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

Optimal. Leaf size=63 \[ -\frac{2 (a+b x)^{3/2}}{\sqrt{x}}+3 b \sqrt{x} \sqrt{a+b x}+3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0503474, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 (a+b x)^{3/2}}{\sqrt{x}}+3 b \sqrt{x} \sqrt{a+b x}+3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.92637, size = 60, normalized size = 0.95 \[ 3 a \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )} + 3 b \sqrt{x} \sqrt{a + b x} - \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0476551, size = 55, normalized size = 0.87 \[ \frac{\sqrt{a+b x} (b x-2 a)}{\sqrt{x}}+3 a \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.045, size = 71, normalized size = 1.1 \[ -{(-bx+2\,a)\sqrt{bx+a}{\frac{1}{\sqrt{x}}}}+{\frac{3\,a}{2}\sqrt{b}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) \sqrt{x \left ( bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/x^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220659, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a \sqrt{b} x \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \, \sqrt{b x + a}{\left (b x - 2 \, a\right )} \sqrt{x}}{2 \, x}, \frac{3 \, a \sqrt{-b} x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) + \sqrt{b x + a}{\left (b x - 2 \, a\right )} \sqrt{x}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.6099, size = 92, normalized size = 1.46 \[ - \frac{2 a^{\frac{3}{2}}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} - \frac{\sqrt{a} b \sqrt{x}}{\sqrt{1 + \frac{b x}{a}}} + 3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} + \frac{b^{2} x^{\frac{3}{2}}}{\sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 14.3209, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x