3.2080 \(\int \frac{a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=140 \[ \frac{15 e^2}{4 \sqrt{d+e x} (b d-a e)^3}-\frac{15 \sqrt{b} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{7/2}}+\frac{5 e}{4 (a+b x) \sqrt{d+e x} (b d-a e)^2}-\frac{1}{2 (a+b x)^2 \sqrt{d+e x} (b d-a e)} \]

[Out]

(15*e^2)/(4*(b*d - a*e)^3*Sqrt[d + e*x]) - 1/(2*(b*d - a*e)*(a + b*x)^2*Sqrt[d +
e*x]) + (5*e)/(4*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x]) - (15*Sqrt[b]*e^2*ArcTa
nh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.192305, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ \frac{15 e^2}{4 \sqrt{d+e x} (b d-a e)^3}-\frac{15 \sqrt{b} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{7/2}}+\frac{5 e}{4 (a+b x) \sqrt{d+e x} (b d-a e)^2}-\frac{1}{2 (a+b x)^2 \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(15*e^2)/(4*(b*d - a*e)^3*Sqrt[d + e*x]) - 1/(2*(b*d - a*e)*(a + b*x)^2*Sqrt[d +
e*x]) + (5*e)/(4*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x]) - (15*Sqrt[b]*e^2*ArcTa
nh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 58.0029, size = 128, normalized size = 0.91 \[ - \frac{15 \sqrt{b} e^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 \left (a e - b d\right )^{\frac{7}{2}}} - \frac{15 b e \sqrt{d + e x}}{4 \left (a + b x\right ) \left (a e - b d\right )^{3}} - \frac{5 e}{2 \left (a + b x\right ) \sqrt{d + e x} \left (a e - b d\right )^{2}} + \frac{1}{2 \left (a + b x\right )^{2} \sqrt{d + e x} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*sqrt(b)*e**2*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(4*(a*e - b*d)**(7/
2)) - 15*b*e*sqrt(d + e*x)/(4*(a + b*x)*(a*e - b*d)**3) - 5*e/(2*(a + b*x)*sqrt(
d + e*x)*(a*e - b*d)**2) + 1/(2*(a + b*x)**2*sqrt(d + e*x)*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.360216, size = 126, normalized size = 0.9 \[ \frac{1}{4} \left (\frac{8 a^2 e^2+a b e (9 d+25 e x)+b^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )}{(a+b x)^2 \sqrt{d+e x} (b d-a e)^3}-\frac{15 \sqrt{b} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((8*a^2*e^2 + a*b*e*(9*d + 25*e*x) + b^2*(-2*d^2 + 5*d*e*x + 15*e^2*x^2))/((b*d
- a*e)^3*(a + b*x)^2*Sqrt[d + e*x]) - (15*Sqrt[b]*e^2*ArcTanh[(Sqrt[b]*Sqrt[d +
e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(7/2))/4

_______________________________________________________________________________________

Maple [A]  time = 0.073, size = 179, normalized size = 1.3 \[ -2\,{\frac{{e}^{2}}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}-{\frac{7\,{b}^{2}{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{9\,ab{e}^{3}}{4\, \left ( ae-bd \right ) ^{3} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{9\,{b}^{2}d{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{15\,b{e}^{2}}{4\, \left ( ae-bd \right ) ^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*e^2/(a*e-b*d)^3/(e*x+d)^(1/2)-7/4*e^2*b^2/(a*e-b*d)^3/(b*e*x+a*e)^2*(e*x+d)^(
3/2)-9/4*e^3*b/(a*e-b*d)^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*a+9/4*e^2*b^2/(a*e-b*d)^3
/(b*e*x+a*e)^2*(e*x+d)^(1/2)*d-15/4*e^2*b/(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((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.310183, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, b^{2} e^{2} x^{2} - 4 \, b^{2} d^{2} + 18 \, a b d e + 16 \, a^{2} e^{2} - 15 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) + 10 \,{\left (b^{2} d e + 5 \, a b e^{2}\right )} x}{8 \,{\left (a^{2} b^{3} d^{3} - 3 \, a^{3} b^{2} d^{2} e + 3 \, a^{4} b d e^{2} - a^{5} e^{3} +{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{4} d^{3} - 3 \, a^{2} b^{3} d^{2} e + 3 \, a^{3} b^{2} d e^{2} - a^{4} b e^{3}\right )} x\right )} \sqrt{e x + d}}, \frac{15 \, b^{2} e^{2} x^{2} - 2 \, b^{2} d^{2} + 9 \, a b d e + 8 \, a^{2} e^{2} - 15 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) + 5 \,{\left (b^{2} d e + 5 \, a b e^{2}\right )} x}{4 \,{\left (a^{2} b^{3} d^{3} - 3 \, a^{3} b^{2} d^{2} e + 3 \, a^{4} b d e^{2} - a^{5} e^{3} +{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{4} d^{3} - 3 \, a^{2} b^{3} d^{2} e + 3 \, a^{3} b^{2} d e^{2} - a^{4} b e^{3}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(30*b^2*e^2*x^2 - 4*b^2*d^2 + 18*a*b*d*e + 16*a^2*e^2 - 15*(b^2*e^2*x^2 + 2
*a*b*e^2*x + a^2*e^2)*sqrt(e*x + d)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e
+ 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 10*(b^2*d*e + 5
*a*b*e^2)*x)/((a^2*b^3*d^3 - 3*a^3*b^2*d^2*e + 3*a^4*b*d*e^2 - a^5*e^3 + (b^5*d^
3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*x^2 + 2*(a*b^4*d^3 - 3*a^2*b^
3*d^2*e + 3*a^3*b^2*d*e^2 - a^4*b*e^3)*x)*sqrt(e*x + d)), 1/4*(15*b^2*e^2*x^2 -
2*b^2*d^2 + 9*a*b*d*e + 8*a^2*e^2 - 15*(b^2*e^2*x^2 + 2*a*b*e^2*x + a^2*e^2)*sqr
t(e*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e
*x + d)*b)) + 5*(b^2*d*e + 5*a*b*e^2)*x)/((a^2*b^3*d^3 - 3*a^3*b^2*d^2*e + 3*a^4
*b*d*e^2 - a^5*e^3 + (b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*x
^2 + 2*(a*b^4*d^3 - 3*a^2*b^3*d^2*e + 3*a^3*b^2*d*e^2 - a^4*b*e^3)*x)*sqrt(e*x +
d))]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.306513, size = 317, normalized size = 2.26 \[ \frac{15 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \, e^{2}}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{x e + d}} + \frac{7 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{2} - 9 \, \sqrt{x e + d} b^{2} d e^{2} + 9 \, \sqrt{x e + d} a b e^{3}}{4 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/4*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^2/((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)) + 2*e^2/((b^3*d^3 - 3*a*b^2*d
^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(x*e + d)) + 1/4*(7*(x*e + d)^(3/2)*b^2*e^2
- 9*sqrt(x*e + d)*b^2*d*e^2 + 9*sqrt(x*e + d)*a*b*e^3)/((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)^2)