3.1646 \(\int \frac{(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=126 \[ -\frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} \sqrt{b d-a e}}-\frac{5 e^2 \sqrt{d+e x}}{8 b^3 (a+b x)}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{5/2}}{3 b (a+b x)^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.177421, antiderivative size = 126, 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 \[ -\frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} \sqrt{b d-a e}}-\frac{5 e^2 \sqrt{d+e x}}{8 b^3 (a+b x)}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{5/2}}{3 b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 45.9103, size = 112, normalized size = 0.89 \[ - \frac{\left (d + e x\right )^{\frac{5}{2}}}{3 b \left (a + b x\right )^{3}} - \frac{5 e \left (d + e x\right )^{\frac{3}{2}}}{12 b^{2} \left (a + b x\right )^{2}} - \frac{5 e^{2} \sqrt{d + e x}}{8 b^{3} \left (a + b x\right )} + \frac{5 e^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{7}{2}} \sqrt{a e - b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.215529, size = 119, normalized size = 0.94 \[ -\frac{\sqrt{d+e x} \left (15 a^2 e^2+10 a b e (d+4 e x)+b^2 \left (8 d^2+26 d e x+33 e^2 x^2\right )\right )}{24 b^3 (a+b x)^3}-\frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} \sqrt{b d-a e}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

-(Sqrt[d + e*x]*(15*a^2*e^2 + 10*a*b*e*(d + 4*e*x) + b^2*(8*d^2 + 26*d*e*x + 33*
e^2*x^2)))/(24*b^3*(a + b*x)^3) - (5*e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*
d - a*e]])/(8*b^(7/2)*Sqrt[b*d - a*e])

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 204, normalized size = 1.6 \[ -{\frac{11\,{e}^{3}}{8\, \left ( bex+ae \right ) ^{3}b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{e}^{4}a}{3\, \left ( bex+ae \right ) ^{3}{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{e}^{3}d}{3\, \left ( bex+ae \right ) ^{3}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{2}{e}^{5}}{8\, \left ( bex+ae \right ) ^{3}{b}^{3}}\sqrt{ex+d}}+{\frac{5\,{e}^{4}ad}{4\, \left ( bex+ae \right ) ^{3}{b}^{2}}\sqrt{ex+d}}-{\frac{5\,{e}^{3}{d}^{2}}{8\, \left ( bex+ae \right ) ^{3}b}\sqrt{ex+d}}+{\frac{5\,{e}^{3}}{8\,{b}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

-11/8*e^3/(b*e*x+a*e)^3/b*(e*x+d)^(5/2)-5/3*e^4/(b*e*x+a*e)^3/b^2*(e*x+d)^(3/2)*
a+5/3*e^3/(b*e*x+a*e)^3/b*(e*x+d)^(3/2)*d-5/8*e^5/(b*e*x+a*e)^3/b^3*(e*x+d)^(1/2
)*a^2+5/4*e^4/(b*e*x+a*e)^3/b^2*(e*x+d)^(1/2)*a*d-5/8*e^3/(b*e*x+a*e)^3/b*(e*x+d
)^(1/2)*d^2+5/8*e^3/b^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((e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.222146, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (33 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} + 10 \, a b d e + 15 \, a^{2} e^{2} + 2 \,{\left (13 \, b^{2} d e + 20 \, a b e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} - 15 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{48 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )} \sqrt{b^{2} d - a b e}}, -\frac{{\left (33 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} + 10 \, a b d e + 15 \, a^{2} e^{2} + 2 \,{\left (13 \, b^{2} d e + 20 \, a b e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} + 15 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{24 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(2*(33*b^2*e^2*x^2 + 8*b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2 + 2*(13*b^2*d*e
+ 20*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)))/((b^6*x^3 + 3*a*b^5*x^2 + 3*a^2*
b^4*x + a^3*b^3)*sqrt(b^2*d - a*b*e)), -1/24*((33*b^2*e^2*x^2 + 8*b^2*d^2 + 10*a
*b*d*e + 15*a^2*e^2 + 2*(13*b^2*d*e + 20*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)*sqrt(e*x + d))))/((b^6*x^3 + 3*a*b^5*x^2 + 3*a^
2*b^4*x + a^3*b^3)*sqrt(-b^2*d + a*b*e))]

_______________________________________________________________________________________

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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220133, size = 223, normalized size = 1.77 \[ \frac{5 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \, \sqrt{-b^{2} d + a b e} b^{3}} - \frac{33 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} - 40 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} + 15 \, \sqrt{x e + d} b^{2} d^{2} e^{3} + 40 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} - 30 \, \sqrt{x e + d} a b d e^{4} + 15 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="giac")

[Out]

5/8*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/(sqrt(-b^2*d + a*b*e)*b^3)
- 1/24*(33*(x*e + d)^(5/2)*b^2*e^3 - 40*(x*e + d)^(3/2)*b^2*d*e^3 + 15*sqrt(x*e
+ d)*b^2*d^2*e^3 + 40*(x*e + d)^(3/2)*a*b*e^4 - 30*sqrt(x*e + d)*a*b*d*e^4 + 15*
sqrt(x*e + d)*a^2*e^5)/(((x*e + d)*b - b*d + a*e)^3*b^3)