3.159 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^3} \, dx\)

Optimal. Leaf size=320 \[ -\frac{3 (2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2} c^4}+\frac{3 d^{5/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{7/2}}+\frac{d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{4 a c^3 \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{3 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{4 a^2 c^3 \sqrt{a+\frac{b}{x}} (b c-a d)^3}+\frac{d (2 b c-3 a d)}{2 a c^2 \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x}{a c \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2} \]

[Out]

(3*b*(2*b*c - a*d)*(2*b^2*c^2 - a*b*c*d + 4*a^2*d^2))/(4*a^2*c^3*(b*c - a*d)^3*S
qrt[a + b/x]) + (d*(2*b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*Sqrt[a + b/x]*(c + d/x)
^2) + (d*(4*b^2*c^2 - 21*a*b*c*d + 12*a^2*d^2))/(4*a*c^3*(b*c - a*d)^2*Sqrt[a +
b/x]*(c + d/x)) + x/(a*c*Sqrt[a + b/x]*(c + d/x)^2) + (3*d^(5/2)*(21*b^2*c^2 - 2
4*a*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*(
b*c - a*d)^(7/2)) - (3*(b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(5/2)*c^
4)

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Rubi [A]  time = 1.52398, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{3 (2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2} c^4}+\frac{3 d^{5/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{7/2}}+\frac{d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{4 a c^3 \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{3 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{4 a^2 c^3 \sqrt{a+\frac{b}{x}} (b c-a d)^3}+\frac{d (2 b c-3 a d)}{2 a c^2 \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x}{a c \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(3/2)*(c + d/x)^3),x]

[Out]

(3*b*(2*b*c - a*d)*(2*b^2*c^2 - a*b*c*d + 4*a^2*d^2))/(4*a^2*c^3*(b*c - a*d)^3*S
qrt[a + b/x]) + (d*(2*b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*Sqrt[a + b/x]*(c + d/x)
^2) + (d*(4*b^2*c^2 - 21*a*b*c*d + 12*a^2*d^2))/(4*a*c^3*(b*c - a*d)^2*Sqrt[a +
b/x]*(c + d/x)) + x/(a*c*Sqrt[a + b/x]*(c + d/x)^2) + (3*d^(5/2)*(21*b^2*c^2 - 2
4*a*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*(
b*c - a*d)^(7/2)) - (3*(b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(5/2)*c^
4)

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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(1/(a+b/x)**(3/2)/(c+d/x)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 1.82433, size = 385, normalized size = 1.2 \[ \frac{-\frac{12 (2 a d+b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{5/2}}+\frac{3 i d^{5/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \log \left (-\frac{8 i c^5 (b c-a d)^{5/2} \left (-2 i \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 a d x+b c x-b d\right )}{3 d^{7/2} (c x+d) \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right )}\right )}{(b c-a d)^{7/2}}+\frac{2 c x \sqrt{a+\frac{b}{x}} \left (2 a^4 d^3 x \left (2 c^2 x^2+9 c d x+6 d^2\right )+a^3 b d^2 \left (-12 c^3 x^3-37 c^2 d x^2-9 c d^2 x+12 d^3\right )+a^2 b^2 c d \left (12 c^3 x^3+12 c^2 d x^2-29 c d^2 x-27 d^3\right )-4 a b^3 c^2 (c x-3 d) (c x+d)^2-12 b^4 c^3 (c x+d)^2\right )}{a^2 (a x+b) (c x+d)^2 (a d-b c)^3}}{8 c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(3/2)*(c + d/x)^3),x]

[Out]

((2*c*Sqrt[a + b/x]*x*(-12*b^4*c^3*(d + c*x)^2 - 4*a*b^3*c^2*(-3*d + c*x)*(d + c
*x)^2 + 2*a^4*d^3*x*(6*d^2 + 9*c*d*x + 2*c^2*x^2) + a^3*b*d^2*(12*d^3 - 9*c*d^2*
x - 37*c^2*d*x^2 - 12*c^3*x^3) + a^2*b^2*c*d*(-27*d^3 - 29*c*d^2*x + 12*c^2*d*x^
2 + 12*c^3*x^3)))/(a^2*(-(b*c) + a*d)^3*(b + a*x)*(d + c*x)^2) - (12*(b*c + 2*a*
d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/a^(5/2) + ((3*I)*d^(5/2)*(21*b^2*
c^2 - 24*a*b*c*d + 8*a^2*d^2)*Log[(((-8*I)/3)*c^5*(b*c - a*d)^(5/2)*(-(b*d) + b*
c*x - 2*a*d*x - (2*I)*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[a + b/x]*x))/(d^(7/2)*(21*b^2
*c^2 - 24*a*b*c*d + 8*a^2*d^2)*(d + c*x))])/(b*c - a*d)^(7/2))/(8*c^4)

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Maple [B]  time = 0.027, size = 5164, normalized size = 16.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/(c+d/x)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*(c + d/x)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.24754, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*(c + d/x)^3),x, algorithm="fricas")

[Out]

[-1/8*(3*(21*a^2*b^2*c^2*d^4 - 24*a^3*b*c*d^5 + 8*a^4*d^6 + (21*a^2*b^2*c^4*d^2
- 24*a^3*b*c^3*d^3 + 8*a^4*c^2*d^4)*x^2 + 2*(21*a^2*b^2*c^3*d^3 - 24*a^3*b*c^2*d
^4 + 8*a^4*c*d^5)*x)*sqrt(a)*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x)*log(-(2*(b*c
- a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x +
d)) - 12*(b^4*c^4*d^2 - a*b^3*c^3*d^3 - 3*a^2*b^2*c^2*d^4 + 5*a^3*b*c*d^5 - 2*a
^4*d^6 + (b^4*c^6 - a*b^3*c^5*d - 3*a^2*b^2*c^4*d^2 + 5*a^3*b*c^3*d^3 - 2*a^4*c^
2*d^4)*x^2 + 2*(b^4*c^5*d - a*b^3*c^4*d^2 - 3*a^2*b^2*c^3*d^3 + 5*a^3*b*c^2*d^4
- 2*a^4*c*d^5)*x)*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*s
qrt(a)) - 2*(12*b^4*c^4*d^2 - 12*a*b^3*c^3*d^3 + 27*a^2*b^2*c^2*d^4 - 12*a^3*b*c
*d^5 + 4*(a*b^3*c^6 - 3*a^2*b^2*c^5*d + 3*a^3*b*c^4*d^2 - a^4*c^3*d^3)*x^3 + (12
*b^4*c^6 - 4*a*b^3*c^5*d - 12*a^2*b^2*c^4*d^2 + 37*a^3*b*c^3*d^3 - 18*a^4*c^2*d^
4)*x^2 + (24*b^4*c^5*d - 20*a*b^3*c^4*d^2 + 29*a^2*b^2*c^3*d^3 + 9*a^3*b*c^2*d^4
- 12*a^4*c*d^5)*x)*sqrt(a))/((a^2*b^3*c^7*d^2 - 3*a^3*b^2*c^6*d^3 + 3*a^4*b*c^5
*d^4 - a^5*c^4*d^5 + (a^2*b^3*c^9 - 3*a^3*b^2*c^8*d + 3*a^4*b*c^7*d^2 - a^5*c^6*
d^3)*x^2 + 2*(a^2*b^3*c^8*d - 3*a^3*b^2*c^7*d^2 + 3*a^4*b*c^6*d^3 - a^5*c^5*d^4)
*x)*sqrt(a)*sqrt((a*x + b)/x)), -1/8*(3*(21*a^2*b^2*c^2*d^4 - 24*a^3*b*c*d^5 + 8
*a^4*d^6 + (21*a^2*b^2*c^4*d^2 - 24*a^3*b*c^3*d^3 + 8*a^4*c^2*d^4)*x^2 + 2*(21*a
^2*b^2*c^3*d^3 - 24*a^3*b*c^2*d^4 + 8*a^4*c*d^5)*x)*sqrt(-a)*sqrt(-d/(b*c - a*d)
)*sqrt((a*x + b)/x)*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x)
- b*d + (b*c - 2*a*d)*x)/(c*x + d)) - 24*(b^4*c^4*d^2 - a*b^3*c^3*d^3 - 3*a^2*b
^2*c^2*d^4 + 5*a^3*b*c*d^5 - 2*a^4*d^6 + (b^4*c^6 - a*b^3*c^5*d - 3*a^2*b^2*c^4*
d^2 + 5*a^3*b*c^3*d^3 - 2*a^4*c^2*d^4)*x^2 + 2*(b^4*c^5*d - a*b^3*c^4*d^2 - 3*a^
2*b^2*c^3*d^3 + 5*a^3*b*c^2*d^4 - 2*a^4*c*d^5)*x)*sqrt((a*x + b)/x)*arctan(a/(sq
rt(-a)*sqrt((a*x + b)/x))) - 2*(12*b^4*c^4*d^2 - 12*a*b^3*c^3*d^3 + 27*a^2*b^2*c
^2*d^4 - 12*a^3*b*c*d^5 + 4*(a*b^3*c^6 - 3*a^2*b^2*c^5*d + 3*a^3*b*c^4*d^2 - a^4
*c^3*d^3)*x^3 + (12*b^4*c^6 - 4*a*b^3*c^5*d - 12*a^2*b^2*c^4*d^2 + 37*a^3*b*c^3*
d^3 - 18*a^4*c^2*d^4)*x^2 + (24*b^4*c^5*d - 20*a*b^3*c^4*d^2 + 29*a^2*b^2*c^3*d^
3 + 9*a^3*b*c^2*d^4 - 12*a^4*c*d^5)*x)*sqrt(-a))/((a^2*b^3*c^7*d^2 - 3*a^3*b^2*c
^6*d^3 + 3*a^4*b*c^5*d^4 - a^5*c^4*d^5 + (a^2*b^3*c^9 - 3*a^3*b^2*c^8*d + 3*a^4*
b*c^7*d^2 - a^5*c^6*d^3)*x^2 + 2*(a^2*b^3*c^8*d - 3*a^3*b^2*c^7*d^2 + 3*a^4*b*c^
6*d^3 - a^5*c^5*d^4)*x)*sqrt(-a)*sqrt((a*x + b)/x)), 1/4*(3*(21*a^2*b^2*c^2*d^4
- 24*a^3*b*c*d^5 + 8*a^4*d^6 + (21*a^2*b^2*c^4*d^2 - 24*a^3*b*c^3*d^3 + 8*a^4*c^
2*d^4)*x^2 + 2*(21*a^2*b^2*c^3*d^3 - 24*a^3*b*c^2*d^4 + 8*a^4*c*d^5)*x)*sqrt(a)*
sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)*arctan(-(b*c - a*d)*sqrt(d/(b*c - a*d))/(d
*sqrt((a*x + b)/x))) + 6*(b^4*c^4*d^2 - a*b^3*c^3*d^3 - 3*a^2*b^2*c^2*d^4 + 5*a^
3*b*c*d^5 - 2*a^4*d^6 + (b^4*c^6 - a*b^3*c^5*d - 3*a^2*b^2*c^4*d^2 + 5*a^3*b*c^3
*d^3 - 2*a^4*c^2*d^4)*x^2 + 2*(b^4*c^5*d - a*b^3*c^4*d^2 - 3*a^2*b^2*c^3*d^3 + 5
*a^3*b*c^2*d^4 - 2*a^4*c*d^5)*x)*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x)
+ (2*a*x + b)*sqrt(a)) + (12*b^4*c^4*d^2 - 12*a*b^3*c^3*d^3 + 27*a^2*b^2*c^2*d^4
- 12*a^3*b*c*d^5 + 4*(a*b^3*c^6 - 3*a^2*b^2*c^5*d + 3*a^3*b*c^4*d^2 - a^4*c^3*d
^3)*x^3 + (12*b^4*c^6 - 4*a*b^3*c^5*d - 12*a^2*b^2*c^4*d^2 + 37*a^3*b*c^3*d^3 -
18*a^4*c^2*d^4)*x^2 + (24*b^4*c^5*d - 20*a*b^3*c^4*d^2 + 29*a^2*b^2*c^3*d^3 + 9*
a^3*b*c^2*d^4 - 12*a^4*c*d^5)*x)*sqrt(a))/((a^2*b^3*c^7*d^2 - 3*a^3*b^2*c^6*d^3
+ 3*a^4*b*c^5*d^4 - a^5*c^4*d^5 + (a^2*b^3*c^9 - 3*a^3*b^2*c^8*d + 3*a^4*b*c^7*d
^2 - a^5*c^6*d^3)*x^2 + 2*(a^2*b^3*c^8*d - 3*a^3*b^2*c^7*d^2 + 3*a^4*b*c^6*d^3 -
a^5*c^5*d^4)*x)*sqrt(a)*sqrt((a*x + b)/x)), 1/4*(3*(21*a^2*b^2*c^2*d^4 - 24*a^3
*b*c*d^5 + 8*a^4*d^6 + (21*a^2*b^2*c^4*d^2 - 24*a^3*b*c^3*d^3 + 8*a^4*c^2*d^4)*x
^2 + 2*(21*a^2*b^2*c^3*d^3 - 24*a^3*b*c^2*d^4 + 8*a^4*c*d^5)*x)*sqrt(-a)*sqrt(d/
(b*c - a*d))*sqrt((a*x + b)/x)*arctan(-(b*c - a*d)*sqrt(d/(b*c - a*d))/(d*sqrt((
a*x + b)/x))) + 12*(b^4*c^4*d^2 - a*b^3*c^3*d^3 - 3*a^2*b^2*c^2*d^4 + 5*a^3*b*c*
d^5 - 2*a^4*d^6 + (b^4*c^6 - a*b^3*c^5*d - 3*a^2*b^2*c^4*d^2 + 5*a^3*b*c^3*d^3 -
2*a^4*c^2*d^4)*x^2 + 2*(b^4*c^5*d - a*b^3*c^4*d^2 - 3*a^2*b^2*c^3*d^3 + 5*a^3*b
*c^2*d^4 - 2*a^4*c*d^5)*x)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x
))) + (12*b^4*c^4*d^2 - 12*a*b^3*c^3*d^3 + 27*a^2*b^2*c^2*d^4 - 12*a^3*b*c*d^5 +
4*(a*b^3*c^6 - 3*a^2*b^2*c^5*d + 3*a^3*b*c^4*d^2 - a^4*c^3*d^3)*x^3 + (12*b^4*c
^6 - 4*a*b^3*c^5*d - 12*a^2*b^2*c^4*d^2 + 37*a^3*b*c^3*d^3 - 18*a^4*c^2*d^4)*x^2
+ (24*b^4*c^5*d - 20*a*b^3*c^4*d^2 + 29*a^2*b^2*c^3*d^3 + 9*a^3*b*c^2*d^4 - 12*
a^4*c*d^5)*x)*sqrt(-a))/((a^2*b^3*c^7*d^2 - 3*a^3*b^2*c^6*d^3 + 3*a^4*b*c^5*d^4
- a^5*c^4*d^5 + (a^2*b^3*c^9 - 3*a^3*b^2*c^8*d + 3*a^4*b*c^7*d^2 - a^5*c^6*d^3)*
x^2 + 2*(a^2*b^3*c^8*d - 3*a^3*b^2*c^7*d^2 + 3*a^4*b*c^6*d^3 - a^5*c^5*d^4)*x)*s
qrt(-a)*sqrt((a*x + b)/x))]

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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(1/(a+b/x)**(3/2)/(c+d/x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.261271, size = 678, normalized size = 2.12 \[ \frac{1}{4} \, b{\left (\frac{3 \,{\left (21 \, b^{2} c^{2} d^{3} - 24 \, a b c d^{4} + 8 \, a^{2} d^{5}\right )} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{4} c^{7} - 3 \, a b^{3} c^{6} d + 3 \, a^{2} b^{2} c^{5} d^{2} - a^{3} b c^{4} d^{3}\right )} \sqrt{b c d - a d^{2}}} + \frac{4 \,{\left (2 \, a b^{3} c^{3} - \frac{3 \,{\left (a x + b\right )} b^{3} c^{3}}{x} + \frac{3 \,{\left (a x + b\right )} a b^{2} c^{2} d}{x} - \frac{3 \,{\left (a x + b\right )} a^{2} b c d^{2}}{x} + \frac{{\left (a x + b\right )} a^{3} d^{3}}{x}\right )}}{{\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3}\right )}{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )}} + \frac{17 \, b^{2} c^{2} d^{3} \sqrt{\frac{a x + b}{x}} - 25 \, a b c d^{4} \sqrt{\frac{a x + b}{x}} + 8 \, a^{2} d^{5} \sqrt{\frac{a x + b}{x}} + \frac{15 \,{\left (a x + b\right )} b c d^{4} \sqrt{\frac{a x + b}{x}}}{x} - \frac{8 \,{\left (a x + b\right )} a d^{5} \sqrt{\frac{a x + b}{x}}}{x}}{{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}{\left (b c - a d + \frac{{\left (a x + b\right )} d}{x}\right )}^{2}} + \frac{12 \,{\left (b c + 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} b c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*(c + d/x)^3),x, algorithm="giac")

[Out]

1/4*b*(3*(21*b^2*c^2*d^3 - 24*a*b*c*d^4 + 8*a^2*d^5)*arctan(d*sqrt((a*x + b)/x)/
sqrt(b*c*d - a*d^2))/((b^4*c^7 - 3*a*b^3*c^6*d + 3*a^2*b^2*c^5*d^2 - a^3*b*c^4*d
^3)*sqrt(b*c*d - a*d^2)) + 4*(2*a*b^3*c^3 - 3*(a*x + b)*b^3*c^3/x + 3*(a*x + b)*
a*b^2*c^2*d/x - 3*(a*x + b)*a^2*b*c*d^2/x + (a*x + b)*a^3*d^3/x)/((a^2*b^3*c^6 -
3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3)*(a*sqrt((a*x + b)/x) - (a*x +
b)*sqrt((a*x + b)/x)/x)) + (17*b^2*c^2*d^3*sqrt((a*x + b)/x) - 25*a*b*c*d^4*sqrt
((a*x + b)/x) + 8*a^2*d^5*sqrt((a*x + b)/x) + 15*(a*x + b)*b*c*d^4*sqrt((a*x + b
)/x)/x - 8*(a*x + b)*a*d^5*sqrt((a*x + b)/x)/x)/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^
2*b*c^4*d^2 - a^3*c^3*d^3)*(b*c - a*d + (a*x + b)*d/x)^2) + 12*(b*c + 2*a*d)*arc
tan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^2*b*c^4))