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

Optimal. Leaf size=68 \[ -\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}+\frac{3 \sqrt{x} \sqrt{a+b x}}{b^2}-\frac{2 x^{3/2}}{b \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.0530922, antiderivative size = 68, 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 \[ -\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}+\frac{3 \sqrt{x} \sqrt{a+b x}}{b^2}-\frac{2 x^{3/2}}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(a + b*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.89946, size = 63, normalized size = 0.93 \[ - \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{b^{\frac{5}{2}}} - \frac{2 x^{\frac{3}{2}}}{b \sqrt{a + b x}} + \frac{3 \sqrt{x} \sqrt{a + b x}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/b**(5/2) - 2*x**(3/2)/(b*sqrt(a + b*x)
) + 3*sqrt(x)*sqrt(a + b*x)/b**2

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Mathematica [A]  time = 0.0745807, size = 58, normalized size = 0.85 \[ \frac{\sqrt{x} (3 a+b x)}{b^2 \sqrt{a+b x}}-\frac{3 a \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.037, size = 106, normalized size = 1.6 \[{\frac{1}{{b}^{2}}\sqrt{x}\sqrt{bx+a}}+{1 \left ( -{\frac{3\,a}{2}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{5}{2}}}}+2\,{\frac{a}{{b}^{3}}\sqrt{b \left ( x+{\frac{a}{b}} \right ) ^{2}- \left ( x+{\frac{a}{b}} \right ) a} \left ( x+{\frac{a}{b}} \right ) ^{-1}} \right ) \sqrt{x \left ( bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(b*x+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237418, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{b x + a} a \sqrt{x} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (b x^{2} + 3 \, a x\right )} \sqrt{b}}{2 \, \sqrt{b x + a} b^{\frac{5}{2}} \sqrt{x}}, -\frac{3 \, \sqrt{b x + a} a \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (b x^{2} + 3 \, a x\right )} \sqrt{-b}}{\sqrt{b x + a} \sqrt{-b} b^{2} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 16.6604, size = 71, normalized size = 1.04 \[ \frac{3 \sqrt{a} \sqrt{x}}{b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} + \frac{x^{\frac{3}{2}}}{\sqrt{a} b \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217619, size = 155, normalized size = 2.28 \[ \frac{{\left (\frac{8 \, a^{2} \sqrt{b}}{{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b} + \frac{3 \, a{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{\sqrt{b}} + \frac{2 \, \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a}}{b}\right )}{\left | b \right |}}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(8*a^2*sqrt(b)/((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b) +
3*a*ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/sqrt(b) + 2*sqrt((b
*x + a)*b - a*b)*sqrt(b*x + a)/b)*abs(b)/b^3