3.513 \(\int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^4} \, dx\)

Optimal. Leaf size=66 \[ -\frac{A \left (a+b x^2\right )^{3/2}}{3 a x^3}-\frac{B \sqrt{a+b x^2}}{x}+\sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0851174, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{A \left (a+b x^2\right )^{3/2}}{3 a x^3}-\frac{B \sqrt{a+b x^2}}{x}+\sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^4,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.9052, size = 56, normalized size = 0.85 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 a x^{3}} + B \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )} - \frac{B \sqrt{a + b x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**4,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0697585, size = 70, normalized size = 1.06 \[ \sqrt{a+b x^2} \left (\frac{-3 a B-A b}{3 a x}-\frac{A}{3 x^3}\right )+\sqrt{b} B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^4,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 75, normalized size = 1.1 \[ -{\frac{A}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{B}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{bBx}{a}\sqrt{b{x}^{2}+a}}+B\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(b*x^2+a)^(1/2)/x^4,x)

[Out]

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

_______________________________________________________________________________________

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((B*x^2 + A)*sqrt(b*x^2 + a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.224419, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, B a \sqrt{b} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left ({\left (3 \, B a + A b\right )} x^{2} + A a\right )} \sqrt{b x^{2} + a}}{6 \, a x^{3}}, \frac{3 \, B a \sqrt{-b} x^{3} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) -{\left ({\left (3 \, B a + A b\right )} x^{2} + A a\right )} \sqrt{b x^{2} + a}}{3 \, a x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^4,x, algorithm="fricas")

[Out]

[1/6*(3*B*a*sqrt(b)*x^3*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*((3*
B*a + A*b)*x^2 + A*a)*sqrt(b*x^2 + a))/(a*x^3), 1/3*(3*B*a*sqrt(-b)*x^3*arctan(b
*x/(sqrt(b*x^2 + a)*sqrt(-b))) - ((3*B*a + A*b)*x^2 + A*a)*sqrt(b*x^2 + a))/(a*x
^3)]

_______________________________________________________________________________________

Sympy [A]  time = 7.74585, size = 107, normalized size = 1.62 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a} - \frac{B \sqrt{a}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + B \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )} - \frac{B b x}{\sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**4,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.248308, size = 204, normalized size = 3.09 \[ -\frac{1}{2} \, B \sqrt{b}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a \sqrt{b} + 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A b^{\frac{3}{2}} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{2} \sqrt{b} + 3 \, B a^{3} \sqrt{b} + A a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^4,x, algorithm="giac")

[Out]

-1/2*B*sqrt(b)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/3*(3*(sqrt(b)*x - sqrt(b*
x^2 + a))^4*B*a*sqrt(b) + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*b^(3/2) - 6*(sqrt(
b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b) + 3*B*a^3*sqrt(b) + A*a^2*b^(3/2))/((sqr
t(b)*x - sqrt(b*x^2 + a))^2 - a)^3