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

Optimal. Leaf size=90 \[ -\frac{b (3 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{\sqrt{a+b x^2} (3 A b-4 a B)}{8 a^2 x^2}-\frac{A \sqrt{a+b x^2}}{4 a x^4} \]

[Out]

-(A*Sqrt[a + b*x^2])/(4*a*x^4) + ((3*A*b - 4*a*B)*Sqrt[a + b*x^2])/(8*a^2*x^2) -
(b*(3*A*b - 4*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(5/2))

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Rubi [A]  time = 0.194912, antiderivative size = 90, 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 \[ -\frac{b (3 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{\sqrt{a+b x^2} (3 A b-4 a B)}{8 a^2 x^2}-\frac{A \sqrt{a+b x^2}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(A*Sqrt[a + b*x^2])/(4*a*x^4) + ((3*A*b - 4*a*B)*Sqrt[a + b*x^2])/(8*a^2*x^2) -
(b*(3*A*b - 4*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(5/2))

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Rubi in Sympy [A]  time = 16.538, size = 82, normalized size = 0.91 \[ - \frac{A \sqrt{a + b x^{2}}}{4 a x^{4}} + \frac{\sqrt{a + b x^{2}} \left (3 A b - 4 B a\right )}{8 a^{2} x^{2}} - \frac{b \left (3 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*sqrt(a + b*x**2)/(4*a*x**4) + sqrt(a + b*x**2)*(3*A*b - 4*B*a)/(8*a**2*x**2)
- b*(3*A*b - 4*B*a)*atanh(sqrt(a + b*x**2)/sqrt(a))/(8*a**(5/2))

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Mathematica [A]  time = 0.123471, size = 99, normalized size = 1.1 \[ \frac{b x^4 \log (x) (3 A b-4 a B)+\sqrt{a} \sqrt{a+b x^2} \left (3 A b x^2-2 a \left (A+2 B x^2\right )\right )+b x^4 (4 a B-3 A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{8 a^{5/2} x^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*Sqrt[a + b*x^2]*(3*A*b*x^2 - 2*a*(A + 2*B*x^2)) + b*(3*A*b - 4*a*B)*x^4
*Log[x] + b*(-3*A*b + 4*a*B)*x^4*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(8*a^(5/2)*x^
4)

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Maple [A]  time = 0.013, size = 119, normalized size = 1.3 \[ -{\frac{A}{4\,a{x}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{3\,Ab}{8\,{a}^{2}{x}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{3\,{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{B}{2\,a{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{Bb}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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

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Fricas [A]  time = 0.244537, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{4} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left ({\left (4 \, B a - 3 \, A b\right )} x^{2} + 2 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{a}}{16 \, a^{\frac{5}{2}} x^{4}}, \frac{{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left ({\left (4 \, B a - 3 \, A b\right )} x^{2} + 2 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{8 \, \sqrt{-a} a^{2} x^{4}}\right ] \]

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 - 3*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 - 3*A*b)*x^2 + 2*A*a)*sqrt(b*x^2 + a)*sqrt(a))/(a^(5/2)*x^4)
, 1/8*((4*B*a*b - 3*A*b^2)*x^4*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - ((4*B*a - 3*A*
b)*x^2 + 2*A*a)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^4)]

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Sympy [A]  time = 49.724, size = 150, normalized size = 1.67 \[ - \frac{A}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A \sqrt{b}}{8 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 A b^{\frac{3}{2}}}{8 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{5}{2}}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 a x} + \frac{B b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241999, size = 163, normalized size = 1.81 \[ -\frac{\frac{{\left (4 \, B a b^{2} - 3 \, A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x^{2} + a} B a^{2} b^{2} - 3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{3} + 5 \, \sqrt{b x^{2} + a} A a b^{3}}{a^{2} b^{2} x^{4}}}{8 \, b} \]

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 - 3*A*b^3)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (4
*(b*x^2 + a)^(3/2)*B*a*b^2 - 4*sqrt(b*x^2 + a)*B*a^2*b^2 - 3*(b*x^2 + a)^(3/2)*A
*b^3 + 5*sqrt(b*x^2 + a)*A*a*b^3)/(a^2*b^2*x^4))/b