B^2*a^2*c^3 + 497*A^4*a*c^4)*d^4*e^5 - 36*(15*A*
B^3*a^3*c^2 - 161*A^3*B*a^2*c^3)*d^3*e^6 + 6*(135*B^4*a^4*c - 414*A^2*B^2*a^3*c^
2 - 625*A^4*a^2*c^3)*d^2*e^7 - 6*(189*A*B^3*a^4*c - 625*A^3*B*a^3*c^2)*d*e^8 + (
81*B^4*a^5 - 625*A^4*a^3*c^2)*e^9)*sqrt(e*x + d) - (30*A^2*B*a^2*c^5*d^8*e^3 + 3
5*(9*A*B^2*a^3*c^4 - A^3*a^2*c^5)*d^7*e^4 + 15*(27*B^3*a^4*c^3 - 59*A^2*B*a^3*c^
4)*d^6*e^5 - 3*(675*A*B^2*a^4*c^3 - 203*A^3*a^3*c^4)*d^5*e^6 + 27*(35*B^3*a^5*c^
2 + 129*A^2*B*a^4*c^3)*d^4*e^7 - 3*(981*A*B^2*a^5*c^2 + 659*A^3*a^4*c^3)*d^3*e^8
+ 3*(117*B^3*a^6*c + 827*A^2*B*a^5*c^2)*d^2*e^9 - (531*A*B^2*a^6*c + 325*A^3*a^
5*c^2)*d*e^10 + 3*(9*B^3*a^7 + 25*A^2*B*a^6*c)*e^11 + (2*A*a^3*c^8*d^14 + 3*B*a^
4*c^7*d^13*e - 19*A*a^4*c^7*d^12*e^2 - 6*B*a^5*c^6*d^11*e^3 + 60*A*a^5*c^6*d^10*
e^4 - 15*B*a^6*c^5*d^9*e^5 - 85*A*a^6*c^5*d^8*e^6 + 60*B*a^7*c^4*d^7*e^7 + 50*A*
a^7*c^4*d^6*e^8 - 75*B*a^8*c^3*d^5*e^9 + 3*A*a^8*c^3*d^4*e^10 + 42*B*a^9*c^2*d^3
*e^11 - 16*A*a^9*c^2*d^2*e^12 - 9*B*a^10*c*d*e^13 + 5*A*a^10*c*e^14)*sqrt((900*A
^2*B^2*c^6*d^10*e^4 + 300*(9*A*B^3*a*c^5 - 7*A^3*B*c^6)*d^9*e^5 + 25*(81*B^4*a^2
*c^4 - 558*A^2*B^2*a*c^5 + 49*A^4*c^6)*d^8*e^6 - 240*(45*A*B^3*a^2*c^4 - 91*A^3*
B*a*c^5)*d^7*e^7 + 20*(405*B^4*a^3*c^3 + 1584*A^2*B^2*a^2*c^4 - 539*A^4*a*c^5)*d
^6*e^8 - 44280*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^9 + 6*(1485*B^4*a^4*c^2 + 1
3170*A^2*B^2*a^3*c^3 + 3661*A^4*a^2*c^4)*d^4*e^10 - 48*(585*A*B^3*a^4*c^2 + 1073
*A^3*B*a^3*c^3)*d^3*e^11 + 4*(405*B^4*a^5*c + 6579*A^2*B^2*a^4*c^2 + 1925*A^4*a^
3*c^3)*d^2*e^12 - 276*(9*A*B^3*a^5*c + 25*A^3*B*a^4*c^2)*d*e^13 + (81*B^4*a^6 +
450*A^2*B^2*a^5*c + 625*A^4*a^4*c^2)*e^14)/(a^3*c^11*d^20 - 10*a^4*c^10*d^18*e^2
+ 45*a^5*c^9*d^16*e^4 - 120*a^6*c^8*d^14*e^6 + 210*a^7*c^7*d^12*e^8 - 252*a^8*c
^6*d^10*e^10 + 210*a^9*c^5*d^8*e^12 - 120*a^10*c^4*d^6*e^14 + 45*a^11*c^3*d^4*e^
16 - 10*a^12*c^2*d^2*e^18 + a^13*c*e^20)))*sqrt((4*A^2*c^4*d^7 + 12*A*B*a*c^3*d^
6*e - 30*A*B*a^2*c^2*d^4*e^3 - 240*A*B*a^3*c*d^2*e^5 - 30*A*B*a^4*e^7 + (9*B^2*a
^2*c^2 - 35*A^2*a*c^3)*d^5*e^2 + 10*(9*B^2*a^3*c + 7*A^2*a^2*c^2)*d^3*e^4 + 15*(
3*B^2*a^4 + 7*A^2*a^3*c)*d*e^6 - (a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*
d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10)*sqrt((900*A^2*B^2*c^6
*d^10*e^4 + 300*(9*A*B^3*a*c^5 - 7*A^3*B*c^6)*d^9*e^5 + 25*(81*B^4*a^2*c^4 - 558
*A^2*B^2*a*c^5 + 49*A^4*c^6)*d^8*e^6 - 240*(45*A*B^3*a^2*c^4 - 91*A^3*B*a*c^5)*d
^7*e^7 + 20*(405*B^4*a^3*c^3 + 1584*A^2*B^2*a^2*c^4 - 539*A^4*a*c^5)*d^6*e^8 - 4
4280*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^9 + 6*(1485*B^4*a^4*c^2 + 13170*A^2*B
^2*a^3*c^3 + 3661*A^4*a^2*c^4)*d^4*e^10 - 48*(585*A*B^3*a^4*c^2 + 1073*A^3*B*a^3
*c^3)*d^3*e^11 + 4*(405*B^4*a^5*c + 6579*A^2*B^2*a^4*c^2 + 1925*A^4*a^3*c^3)*d^2
*e^12 - 276*(9*A*B^3*a^5*c + 25*A^3*B*a^4*c^2)*d*e^13 + (81*B^4*a^6 + 450*A^2*B^
2*a^5*c + 625*A^4*a^4*c^2)*e^14)/(a^3*c^11*d^20 - 10*a^4*c^10*d^18*e^2 + 45*a^5*
c^9*d^16*e^4 - 120*a^6*c^8*d^14*e^6 + 210*a^7*c^7*d^12*e^8 - 252*a^8*c^6*d^10*e^
10 + 210*a^9*c^5*d^8*e^12 - 120*a^10*c^4*d^6*e^14 + 45*a^11*c^3*d^4*e^16 - 10*a^
12*c^2*d^2*e^18 + a^13*c*e^20)))/(a^3*c^5*d^10 - 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*
d^6*e^4 - 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 - a^8*e^10))) + 4*(A*c^2*d^3 - B*
a*c*d^2*e - A*a*c*d*e^2 + B*a^2*e^3)*x)/((a^2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^
4 - (a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x + d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(e*x+d)**(3/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 - a)^2*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out