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

Optimal. Leaf size=72 \[ \frac{A b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+b x^2}}{2 a x^2}-\frac{B \sqrt{a+b x^2}}{a x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.192931, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{A b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+b x^2}}{2 a x^2}-\frac{B \sqrt{a+b x^2}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.1556, size = 60, normalized size = 0.83 \[ - \frac{A \sqrt{a + b x^{2}}}{2 a x^{2}} + \frac{A b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} - \frac{B \sqrt{a + b x^{2}}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0795756, size = 72, normalized size = 1. \[ \frac{-\sqrt{a} \sqrt{a+b x^2} (A+2 B x)+A b x^2 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )-A b x^2 \log (x)}{2 a^{3/2} x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 68, normalized size = 0.9 \[ -{\frac{A}{2\,a{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B}{ax}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^3/(b*x^2+a)^(1/2),x)

[Out]

-1/2*A*(b*x^2+a)^(1/2)/a/x^2+1/2*A*b/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/
x)-B*(b*x^2+a)^(1/2)/a/x

_______________________________________________________________________________________

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(b*x^2 + a)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.273409, size = 1, normalized size = 0.01 \[ \left [\frac{A b x^{2} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \, \sqrt{b x^{2} + a}{\left (2 \, B x + A\right )} \sqrt{a}}{4 \, a^{\frac{3}{2}} x^{2}}, \frac{A b x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) - \sqrt{b x^{2} + a}{\left (2 \, B x + A\right )} \sqrt{-a}}{2 \, \sqrt{-a} a x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 5.06845, size = 66, normalized size = 0.92 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 a x} + \frac{A b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{3}{2}}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*sqrt(b)*sqrt(a/(b*x**2) + 1)/(2*a*x) + A*b*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(
3/2)) - B*sqrt(b)*sqrt(a/(b*x**2) + 1)/a

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225065, size = 197, normalized size = 2.74 \[ -\frac{A b \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{3} A b + 2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a \sqrt{b} +{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{2} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*b*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + ((sqrt(b)*x
- sqrt(b*x^2 + a))^3*A*b + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a*sqrt(b) + (sqrt
(b)*x - sqrt(b*x^2 + a))*A*a*b - 2*B*a^2*sqrt(b))/(((sqrt(b)*x - sqrt(b*x^2 + a)
)^2 - a)^2*a)