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

Optimal. Leaf size=62 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{(a+b x)^{3/2}}{2 x^2}-\frac{3 b \sqrt{a+b x}}{4 x} \]

[Out]

(-3*b*Sqrt[a + b*x])/(4*x) - (a + b*x)^(3/2)/(2*x^2) - (3*b^2*ArcTanh[Sqrt[a + b
*x]/Sqrt[a]])/(4*Sqrt[a])

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Rubi [A]  time = 0.053313, antiderivative size = 62, 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 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{(a+b x)^{3/2}}{2 x^2}-\frac{3 b \sqrt{a+b x}}{4 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/x^3,x]

[Out]

(-3*b*Sqrt[a + b*x])/(4*x) - (a + b*x)^(3/2)/(2*x^2) - (3*b^2*ArcTanh[Sqrt[a + b
*x]/Sqrt[a]])/(4*Sqrt[a])

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Rubi in Sympy [A]  time = 7.05687, size = 56, normalized size = 0.9 \[ - \frac{3 b \sqrt{a + b x}}{4 x} - \frac{\left (a + b x\right )^{\frac{3}{2}}}{2 x^{2}} - \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*b*sqrt(a + b*x)/(4*x) - (a + b*x)**(3/2)/(2*x**2) - 3*b**2*atanh(sqrt(a + b*x
)/sqrt(a))/(4*sqrt(a))

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Mathematica [A]  time = 0.0521713, size = 53, normalized size = 0.85 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{\sqrt{a+b x} (2 a+5 b x)}{4 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]*(2*a + 5*b*x))/(4*x^2) - (3*b^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/
(4*Sqrt[a])

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Maple [A]  time = 0.016, size = 51, normalized size = 0.8 \[ 2\,{b}^{2} \left ({\frac{-5/8\, \left ( bx+a \right ) ^{3/2}+3/8\,a\sqrt{bx+a}}{{b}^{2}{x}^{2}}}-3/8\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/x^3,x)

[Out]

2*b^2*((-5/8*(b*x+a)^(3/2)+3/8*a*(b*x+a)^(1/2))/x^2/b^2-3/8*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)^(3/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236093, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b^{2} x^{2} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \,{\left (5 \, b x + 2 \, a\right )} \sqrt{b x + a} \sqrt{a}}{8 \, \sqrt{a} x^{2}}, \frac{3 \, b^{2} x^{2} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (5 \, b x + 2 \, a\right )} \sqrt{b x + a} \sqrt{-a}}{4 \, \sqrt{-a} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*b^2*x^2*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) - 2*(5*b*x + 2*
a)*sqrt(b*x + a)*sqrt(a))/(sqrt(a)*x^2), 1/4*(3*b^2*x^2*arctan(a/(sqrt(b*x + a)*
sqrt(-a))) - (5*b*x + 2*a)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*x^2)]

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Sympy [A]  time = 10.1924, size = 76, normalized size = 1.23 \[ - \frac{a \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{2 x^{\frac{3}{2}}} - \frac{5 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{4 \sqrt{x}} - \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(b)*sqrt(a/(b*x) + 1)/(2*x**(3/2)) - 5*b**(3/2)*sqrt(a/(b*x) + 1)/(4*sqrt
(x)) - 3*b**2*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(4*sqrt(a))

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GIAC/XCAS [A]  time = 0.206905, size = 86, normalized size = 1.39 \[ \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3} - 3 \, \sqrt{b x + a} a b^{3}}{b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(3*b^3*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) - (5*(b*x + a)^(3/2)*b^3 - 3*
sqrt(b*x + a)*a*b^3)/(b^2*x^2))/b