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3.1648 \(\int \frac{\sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=146 \[ -\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2}}+\frac{e^2 \sqrt{d+e x}}{8 b (a+b x) (b d-a e)^2}-\frac{e \sqrt{d+e x}}{12 b (a+b x)^2 (b d-a e)}-\frac{\sqrt{d+e x}}{3 b (a+b x)^3} \]

[Out]

-Sqrt[d + e*x]/(3*b*(a + b*x)^3) - (e*Sqrt[d + e*x])/(12*b*(b*d - a*e)*(a + b*x)
^2) + (e^2*Sqrt[d + e*x])/(8*b*(b*d - a*e)^2*(a + b*x)) - (e^3*ArcTanh[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(3/2)*(b*d - a*e)^(5/2))

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Rubi [A]  time = 0.21483, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2}}+\frac{e^2 \sqrt{d+e x}}{8 b (a+b x) (b d-a e)^2}-\frac{e \sqrt{d+e x}}{12 b (a+b x)^2 (b d-a e)}-\frac{\sqrt{d+e x}}{3 b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[d + e*x]/(3*b*(a + b*x)^3) - (e*Sqrt[d + e*x])/(12*b*(b*d - a*e)*(a + b*x)
^2) + (e^2*Sqrt[d + e*x])/(8*b*(b*d - a*e)^2*(a + b*x)) - (e^3*ArcTanh[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(3/2)*(b*d - a*e)^(5/2))

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Rubi in Sympy [A]  time = 58.7804, size = 119, normalized size = 0.82 \[ \frac{e^{2} \sqrt{d + e x}}{8 b \left (a + b x\right ) \left (a e - b d\right )^{2}} + \frac{e \sqrt{d + e x}}{12 b \left (a + b x\right )^{2} \left (a e - b d\right )} - \frac{\sqrt{d + e x}}{3 b \left (a + b x\right )^{3}} + \frac{e^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{3}{2}} \left (a e - b d\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**2*sqrt(d + e*x)/(8*b*(a + b*x)*(a*e - b*d)**2) + e*sqrt(d + e*x)/(12*b*(a + b
*x)**2*(a*e - b*d)) - sqrt(d + e*x)/(3*b*(a + b*x)**3) + e**3*atan(sqrt(b)*sqrt(
d + e*x)/sqrt(a*e - b*d))/(8*b**(3/2)*(a*e - b*d)**(5/2))

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Mathematica [A]  time = 0.188566, size = 130, normalized size = 0.89 \[ \sqrt{d+e x} \left (\frac{e^2}{8 b (a+b x) (b d-a e)^2}-\frac{e}{12 b (a+b x)^2 (b d-a e)}-\frac{1}{3 b (a+b x)^3}\right )-\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x]*(-1/(3*b*(a + b*x)^3) - e/(12*b*(b*d - a*e)*(a + b*x)^2) + e^2/(8*
b*(b*d - a*e)^2*(a + b*x))) - (e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*
e]])/(8*b^(3/2)*(b*d - a*e)^(5/2))

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Maple [A]  time = 0.021, size = 170, normalized size = 1.2 \[{\frac{{e}^{3}b}{8\, \left ( bex+ae \right ) ^{3} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{3}}{3\, \left ( bex+ae \right ) ^{3} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{3}}{8\, \left ( bex+ae \right ) ^{3}b}\sqrt{ex+d}}+{\frac{{e}^{3}}{8\,b \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*e^3/(b*e*x+a*e)^3*b/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(5/2)+1/3*e^3/(b*e*x
+a*e)^3/(a*e-b*d)*(e*x+d)^(3/2)-1/8*e^3/(b*e*x+a*e)^3/b*(e*x+d)^(1/2)+1/8*e^3/b/
(a^2*e^2-2*a*b*d*e+b^2*d^2)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b
*d))^(1/2))

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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)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224232, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} - 8 \, b^{2} d^{2} + 14 \, a b d e - 3 \, a^{2} e^{2} - 2 \,{\left (b^{2} d e - 4 \, a b e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} + 3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{48 \,{\left (a^{3} b^{3} d^{2} - 2 \, a^{4} b^{2} d e + a^{5} b e^{2} +{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} x^{3} + 3 \,{\left (a b^{5} d^{2} - 2 \, a^{2} b^{4} d e + a^{3} b^{3} e^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d^{2} - 2 \, a^{3} b^{3} d e + a^{4} b^{2} e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e}}, \frac{{\left (3 \, b^{2} e^{2} x^{2} - 8 \, b^{2} d^{2} + 14 \, a b d e - 3 \, a^{2} e^{2} - 2 \,{\left (b^{2} d e - 4 \, a b e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} - 3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{24 \,{\left (a^{3} b^{3} d^{2} - 2 \, a^{4} b^{2} d e + a^{5} b e^{2} +{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} x^{3} + 3 \,{\left (a b^{5} d^{2} - 2 \, a^{2} b^{4} d e + a^{3} b^{3} e^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d^{2} - 2 \, a^{3} b^{3} d e + a^{4} b^{2} e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(2*(3*b^2*e^2*x^2 - 8*b^2*d^2 + 14*a*b*d*e - 3*a^2*e^2 - 2*(b^2*d*e - 4*a*
b*e^2)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 3*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 +
 3*a^2*b*e^3*x + a^3*e^3)*log((sqrt(b^2*d - a*b*e)*(b*e*x + 2*b*d - a*e) - 2*(b^
2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)))/((a^3*b^3*d^2 - 2*a^4*b^2*d*e + a^5*b*e^
2 + (b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*x^3 + 3*(a*b^5*d^2 - 2*a^2*b^4*d*e + a
^3*b^3*e^2)*x^2 + 3*(a^2*b^4*d^2 - 2*a^3*b^3*d*e + a^4*b^2*e^2)*x)*sqrt(b^2*d -
a*b*e)), 1/24*((3*b^2*e^2*x^2 - 8*b^2*d^2 + 14*a*b*d*e - 3*a^2*e^2 - 2*(b^2*d*e
- 4*a*b*e^2)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) - 3*(b^3*e^3*x^3 + 3*a*b^2*e^
3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(
e*x + d))))/((a^3*b^3*d^2 - 2*a^4*b^2*d*e + a^5*b*e^2 + (b^6*d^2 - 2*a*b^5*d*e +
 a^2*b^4*e^2)*x^3 + 3*(a*b^5*d^2 - 2*a^2*b^4*d*e + a^3*b^3*e^2)*x^2 + 3*(a^2*b^4
*d^2 - 2*a^3*b^3*d*e + a^4*b^2*e^2)*x)*sqrt(-b^2*d + a*b*e))]

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Sympy [A]  time = 46.2713, size = 4592, normalized size = 31.45 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-66*a**3*e**6*sqrt(d + e*x)/(48*a**6*b*e**6 - 144*a**5*b**2*d*e**5 + 144*a**5*b*
*2*e**6*x - 720*a**4*b**3*d*e**5*x + 144*a**4*b**3*e**4*(d + e*x)**2 + 480*a**3*
b**4*d**3*e**3 + 1440*a**3*b**4*d**2*e**4*x - 576*a**3*b**4*d*e**3*(d + e*x)**2
+ 48*a**3*b**4*e**3*(d + e*x)**3 - 720*a**2*b**5*d**4*e**2 - 1440*a**2*b**5*d**3
*e**3*x + 864*a**2*b**5*d**2*e**2*(d + e*x)**2 - 144*a**2*b**5*d*e**2*(d + e*x)*
*3 + 432*a*b**6*d**5*e + 720*a*b**6*d**4*e**2*x - 576*a*b**6*d**3*e*(d + e*x)**2
 + 144*a*b**6*d**2*e*(d + e*x)**3 - 96*b**7*d**6 - 144*b**7*d**5*e*x + 144*b**7*
d**4*(d + e*x)**2 - 48*b**7*d**3*(d + e*x)**3) + 198*a**2*d*e**5*sqrt(d + e*x)/(
48*a**6*e**6 - 144*a**5*b*d*e**5 + 144*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x +
144*a**4*b**2*e**4*(d + e*x)**2 + 480*a**3*b**3*d**3*e**3 + 1440*a**3*b**3*d**2*
e**4*x - 576*a**3*b**3*d*e**3*(d + e*x)**2 + 48*a**3*b**3*e**3*(d + e*x)**3 - 72
0*a**2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**3*x + 864*a**2*b**4*d**2*e**2*(d
+ e*x)**2 - 144*a**2*b**4*d*e**2*(d + e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*d
**4*e**2*x - 576*a*b**5*d**3*e*(d + e*x)**2 + 144*a*b**5*d**2*e*(d + e*x)**3 - 9
6*b**6*d**6 - 144*b**6*d**5*e*x + 144*b**6*d**4*(d + e*x)**2 - 48*b**6*d**3*(d +
 e*x)**3) - 80*a**2*e**5*(d + e*x)**(3/2)/(48*a**6*e**6 - 144*a**5*b*d*e**5 + 14
4*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x + 144*a**4*b**2*e**4*(d + e*x)**2 + 480
*a**3*b**3*d**3*e**3 + 1440*a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e**3*(d + e*
x)**2 + 48*a**3*b**3*e**3*(d + e*x)**3 - 720*a**2*b**4*d**4*e**2 - 1440*a**2*b**
4*d**3*e**3*x + 864*a**2*b**4*d**2*e**2*(d + e*x)**2 - 144*a**2*b**4*d*e**2*(d +
 e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*d**4*e**2*x - 576*a*b**5*d**3*e*(d + e
*x)**2 + 144*a*b**5*d**2*e*(d + e*x)**3 - 96*b**6*d**6 - 144*b**6*d**5*e*x + 144
*b**6*d**4*(d + e*x)**2 - 48*b**6*d**3*(d + e*x)**3) - 198*a*b*d**2*e**4*sqrt(d
+ e*x)/(48*a**6*e**6 - 144*a**5*b*d*e**5 + 144*a**5*b*e**6*x - 720*a**4*b**2*d*e
**5*x + 144*a**4*b**2*e**4*(d + e*x)**2 + 480*a**3*b**3*d**3*e**3 + 1440*a**3*b*
*3*d**2*e**4*x - 576*a**3*b**3*d*e**3*(d + e*x)**2 + 48*a**3*b**3*e**3*(d + e*x)
**3 - 720*a**2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**3*x + 864*a**2*b**4*d**2*
e**2*(d + e*x)**2 - 144*a**2*b**4*d*e**2*(d + e*x)**3 + 432*a*b**5*d**5*e + 720*
a*b**5*d**4*e**2*x - 576*a*b**5*d**3*e*(d + e*x)**2 + 144*a*b**5*d**2*e*(d + e*x
)**3 - 96*b**6*d**6 - 144*b**6*d**5*e*x + 144*b**6*d**4*(d + e*x)**2 - 48*b**6*d
**3*(d + e*x)**3) + 160*a*b*d*e**4*(d + e*x)**(3/2)/(48*a**6*e**6 - 144*a**5*b*d
*e**5 + 144*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x + 144*a**4*b**2*e**4*(d + e*x
)**2 + 480*a**3*b**3*d**3*e**3 + 1440*a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e*
*3*(d + e*x)**2 + 48*a**3*b**3*e**3*(d + e*x)**3 - 720*a**2*b**4*d**4*e**2 - 144
0*a**2*b**4*d**3*e**3*x + 864*a**2*b**4*d**2*e**2*(d + e*x)**2 - 144*a**2*b**4*d
*e**2*(d + e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*d**4*e**2*x - 576*a*b**5*d**
3*e*(d + e*x)**2 + 144*a*b**5*d**2*e*(d + e*x)**3 - 96*b**6*d**6 - 144*b**6*d**5
*e*x + 144*b**6*d**4*(d + e*x)**2 - 48*b**6*d**3*(d + e*x)**3) - 30*a*b*e**4*(d
+ e*x)**(5/2)/(48*a**6*e**6 - 144*a**5*b*d*e**5 + 144*a**5*b*e**6*x - 720*a**4*b
**2*d*e**5*x + 144*a**4*b**2*e**4*(d + e*x)**2 + 480*a**3*b**3*d**3*e**3 + 1440*
a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e**3*(d + e*x)**2 + 48*a**3*b**3*e**3*(d
 + e*x)**3 - 720*a**2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**3*x + 864*a**2*b**
4*d**2*e**2*(d + e*x)**2 - 144*a**2*b**4*d*e**2*(d + e*x)**3 + 432*a*b**5*d**5*e
 + 720*a*b**5*d**4*e**2*x - 576*a*b**5*d**3*e*(d + e*x)**2 + 144*a*b**5*d**2*e*(
d + e*x)**3 - 96*b**6*d**6 - 144*b**6*d**5*e*x + 144*b**6*d**4*(d + e*x)**2 - 48
*b**6*d**3*(d + e*x)**3) + 10*a*e**4*sqrt(d + e*x)/(8*a**4*b*e**4 - 16*a**3*b**2
*d*e**3 + 16*a**3*b**2*e**4*x - 48*a**2*b**3*d*e**3*x + 8*a**2*b**3*e**2*(d + e*
x)**2 + 16*a*b**4*d**3*e + 48*a*b**4*d**2*e**2*x - 16*a*b**4*d*e*(d + e*x)**2 -
8*b**5*d**4 - 16*b**5*d**3*e*x + 8*b**5*d**2*(d + e*x)**2) + 5*a*e**4*sqrt(-1/(b
*(a*e - b*d)**7))*log(-a**4*e**4*sqrt(-1/(b*(a*e - b*d)**7)) + 4*a**3*b*d*e**3*s
qrt(-1/(b*(a*e - b*d)**7)) - 6*a**2*b**2*d**2*e**2*sqrt(-1/(b*(a*e - b*d)**7)) +
 4*a*b**3*d**3*e*sqrt(-1/(b*(a*e - b*d)**7)) - b**4*d**4*sqrt(-1/(b*(a*e - b*d)*
*7)) + sqrt(d + e*x))/(16*b) - 5*a*e**4*sqrt(-1/(b*(a*e - b*d)**7))*log(a**4*e**
4*sqrt(-1/(b*(a*e - b*d)**7)) - 4*a**3*b*d*e**3*sqrt(-1/(b*(a*e - b*d)**7)) + 6*
a**2*b**2*d**2*e**2*sqrt(-1/(b*(a*e - b*d)**7)) - 4*a*b**3*d**3*e*sqrt(-1/(b*(a*
e - b*d)**7)) + b**4*d**4*sqrt(-1/(b*(a*e - b*d)**7)) + sqrt(d + e*x))/(16*b) +
66*b**2*d**3*e**3*sqrt(d + e*x)/(48*a**6*e**6 - 144*a**5*b*d*e**5 + 144*a**5*b*e
**6*x - 720*a**4*b**2*d*e**5*x + 144*a**4*b**2*e**4*(d + e*x)**2 + 480*a**3*b**3
*d**3*e**3 + 1440*a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e**3*(d + e*x)**2 + 48
*a**3*b**3*e**3*(d + e*x)**3 - 720*a**2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**
3*x + 864*a**2*b**4*d**2*e**2*(d + e*x)**2 - 144*a**2*b**4*d*e**2*(d + e*x)**3 +
 432*a*b**5*d**5*e + 720*a*b**5*d**4*e**2*x - 576*a*b**5*d**3*e*(d + e*x)**2 + 1
44*a*b**5*d**2*e*(d + e*x)**3 - 96*b**6*d**6 - 144*b**6*d**5*e*x + 144*b**6*d**4
*(d + e*x)**2 - 48*b**6*d**3*(d + e*x)**3) - 80*b**2*d**2*e**3*(d + e*x)**(3/2)/
(48*a**6*e**6 - 144*a**5*b*d*e**5 + 144*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x +
 144*a**4*b**2*e**4*(d + e*x)**2 + 480*a**3*b**3*d**3*e**3 + 1440*a**3*b**3*d**2
*e**4*x - 576*a**3*b**3*d*e**3*(d + e*x)**2 + 48*a**3*b**3*e**3*(d + e*x)**3 - 7
20*a**2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**3*x + 864*a**2*b**4*d**2*e**2*(d
 + e*x)**2 - 144*a**2*b**4*d*e**2*(d + e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*
d**4*e**2*x - 576*a*b**5*d**3*e*(d + e*x)**2 + 144*a*b**5*d**2*e*(d + e*x)**3 -
96*b**6*d**6 - 144*b**6*d**5*e*x + 144*b**6*d**4*(d + e*x)**2 - 48*b**6*d**3*(d
+ e*x)**3) + 30*b**2*d*e**3*(d + e*x)**(5/2)/(48*a**6*e**6 - 144*a**5*b*d*e**5 +
 144*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x + 144*a**4*b**2*e**4*(d + e*x)**2 +
480*a**3*b**3*d**3*e**3 + 1440*a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e**3*(d +
 e*x)**2 + 48*a**3*b**3*e**3*(d + e*x)**3 - 720*a**2*b**4*d**4*e**2 - 1440*a**2*
b**4*d**3*e**3*x + 864*a**2*b**4*d**2*e**2*(d + e*x)**2 - 144*a**2*b**4*d*e**2*(
d + e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*d**4*e**2*x - 576*a*b**5*d**3*e*(d
+ e*x)**2 + 144*a*b**5*d**2*e*(d + e*x)**3 - 96*b**6*d**6 - 144*b**6*d**5*e*x +
144*b**6*d**4*(d + e*x)**2 - 48*b**6*d**3*(d + e*x)**3) - 5*d*e**3*sqrt(-1/(b*(a
*e - b*d)**7))*log(-a**4*e**4*sqrt(-1/(b*(a*e - b*d)**7)) + 4*a**3*b*d*e**3*sqrt
(-1/(b*(a*e - b*d)**7)) - 6*a**2*b**2*d**2*e**2*sqrt(-1/(b*(a*e - b*d)**7)) + 4*
a*b**3*d**3*e*sqrt(-1/(b*(a*e - b*d)**7)) - b**4*d**4*sqrt(-1/(b*(a*e - b*d)**7)
) + sqrt(d + e*x))/16 + 5*d*e**3*sqrt(-1/(b*(a*e - b*d)**7))*log(a**4*e**4*sqrt(
-1/(b*(a*e - b*d)**7)) - 4*a**3*b*d*e**3*sqrt(-1/(b*(a*e - b*d)**7)) + 6*a**2*b*
*2*d**2*e**2*sqrt(-1/(b*(a*e - b*d)**7)) - 4*a*b**3*d**3*e*sqrt(-1/(b*(a*e - b*d
)**7)) + b**4*d**4*sqrt(-1/(b*(a*e - b*d)**7)) + sqrt(d + e*x))/16 - 10*d*e**3*s
qrt(d + e*x)/(8*a**4*e**4 - 16*a**3*b*d*e**3 + 16*a**3*b*e**4*x - 48*a**2*b**2*d
*e**3*x + 8*a**2*b**2*e**2*(d + e*x)**2 + 16*a*b**3*d**3*e + 48*a*b**3*d**2*e**2
*x - 16*a*b**3*d*e*(d + e*x)**2 - 8*b**4*d**4 - 16*b**4*d**3*e*x + 8*b**4*d**2*(
d + e*x)**2) + 6*e**3*(d + e*x)**(3/2)/(8*a**4*e**4 - 16*a**3*b*d*e**3 + 16*a**3
*b*e**4*x - 48*a**2*b**2*d*e**3*x + 8*a**2*b**2*e**2*(d + e*x)**2 + 16*a*b**3*d*
*3*e + 48*a*b**3*d**2*e**2*x - 16*a*b**3*d*e*(d + e*x)**2 - 8*b**4*d**4 - 16*b**
4*d**3*e*x + 8*b**4*d**2*(d + e*x)**2) - 3*e**3*sqrt(-1/(b*(a*e - b*d)**5))*log(
-a**3*e**3*sqrt(-1/(b*(a*e - b*d)**5)) + 3*a**2*b*d*e**2*sqrt(-1/(b*(a*e - b*d)*
*5)) - 3*a*b**2*d**2*e*sqrt(-1/(b*(a*e - b*d)**5)) + b**3*d**3*sqrt(-1/(b*(a*e -
 b*d)**5)) + sqrt(d + e*x))/(8*b) + 3*e**3*sqrt(-1/(b*(a*e - b*d)**5))*log(a**3*
e**3*sqrt(-1/(b*(a*e - b*d)**5)) - 3*a**2*b*d*e**2*sqrt(-1/(b*(a*e - b*d)**5)) +
 3*a*b**2*d**2*e*sqrt(-1/(b*(a*e - b*d)**5)) - b**3*d**3*sqrt(-1/(b*(a*e - b*d)*
*5)) + sqrt(d + e*x))/(8*b)

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GIAC/XCAS [A]  time = 0.215293, size = 285, normalized size = 1.95 \[ \frac{\arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} \sqrt{-b^{2} d + a b e}} + \frac{3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} - 8 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} - 3 \, \sqrt{x e + d} b^{2} d^{2} e^{3} + 8 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} + 6 \, \sqrt{x e + d} a b d e^{4} - 3 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^3*d^2 - 2*a*b^2*d*e + a
^2*b*e^2)*sqrt(-b^2*d + a*b*e)) + 1/24*(3*(x*e + d)^(5/2)*b^2*e^3 - 8*(x*e + d)^
(3/2)*b^2*d*e^3 - 3*sqrt(x*e + d)*b^2*d^2*e^3 + 8*(x*e + d)^(3/2)*a*b*e^4 + 6*sq
rt(x*e + d)*a*b*d*e^4 - 3*sqrt(x*e + d)*a^2*e^5)/((b^3*d^2 - 2*a*b^2*d*e + a^2*b
*e^2)*((x*e + d)*b - b*d + a*e)^3)

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