3.1649 \(\int \frac{1}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=147 \[ \frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{7/2}}-\frac{5 e^2 \sqrt{d+e x}}{8 (a+b x) (b d-a e)^3}+\frac{5 e \sqrt{d+e x}}{12 (a+b x)^2 (b d-a e)^2}-\frac{\sqrt{d+e x}}{3 (a+b x)^3 (b d-a e)} \]

[Out]

-Sqrt[d + e*x]/(3*(b*d - a*e)*(a + b*x)^3) + (5*e*Sqrt[d + e*x])/(12*(b*d - a*e)
^2*(a + b*x)^2) - (5*e^2*Sqrt[d + e*x])/(8*(b*d - a*e)^3*(a + b*x)) + (5*e^3*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*Sqrt[b]*(b*d - a*e)^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.212139, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{7/2}}-\frac{5 e^2 \sqrt{d+e x}}{8 (a+b x) (b d-a e)^3}+\frac{5 e \sqrt{d+e x}}{12 (a+b x)^2 (b d-a e)^2}-\frac{\sqrt{d+e x}}{3 (a+b x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[d + e*x]/(3*(b*d - a*e)*(a + b*x)^3) + (5*e*Sqrt[d + e*x])/(12*(b*d - a*e)
^2*(a + b*x)^2) - (5*e^2*Sqrt[d + e*x])/(8*(b*d - a*e)^3*(a + b*x)) + (5*e^3*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*Sqrt[b]*(b*d - a*e)^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 61.9029, size = 128, normalized size = 0.87 \[ \frac{5 e^{2} \sqrt{d + e x}}{8 \left (a + b x\right ) \left (a e - b d\right )^{3}} + \frac{5 e \sqrt{d + e x}}{12 \left (a + b x\right )^{2} \left (a e - b d\right )^{2}} + \frac{\sqrt{d + e x}}{3 \left (a + b x\right )^{3} \left (a e - b d\right )} + \frac{5 e^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 \sqrt{b} \left (a e - b d\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*e**2*sqrt(d + e*x)/(8*(a + b*x)*(a*e - b*d)**3) + 5*e*sqrt(d + e*x)/(12*(a + b
*x)**2*(a*e - b*d)**2) + sqrt(d + e*x)/(3*(a + b*x)**3*(a*e - b*d)) + 5*e**3*ata
n(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(8*sqrt(b)*(a*e - b*d)**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.219211, size = 128, normalized size = 0.87 \[ \frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{7/2}}-\frac{\sqrt{d+e x} \left (33 a^2 e^2+2 a b e (20 e x-13 d)+b^2 \left (8 d^2-10 d e x+15 e^2 x^2\right )\right )}{24 (a+b x)^3 (b d-a e)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(33*a^2*e^2 + 2*a*b*e*(-13*d + 20*e*x) + b^2*(8*d^2 - 10*d*e*x +
15*e^2*x^2)))/(24*(b*d - a*e)^3*(a + b*x)^3) + (5*e^3*ArcTanh[(Sqrt[b]*Sqrt[d +
e*x])/Sqrt[b*d - a*e]])/(8*Sqrt[b]*(b*d - a*e)^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 147, normalized size = 1. \[{\frac{{e}^{3}}{ \left ( 3\,ae-3\,bd \right ) \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{5\,{e}^{3}}{12\, \left ( ae-bd \right ) ^{2} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{5\,{e}^{3}}{8\, \left ( ae-bd \right ) ^{3} \left ( bex+ae \right ) }\sqrt{ex+d}}+{\frac{5\,{e}^{3}}{8\, \left ( ae-bd \right ) ^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*e^3*(e*x+d)^(1/2)/(a*e-b*d)/(b*e*x+a*e)^3+5/12*e^3/(a*e-b*d)^2*(e*x+d)^(1/2)
/(b*e*x+a*e)^2+5/8*e^3/(a*e-b*d)^3*(e*x+d)^(1/2)/(b*e*x+a*e)+5/8*e^3/(a*e-b*d)^3
/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.22412, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (15 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 26 \, a b d e + 33 \, a^{2} e^{2} - 10 \,{\left (b^{2} d e - 4 \, a b e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} + 15 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{48 \,{\left (a^{3} b^{3} d^{3} - 3 \, a^{4} b^{2} d^{2} e + 3 \, a^{5} b d e^{2} - a^{6} e^{3} +{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{3} + 3 \,{\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3}\right )} x\right )} \sqrt{b^{2} d - a b e}}, -\frac{{\left (15 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 26 \, a b d e + 33 \, a^{2} e^{2} - 10 \,{\left (b^{2} d e - 4 \, a b e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} - 15 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{24 \,{\left (a^{3} b^{3} d^{3} - 3 \, a^{4} b^{2} d^{2} e + 3 \, a^{5} b d e^{2} - a^{6} e^{3} +{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{3} + 3 \,{\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3}\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(2*(15*b^2*e^2*x^2 + 8*b^2*d^2 - 26*a*b*d*e + 33*a^2*e^2 - 10*(b^2*d*e -
4*a*b*e^2)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 15*(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^3 - 3*a^4*b^2*d^2*e + 3
*a^5*b*d*e^2 - a^6*e^3 + (b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^
3)*x^3 + 3*(a*b^5*d^3 - 3*a^2*b^4*d^2*e + 3*a^3*b^3*d*e^2 - a^4*b^2*e^3)*x^2 + 3
*(a^2*b^4*d^3 - 3*a^3*b^3*d^2*e + 3*a^4*b^2*d*e^2 - a^5*b*e^3)*x)*sqrt(b^2*d - a
*b*e)), -1/24*((15*b^2*e^2*x^2 + 8*b^2*d^2 - 26*a*b*d*e + 33*a^2*e^2 - 10*(b^2*d
*e - 4*a*b*e^2)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) - 15*(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)*s
qrt(e*x + d))))/((a^3*b^3*d^3 - 3*a^4*b^2*d^2*e + 3*a^5*b*d*e^2 - a^6*e^3 + (b^6
*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*x^3 + 3*(a*b^5*d^3 - 3*a^2
*b^4*d^2*e + 3*a^3*b^3*d*e^2 - a^4*b^2*e^3)*x^2 + 3*(a^2*b^4*d^3 - 3*a^3*b^3*d^2
*e + 3*a^4*b^2*d*e^2 - a^5*b*e^3)*x)*sqrt(-b^2*d + a*b*e))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{4} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + b*x)**4*sqrt(d + e*x)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212796, size = 315, normalized size = 2.14 \[ -\frac{5 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{15 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} - 40 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} + 33 \, \sqrt{x e + d} b^{2} d^{2} e^{3} + 40 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} - 66 \, \sqrt{x e + d} a b d e^{4} + 33 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/8*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^3*d^3 - 3*a*b^2*d^2*e
+ 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(-b^2*d + a*b*e)) - 1/24*(15*(x*e + d)^(5/2)*b^2*
e^3 - 40*(x*e + d)^(3/2)*b^2*d*e^3 + 33*sqrt(x*e + d)*b^2*d^2*e^3 + 40*(x*e + d)
^(3/2)*a*b*e^4 - 66*sqrt(x*e + d)*a*b*d*e^4 + 33*sqrt(x*e + d)*a^2*e^5)/((b^3*d^
3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*((x*e + d)*b - b*d + a*e)^3)