3.1452 \(\int \frac{A+B x}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx\)

Optimal. Leaf size=243 \[ -\frac{2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{2 \left (a B e^2-2 A c d e+B c d^2\right )}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 1.16915, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{2 \left (a B e^2-2 A c d e+B c d^2\right )}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^(5/2)*(a - c*x^2)),x]

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 1.4287, size = 269, normalized size = 1.11 \[ -\frac{2 \left (a A e^3+a B e^2 (2 d+3 e x)-A c d e (7 d+6 e x)+B c d^2 (4 d+3 e x)\right )}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{\left (A \sqrt{c}-\sqrt{a} B\right ) \sqrt{c d-\sqrt{a} \sqrt{c} e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} \left (\sqrt{a} e-\sqrt{c} d\right )^3}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \sqrt{\sqrt{a} \sqrt{c} e+c d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^(5/2)*(a - c*x^2)),x]

[Out]

(-2*(a*A*e^3 + a*B*e^2*(2*d + 3*e*x) + B*c*d^2*(4*d + 3*e*x) - A*c*d*e*(7*d + 6*
e*x)))/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + ((-(Sqrt[a]*B) + A*Sqrt[c])*Sqrt[
c*d - Sqrt[a]*Sqrt[c]*e]*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt
[c]*e]])/(Sqrt[a]*(-(Sqrt[c]*d) + Sqrt[a]*e)^3) + ((Sqrt[a]*B + A*Sqrt[c])*Sqrt[
c*d + Sqrt[a]*Sqrt[c]*e]*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt
[c]*e]])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)^3)

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Maple [B]  time = 0.051, size = 973, normalized size = 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)^(5/2)/(-c*x^2+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(B*x + A)/((c*x^2 - a)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

-integrate((B*x + A)/((c*x^2 - a)*(e*x + d)^(5/2)), x)

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Fricas [A]  time = 15.5275, size = 14954, normalized size = 61.54 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(B*x + A)/((c*x^2 - a)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

-1/6*(16*B*c*d^3 - 28*A*c*d^2*e + 8*B*a*d*e^2 + 4*A*a*e^3 - 3*(c^2*d^5 - 2*a*c*d
^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*x + d)*sqrt
(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 +
A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2
)*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*
e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A
^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240
*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5
+ 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^
4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 +
A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*
e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2
+ A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 -
120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^
4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a
^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^
4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log((2*(A*B^3*a*c^4 - A^3*B*c^5)*d^5 - 5*(B
^4*a^2*c^3 - A^4*c^5)*d^4*e + 20*(A*B^3*a^2*c^3 - A^3*B*a*c^4)*d^3*e^2 - 10*(B^4
*a^3*c^2 - A^4*a*c^4)*d^2*e^3 + 10*(A*B^3*a^3*c^2 - A^3*B*a^2*c^3)*d*e^4 - (B^4*
a^4*c - A^4*a^2*c^3)*e^5)*sqrt(e*x + d) + (2*A*B^2*a*c^5*d^8 - (5*B^3*a^2*c^4 +
11*A^2*B*a*c^5)*d^7*e + (41*A*B^2*a^2*c^4 + 15*A^3*a*c^5)*d^6*e^2 - (25*B^3*a^3*
c^3 + 87*A^2*B*a^2*c^4)*d^5*e^3 + 35*(3*A*B^2*a^3*c^3 + A^3*a^2*c^4)*d^4*e^4 - (
31*B^3*a^4*c^2 + 81*A^2*B*a^3*c^3)*d^3*e^5 + (43*A*B^2*a^4*c^2 + 13*A^3*a^3*c^3)
*d^2*e^6 - (3*B^3*a^5*c + 13*A^2*B*a^4*c^2)*d*e^7 + (A*B^2*a^5*c + A^3*a^4*c^2)*
e^8 + (A*a*c^7*d^13 - 3*B*a^2*c^6*d^12*e - 2*A*a^2*c^6*d^11*e^2 + 14*B*a^3*c^5*d
^10*e^3 - 5*A*a^3*c^5*d^9*e^4 - 25*B*a^4*c^4*d^8*e^5 + 20*A*a^4*c^4*d^7*e^6 + 20
*B*a^5*c^3*d^6*e^7 - 25*A*a^5*c^3*d^5*e^8 - 5*B*a^6*c^2*d^4*e^9 + 14*A*a^6*c^2*d
^3*e^10 - 2*B*a^7*c*d^2*e^11 - 3*A*a^7*c*d*e^12 + B*a^8*e^13)*sqrt((4*A^2*B^2*c^
7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^
6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a
^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*
a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4
*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2
*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^
4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*
e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^
6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e
^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c
^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^2
*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^
8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sq
rt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 +
26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e
^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*
a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*
A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5
*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*
c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 -
10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d
^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 +
45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*
d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))
) + 3*(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^
5)*x)*sqrt(e*x + d)*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a
^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*
(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d
^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^
10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 +
5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c
^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*
c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6
- 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4
*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^
6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2
+ 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^
5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16
- 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3
*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log((2*(A*B^3*a*c^4
- A^3*B*c^5)*d^5 - 5*(B^4*a^2*c^3 - A^4*c^5)*d^4*e + 20*(A*B^3*a^2*c^3 - A^3*B*
a*c^4)*d^3*e^2 - 10*(B^4*a^3*c^2 - A^4*a*c^4)*d^2*e^3 + 10*(A*B^3*a^3*c^2 - A^3*
B*a^2*c^3)*d*e^4 - (B^4*a^4*c - A^4*a^2*c^3)*e^5)*sqrt(e*x + d) - (2*A*B^2*a*c^5
*d^8 - (5*B^3*a^2*c^4 + 11*A^2*B*a*c^5)*d^7*e + (41*A*B^2*a^2*c^4 + 15*A^3*a*c^5
)*d^6*e^2 - (25*B^3*a^3*c^3 + 87*A^2*B*a^2*c^4)*d^5*e^3 + 35*(3*A*B^2*a^3*c^3 +
A^3*a^2*c^4)*d^4*e^4 - (31*B^3*a^4*c^2 + 81*A^2*B*a^3*c^3)*d^3*e^5 + (43*A*B^2*a
^4*c^2 + 13*A^3*a^3*c^3)*d^2*e^6 - (3*B^3*a^5*c + 13*A^2*B*a^4*c^2)*d*e^7 + (A*B
^2*a^5*c + A^3*a^4*c^2)*e^8 + (A*a*c^7*d^13 - 3*B*a^2*c^6*d^12*e - 2*A*a^2*c^6*d
^11*e^2 + 14*B*a^3*c^5*d^10*e^3 - 5*A*a^3*c^5*d^9*e^4 - 25*B*a^4*c^4*d^8*e^5 + 2
0*A*a^4*c^4*d^7*e^6 + 20*B*a^5*c^3*d^6*e^7 - 25*A*a^5*c^3*d^5*e^8 - 5*B*a^6*c^2*
d^4*e^9 + 14*A*a^6*c^2*d^3*e^10 - 2*B*a^7*c*d^2*e^11 - 3*A*a^7*c*d*e^12 + B*a^8*
e^13)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a
^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^
6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504
*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c
^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*
(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^
3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10
*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a
^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6
*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(10*A*B*a*
c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5
- 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 + (a
*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*
c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^
9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*
c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^
6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 +
62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^
4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*
B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^
3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*
d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 -
120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/
(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^
5*c*d^2*e^8 - a^6*e^10))) - 3*(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e
- 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*x + d)*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a
^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 +
A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 - (a*c^5*d^10 - 5*a^2*c^
4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10
)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c
^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d
^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*
B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 +
11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4
*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*
a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^2
0 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c
^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^1
4 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*
c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^
10))*log((2*(A*B^3*a*c^4 - A^3*B*c^5)*d^5 - 5*(B^4*a^2*c^3 - A^4*c^5)*d^4*e + 20
*(A*B^3*a^2*c^3 - A^3*B*a*c^4)*d^3*e^2 - 10*(B^4*a^3*c^2 - A^4*a*c^4)*d^2*e^3 +
10*(A*B^3*a^3*c^2 - A^3*B*a^2*c^3)*d*e^4 - (B^4*a^4*c - A^4*a^2*c^3)*e^5)*sqrt(e
*x + d) + (2*A*B^2*a*c^5*d^8 - (5*B^3*a^2*c^4 + 11*A^2*B*a*c^5)*d^7*e + (41*A*B^
2*a^2*c^4 + 15*A^3*a*c^5)*d^6*e^2 - (25*B^3*a^3*c^3 + 87*A^2*B*a^2*c^4)*d^5*e^3
+ 35*(3*A*B^2*a^3*c^3 + A^3*a^2*c^4)*d^4*e^4 - (31*B^3*a^4*c^2 + 81*A^2*B*a^3*c^
3)*d^3*e^5 + (43*A*B^2*a^4*c^2 + 13*A^3*a^3*c^3)*d^2*e^6 - (3*B^3*a^5*c + 13*A^2
*B*a^4*c^2)*d*e^7 + (A*B^2*a^5*c + A^3*a^4*c^2)*e^8 - (A*a*c^7*d^13 - 3*B*a^2*c^
6*d^12*e - 2*A*a^2*c^6*d^11*e^2 + 14*B*a^3*c^5*d^10*e^3 - 5*A*a^3*c^5*d^9*e^4 -
25*B*a^4*c^4*d^8*e^5 + 20*A*a^4*c^4*d^7*e^6 + 20*B*a^5*c^3*d^6*e^7 - 25*A*a^5*c^
3*d^5*e^8 - 5*B*a^6*c^2*d^4*e^9 + 14*A*a^6*c^2*d^3*e^10 - 2*B*a^7*c*d^2*e^11 - 3
*A*a^7*c*d*e^12 + B*a^8*e^13)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B
*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*
B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*
A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^
4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*
B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8
- 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^
4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*
a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^
8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*
e^20)))*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (
B^2*a*c^3 + A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c +
A^2*a^2*c^2)*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*
a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B
^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d
^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*
B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5
+ 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^
3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a
^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*
B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8
*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10
+ 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*
d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 1
0*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) + 3*(c^2*d^5 - 2*a*c*d^3*e^2 +
a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*x + d)*sqrt(-(10*A*
B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)
*d^5 - 10*(B^2*a^2*c^2 + A^2*a*c^3)*d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4
- (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*
a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7
)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*
a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*
a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^
3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^
3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20
*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^
4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*
c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^
12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20
)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 +
5*a^5*c*d^2*e^8 - a^6*e^10))*log((2*(A*B^3*a*c^4 - A^3*B*c^5)*d^5 - 5*(B^4*a^2*c
^3 - A^4*c^5)*d^4*e + 20*(A*B^3*a^2*c^3 - A^3*B*a*c^4)*d^3*e^2 - 10*(B^4*a^3*c^2
- A^4*a*c^4)*d^2*e^3 + 10*(A*B^3*a^3*c^2 - A^3*B*a^2*c^3)*d*e^4 - (B^4*a^4*c -
A^4*a^2*c^3)*e^5)*sqrt(e*x + d) - (2*A*B^2*a*c^5*d^8 - (5*B^3*a^2*c^4 + 11*A^2*B
*a*c^5)*d^7*e + (41*A*B^2*a^2*c^4 + 15*A^3*a*c^5)*d^6*e^2 - (25*B^3*a^3*c^3 + 87
*A^2*B*a^2*c^4)*d^5*e^3 + 35*(3*A*B^2*a^3*c^3 + A^3*a^2*c^4)*d^4*e^4 - (31*B^3*a
^4*c^2 + 81*A^2*B*a^3*c^3)*d^3*e^5 + (43*A*B^2*a^4*c^2 + 13*A^3*a^3*c^3)*d^2*e^6
- (3*B^3*a^5*c + 13*A^2*B*a^4*c^2)*d*e^7 + (A*B^2*a^5*c + A^3*a^4*c^2)*e^8 - (A
*a*c^7*d^13 - 3*B*a^2*c^6*d^12*e - 2*A*a^2*c^6*d^11*e^2 + 14*B*a^3*c^5*d^10*e^3
- 5*A*a^3*c^5*d^9*e^4 - 25*B*a^4*c^4*d^8*e^5 + 20*A*a^4*c^4*d^7*e^6 + 20*B*a^5*c
^3*d^6*e^7 - 25*A*a^5*c^3*d^5*e^8 - 5*B*a^6*c^2*d^4*e^9 + 14*A*a^6*c^2*d^3*e^10
- 2*B*a^7*c*d^2*e^11 - 3*A*a^7*c*d*e^12 + B*a^8*e^13)*sqrt((4*A^2*B^2*c^7*d^10 -
20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*B^2*a*c^6 + 5*A^
4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*(5*B^4*a^3*c^4 +
32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4 + A^3*B*a^2*c^5)
*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*c^5)*d^4*e^6 - 2
40*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7*A^2*B^2*a^4*c^3
+ A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e^9 + (B^4*a^6*c
+ 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45
*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^
10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10
*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(10*A*B*a*c^3*d^4*e + 20*A*B*a^2*c^2*d^2*e
^3 + 2*A*B*a^3*c*e^5 - (B^2*a*c^3 + A^2*c^4)*d^5 - 10*(B^2*a^2*c^2 + A^2*a*c^3)*
d^3*e^2 - 5*(B^2*a^3*c + A^2*a^2*c^2)*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 +
10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((4*A^
2*B^2*c^7*d^10 - 20*(A*B^3*a*c^6 + A^3*B*c^7)*d^9*e + 5*(5*B^4*a^2*c^5 + 26*A^2*
B^2*a*c^6 + 5*A^4*c^7)*d^8*e^2 - 240*(A*B^3*a^2*c^5 + A^3*B*a*c^6)*d^7*e^3 + 20*
(5*B^4*a^3*c^4 + 32*A^2*B^2*a^2*c^5 + 5*A^4*a*c^6)*d^6*e^4 - 504*(A*B^3*a^3*c^4
+ A^3*B*a^2*c^5)*d^5*e^5 + 10*(11*B^4*a^4*c^3 + 62*A^2*B^2*a^3*c^4 + 11*A^4*a^2*
c^5)*d^4*e^6 - 240*(A*B^3*a^4*c^3 + A^3*B*a^3*c^4)*d^3*e^7 + 20*(B^4*a^5*c^2 + 7
*A^2*B^2*a^4*c^3 + A^4*a^3*c^4)*d^2*e^8 - 20*(A*B^3*a^5*c^2 + A^3*B*a^4*c^3)*d*e
^9 + (B^4*a^6*c + 2*A^2*B^2*a^5*c^2 + A^4*a^4*c^3)*e^10)/(a*c^10*d^20 - 10*a^2*c
^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8
- 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c
^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2
+ 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) + 12*(
B*c*d^2*e - 2*A*c*d*e^2 + B*a*e^3)*x)/((c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c
^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*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)**(5/2)/(-c*x**2+a),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)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Timed out