3.514 \(\int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^5} \, dx\)

Optimal. Leaf size=88 \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{\sqrt{a+b x^2} (A b-4 a B)}{8 a x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{4 a x^4} \]

[Out]

((A*b - 4*a*B)*Sqrt[a + b*x^2])/(8*a*x^2) - (A*(a + b*x^2)^(3/2))/(4*a*x^4) + (b
*(A*b - 4*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.195805, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{\sqrt{a+b x^2} (A b-4 a B)}{8 a x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^5,x]

[Out]

((A*b - 4*a*B)*Sqrt[a + b*x^2])/(8*a*x^2) - (A*(a + b*x^2)^(3/2))/(4*a*x^4) + (b
*(A*b - 4*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.3015, size = 76, normalized size = 0.86 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{4 a x^{4}} + \frac{\sqrt{a + b x^{2}} \left (A b - 4 B a\right )}{8 a x^{2}} + \frac{b \left (A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**5,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.110759, size = 99, normalized size = 1.12 \[ \frac{b (A b-4 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{8 a^{3/2}}-\frac{b \log (x) (A b-4 a B)}{8 a^{3/2}}+\sqrt{a+b x^2} \left (\frac{-4 a B-A b}{8 a x^2}-\frac{A}{4 x^4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^5,x]

[Out]

(-A/(4*x^4) + (-(A*b) - 4*a*B)/(8*a*x^2))*Sqrt[a + b*x^2] - (b*(A*b - 4*a*B)*Log
[x])/(8*a^(3/2)) + (b*(A*b - 4*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(8*a^(3/2)
)

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 153, normalized size = 1.7 \[ -{\frac{A}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ab}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{b}^{2}A}{8\,{a}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{B}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Bb}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{Bb}{2\,a}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(b*x^2+a)^(1/2)/x^5,x)

[Out]

-1/4*A*(b*x^2+a)^(3/2)/a/x^4+1/8*A*b/a^2/x^2*(b*x^2+a)^(3/2)+1/8*A*b^2/a^(3/2)*l
n((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-1/8*A*b^2/a^2*(b*x^2+a)^(1/2)-1/2*B/a/x^2*(
b*x^2+a)^(3/2)-1/2*B*b/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+1/2*B*b/a*(
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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.227569, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B a b - A b^{2}\right )} x^{4} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left ({\left (4 \, B a + A b\right )} x^{2} + 2 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{a}}{16 \, a^{\frac{3}{2}} x^{4}}, -\frac{{\left (4 \, B a b - A b^{2}\right )} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left ({\left (4 \, B a + A b\right )} x^{2} + 2 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{8 \, \sqrt{-a} a x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^5,x, algorithm="fricas")

[Out]

[-1/16*((4*B*a*b - A*b^2)*x^4*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2 + a)*a)
/x^2) + 2*((4*B*a + A*b)*x^2 + 2*A*a)*sqrt(b*x^2 + a)*sqrt(a))/(a^(3/2)*x^4), -1
/8*((4*B*a*b - A*b^2)*x^4*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + ((4*B*a + A*b)*x^2
+ 2*A*a)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*x^4)]

_______________________________________________________________________________________

Sympy [A]  time = 70.4917, size = 144, normalized size = 1.64 \[ - \frac{A a}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 A \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A b^{\frac{3}{2}}}{8 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{3}{2}}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{B b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**5,x)

[Out]

-A*a/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*A*sqrt(b)/(8*x**3*sqrt(a/(b*x**2)
+ 1)) - A*b**(3/2)/(8*a*x*sqrt(a/(b*x**2) + 1)) + A*b**2*asinh(sqrt(a)/(sqrt(b)
*x))/(8*a**(3/2)) - B*sqrt(b)*sqrt(a/(b*x**2) + 1)/(2*x) - B*b*asinh(sqrt(a)/(sq
rt(b)*x))/(2*sqrt(a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.231908, size = 162, normalized size = 1.84 \[ \frac{\frac{{\left (4 \, B a b^{2} - A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x^{2} + a} B a^{2} b^{2} +{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{3} + \sqrt{b x^{2} + a} A a b^{3}}{a b^{2} x^{4}}}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^5,x, algorithm="giac")

[Out]

1/8*((4*B*a*b^2 - A*b^3)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) - (4*(b*x
^2 + a)^(3/2)*B*a*b^2 - 4*sqrt(b*x^2 + a)*B*a^2*b^2 + (b*x^2 + a)^(3/2)*A*b^3 +
sqrt(b*x^2 + a)*A*a*b^3)/(a*b^2*x^4))/b