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

Optimal. Leaf size=50 \[ \frac{2 (A b-a B)}{a b \sqrt{a+b x}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0651117, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 (A b-a B)}{a b \sqrt{a+b x}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.93303, size = 44, normalized size = 0.88 \[ - \frac{2 A \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} + \frac{2 \left (A b - B a\right )}{a b \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0787094, size = 50, normalized size = 1. \[ -\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2 (a B-A b)}{a b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 46, normalized size = 0.9 \[ 2\,{\frac{1}{b} \left ( -{\frac{-Ab+Ba}{a\sqrt{bx+a}}}-{\frac{Ab}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x/(b*x+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23013, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x + a} A b \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \,{\left (B a - A b\right )} \sqrt{a}}{\sqrt{b x + a} a^{\frac{3}{2}} b}, \frac{2 \,{\left (\sqrt{b x + a} A b \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (B a - A b\right )} \sqrt{-a}\right )}}{\sqrt{b x + a} \sqrt{-a} a b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.1826, size = 162, normalized size = 3.24 \[ A \left (\frac{2 a^{3} \sqrt{1 + \frac{b x}{a}}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{a^{3} \log{\left (\frac{b x}{a} \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{a^{2} b x \log{\left (\frac{b x}{a} \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 a^{2} b x \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x}\right ) - \frac{2 B}{b \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.233856, size = 66, normalized size = 1.32 \[ \frac{2 \, A \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{2 \,{\left (B a - A b\right )}}{\sqrt{b x + a} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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