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

Optimal. Leaf size=53 \[ \frac{A b-a B}{a b \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.128484, antiderivative size = 53, 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 \[ \frac{A b-a B}{a b \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.1833, size = 42, normalized size = 0.79 \[ - \frac{A \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} + \frac{A b - B a}{a b \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.118918, size = 64, normalized size = 1.21 \[ -\frac{A \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{a^{3/2}}+\frac{A \log (x)}{a^{3/2}}+\frac{A b-a B}{a b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 60, normalized size = 1.1 \[{\frac{A}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{A\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239646, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, \sqrt{b x^{2} + a}{\left (B a - A b\right )} \sqrt{a} -{\left (A b^{2} x^{2} + A a b\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{2 \,{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{a}}, -\frac{\sqrt{b x^{2} + a}{\left (B a - A b\right )} \sqrt{-a} +{\left (A b^{2} x^{2} + A a b\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 15.0147, size = 212, normalized size = 4. \[ A \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{3} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{2} b x^{2} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{2} b x^{2} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}}\right ) + B \left (\begin{cases} - \frac{1}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(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)) + B*Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True
))

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GIAC/XCAS [A]  time = 0.226369, size = 70, normalized size = 1.32 \[ \frac{A \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{B a - A b}{\sqrt{b x^{2} + a} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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