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

Optimal. Leaf size=71 \[ \frac{\sqrt{a+b x} (2 a B+A b)}{a}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{A (a+b x)^{3/2}}{a x} \]

[Out]

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

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Rubi [A]  time = 0.104875, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\sqrt{a+b x} (2 a B+A b)}{a}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{A (a+b x)^{3/2}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.45249, size = 63, normalized size = 0.89 \[ - \frac{A \left (a + b x\right )^{\frac{3}{2}}}{a x} + \frac{2 \sqrt{a + b x} \left (\frac{A b}{2} + B a\right )}{a} - \frac{2 \left (\frac{A b}{2} + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0652823, size = 52, normalized size = 0.73 \[ \sqrt{a+b x} \left (2 B-\frac{A}{x}\right )-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 50, normalized size = 0.7 \[ 2\,B\sqrt{bx+a}-{\frac{A}{x}\sqrt{bx+a}}-{(Ab+2\,Ba){\it Artanh} \left ({1\sqrt{bx+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219278, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, B a + A b\right )} x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (2 \, B x - A\right )} \sqrt{b x + a} \sqrt{a}}{2 \, \sqrt{a} x}, \frac{{\left (2 \, B a + A b\right )} x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (2 \, B x - A\right )} \sqrt{b x + a} \sqrt{-a}}{\sqrt{-a} x}\right ] \]

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

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Sympy [A]  time = 13.5127, size = 267, normalized size = 3.76 \[ - \frac{A a b \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{A a b \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} - 2 A b \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) - \frac{A \sqrt{a + b x}}{x} - 2 B a \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) + 2 B \sqrt{a + b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226117, size = 82, normalized size = 1.15 \[ \frac{2 \, \sqrt{b x + a} B b - \frac{\sqrt{b x + a} A b}{x} + \frac{{\left (2 \, B a b + A b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}}}{b} \]

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]

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