3.18 \(\int \frac{\sqrt{a+b x} (e+f x)}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0730163, 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 \[ 2 e \sqrt{a+b x}-2 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 f (a+b x)^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(e + f*x))/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.57568, size = 49, normalized size = 0.91 \[ - 2 \sqrt{a} e \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 e \sqrt{a + b x} + \frac{2 f \left (a + b x\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x+e)*(b*x+a)**(1/2)/x,x)

[Out]

-2*sqrt(a)*e*atanh(sqrt(a + b*x)/sqrt(a)) + 2*e*sqrt(a + b*x) + 2*f*(a + b*x)**(
3/2)/(3*b)

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Mathematica [A]  time = 0.0744661, size = 53, normalized size = 0.98 \[ \frac{2 \sqrt{a+b x} (a f+3 b e+b f x)}{3 b}-2 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(e + f*x))/x,x]

[Out]

(2*Sqrt[a + b*x]*(3*b*e + a*f + b*f*x))/(3*b) - 2*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x
]/Sqrt[a]]

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Maple [A]  time = 0.01, size = 46, normalized size = 0.9 \[ 2\,{\frac{1}{b} \left ( 1/3\,f \left ( bx+a \right ) ^{3/2}+be\sqrt{bx+a}-\sqrt{a}be{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)*(b*x+a)^(1/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225641, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{a} b e \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (b f x + 3 \, b e + a f\right )} \sqrt{b x + a}}{3 \, b}, -\frac{2 \,{\left (3 \, \sqrt{-a} b e \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (b f x + 3 \, b e + a f\right )} \sqrt{b x + a}\right )}}{3 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.01522, size = 110, normalized size = 2.04 \[ - 2 a e \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) + 2 e \sqrt{a + b x} + \frac{2 f \left (a + b x\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x+e)*(b*x+a)**(1/2)/x,x)

[Out]

-2*a*e*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*e*sqrt(a + b*x) + 2*f*(a + b*x)**(3/2
)/(3*b)

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GIAC/XCAS [A]  time = 0.209106, size = 77, normalized size = 1.43 \[ \frac{2 \, a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) e}{\sqrt{-a}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} b^{2} f + 3 \, \sqrt{b x + a} b^{3} e\right )}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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