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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0485088, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+2 a \sqrt{a+b x}+\frac{2}{3} (a+b x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/x,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.47086, size = 44, normalized size = 0.9 \[ - 2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 a \sqrt{a + b x} + \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0281831, size = 46, normalized size = 0.94 \[ \left (\frac{8 a}{3}+\frac{2 b x}{3}\right ) \sqrt{a+b x}-2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)/x,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 38, normalized size = 0.8 \[{\frac{2}{3} \left ( bx+a \right ) ^{{\frac{3}{2}}}}-2\,{a}^{3/2}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) +2\,a\sqrt{bx+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/x,x)

[Out]

2/3*(b*x+a)^(3/2)-2*a^(3/2)*arctanh((b*x+a)^(1/2)/a^(1/2))+2*a*(b*x+a)^(1/2)

_______________________________________________________________________________________

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)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.234987, size = 1, normalized size = 0.02 \[ \left [a^{\frac{3}{2}} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \frac{2}{3} \,{\left (b x + 4 \, a\right )} \sqrt{b x + a}, -2 \, \sqrt{-a} a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) + \frac{2}{3} \,{\left (b x + 4 \, a\right )} \sqrt{b x + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[a^(3/2)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2/3*(b*x + 4*a)*sqrt(b*x
+ a), -2*sqrt(-a)*a*arctan(sqrt(b*x + a)/sqrt(-a)) + 2/3*(b*x + 4*a)*sqrt(b*x +
a)]

_______________________________________________________________________________________

Sympy [A]  time = 6.95833, size = 71, normalized size = 1.45 \[ \frac{8 a^{\frac{3}{2}} \sqrt{1 + \frac{b x}{a}}}{3} + a^{\frac{3}{2}} \log{\left (\frac{b x}{a} \right )} - 2 a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )} + \frac{2 \sqrt{a} b x \sqrt{1 + \frac{b x}{a}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)/x,x)

[Out]

8*a**(3/2)*sqrt(1 + b*x/a)/3 + a**(3/2)*log(b*x/a) - 2*a**(3/2)*log(sqrt(1 + b*x
/a) + 1) + 2*sqrt(a)*b*x*sqrt(1 + b*x/a)/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.209605, size = 59, normalized size = 1.2 \[ \frac{2 \, a^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2}{3} \,{\left (b x + a\right )}^{\frac{3}{2}} + 2 \, \sqrt{b x + a} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a^2*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 2/3*(b*x + a)^(3/2) + 2*sqrt(b*x
+ a)*a