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

Optimal. Leaf size=47 \[ \frac{A+B x}{a \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.1346, antiderivative size = 47, 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 \[ \frac{A+B x}{a \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(A + B*x)/(a*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 = 13.4371, size = 39, normalized size = 0.83 \[ - \frac{A \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} + \frac{A + B x}{a \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.101972, size = 58, normalized size = 1.23 \[ -\frac{A \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{a^{3/2}}+\frac{A \log (x)}{a^{3/2}}+\frac{A+B x}{a \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(A + B*x)/(a*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 = 0.01, size = 60, normalized size = 1.3 \[{\frac{Bx}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{A}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{A\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/a/(b*x^2+a)^(1/2)+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)

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259789, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{b x^{2} + a}{\left (B x + A\right )} \sqrt{a} +{\left (A b x^{2} + A a\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{2 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{a}}, \frac{\sqrt{b x^{2} + a}{\left (B x + A\right )} \sqrt{-a} -{\left (A b x^{2} + A a\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{{\left (a b x^{2} + a^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.88998, size = 206, normalized size = 4.38 \[ A \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{3} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{2} b x^{2} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{2} b x^{2} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}}\right ) + \frac{B x}{a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+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*x/(a**(3/2)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.216938, size = 80, normalized size = 1.7 \[ \frac{\frac{B x}{a} + \frac{A}{a}}{\sqrt{b x^{2} + a}} + \frac{2 \, A \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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