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

Optimal. Leaf size=77 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{x} \sqrt{a+b x}}{4 b^2}+\frac{x^{3/2} \sqrt{a+b x}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0540794, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{x} \sqrt{a+b x}}{4 b^2}+\frac{x^{3/2} \sqrt{a+b x}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/Sqrt[a + b*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 8.02062, size = 70, normalized size = 0.91 \[ \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{5}{2}}} - \frac{3 a \sqrt{x} \sqrt{a + b x}}{4 b^{2}} + \frac{x^{\frac{3}{2}} \sqrt{a + b x}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)/(b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.041313, size = 67, normalized size = 0.87 \[ \frac{3 a^2 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} \sqrt{x} \sqrt{a+b x} (2 b x-3 a)}{4 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 84, normalized size = 1.1 \[{\frac{1}{2\,b}{x}^{{\frac{3}{2}}}\sqrt{bx+a}}-{\frac{3\,a}{4\,{b}^{2}}\sqrt{x}\sqrt{bx+a}}+{\frac{3\,{a}^{2}}{8}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(b*x+a)^(1/2),x)

[Out]

1/2*x^(3/2)*(b*x+a)^(1/2)/b-3/4*a*x^(1/2)*(b*x+a)^(1/2)/b^2+3/8*a^2/b^(5/2)*(x*(
b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(1/2)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(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)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226435, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (2 \, b x - 3 \, a\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{8 \, b^{\frac{5}{2}}}, \frac{3 \, a^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (2 \, b x - 3 \, a\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 17.3327, size = 100, normalized size = 1.3 \[ - \frac{3 a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{\sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{1 + \frac{b x}{a}}} + \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)/(b*x+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 12.7125, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x