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

Optimal. Leaf size=54 \[ 2 A \sqrt{a+b x}-2 \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 B (a+b x)^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0675193, antiderivative size = 54, 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 \[ 2 A \sqrt{a+b x}-2 \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 B (a+b x)^{3/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.61309, size = 49, normalized size = 0.91 \[ - 2 A \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 A \sqrt{a + b x} + \frac{2 B \left (a + b x\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0721094, size = 53, normalized size = 0.98 \[ \frac{2 \sqrt{a+b x} (B (a+b x)+3 A b)}{3 b}-2 \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 46, normalized size = 0.9 \[ 2\,{\frac{1}{b} \left ( 1/3\,B \left ( bx+a \right ) ^{3/2}+Ab\sqrt{bx+a}-A\sqrt{a}b{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218239, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, A \sqrt{a} b \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (B b x + B a + 3 \, A b\right )} \sqrt{b x + a}}{3 \, b}, -\frac{2 \,{\left (3 \, A \sqrt{-a} b \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (B b x + B a + 3 \, A b\right )} \sqrt{b x + a}\right )}}{3 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.18803, size = 110, normalized size = 2.04 \[ - 2 A a \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) + 2 A \sqrt{a + b x} + \frac{2 B \left (a + b x\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227568, size = 74, normalized size = 1.37 \[ \frac{2 \, A a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} B b^{2} + 3 \, \sqrt{b x + a} A b^{3}\right )}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 2/3*((b*x + a)^(3/2)*B*b^2 + 3*s
qrt(b*x + a)*A*b^3)/b^3