3.2078 \(\int \frac{(a+b x) \sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=110 \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{3/2}}-\frac{e \sqrt{d+e x}}{4 b (a+b x) (b d-a e)}-\frac{\sqrt{d+e x}}{2 b (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.138786, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{3/2}}-\frac{e \sqrt{d+e x}}{4 b (a+b x) (b d-a e)}-\frac{\sqrt{d+e x}}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.0294, size = 88, normalized size = 0.8 \[ \frac{e \sqrt{d + e x}}{4 b \left (a + b x\right ) \left (a e - b d\right )} - \frac{\sqrt{d + e x}}{2 b \left (a + b x\right )^{2}} + \frac{e^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 b^{\frac{3}{2}} \left (a e - b d\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.150471, size = 99, normalized size = 0.9 \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} (-a e+2 b d+b e x)}{4 b (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(2*b*d - a*e + b*e*x))/(4*b*(b*d - a*e)*(a + b*x)^2) + (e^2*ArcT
anh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(3/2)*(b*d - a*e)^(3/2))

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Maple [A]  time = 0.019, size = 111, normalized size = 1. \[{\frac{{e}^{2}}{4\, \left ( bex+ae \right ) ^{2} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{2}}{4\, \left ( bex+ae \right ) ^{2}b}\sqrt{ex+d}}+{\frac{{e}^{2}}{ \left ( 4\,ae-4\,bd \right ) b}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*e^2/(b*e*x+a*e)^2/(a*e-b*d)*(e*x+d)^(3/2)-1/4*e^2/(b*e*x+a*e)^2/b*(e*x+d)^(1
/2)+1/4*e^2/(a*e-b*d)/b/(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((b*x + a)*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 = 0.301403, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} \sqrt{e x + d} +{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{8 \,{\left (a^{2} b^{2} d - a^{3} b e +{\left (b^{4} d - a b^{3} e\right )} x^{2} + 2 \,{\left (a b^{3} d - a^{2} b^{2} e\right )} x\right )} \sqrt{b^{2} d - a b e}}, -\frac{\sqrt{-b^{2} d + a b e}{\left (b e x + 2 \, b d - a e\right )} \sqrt{e x + d} -{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{4 \,{\left (a^{2} b^{2} d - a^{3} b e +{\left (b^{4} d - a b^{3} e\right )} x^{2} + 2 \,{\left (a b^{3} d - a^{2} b^{2} e\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(2*sqrt(b^2*d - a*b*e)*(b*e*x + 2*b*d - a*e)*sqrt(e*x + d) + (b^2*e^2*x^2
+ 2*a*b*e^2*x + a^2*e^2)*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^2*b^2*d - a^3*b*e + (b^4*d - a*b^3*e)
*x^2 + 2*(a*b^3*d - a^2*b^2*e)*x)*sqrt(b^2*d - a*b*e)), -1/4*(sqrt(-b^2*d + a*b*
e)*(b*e*x + 2*b*d - a*e)*sqrt(e*x + d) - (b^2*e^2*x^2 + 2*a*b*e^2*x + a^2*e^2)*a
rctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((a^2*b^2*d - a^3*b*e
+ (b^4*d - a*b^3*e)*x^2 + 2*(a*b^3*d - a^2*b^2*e)*x)*sqrt(-b^2*d + a*b*e))]

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Sympy [A]  time = 72.823, size = 1658, normalized size = 15.07 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*a**2*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) + 20*a*d*e**3*sqrt(d + e*x)/(8*a**4*e**4 - 1
6*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*a*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*a*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*sq
rt(-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*a*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) - 10*b*d**2*e**2*sqrt(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*b*d*e**2*(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*d*e**2*sqr
t(-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 + 3*d*e**2*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*sq
rt(-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 + 2*e**2*sqrt(d + e*x)/(2*a**
2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b**3*d*e*x) - e**2*sqrt(-1/(b*(a*e
- b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(
a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) +
e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*
a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sq
rt(d + e*x))/(2*b)

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GIAC/XCAS [A]  time = 0.314691, size = 178, normalized size = 1.62 \[ -\frac{\arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \,{\left (b^{2} d - a b e\right )} \sqrt{-b^{2} d + a b e}} - \frac{{\left (x e + d\right )}^{\frac{3}{2}} b e^{2} + \sqrt{x e + d} b d e^{2} - \sqrt{x e + d} a e^{3}}{4 \,{\left (b^{2} d - a b e\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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