3.769 \(\int \frac{\sqrt{x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=127 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2}}+\frac{\sqrt{x} (a B+A b)}{8 a^2 b^2 (a+b x)}-\frac{\sqrt{x} (a B+A b)}{4 a b^2 (a+b x)^2}+\frac{x^{3/2} (A b-a B)}{3 a b (a+b x)^3} \]

[Out]

((A*b - a*B)*x^(3/2))/(3*a*b*(a + b*x)^3) - ((A*b + a*B)*Sqrt[x])/(4*a*b^2*(a +
b*x)^2) + ((A*b + a*B)*Sqrt[x])/(8*a^2*b^2*(a + b*x)) + ((A*b + a*B)*ArcTan[(Sqr
t[b]*Sqrt[x])/Sqrt[a]])/(8*a^(5/2)*b^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.141792, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2}}+\frac{\sqrt{x} (a B+A b)}{8 a^2 b^2 (a+b x)}-\frac{\sqrt{x} (a B+A b)}{4 a b^2 (a+b x)^2}+\frac{x^{3/2} (A b-a B)}{3 a b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*x^(3/2))/(3*a*b*(a + b*x)^3) - ((A*b + a*B)*Sqrt[x])/(4*a*b^2*(a +
b*x)^2) + ((A*b + a*B)*Sqrt[x])/(8*a^2*b^2*(a + b*x)) + ((A*b + a*B)*ArcTan[(Sqr
t[b]*Sqrt[x])/Sqrt[a]])/(8*a^(5/2)*b^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.5894, size = 109, normalized size = 0.86 \[ \frac{x^{\frac{3}{2}} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{3}} - \frac{\sqrt{x} \left (A b + B a\right )}{4 a b^{2} \left (a + b x\right )^{2}} + \frac{\sqrt{x} \left (A b + B a\right )}{8 a^{2} b^{2} \left (a + b x\right )} + \frac{\left (A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*x**(1/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.215132, size = 106, normalized size = 0.83 \[ \frac{\frac{3 (a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{\sqrt{b} \sqrt{x} \left (-3 a^3 B-a^2 b (3 A+8 B x)+a b^2 x (8 A+3 B x)+3 A b^3 x^2\right )}{a^2 (a+b x)^3}}{24 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[b]*Sqrt[x]*(-3*a^3*B + 3*A*b^3*x^2 + a*b^2*x*(8*A + 3*B*x) - a^2*b*(3*A +
8*B*x)))/(a^2*(a + b*x)^3) + (3*(A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/
a^(5/2))/(24*b^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 111, normalized size = 0.9 \[ 2\,{\frac{1}{ \left ( bx+a \right ) ^{3}} \left ( 1/16\,{\frac{ \left ( Ab+Ba \right ){x}^{5/2}}{{a}^{2}}}+1/6\,{\frac{ \left ( Ab-Ba \right ){x}^{3/2}}{ab}}-1/16\,{\frac{ \left ( Ab+Ba \right ) \sqrt{x}}{{b}^{2}}} \right ) }+{\frac{A}{8\,{a}^{2}b}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{8\,a{b}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(1/16*(A*b+B*a)/a^2*x^(5/2)+1/6*(A*b-B*a)/a/b*x^(3/2)-1/16*(A*b+B*a)/b^2*x^(1/
2))/(b*x+a)^3+1/8/a^2/b/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+1/8/a/b^2/(a
*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.294718, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, B a^{3} + 3 \, A a^{2} b - 3 \,{\left (B a b^{2} + A b^{3}\right )} x^{2} + 8 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 3 \,{\left (B a^{4} + A a^{3} b +{\left (B a b^{3} + A b^{4}\right )} x^{3} + 3 \,{\left (B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 3 \,{\left (B a^{3} b + A a^{2} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{48 \,{\left (a^{2} b^{5} x^{3} + 3 \, a^{3} b^{4} x^{2} + 3 \, a^{4} b^{3} x + a^{5} b^{2}\right )} \sqrt{-a b}}, -\frac{{\left (3 \, B a^{3} + 3 \, A a^{2} b - 3 \,{\left (B a b^{2} + A b^{3}\right )} x^{2} + 8 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 3 \,{\left (B a^{4} + A a^{3} b +{\left (B a b^{3} + A b^{4}\right )} x^{3} + 3 \,{\left (B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 3 \,{\left (B a^{3} b + A a^{2} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{24 \,{\left (a^{2} b^{5} x^{3} + 3 \, a^{3} b^{4} x^{2} + 3 \, a^{4} b^{3} x + a^{5} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(2*(3*B*a^3 + 3*A*a^2*b - 3*(B*a*b^2 + A*b^3)*x^2 + 8*(B*a^2*b - A*a*b^2)
*x)*sqrt(-a*b)*sqrt(x) - 3*(B*a^4 + A*a^3*b + (B*a*b^3 + A*b^4)*x^3 + 3*(B*a^2*b
^2 + A*a*b^3)*x^2 + 3*(B*a^3*b + A*a^2*b^2)*x)*log((2*a*b*sqrt(x) + sqrt(-a*b)*(
b*x - a))/(b*x + a)))/((a^2*b^5*x^3 + 3*a^3*b^4*x^2 + 3*a^4*b^3*x + a^5*b^2)*sqr
t(-a*b)), -1/24*((3*B*a^3 + 3*A*a^2*b - 3*(B*a*b^2 + A*b^3)*x^2 + 8*(B*a^2*b - A
*a*b^2)*x)*sqrt(a*b)*sqrt(x) + 3*(B*a^4 + A*a^3*b + (B*a*b^3 + A*b^4)*x^3 + 3*(B
*a^2*b^2 + A*a*b^3)*x^2 + 3*(B*a^3*b + A*a^2*b^2)*x)*arctan(a/(sqrt(a*b)*sqrt(x)
)))/((a^2*b^5*x^3 + 3*a^3*b^4*x^2 + 3*a^4*b^3*x + a^5*b^2)*sqrt(a*b))]

_______________________________________________________________________________________

Sympy [A]  time = 24.4823, size = 915, normalized size = 7.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*x**(1/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-66*A*a**3*sqrt(x)/(48*a**6*b + 144*a**5*b**2*x + 144*a**4*b**3*x**2 + 48*a**3*b
**4*x**3) - 80*A*a**2*x**(3/2)/(48*a**6 + 144*a**5*b*x + 144*a**4*b**2*x**2 + 48
*a**3*b**3*x**3) - 30*A*a*b*x**(5/2)/(48*a**6 + 144*a**5*b*x + 144*a**4*b**2*x**
2 + 48*a**3*b**3*x**3) + 10*A*a*sqrt(x)/(8*a**4*b + 16*a**3*b**2*x + 8*a**2*b**3
*x**2) + 5*A*a*sqrt(-1/(a**7*b))*log(-a**4*sqrt(-1/(a**7*b)) + sqrt(x))/(16*b) -
5*A*a*sqrt(-1/(a**7*b))*log(a**4*sqrt(-1/(a**7*b)) + sqrt(x))/(16*b) + 6*A*x**(
3/2)/(8*a**4 + 16*a**3*b*x + 8*a**2*b**2*x**2) - 3*A*sqrt(-1/(a**5*b))*log(-a**3
*sqrt(-1/(a**5*b)) + sqrt(x))/(8*b) + 3*A*sqrt(-1/(a**5*b))*log(a**3*sqrt(-1/(a*
*5*b)) + sqrt(x))/(8*b) + 66*B*a**4*sqrt(x)/(48*a**6*b**2 + 144*a**5*b**3*x + 14
4*a**4*b**4*x**2 + 48*a**3*b**5*x**3) + 80*B*a**3*x**(3/2)/(48*a**6*b + 144*a**5
*b**2*x + 144*a**4*b**3*x**2 + 48*a**3*b**4*x**3) + 30*B*a**2*x**(5/2)/(48*a**6
+ 144*a**5*b*x + 144*a**4*b**2*x**2 + 48*a**3*b**3*x**3) - 20*B*a**2*sqrt(x)/(8*
a**4*b**2 + 16*a**3*b**3*x + 8*a**2*b**4*x**2) - 5*B*a**2*sqrt(-1/(a**7*b))*log(
-a**4*sqrt(-1/(a**7*b)) + sqrt(x))/(16*b**2) + 5*B*a**2*sqrt(-1/(a**7*b))*log(a*
*4*sqrt(-1/(a**7*b)) + sqrt(x))/(16*b**2) - 12*B*a*x**(3/2)/(8*a**4*b + 16*a**3*
b**2*x + 8*a**2*b**3*x**2) + 3*B*a*sqrt(-1/(a**5*b))*log(-a**3*sqrt(-1/(a**5*b))
+ sqrt(x))/(4*b**2) - 3*B*a*sqrt(-1/(a**5*b))*log(a**3*sqrt(-1/(a**5*b)) + sqrt
(x))/(4*b**2) + 2*B*sqrt(x)/(2*a**2*b**2 + 2*a*b**3*x) - B*sqrt(-1/(a**3*b))*log
(-a**2*sqrt(-1/(a**3*b)) + sqrt(x))/(2*b**2) + B*sqrt(-1/(a**3*b))*log(a**2*sqrt
(-1/(a**3*b)) + sqrt(x))/(2*b**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.272686, size = 143, normalized size = 1.13 \[ \frac{{\left (B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b^{2}} + \frac{3 \, B a b^{2} x^{\frac{5}{2}} + 3 \, A b^{3} x^{\frac{5}{2}} - 8 \, B a^{2} b x^{\frac{3}{2}} + 8 \, A a b^{2} x^{\frac{3}{2}} - 3 \, B a^{3} \sqrt{x} - 3 \, A a^{2} b \sqrt{x}}{24 \,{\left (b x + a\right )}^{3} a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(B*a + A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2*b^2) + 1/24*(3*B*a*b^
2*x^(5/2) + 3*A*b^3*x^(5/2) - 8*B*a^2*b*x^(3/2) + 8*A*a*b^2*x^(3/2) - 3*B*a^3*sq
rt(x) - 3*A*a^2*b*sqrt(x))/((b*x + a)^3*a^2*b^2)