3.382 \(\int \frac{\sqrt{d+e x}}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=286 \[ \frac{c \sqrt{d+e x} (12 c d-b e)}{4 b^3 d (b+c x)^2}+\frac{\sqrt{d+e x} (8 c d-b e)}{4 b^2 d x (b+c x)^2}-\frac{\left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{3/2}}+\frac{c^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}+\frac{c \sqrt{d+e x} \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )}{4 b^4 d (b+c x) (c d-b e)}-\frac{\sqrt{d+e x}}{2 b x^2 (b+c x)^2} \]

[Out]

(c*(12*c*d - b*e)*Sqrt[d + e*x])/(4*b^3*d*(b + c*x)^2) - Sqrt[d + e*x]/(2*b*x^2*
(b + c*x)^2) + ((8*c*d - b*e)*Sqrt[d + e*x])/(4*b^2*d*x*(b + c*x)^2) + (c*(24*c^
2*d^2 - 24*b*c*d*e + b^2*e^2)*Sqrt[d + e*x])/(4*b^4*d*(c*d - b*e)*(b + c*x)) - (
(48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(3/
2)) + (c^(3/2)*(48*c^2*d^2 - 84*b*c*d*e + 35*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d +
e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(3/2))

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Rubi [A]  time = 1.19822, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{c \sqrt{d+e x} (12 c d-b e)}{4 b^3 d (b+c x)^2}+\frac{\sqrt{d+e x} (8 c d-b e)}{4 b^2 d x (b+c x)^2}-\frac{\left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{3/2}}+\frac{c^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}+\frac{c \sqrt{d+e x} \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )}{4 b^4 d (b+c x) (c d-b e)}-\frac{\sqrt{d+e x}}{2 b x^2 (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(c*(12*c*d - b*e)*Sqrt[d + e*x])/(4*b^3*d*(b + c*x)^2) - Sqrt[d + e*x]/(2*b*x^2*
(b + c*x)^2) + ((8*c*d - b*e)*Sqrt[d + e*x])/(4*b^2*d*x*(b + c*x)^2) + (c*(24*c^
2*d^2 - 24*b*c*d*e + b^2*e^2)*Sqrt[d + e*x])/(4*b^4*d*(c*d - b*e)*(b + c*x)) - (
(48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(3/
2)) + (c^(3/2)*(48*c^2*d^2 - 84*b*c*d*e + 35*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d +
e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(3/2))

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Rubi in Sympy [A]  time = 174.936, size = 258, normalized size = 0.9 \[ - \frac{\sqrt{d + e x}}{2 b x^{2} \left (b + c x\right )^{2}} - \frac{\sqrt{d + e x} \left (b e - 8 c d\right )}{4 b^{2} d x \left (b + c x\right )^{2}} - \frac{c \sqrt{d + e x} \left (b e - 12 c d\right )}{4 b^{3} d \left (b + c x\right )^{2}} - \frac{c \sqrt{d + e x} \left (b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right )}{4 b^{4} d \left (b + c x\right ) \left (b e - c d\right )} + \frac{c^{\frac{3}{2}} \left (35 b^{2} e^{2} - 84 b c d e + 48 c^{2} d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{4 b^{5} \left (b e - c d\right )^{\frac{3}{2}}} + \frac{\left (b^{2} e^{2} + 12 b c d e - 48 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{4 b^{5} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)/(c*x**2+b*x)**3,x)

[Out]

-sqrt(d + e*x)/(2*b*x**2*(b + c*x)**2) - sqrt(d + e*x)*(b*e - 8*c*d)/(4*b**2*d*x
*(b + c*x)**2) - c*sqrt(d + e*x)*(b*e - 12*c*d)/(4*b**3*d*(b + c*x)**2) - c*sqrt
(d + e*x)*(b**2*e**2 - 24*b*c*d*e + 24*c**2*d**2)/(4*b**4*d*(b + c*x)*(b*e - c*d
)) + c**(3/2)*(35*b**2*e**2 - 84*b*c*d*e + 48*c**2*d**2)*atan(sqrt(c)*sqrt(d + e
*x)/sqrt(b*e - c*d))/(4*b**5*(b*e - c*d)**(3/2)) + (b**2*e**2 + 12*b*c*d*e - 48*
c**2*d**2)*atanh(sqrt(d + e*x)/sqrt(d))/(4*b**5*d**(3/2))

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Mathematica [A]  time = 1.47802, size = 198, normalized size = 0.69 \[ \frac{\frac{\left (b^2 e^2+12 b c d e-48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{c^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{(c d-b e)^{3/2}}+b \sqrt{d+e x} \left (\frac{c^2 (12 c d-11 b e)}{(b+c x) (c d-b e)}+\frac{2 b c^2}{(b+c x)^2}+\frac{12 c d-b e}{d x}-\frac{2 b}{x^2}\right )}{4 b^5} \]

Antiderivative was successfully verified.

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

[Out]

(b*Sqrt[d + e*x]*((-2*b)/x^2 + (12*c*d - b*e)/(d*x) + (2*b*c^2)/(b + c*x)^2 + (c
^2*(12*c*d - 11*b*e))/((c*d - b*e)*(b + c*x))) + ((-48*c^2*d^2 + 12*b*c*d*e + b^
2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(3/2) + (c^(3/2)*(48*c^2*d^2 - 84*b*c*d
*e + 35*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(c*d - b*e)^(
3/2))/(4*b^5)

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Maple [A]  time = 0.032, size = 436, normalized size = 1.5 \[{\frac{11\,{e}^{2}{c}^{3}}{4\,{b}^{3} \left ( cex+be \right ) ^{2} \left ( be-cd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-3\,{\frac{e{c}^{4} \left ( ex+d \right ) ^{3/2}d}{{b}^{4} \left ( cex+be \right ) ^{2} \left ( be-cd \right ) }}+{\frac{13\,{e}^{2}{c}^{2}}{4\,{b}^{3} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-3\,{\frac{e{c}^{3}\sqrt{ex+d}d}{{b}^{4} \left ( cex+be \right ) ^{2}}}+{\frac{35\,{e}^{2}{c}^{2}}{4\,{b}^{3} \left ( be-cd \right ) }\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-21\,{\frac{e{c}^{3}d}{{b}^{4} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+12\,{\frac{{c}^{4}{d}^{2}}{{b}^{5} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{1}{4\,{b}^{3}{x}^{2}d} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{ \left ( ex+d \right ) ^{3/2}c}{e{b}^{4}{x}^{2}}}-3\,{\frac{c\sqrt{ex+d}d}{e{b}^{4}{x}^{2}}}-{\frac{1}{4\,{b}^{3}{x}^{2}}\sqrt{ex+d}}+{\frac{{e}^{2}}{4\,{b}^{3}}{\it Artanh} \left ({1\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{3}{2}}}}+3\,{\frac{ce}{{b}^{4}\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{\sqrt{d}{c}^{2}}{{b}^{5}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/4*e^2/b^3*c^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(3/2)-3*e/b^4*c^4/(c*e*x+b*e)^2
/(b*e-c*d)*(e*x+d)^(3/2)*d+13/4*e^2/b^3*c^2/(c*e*x+b*e)^2*(e*x+d)^(1/2)-3*e/b^4*
c^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d+35/4*e^2/b^3*c^2/(b*e-c*d)/((b*e-c*d)*c)^(1/2)
*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))-21*e/b^4*c^3/(b*e-c*d)/((b*e-c*d)*c
)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d+12/b^5*c^4/(b*e-c*d)/((b*e
-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d^2-1/4/b^3/x^2/d*(e*
x+d)^(3/2)+3/e/b^4/x^2*(e*x+d)^(3/2)*c-3/e/b^4/x^2*(e*x+d)^(1/2)*c*d-1/4/b^3/x^2
*(e*x+d)^(1/2)+1/4*e^2/b^3/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))+3*e/b^4/d^(1/2
)*arctanh((e*x+d)^(1/2)/d^(1/2))*c-12/b^5*d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))
*c^2

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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(sqrt(e*x + d)/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.56515, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(((48*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*c^3*d*e^2)*x^4 + 2*(48*b*c^4*d^3 -
84*b^2*c^3*d^2*e + 35*b^3*c^2*d*e^2)*x^3 + (48*b^2*c^3*d^3 - 84*b^3*c^2*d^2*e +
35*b^4*c*d*e^2)*x^2)*sqrt(d)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e - 2*(
c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 2*(2*b^4*c*d^2 - 2*b^
5*d*e - (24*b*c^4*d^2 - 24*b^2*c^3*d*e + b^3*c^2*e^2)*x^3 - (36*b^2*c^3*d^2 - 37
*b^3*c^2*d*e + 2*b^4*c*e^2)*x^2 - (8*b^3*c^2*d^2 - 9*b^4*c*d*e + b^5*e^2)*x)*sqr
t(e*x + d)*sqrt(d) + ((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*
e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^
3 + (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*log(((e*
x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)*d)/x))/(((b^5*c^3*d^2 - b^6*c^2*d*e)*x^4 + 2*
(b^6*c^2*d^2 - b^7*c*d*e)*x^3 + (b^7*c*d^2 - b^8*d*e)*x^2)*sqrt(d)), 1/8*(2*((48
*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*c^3*d*e^2)*x^4 + 2*(48*b*c^4*d^3 - 84*b^2*c^3
*d^2*e + 35*b^3*c^2*d*e^2)*x^3 + (48*b^2*c^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d
*e^2)*x^2)*sqrt(d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))
/(sqrt(e*x + d)*c)) - 2*(2*b^4*c*d^2 - 2*b^5*d*e - (24*b*c^4*d^2 - 24*b^2*c^3*d*
e + b^3*c^2*e^2)*x^3 - (36*b^2*c^3*d^2 - 37*b^3*c^2*d*e + 2*b^4*c*e^2)*x^2 - (8*
b^3*c^2*d^2 - 9*b^4*c*d*e + b^5*e^2)*x)*sqrt(e*x + d)*sqrt(d) - ((48*c^5*d^3 - 6
0*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c
^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*
e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*log(((e*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)*d)
/x))/(((b^5*c^3*d^2 - b^6*c^2*d*e)*x^4 + 2*(b^6*c^2*d^2 - b^7*c*d*e)*x^3 + (b^7*
c*d^2 - b^8*d*e)*x^2)*sqrt(d)), -1/8*(((48*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*c^3
*d*e^2)*x^4 + 2*(48*b*c^4*d^3 - 84*b^2*c^3*d^2*e + 35*b^3*c^2*d*e^2)*x^3 + (48*b
^2*c^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d*e^2)*x^2)*sqrt(-d)*sqrt(c/(c*d - b*e)
)*log((c*e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c
*x + b)) + 2*(2*b^4*c*d^2 - 2*b^5*d*e - (24*b*c^4*d^2 - 24*b^2*c^3*d*e + b^3*c^2
*e^2)*x^3 - (36*b^2*c^3*d^2 - 37*b^3*c^2*d*e + 2*b^4*c*e^2)*x^2 - (8*b^3*c^2*d^2
- 9*b^4*c*d*e + b^5*e^2)*x)*sqrt(e*x + d)*sqrt(-d) - 2*((48*c^5*d^3 - 60*b*c^4*
d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e
+ 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b
^4*c*d*e^2 + b^5*e^3)*x^2)*arctan(d/(sqrt(e*x + d)*sqrt(-d))))/(((b^5*c^3*d^2 -
b^6*c^2*d*e)*x^4 + 2*(b^6*c^2*d^2 - b^7*c*d*e)*x^3 + (b^7*c*d^2 - b^8*d*e)*x^2)*
sqrt(-d)), 1/4*(((48*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*c^3*d*e^2)*x^4 + 2*(48*b*
c^4*d^3 - 84*b^2*c^3*d^2*e + 35*b^3*c^2*d*e^2)*x^3 + (48*b^2*c^3*d^3 - 84*b^3*c^
2*d^2*e + 35*b^4*c*d*e^2)*x^2)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)
*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) - (2*b^4*c*d^2 - 2*b^5*d*e - (24*b*c^4*
d^2 - 24*b^2*c^3*d*e + b^3*c^2*e^2)*x^3 - (36*b^2*c^3*d^2 - 37*b^3*c^2*d*e + 2*b
^4*c*e^2)*x^2 - (8*b^3*c^2*d^2 - 9*b^4*c*d*e + b^5*e^2)*x)*sqrt(e*x + d)*sqrt(-d
) + ((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*
b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d
^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*arctan(d/(sqrt(e*x + d)*s
qrt(-d))))/(((b^5*c^3*d^2 - b^6*c^2*d*e)*x^4 + 2*(b^6*c^2*d^2 - b^7*c*d*e)*x^3 +
(b^7*c*d^2 - b^8*d*e)*x^2)*sqrt(-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((e*x+d)**(1/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230308, size = 686, normalized size = 2.4 \[ -\frac{{\left (48 \, c^{4} d^{2} - 84 \, b c^{3} d e + 35 \, b^{2} c^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{4 \,{\left (b^{5} c d - b^{6} e\right )} \sqrt{-c^{2} d + b c e}} + \frac{24 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{4} d^{2} e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{4} d^{3} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{4} d^{4} e - 24 \, \sqrt{x e + d} c^{4} d^{5} e - 24 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{3} d e^{2} + 108 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{3} d^{2} e^{2} - 144 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{3} d^{3} e^{2} + 60 \, \sqrt{x e + d} b c^{3} d^{4} e^{2} +{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c^{2} e^{3} - 40 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c^{2} d e^{3} + 85 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{2} d^{2} e^{3} - 46 \, \sqrt{x e + d} b^{2} c^{2} d^{3} e^{3} + 2 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} c e^{4} - 13 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c d e^{4} + 9 \, \sqrt{x e + d} b^{3} c d^{2} e^{4} +{\left (x e + d\right )}^{\frac{3}{2}} b^{4} e^{5} + \sqrt{x e + d} b^{4} d e^{5}}{4 \,{\left (b^{4} c d^{2} - b^{5} d e\right )}{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}^{2}} + \frac{{\left (48 \, c^{2} d^{2} - 12 \, b c d e - b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{4 \, b^{5} \sqrt{-d} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(48*c^4*d^2 - 84*b*c^3*d*e + 35*b^2*c^2*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c
^2*d + b*c*e))/((b^5*c*d - b^6*e)*sqrt(-c^2*d + b*c*e)) + 1/4*(24*(x*e + d)^(7/2
)*c^4*d^2*e - 72*(x*e + d)^(5/2)*c^4*d^3*e + 72*(x*e + d)^(3/2)*c^4*d^4*e - 24*s
qrt(x*e + d)*c^4*d^5*e - 24*(x*e + d)^(7/2)*b*c^3*d*e^2 + 108*(x*e + d)^(5/2)*b*
c^3*d^2*e^2 - 144*(x*e + d)^(3/2)*b*c^3*d^3*e^2 + 60*sqrt(x*e + d)*b*c^3*d^4*e^2
+ (x*e + d)^(7/2)*b^2*c^2*e^3 - 40*(x*e + d)^(5/2)*b^2*c^2*d*e^3 + 85*(x*e + d)
^(3/2)*b^2*c^2*d^2*e^3 - 46*sqrt(x*e + d)*b^2*c^2*d^3*e^3 + 2*(x*e + d)^(5/2)*b^
3*c*e^4 - 13*(x*e + d)^(3/2)*b^3*c*d*e^4 + 9*sqrt(x*e + d)*b^3*c*d^2*e^4 + (x*e
+ d)^(3/2)*b^4*e^5 + sqrt(x*e + d)*b^4*d*e^5)/((b^4*c*d^2 - b^5*d*e)*((x*e + d)^
2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2) + 1/4*(48*c^2*d^2 - 12
*b*c*d*e - b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d)