3.417 \(\int \frac{A+B x}{x^2 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=49 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{A \sqrt{a+b x}}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0742844, antiderivative size = 49, 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 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{A \sqrt{a+b x}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.97793, size = 42, normalized size = 0.86 \[ - \frac{A \sqrt{a + b x}}{a x} + \frac{2 \left (\frac{A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**2/(b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0657315, size = 51, normalized size = 1.04 \[ -\frac{(2 a B-A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{A \sqrt{a+b x}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 42, normalized size = 0.9 \[{(Ab-2\,Ba){\it Artanh} \left ({1\sqrt{bx+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}-{\frac{A}{ax}\sqrt{bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^2/(b*x+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23151, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (2 \, B a - A b\right )} x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \, \sqrt{b x + a} A \sqrt{a}}{2 \, a^{\frac{3}{2}} x}, \frac{{\left (2 \, B a - A b\right )} x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - \sqrt{b x + a} A \sqrt{-a}}{\sqrt{-a} a x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 26.1596, size = 165, normalized size = 3.37 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a \sqrt{x}} + \frac{A b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{3}{2}}} + 2 B \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + b x}} \right )}}{a \sqrt{- \frac{1}{a}}} & \text{for}\: - \frac{1}{a} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{a + b x} \sqrt{\frac{1}{a}}} \right )}}{a \sqrt{\frac{1}{a}}} & \text{for}\: - \frac{1}{a} < 0 \wedge \frac{1}{a} < \frac{1}{a + b x} \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{a + b x} \sqrt{\frac{1}{a}}} \right )}}{a \sqrt{\frac{1}{a}}} & \text{for}\: \frac{1}{a} > \frac{1}{a + b x} \wedge - \frac{1}{a} < 0 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**2/(b*x+a)**(1/2),x)

[Out]

-A*sqrt(b)*sqrt(a/(b*x) + 1)/(a*sqrt(x)) + A*b*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/
a**(3/2) + 2*B*Piecewise((atan(1/(sqrt(-1/a)*sqrt(a + b*x)))/(a*sqrt(-1/a)), -1/
a > 0), (-acoth(1/(sqrt(a + b*x)*sqrt(1/a)))/(a*sqrt(1/a)), (-1/a < 0) & (1/a <
1/(a + b*x))), (-atanh(1/(sqrt(a + b*x)*sqrt(1/a)))/(a*sqrt(1/a)), (-1/a < 0) &
(1/a > 1/(a + b*x))))

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GIAC/XCAS [A]  time = 0.213989, size = 78, normalized size = 1.59 \[ -\frac{\frac{\sqrt{b x + a} A b}{a x} - \frac{{\left (2 \, B a b - A b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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