3.1647 \(\int \frac{(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=136 \[ \frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{3/2}}-\frac{e^2 \sqrt{d+e x}}{8 b^2 (a+b x) (b d-a e)}-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{3/2}}{3 b (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.1999, antiderivative size = 136, 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 \[ \frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{3/2}}-\frac{e^2 \sqrt{d+e x}}{8 b^2 (a+b x) (b d-a e)}-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{3/2}}{3 b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 52.7694, size = 114, normalized size = 0.84 \[ - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3 b \left (a + b x\right )^{3}} + \frac{e^{2} \sqrt{d + e x}}{8 b^{2} \left (a + b x\right ) \left (a e - b d\right )} - \frac{e \sqrt{d + e x}}{4 b^{2} \left (a + b x\right )^{2}} + \frac{e^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{5}{2}} \left (a e - b d\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)**(3/2)/(3*b*(a + b*x)**3) + e**2*sqrt(d + e*x)/(8*b**2*(a + b*x)*(a*e
- b*d)) - e*sqrt(d + e*x)/(4*b**2*(a + b*x)**2) + e**3*atan(sqrt(b)*sqrt(d + e*
x)/sqrt(a*e - b*d))/(8*b**(5/2)*(a*e - b*d)**(3/2))

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Mathematica [A]  time = 0.201131, size = 128, normalized size = 0.94 \[ \frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{3/2}}+\sqrt{d+e x} \left (-\frac{e^2}{8 b^2 (a+b x) (b d-a e)}+\frac{a e-b d}{3 b^2 (a+b x)^3}-\frac{7 e}{12 b^2 (a+b x)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 163, normalized size = 1.2 \[{\frac{{e}^{3}}{8\, \left ( bex+ae \right ) ^{3} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{3}}{3\, \left ( bex+ae \right ) ^{3}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{4}a}{8\, \left ( bex+ae \right ) ^{3}{b}^{2}}\sqrt{ex+d}}+{\frac{{e}^{3}d}{8\, \left ( bex+ae \right ) ^{3}b}\sqrt{ex+d}}+{\frac{{e}^{3}}{ \left ( 8\,ae-8\,bd \right ){b}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*e^3/(b*e*x+a*e)^3/(a*e-b*d)*(e*x+d)^(5/2)-1/3*e^3/(b*e*x+a*e)^3/b*(e*x+d)^(3
/2)-1/8*e^4/(b*e*x+a*e)^3/b^2*(e*x+d)^(1/2)*a+1/8*e^3/(b*e*x+a*e)^3/b*(e*x+d)^(1
/2)*d+1/8*e^3/(a*e-b*d)/b^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((e*x + d)^(3/2)/(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 = 0.223388, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 2 \, a b d e - 3 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 4 \, a b e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} + 3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{48 \,{\left (a^{3} b^{3} d - a^{4} b^{2} e +{\left (b^{6} d - a b^{5} e\right )} x^{3} + 3 \,{\left (a b^{5} d - a^{2} b^{4} e\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d - a^{3} b^{3} e\right )} x\right )} \sqrt{b^{2} d - a b e}}, -\frac{{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 2 \, a b d e - 3 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 4 \, a b e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} - 3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{24 \,{\left (a^{3} b^{3} d - a^{4} b^{2} e +{\left (b^{6} d - a b^{5} e\right )} x^{3} + 3 \,{\left (a b^{5} d - a^{2} b^{4} e\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d - a^{3} b^{3} e\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(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 - 2*a*b*d*e - 3*a^2*e^2 + 2*(7*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 - a^4*b^2*e + (b^6*d - a*b
^5*e)*x^3 + 3*(a*b^5*d - a^2*b^4*e)*x^2 + 3*(a^2*b^4*d - a^3*b^3*e)*x)*sqrt(b^2*
d - a*b*e)), -1/24*((3*b^2*e^2*x^2 + 8*b^2*d^2 - 2*a*b*d*e - 3*a^2*e^2 + 2*(7*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 - a^4*b^2*e + (b^6*d - a*b^5*e)*x^3 + 3*(a*b^5*d -
a^2*b^4*e)*x^2 + 3*(a^2*b^4*d - a^3*b^3*e)*x)*sqrt(-b^2*d + a*b*e))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217358, size = 258, normalized size = 1.9 \[ -\frac{\arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{3} d - a b^{2} e\right )} \sqrt{-b^{2} d + a b e}} - \frac{3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} + 8 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} - 3 \, \sqrt{x e + d} b^{2} d^{2} e^{3} - 8 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} + 6 \, \sqrt{x e + d} a b d e^{4} - 3 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left (b^{3} d - a b^{2} e\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(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 - a*b^2*e)*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*sqrt(x*e + d)*a*b*d
*e^4 - 3*sqrt(x*e + d)*a^2*e^5)/((b^3*d - a*b^2*e)*((x*e + d)*b - b*d + a*e)^3)