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

Optimal. Leaf size=70 \[ -\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2 A \sqrt{a+b x^2}}{a^2 x}+\frac{A+B x}{a x \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.212819, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2 A \sqrt{a+b x^2}}{a^2 x}+\frac{A+B x}{a x \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.0147, size = 60, normalized size = 0.86 \[ - \frac{2 A \sqrt{a + b x^{2}}}{a^{2} x} - \frac{B \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} + \frac{A + B x}{a x \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.161076, size = 73, normalized size = 1.04 \[ \frac{\frac{-a A+a B x-2 A b x^2}{x \sqrt{a+b x^2}}-\sqrt{a} B \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\sqrt{a} B \log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 80, normalized size = 1.1 \[ -{\frac{A}{ax}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-2\,{\frac{Axb}{{a}^{2}\sqrt{b{x}^{2}+a}}}+{\frac{B}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26437, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (2 \, A b x^{2} - B a x + A a\right )} \sqrt{b x^{2} + a} \sqrt{a} -{\left (B a b x^{3} + B a^{2} x\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{2 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{a}}, -\frac{{\left (2 \, A b x^{2} - B a x + A a\right )} \sqrt{b x^{2} + a} \sqrt{-a} +{\left (B a b x^{3} + B a^{2} x\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.37371, size = 235, normalized size = 3.36 \[ A \left (- \frac{1}{a \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{2 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x^{2}} + 1}}\right ) + B \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{3} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{2} b x^{2} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{2} b x^{2} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-1/(a*sqrt(b)*x**2*sqrt(a/(b*x**2) + 1)) - 2*sqrt(b)/(a**2*sqrt(a/(b*x**2) +
1))) + 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 = 0.222788, size = 130, normalized size = 1.86 \[ -\frac{\frac{A b x}{a^{2}} - \frac{B}{a}}{\sqrt{b x^{2} + a}} + \frac{2 \, B \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{2 \, A \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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