3.1689 \(\int \sqrt{a+\frac{b}{x}} x^3 \, dx\)

Optimal. Leaf size=117 \[ -\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{7/2}}+\frac{5 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^3}-\frac{5 b^2 x^2 \sqrt{a+\frac{b}{x}}}{96 a^2}+\frac{1}{4} x^4 \sqrt{a+\frac{b}{x}}+\frac{b x^3 \sqrt{a+\frac{b}{x}}}{24 a} \]

[Out]

(5*b^3*Sqrt[a + b/x]*x)/(64*a^3) - (5*b^2*Sqrt[a + b/x]*x^2)/(96*a^2) + (b*Sqrt[
a + b/x]*x^3)/(24*a) + (Sqrt[a + b/x]*x^4)/4 - (5*b^4*ArcTanh[Sqrt[a + b/x]/Sqrt
[a]])/(64*a^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.170805, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{7/2}}+\frac{5 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^3}-\frac{5 b^2 x^2 \sqrt{a+\frac{b}{x}}}{96 a^2}+\frac{1}{4} x^4 \sqrt{a+\frac{b}{x}}+\frac{b x^3 \sqrt{a+\frac{b}{x}}}{24 a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]*x^3,x]

[Out]

(5*b^3*Sqrt[a + b/x]*x)/(64*a^3) - (5*b^2*Sqrt[a + b/x]*x^2)/(96*a^2) + (b*Sqrt[
a + b/x]*x^3)/(24*a) + (Sqrt[a + b/x]*x^4)/4 - (5*b^4*ArcTanh[Sqrt[a + b/x]/Sqrt
[a]])/(64*a^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.068, size = 99, normalized size = 0.85 \[ \frac{x^{4} \sqrt{a + \frac{b}{x}}}{4} + \frac{b x^{3} \sqrt{a + \frac{b}{x}}}{24 a} - \frac{5 b^{2} x^{2} \sqrt{a + \frac{b}{x}}}{96 a^{2}} + \frac{5 b^{3} x \sqrt{a + \frac{b}{x}}}{64 a^{3}} - \frac{5 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{64 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*sqrt(a + b/x)/4 + b*x**3*sqrt(a + b/x)/(24*a) - 5*b**2*x**2*sqrt(a + b/x)/(
96*a**2) + 5*b**3*x*sqrt(a + b/x)/(64*a**3) - 5*b**4*atanh(sqrt(a + b/x)/sqrt(a)
)/(64*a**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.13792, size = 90, normalized size = 0.77 \[ \frac{2 \sqrt{a} x \sqrt{a+\frac{b}{x}} \left (48 a^3 x^3+8 a^2 b x^2-10 a b^2 x+15 b^3\right )-15 b^4 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{384 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]*x^3,x]

[Out]

(2*Sqrt[a]*Sqrt[a + b/x]*x*(15*b^3 - 10*a*b^2*x + 8*a^2*b*x^2 + 48*a^3*x^3) - 15
*b^4*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(384*a^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 135, normalized size = 1.2 \[ -{\frac{x}{384}\sqrt{{\frac{ax+b}{x}}} \left ( -96\,x \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{7/2}+80\,{a}^{5/2} \left ( a{x}^{2}+bx \right ) ^{3/2}b-60\,{a}^{5/2}\sqrt{a{x}^{2}+bx}x{b}^{2}-30\,{a}^{3/2}\sqrt{a{x}^{2}+bx}{b}^{3}+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{4} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b/x)^(1/2),x)

[Out]

-1/384*((a*x+b)/x)^(1/2)*x*(-96*x*(a*x^2+b*x)^(3/2)*a^(7/2)+80*a^(5/2)*(a*x^2+b*
x)^(3/2)*b-60*a^(5/2)*(a*x^2+b*x)^(1/2)*x*b^2-30*a^(3/2)*(a*x^2+b*x)^(1/2)*b^3+1
5*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*b^4)/(x*(a*x+b))^(1/2)
/a^(9/2)

_______________________________________________________________________________________

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(a + b/x)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.245576, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{4} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (48 \, a^{3} x^{4} + 8 \, a^{2} b x^{3} - 10 \, a b^{2} x^{2} + 15 \, b^{3} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{384 \, a^{\frac{7}{2}}}, \frac{15 \, b^{4} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (48 \, a^{3} x^{4} + 8 \, a^{2} b x^{3} - 10 \, a b^{2} x^{2} + 15 \, b^{3} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{192 \, \sqrt{-a} a^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(15*b^4*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(48*a^3*x
^4 + 8*a^2*b*x^3 - 10*a*b^2*x^2 + 15*b^3*x)*sqrt(a)*sqrt((a*x + b)/x))/a^(7/2),
1/192*(15*b^4*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) + (48*a^3*x^4 + 8*a^2*b*x^3
- 10*a*b^2*x^2 + 15*b^3*x)*sqrt(-a)*sqrt((a*x + b)/x))/(sqrt(-a)*a^3)]

_______________________________________________________________________________________

Sympy [A]  time = 29.5256, size = 153, normalized size = 1.31 \[ \frac{a x^{\frac{9}{2}}}{4 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{7 \sqrt{b} x^{\frac{7}{2}}}{24 \sqrt{\frac{a x}{b} + 1}} - \frac{b^{\frac{3}{2}} x^{\frac{5}{2}}}{96 a \sqrt{\frac{a x}{b} + 1}} + \frac{5 b^{\frac{5}{2}} x^{\frac{3}{2}}}{192 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{5 b^{\frac{7}{2}} \sqrt{x}}{64 a^{3} \sqrt{\frac{a x}{b} + 1}} - \frac{5 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{64 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**(9/2)/(4*sqrt(b)*sqrt(a*x/b + 1)) + 7*sqrt(b)*x**(7/2)/(24*sqrt(a*x/b + 1))
- b**(3/2)*x**(5/2)/(96*a*sqrt(a*x/b + 1)) + 5*b**(5/2)*x**(3/2)/(192*a**2*sqrt
(a*x/b + 1)) + 5*b**(7/2)*sqrt(x)/(64*a**3*sqrt(a*x/b + 1)) - 5*b**4*asinh(sqrt(
a)*sqrt(x)/sqrt(b))/(64*a**(7/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.240663, size = 146, normalized size = 1.25 \[ \frac{5 \, b^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ){\rm sign}\left (x\right )}{128 \, a^{\frac{7}{2}}} - \frac{5 \, b^{4}{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{128 \, a^{\frac{7}{2}}} + \frac{1}{192} \, \sqrt{a x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, x{\rm sign}\left (x\right ) + \frac{b{\rm sign}\left (x\right )}{a}\right )} x - \frac{5 \, b^{2}{\rm sign}\left (x\right )}{a^{2}}\right )} x + \frac{15 \, b^{3}{\rm sign}\left (x\right )}{a^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/128*b^4*ln(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))*sign(x)/a^(7/2
) - 5/128*b^4*ln(abs(b))*sign(x)/a^(7/2) + 1/192*sqrt(a*x^2 + b*x)*(2*(4*(6*x*si
gn(x) + b*sign(x)/a)*x - 5*b^2*sign(x)/a^2)*x + 15*b^3*sign(x)/a^3)