3.290 \(\int \frac{\sqrt{a+b x}}{x^3} \, dx\)

Optimal. Leaf size=65 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x}}{2 x^2}-\frac{b \sqrt{a+b x}}{4 a x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0575851, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x}}{2 x^2}-\frac{b \sqrt{a+b x}}{4 a x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/x^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.15407, size = 53, normalized size = 0.82 \[ - \frac{\sqrt{a + b x}}{2 x^{2}} - \frac{b \sqrt{a + b x}}{4 a x} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/x**3,x)

[Out]

-sqrt(a + b*x)/(2*x**2) - b*sqrt(a + b*x)/(4*a*x) + b**2*atanh(sqrt(a + b*x)/sqr
t(a))/(4*a**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0435298, size = 55, normalized size = 0.85 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x} (2 a+b x)}{4 a x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/x^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 53, normalized size = 0.8 \[ 2\,{b}^{2} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( bx+a \right ) ^{3/2}}{a}}-1/8\,\sqrt{bx+a} \right ) }+1/8\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x^3,x)

[Out]

2*b^2*((-1/8/a*(b*x+a)^(3/2)-1/8*(b*x+a)^(1/2))/x^2/b^2+1/8*arctanh((b*x+a)^(1/2
)/a^(1/2))/a^(3/2))

_______________________________________________________________________________________

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(sqrt(b*x + a)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.227193, size = 1, normalized size = 0.02 \[ \left [\frac{b^{2} x^{2} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \,{\left (b x + 2 \, a\right )} \sqrt{b x + a} \sqrt{a}}{8 \, a^{\frac{3}{2}} x^{2}}, -\frac{b^{2} x^{2} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (b x + 2 \, a\right )} \sqrt{b x + a} \sqrt{-a}}{4 \, \sqrt{-a} a x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(b^2*x^2*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x) - 2*(b*x + 2*a)*s
qrt(b*x + a)*sqrt(a))/(a^(3/2)*x^2), -1/4*(b^2*x^2*arctan(a/(sqrt(b*x + a)*sqrt(
-a))) + (b*x + 2*a)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a*x^2)]

_______________________________________________________________________________________

Sympy [A]  time = 11.6594, size = 97, normalized size = 1.49 \[ - \frac{a}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{3 \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{b^{\frac{3}{2}}}{4 a \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)/x**3,x)

[Out]

-a/(2*sqrt(b)*x**(5/2)*sqrt(a/(b*x) + 1)) - 3*sqrt(b)/(4*x**(3/2)*sqrt(a/(b*x) +
1)) - b**(3/2)/(4*a*sqrt(x)*sqrt(a/(b*x) + 1)) + b**2*asinh(sqrt(a)/(sqrt(b)*sq
rt(x)))/(4*a**(3/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207896, size = 89, normalized size = 1.37 \[ -\frac{\frac{b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b x + a\right )}^{\frac{3}{2}} b^{3} + \sqrt{b x + a} a b^{3}}{a b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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