3.578 \(\int \frac{A+B x^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 A b-2 a B}{2 a^2 \sqrt{a+b x^2}}-\frac{A}{2 a x^2 \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.190206, antiderivative size = 86, 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 \[ \frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 A b-2 a B}{2 a^2 \sqrt{a+b x^2}}-\frac{A}{2 a x^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.8769, size = 73, normalized size = 0.85 \[ - \frac{A}{2 a x^{2} \sqrt{a + b x^{2}}} - \frac{\frac{3 A b}{2} - B a}{a^{2} \sqrt{a + b x^{2}}} + \frac{\left (\frac{3 A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.227495, size = 91, normalized size = 1.06 \[ \frac{\frac{\sqrt{a} \left (-a A+2 a B x^2-3 A b x^2\right )}{x^2 \sqrt{a+b x^2}}+(3 A b-2 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\log (x) (2 a B-3 A b)}{2 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 109, normalized size = 1.3 \[ -{\frac{A}{2\,a{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,Ab}{2\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{B}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246394, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left ({\left (2 \, B a - 3 \, A b\right )} x^{2} - A a\right )} \sqrt{b x^{2} + a} \sqrt{a} -{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{4 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \sqrt{a}}, \frac{{\left ({\left (2 \, B a - 3 \, A b\right )} x^{2} - A a\right )} \sqrt{b x^{2} + a} \sqrt{-a} -{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 29.4305, size = 262, normalized size = 3.05 \[ A \left (- \frac{1}{2 a \sqrt{b} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 \sqrt{b}}{2 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{5}{2}}}\right ) + B \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{3} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{2} b x^{2} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{2} b x^{2} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-1/(2*a*sqrt(b)*x**3*sqrt(a/(b*x**2) + 1)) - 3*sqrt(b)/(2*a**2*x*sqrt(a/(b*x*
*2) + 1)) + 3*b*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(5/2))) + B*(2*a**3*sqrt(1 + b*
x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) + a**3*log(b*x**2/a)/(2*a**(9/2) + 2*a*
*(7/2)*b*x**2) - 2*a**3*log(sqrt(1 + b*x**2/a) + 1)/(2*a**(9/2) + 2*a**(7/2)*b*x
**2) + a**2*b*x**2*log(b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) - 2*a**2*b*x**
2*log(sqrt(1 + b*x**2/a) + 1)/(2*a**(9/2) + 2*a**(7/2)*b*x**2))

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GIAC/XCAS [A]  time = 0.2282, size = 134, normalized size = 1.56 \[ \frac{{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a} a^{2}} + \frac{2 \,{\left (b x^{2} + a\right )} B a - 2 \, B a^{2} - 3 \,{\left (b x^{2} + a\right )} A b + 2 \, A a b}{2 \,{\left ({\left (b x^{2} + a\right )}^{\frac{3}{2}} - \sqrt{b x^{2} + a} a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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