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

Optimal. Leaf size=98 \[ \frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{5/2}}-\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b^2}+\frac{a x^{3/2} \sqrt{a+b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a+b x} \]

[Out]

-(a^2*Sqrt[x]*Sqrt[a + b*x])/(8*b^2) + (a*x^(3/2)*Sqrt[a + b*x])/(12*b) + (x^(5/
2)*Sqrt[a + b*x])/3 + (a^3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0775024, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{5/2}}-\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b^2}+\frac{a x^{3/2} \sqrt{a+b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a+b x} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*Sqrt[a + b*x],x]

[Out]

-(a^2*Sqrt[x]*Sqrt[a + b*x])/(8*b^2) + (a*x^(3/2)*Sqrt[a + b*x])/(12*b) + (x^(5/
2)*Sqrt[a + b*x])/3 + (a^3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.855, size = 88, normalized size = 0.9 \[ \frac{a^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{5}{2}}} + \frac{a^{2} \sqrt{x} \sqrt{a + b x}}{8 b^{2}} - \frac{a \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{4 b^{2}} + \frac{x^{\frac{3}{2}} \left (a + b x\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*atanh(sqrt(a + b*x)/(sqrt(b)*sqrt(x)))/(8*b**(5/2)) + a**2*sqrt(x)*sqrt(a +
b*x)/(8*b**2) - a*sqrt(x)*(a + b*x)**(3/2)/(4*b**2) + x**(3/2)*(a + b*x)**(3/2)
/(3*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0481184, size = 78, normalized size = 0.8 \[ \frac{3 a^3 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (-3 a^2+2 a b x+8 b^2 x^2\right )}{24 b^{5/2}} \]

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 + 2*a*b*x + 8*b^2*x^2) + 3*a^3*Log[b*Sqrt
[x] + Sqrt[b]*Sqrt[a + b*x]])/(24*b^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 102, normalized size = 1. \[{\frac{1}{3\,b}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{a}{4\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{{a}^{2}}{8\,{b}^{2}}\sqrt{x}\sqrt{bx+a}}+{\frac{{a}^{3}}{16}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(b*x+a)^(1/2),x)

[Out]

1/3/b*x^(3/2)*(b*x+a)^(3/2)-1/4*a/b^2*x^(1/2)*(b*x+a)^(3/2)+1/8*a^2*x^(1/2)*(b*x
+a)^(1/2)/b^2+1/16*a^3/b^(5/2)*(x*(b*x+a))^(1/2)/(b*x+a)^(1/2)/x^(1/2)*ln((1/2*a
+b*x)/b^(1/2)+(b*x^2+a*x)^(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(sqrt(b*x + a)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.221342, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, b^{2} x^{2} + 2 \, a b x - 3 \, a^{2}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{5}{2}}}, \frac{3 \, a^{3} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (8 \, b^{2} x^{2} + 2 \, a b x - 3 \, a^{2}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(3*a^3*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) + 2*(8*b^2*x^2
+ 2*a*b*x - 3*a^2)*sqrt(b*x + a)*sqrt(b)*sqrt(x))/b^(5/2), 1/24*(3*a^3*arctan(s
qrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (8*b^2*x^2 + 2*a*b*x - 3*a^2)*sqrt(b*x + a)
*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^2)]

_______________________________________________________________________________________

Sympy [A]  time = 26.6423, size = 122, normalized size = 1.24 \[ - \frac{a^{\frac{5}{2}} \sqrt{x}}{8 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b \sqrt{1 + \frac{b x}{a}}} + \frac{5 \sqrt{a} x^{\frac{5}{2}}}{12 \sqrt{1 + \frac{b x}{a}}} + \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} + \frac{b x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(5/2)*sqrt(x)/(8*b**2*sqrt(1 + b*x/a)) - a**(3/2)*x**(3/2)/(24*b*sqrt(1 + b*
x/a)) + 5*sqrt(a)*x**(5/2)/(12*sqrt(1 + b*x/a)) + a**3*asinh(sqrt(b)*sqrt(x)/sqr
t(a))/(8*b**(5/2)) + b*x**(7/2)/(3*sqrt(a)*sqrt(1 + b*x/a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 12.4489, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x