3.319 \(\int \frac{A+B x}{x^{3/2} (a+b x)} \, dx\)

Optimal. Leaf size=49 \[ -\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{2 A}{a \sqrt{x}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0701896, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{2 A}{a \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.35686, size = 46, normalized size = 0.94 \[ - \frac{2 A}{a \sqrt{x}} - \frac{2 \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(3/2)/(b*x+a),x)

[Out]

-2*A/(a*sqrt(x)) - 2*(A*b - B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(a**(3/2)*sqrt(b)
)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0517566, size = 49, normalized size = 1. \[ \frac{2 (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{2 A}{a \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 53, normalized size = 1.1 \[ -2\,{\frac{A}{a\sqrt{x}}}-2\,{\frac{Ab}{a\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+2\,{\frac{B}{\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(3/2)/(b*x+a),x)

[Out]

-2*A/a/x^(1/2)-2/a/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A*b+2/(a*b)^(1/2)*a
rctan(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)/((b*x + a)*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.222826, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (B a - A b\right )} \sqrt{x} \log \left (-\frac{2 \, a b \sqrt{x} - \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right ) + 2 \, \sqrt{-a b} A}{\sqrt{-a b} a \sqrt{x}}, -\frac{2 \,{\left ({\left (B a - A b\right )} \sqrt{x} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right ) + \sqrt{a b} A\right )}}{\sqrt{a b} a \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-((B*a - A*b)*sqrt(x)*log(-(2*a*b*sqrt(x) - sqrt(-a*b)*(b*x - a))/(b*x + a)) +
2*sqrt(-a*b)*A)/(sqrt(-a*b)*a*sqrt(x)), -2*((B*a - A*b)*sqrt(x)*arctan(a/(sqrt(a
*b)*sqrt(x))) + sqrt(a*b)*A)/(sqrt(a*b)*a*sqrt(x))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{\frac{3}{2}} \left (a + b x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**(3/2)/(b*x+a),x)

[Out]

Integral((A + B*x)/(x**(3/2)*(a + b*x)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213324, size = 53, normalized size = 1.08 \[ \frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{2 \, A}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(B*a - A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a) - 2*A/(a*sqrt(x))